*143*
*13*

*English*
*Pages [410]*
*Year 2024*

- Author / Uploaded
- Philippe G. LeFloch
- Mai Duc Thanh

*Table of contents : Contents1 Overview of this monograph2 Models arising in fluid and solid dynamics 2.1 Derivation of the equations governing fluid flows 2.1.1 Material derivative 2.1.2 Reynolds' transport theorem 2.1.3 Conservation law of mass 2.1.4 Balance law of momentum 2.1.5 Balance law of energy 2.1.6 Governing equations and systems of conservation laws 2.2 Fundamental notions from thermodynamics 2.2.1 Equation of state and thermodynamic identity 2.2.2 Helmholtz free energy 2.2.3 The class of ideal fluids 2.2.4 The class of stiffened fluids 2.2.5 The class of Hayes fluids 2.2.6 The class of Van der Waals fluids 2.3 Physical models of particular interest 2.3.1 Fluid equations in Eulerian coordinates 2.3.2 Fluid equations in Lagrangian coordinates 2.3.3 Viscous-capillary models 2.3.4 Fluid flows in a nozzle with variable cross-section 2.3.5 Shallow water model for variable topography 2.3.6 A model of nonlinear elastodynamics with phase transitions 2.4 Bibliographical notes3 Nonlinear hyperbolic systems of balance laws 3.1 Nonlinear hyperbolic systems of interest 3.1.1 Balance laws in conservative or nonconservative forms 3.1.2 Hyperbolicity 3.1.3 Examples of conservative models 3.1.4 Examples of nonconservative models 3.2 Nonlinearity conditions and examples 3.2.1 Nonlinearity conditions 3.2.2 Examples 3.3 Convex and nonconvex equations of state 3.3.1 Fluid dynamics equations: ideal gas 3.3.2 Fluid dynamics equations: stiffened fluids 3.3.3 Fluid dynamics equations: Hayes fluids 3.3.4 Fluid dynamics equations: van der Waals fluids 3.4 Weak solutions and elementary waves 3.4.1 Weak solutions of systems of conservation laws 3.4.2 Weak solutions of nonconservative systems of balance laws 3.4.3 Integral curves and rarefaction waves 3.4.4 Shock waves and Hugoniot curves 3.4.5 Non-uniqueness of weak solutions 3.5 Admissibility criteria and Riemann problem 3.5.1 Entropy conditions 3.5.2 Entropy inequality and the concept of nonclassical shocks 3.5.3 Illustration of admissibility criteria: isothermal van der Waals fluids 3.5.4 Entropy condition for nonconservative systems 3.5.5 The Riemann problem 3.6 Bibliographical notes4 Riemann problem for ideal fluid 4.1 Introduction 4.2 Riemann problem for isentropic ideal flows 4.2.1 Hyperbolicity and genuine nonlinearity 4.2.2 Rarefaction waves 4.2.3 Shock waves 4.2.4 Wave curves and Riemann problem 4.3 Riemann problem for polytropic ideal fluids 4.3.1 Basic properties 4.3.2 Rarefaction waves 4.3.3 Hugoniot curves 4.3.4 Admissible shock waves 4.3.5 Solutions of the Riemann problem 4.4 Bibliographical notes5 Compressible fluids governed by a general equation of state 5.1 Introduction 5.2 Riemann problem for isentropic van der Waals fluids 5.2.1 Preliminaries 5.2.2 Rarefaction waves 5.2.3 Shock waves 5.2.4 Composite waves and Riemann problem 5.3 General EOS: rarefaction curves 5.3.1 Parameterization by the specific volume 5.3.2 Parameterization by the pressure 5.4 General EOS: shock curves 5.4.1 Hugoniot curves 5.4.2 Admissibility criteria 5.5 Wave curves and Riemann problem 5.5.1 Wave curves 5.5.2 Riemann problem 5.6 Bibliographical notes6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime 6.1 Introduction 6.2 Background 6.2.1 Strict hyperbolicity and non-genuine nonlinearity 6.2.2 Hugoniot curves 6.2.3 Lax's shock inequalities and Lax shocks 6.2.4 Integral curves and rarefaction waves 6.3 Admissible shocks satisfying a single entropy inequality 6.3.1 The entropy inequality 6.3.2 Sets of admissible shock waves 6.3.3 Non-Lax shocks and composite waves 6.3.4 Two-parameter sets of waves 6.4 General Riemann solver based on a kinetic relation 6.4.1 Kinetic relation imposed from left-to-right of the shock 6.4.2 Construction of wave curves 6.4.3 General Riemann solvers 6.4.4 Classical Riemann solver based on Wendroff's construction 6.5 Bibliographical notes7 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic-elliptic regime 7.1 Introduction 7.2 Background 7.2.1 Elliptic-hyperbolic model 7.2.2 Hugoniot curves and integral curves 7.2.3 Tangent functions 7.2.4 Lax shocks and rarefaction waves 7.3 Admissible shocks satisfying an entropy inequality 7.3.1 The entropy inequality 7.3.2 Properties of entropy dissipation and the set of admissible shocks 7.3.3 Subsonic phase boundaries and composite waves 7.3.4 Two-parameter sets of waves 7.4 Riemann solver based on a kinetic relation 7.4.1 Kinetic relation imposed from back side to front side of the shock 7.4.2 Construction of wave curves 7.4.3 Riemann solver based on a kinetic relation 7.4.4 Classical Riemann solver based on stationary phase boundaries 7.5 Bibliographical notes8 Compressible fluids in a nozzle with discontinuous cross-section: Isentropic flows 8.1 Introduction 8.2 Basic properties 8.2.1 Model reduced to a system in nonconservative form 8.2.2 A non-strictly hyperbolic system 8.2.3 Rarefaction waves 8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves 8.3.1 Shock waves with constant cross-section 8.3.2 Shock waves with zero speed: stationary contact waves 8.3.3 Two-parameter sets of composite waves 8.3.4 Admissibility Criterion for stationary contacts 8.4 Solutions of the Riemann problem 8.4.1 Construction 1: supersonic/supersonic 8.4.2 Construction 2: supersonic/subsonic 8.4.3 Construction 3: resonant waves for supersonic regime 8.4.4 Existence and uniqueness properties: supersonic regime 8.4.5 Construction 4: subsonic/supersonic 8.4.6 Construction 5: subsonic/subsonic 8.4.7 Construction 6: resonant waves for subsonic regime 8.4.8 Existence and uniqueness properties for subsonic regime 8.5 Bibliographical notes9 Compressible fluids in a nozzle with discontinuous cross-section—General flows 9.1 Basic properties 9.1.1 Hyperbolicity and non-strict hyperbolicity 9.1.2 Nonlinearity and linear degeneracy of characteristic fields 9.2 Elementary waves 9.2.1 Rarefaction waves 9.2.2 Shock waves 9.2.3 Stationary contact waves 9.2.4 Example: selection of admissible stationary contacts 9.3 Riemann problem 9.3.1 Construction 1: supersonic/supersonic 9.3.2 Construction 2: supersonic/subsonic 9.3.3 Construction 3: resonant waves for supersonic regime 9.3.4 Existence and uniqueness properties: supersonic regime 9.3.5 Construction 4: subsonic/supersonic 9.3.6 Construction 5: subsonic/subsonic 9.3.7 Construction 6: resonant waves for subsonic regime 9.3.8 Existence and uniqueness properties: subsonic regime 9.4 Quantitative properties 9.4.1 Entropy inequality in divergence form 9.4.2 Minimum entropy principle 9.5 Bibliographical notes10 Shallow water flows with discontinuous topography 10.1 Basic properties 10.1.1 System in nonconservative form 10.1.2 Non-hyperbolicity and genuine nonlinearity of characteristic fields 10.2 Elementary waves 10.2.1 Rarefaction waves 10.2.2 Shock waves 10.2.3 Stationary contact waves 10.2.4 Monotonicity of the composite wave curves 10.3 Riemann problem 10.3.1 Construction 1: supercritical/supercritical 10.3.2 Construction 2: supercritical/subcritical 10.3.3 Construction 3: resonant waves for supercritical regime 10.3.4 Existence and uniqueness properties for supercritical regime 10.3.5 Construction 4: subcritical/supercritical 10.3.6 Construction 5: subcritical/subcritical 10.3.7 Construction 6: resonant waves in the subcritical regime 10.3.8 Existence and uniqueness properties for the subcritical regime 10.4 Bibliographical notes11 Shallow water flows with temperature gradient 11.1 Basic properties and elementary waves 11.1.1 Hyperbolicity and genuine nonlinearity 11.1.2 Rarefaction waves 11.1.3 Shock waves and material contact discontinuities 11.1.4 Stationary contact discontinuities 11.2 Admissible stationary contact discontinuities 11.3 Riemann problem 11.3.1 Riemann problem for flat bottom 11.3.2 Riemann problem for discontinuous bottom 11.3.3 Construction 1: supercritical/supercritical 11.3.4 Construction 2: supercritical/subcritical 11.3.5 Construction 3: resonant waves in supercritical regime 11.3.6 Construction 4: subcritical/supercritical 11.3.7 Construction 5: subcritical/subcritical 11.3.8 Construction 6: resonant waves in subcritical regime 11.4 Bibliographical notes12 Baer-Nunziato model of two-phase flows 12.1 Introduction 12.2 Preliminaries 12.2.1 Two-phase flow models 12.2.2 Properties of the isentropic Baer-Nunziato model 12.2.3 Rarefaction waves 12.3 Shock waves and solid contacts 12.3.1 Shock waves 12.3.2 Solid contact waves 12.4 Riemann problem based on a phase decomposition 12.4.1 Phase decomposition 12.4.2 Solutions containing a supersonic solid contact 12.4.3 Solutions containing a subsonic solid contact 12.5 Bibliographical notesReferencesIndex*

Applied Mathematical Sciences

Philippe G. LeFloch Mai Duc Thanh

The Riemann Problem in Continuum Physics

Applied Mathematical Sciences Founding Editors F. John J. P. LaSalle L. Sirovich

Volume 219

Series Editors Anthony Bloch, Department of Mathematics, University of Michigan, Ann Arbor, MI, USA C. L. Epstein, Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, Department of Mathematics, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, NY, USA Advisory Editors J. Bell, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA, USA P. Constantin, Department of Mathematics, Princeton University, Princeton, NJ, USA R. Durrett, Department of Mathematics, Duke University, Durham, CA, USA R. Kohn, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA R. Pego, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA L. Ryzhik, Department of Mathematics, Stanford University, Stanford, CA, USA A. Singer, Department of Mathematics, Princeton University, Princeton, NJ, USA A. Stevens, Department of Applied Mathematics, University of M¨unster, M¨unster, Germany S. Wright, Computer Sciences Department, University of Wisconsin, Madison, WI, USA

The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of computer modeling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will provide an outlet for topics of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. These books should be accessible to readers versed in mathematics or science and engineering, and will feature a lively tutorial style, a focus on topics of current interest, and present clear exposition of broad appeal. A compliment to the Applied Mathematical Sciences series is the Texts in Applied Mathematics series, which publishes textbooks suitable for advanced undergraduate and beginning graduate courses.

Philippe G. LeFloch · Mai Duc Thanh

The Riemann Problem in Continuum Physics

Philippe G. LeFloch Laboratoire Jacques-Louis Lions Sorbonne University Paris, France

Mai Duc Thanh Department of Mathematics International University Ho Chi Minh City, Vietnam

ISSN 0066-5452 ISSN 2196-968X (electronic) Applied Mathematical Sciences ISBN 978-3-031-42524-0 ISBN 978-3-031-42525-7 (eBook) https://doi.org/10.1007/978-3-031-42525-7 Mathematics Subject Classification: 35-XX, 76-XX, 65-XX

© Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Contents

1

Overview of this monograph . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Models arising in fluid and solid dynamics . . . . . . . . . . . . . 2.1 Derivation of the equations governing ﬂuid ﬂows . . . . . . . . . 2.1.1 Material derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Reynolds’ transport theorem . . . . . . . . . . . . . . . . . . . . 2.1.3 Conservation law of mass . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Balance law of momentum . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Balance law of energy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Governing equations and systems of conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fundamental notions from thermodynamics . . . . . . . . . . . . . 2.2.1 Equation of state and thermodynamic identity . . . . 2.2.2 Helmholtz free energy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The class of ideal ﬂuids . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 The class of stiﬀened ﬂuids . . . . . . . . . . . . . . . . . . . . . . 2.2.5 The class of Hayes ﬂuids . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 The class of Van der Waals ﬂuids . . . . . . . . . . . . . . . . 2.3 Physical models of particular interest . . . . . . . . . . . . . . . . . . . 2.3.1 Fluid equations in Eulerian coordinates . . . . . . . . . . . 2.3.2 Fluid equations in Lagrangian coordinates . . . . . . . . 2.3.3 Viscous-capillary models . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Fluid ﬂows in a nozzle with variable cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Shallow water model for variable topography . . . . . . 2.3.6 A model of nonlinear elastodynamics with phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 9 11 16 17 19

3

Nonlinear hyperbolic systems of balance laws . . . . . . . . . . 3.1 Nonlinear hyperbolic systems of interest . . . . . . . . . . . . . . . . 3.1.1 Balance laws in conservative or nonconservative forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Examples of conservative models . . . . . . . . . . . . . . . . . 3.1.4 Examples of nonconservative models . . . . . . . . . . . . . 3.2 Nonlinearity conditions and examples . . . . . . . . . . . . . . . . . . 3.2.1 Nonlinearity conditions . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 23 23 24 25 25 26 27 27 27 28 30 31 34 38 40 41 41 41 42 43 47 51 51 52

v

vi

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3.3

Convex and nonconvex equations of state . . . . . . . . . . . . . . . 3.3.1 Fluid dynamics equations: ideal gas . . . . . . . . . . . . . . 3.3.2 Fluid dynamics equations: stiﬀened ﬂuids . . . . . . . . . 3.3.3 Fluid dynamics equations: Hayes ﬂuids . . . . . . . . . . . 3.3.4 Fluid dynamics equations: van der Waals ﬂuids . . . . Weak solutions and elementary waves . . . . . . . . . . . . . . . . . . 3.4.1 Weak solutions of systems of conservation laws . . . . 3.4.2 Weak solutions of nonconservative systems of balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Integral curves and rarefaction waves . . . . . . . . . . . . . 3.4.4 Shock waves and Hugoniot curves . . . . . . . . . . . . . . . . 3.4.5 Non-uniqueness of weak solutions . . . . . . . . . . . . . . . . Admissibility criteria and Riemann problem . . . . . . . . . . . . . 3.5.1 Entropy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Entropy inequality and the concept of nonclassical shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Illustration of admissibility criteria: isothermal van der Waals ﬂuids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Entropy condition for nonconservative systems . . . . 3.5.5 The Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 56 57 60 60

4

Riemann problem for ideal fluid . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Riemann problem for isentropic ideal ﬂows . . . . . . . . . . . . . . 4.2.1 Hyperbolicity and genuine nonlinearity . . . . . . . . . . . 4.2.2 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Wave curves and Riemann problem . . . . . . . . . . . . . . 4.3 Riemann problem for polytropic ideal ﬂuids . . . . . . . . . . . . . 4.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Hugoniot curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Admissible shock waves . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Solutions of the Riemann problem . . . . . . . . . . . . . . . 4.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 87 87 89 90 92 94 94 95 97 100 103 108

5

Compressible fluids governed by a general equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Riemann problem for isentropic van der Waals ﬂuids . . . . . 5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Composite waves and Riemann problem . . . . . . . . . .

109 109 110 110 111 113 115

3.4

3.5

3.6

63 64 66 71 72 72 74 76 83 83 85

Contents

5.3

5.4

5.5

5.6 6

7

vii

General EOS: rarefaction curves . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Parameterization by the speciﬁc volume . . . . . . . . . . 5.3.2 Parameterization by the pressure . . . . . . . . . . . . . . . . General EOS: shock curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Hugoniot curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Admissibility criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . Wave curves and Riemann problem . . . . . . . . . . . . . . . . . . . . 5.5.1 Wave curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Strict hyperbolicity and non-genuine nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Hugoniot curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Lax’s shock inequalities and Lax shocks . . . . . . . . . . 6.2.4 Integral curves and rarefaction waves . . . . . . . . . . . . . 6.3 Admissible shocks satisfying a single entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Sets of admissible shock waves . . . . . . . . . . . . . . . . . . . 6.3.3 Non-Lax shocks and composite waves . . . . . . . . . . . . 6.3.4 Two-parameter sets of waves . . . . . . . . . . . . . . . . . . . . 6.4 General Riemann solver based on a kinetic relation . . . . . . 6.4.1 Kinetic relation imposed from left-to-right of the shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Construction of wave curves . . . . . . . . . . . . . . . . . . . . . 6.4.3 General Riemann solvers . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Classical Riemann solver based on Wendroﬀ’s construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonclassical Riemann solver with prescribed kinetics—The hyperbolic-elliptic regime . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Elliptic-hyperbolic model . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Hugoniot curves and integral curves . . . . . . . . . . . . . . 7.2.3 Tangent functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Lax shocks and rarefaction waves . . . . . . . . . . . . . . . . 7.3 Admissible shocks satisfying an entropy inequality . . . . . . . 7.3.1 The entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . .

116 116 120 121 121 126 133 133 137 138 141 141 142 142 143 144 147 148 148 149 151 154 155 155 156 160 162 164 167 167 167 167 169 171 173 174 174

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Contents

7.4

7.5 8

7.3.2 Properties of entropy dissipation and the set of admissible shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Subsonic phase boundaries and composite waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Two-parameter sets of waves . . . . . . . . . . . . . . . . . . . . Riemann solver based on a kinetic relation . . . . . . . . . . . . . . 7.4.1 Kinetic relation imposed from back side to front side of the shock . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Construction of wave curves . . . . . . . . . . . . . . . . . . . . . 7.4.3 Riemann solver based on a kinetic relation . . . . . . . . 7.4.4 Classical Riemann solver based on stationary phase boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Compressible fluids in a nozzle with discontinuous cross-section: Isentropic flows . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Model reduced to a system in nonconservative form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 A non-strictly hyperbolic system . . . . . . . . . . . . . . . . . 8.2.3 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Shock waves with constant cross-section . . . . . . . . . . 8.3.2 Shock waves with zero speed: stationary contact waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Two-parameter sets of composite waves . . . . . . . . . . . 8.3.4 Admissibility Criterion for stationary contacts . . . . . 8.4 Solutions of the Riemann problem . . . . . . . . . . . . . . . . . . . . . 8.4.1 Construction 1: supersonic/supersonic . . . . . . . . . . . . 8.4.2 Construction 2: supersonic/subsonic . . . . . . . . . . . . . . 8.4.3 Construction 3: resonant waves for supersonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Existence and uniqueness properties: supersonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Construction 4: subsonic/supersonic . . . . . . . . . . . . . . 8.4.6 Construction 5: subsonic/subsonic . . . . . . . . . . . . . . . 8.4.7 Construction 6: resonant waves for subsonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.8 Existence and uniqueness properties for subsonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 181 183 184 184 185 191 193 194 195 195 197 197 198 200 203 203 205 213 215 215 216 218 221 222 225 228 231 232 234

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10

Compressible fluids in a nozzle with discontinuous cross-section—General flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Hyperbolicity and non-strict hyperbolicity . . . . . . . . 9.1.2 Nonlinearity and linear degeneracy of characteristic ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Elementary waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Stationary contact waves . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Example: selection of admissible stationary contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Construction 1: supersonic/supersonic . . . . . . . . . . . . 9.3.2 Construction 2: supersonic/subsonic . . . . . . . . . . . . . . 9.3.3 Construction 3: resonant waves for supersonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Existence and uniqueness properties: supersonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Construction 4: subsonic/supersonic . . . . . . . . . . . . . . 9.3.6 Construction 5: subsonic/subsonic . . . . . . . . . . . . . . . 9.3.7 Construction 6: resonant waves for subsonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.8 Existence and uniqueness properties: subsonic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Quantitative properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Entropy inequality in divergence form . . . . . . . . . . . . 9.4.2 Minimum entropy principle . . . . . . . . . . . . . . . . . . . . . 9.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shallow water flows with discontinuous topography . . . . 10.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 System in nonconservative form . . . . . . . . . . . . . . . . . 10.1.2 Non-hyperbolicity and genuine nonlinearity of characteristic ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Elementary waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Stationary contact waves . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Monotonicity of the composite wave curves . . . . . . . 10.3 Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Construction 1: supercritical/supercritical . . . . . . . . 10.3.2 Construction 2: supercritical/subcritical . . . . . . . . . .

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235 236 236 241 242 242 246 249 253 257 258 260 263 265 267 271 272 275 276 276 281 283 285 286 286 286 288 288 290 293 298 305 306 309

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10.3.3 Construction 3: resonant waves for supercritical regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 Existence and uniqueness properties for supercritical regime . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Construction 4: subcritical/supercritical . . . . . . . . . . 10.3.6 Construction 5: subcritical/subcritical . . . . . . . . . . . . 10.3.7 Construction 6: resonant waves in the subcritical regime . . . . . . . . . . . . . . . . . . . . . . . . 10.3.8 Existence and uniqueness properties for the subcritical regime . . . . . . . . . . . . . . . . . . . . . . . 10.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

12

311 314 316 319 322 324 326

Shallow water flows with temperature gradient . . . . . . . . 11.1 Basic properties and elementary waves . . . . . . . . . . . . . . . . . 11.1.1 Hyperbolicity and genuine nonlinearity . . . . . . . . . . . 11.1.2 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Shock waves and material contact discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Stationary contact discontinuities . . . . . . . . . . . . . . . . 11.2 Admissible stationary contact discontinuities . . . . . . . . . . . . 11.3 Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Riemann problem for ﬂat bottom . . . . . . . . . . . . . . . . 11.3.2 Riemann problem for discontinuous bottom . . . . . . . 11.3.3 Construction 1: supercritical/supercritical . . . . . . . . 11.3.4 Construction 2: supercritical/subcritical . . . . . . . . . . 11.3.5 Construction 3: resonant waves in supercritical regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Construction 4: subcritical/supercritical . . . . . . . . . . 11.3.7 Construction 5: subcritical/subcritical . . . . . . . . . . . . 11.3.8 Construction 6: resonant waves in subcritical regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327 328 328 330

Baer-Nunziato model of two-phase flows . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Two-phase ﬂow models . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Properties of the isentropic Baer-Nunziato model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Shock waves and solid contacts . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Solid contact waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 361 362 362

331 335 337 340 341 343 344 345 345 346 348 348 349

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12.4 Riemann problem based on a phase decomposition . . . . . . . 12.4.1 Phase decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Solutions containing a supersonic solid contact . . . . 12.4.3 Solutions containing a subsonic solid contact . . . . . . 12.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

374 374 376 379 381

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Overview of this monograph

The Riemann problem In the realm of physical phenomena, one often encounters events such as gas pipe explosions, or the potential rupture of a dam in a hydroelectric plant. The common thread in these scenarios is the breakdown of materials under fluid pressure. Our interest lies in the mathematical analysis of such phenomena, which naturally leads us to the study of the Riemann problem in continuum physics. The Riemann problem, named after B. Riemann who first investigated it in his “Gesammelte Werke” (1896, pp.149ff), poses the following scenario: a tube, containing gas at rest and divided by a thin membrane, suddenly has the membrane ruptured. The question then is, what becomes of the gas? The Riemann problem can be more generally viewed as an initial-value problem for a hyperbolic system of nonlinear conservation laws with (vector-valued) unknown u = u(x,t), spatial variable x, and time variable t > 0 and the following initial data: uL , x < 0, (1.0.1) u(x, 0) = uR , x > 0. Here, uL and uR are prescribed constant states. In continuum physics, balance laws arise from fundamental principles concerning the mass, momentum, and energy of a fluid flow. They may take the form of conservation laws or, more generally, they can handle evolution phenomena where the quantity in question is not conserved in time. For instance, the fluid equations include suitable source terms when the fluid travels through a tube with non-uniform cross-section, or when the riverbed beneath the flow of water (for instance) is irregular, and in turn nonconservative effects take place. Furthermore, in multi-phase flows, different fluids may exchange momentum and energy during propagation. Balance laws often need to be studied as a coupled system, while a single balance law may be relevant to provide a simplified model which helps to develop the © Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 1

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1 Overview of this monograph

analytical and numerical techniques that are extended next to more realistic models. The systems of balance laws of interest can often be represented as

∂ u ∂ f (u) ∂u + = S(u) , ∂t ∂x ∂x

x∈R I , t > 0.

(1.0.2)

In this system of equations, the left-hand side characterizes the conservative aspect of the system, while the right-hand side describes the sources. One of the most intriguing phenomena in the study of compressible fluids and related materials is the formation and propagation of shock waves. A shock wave is a time-dependent, discontinuous solution of the typical form u− , x < σ t, u(x,t) = (1.0.3) u+ , x > σ t. In this expression, u− , u+ are the left-hand and right-hand constant states, respectively, and σ = σ (u− , u+ ) is a real number, denoting the shock speed. A shock wave is a propagating jump discontinuity, which arises at a time and location of breakdown in the fluid and, in a description in one space dimension, may be moving either to the left or right from this point. However, not every initial discontinuity will evolve simply into a shock wave. In observing a dam rupture, for instance, one might witness areas where the water flows smoothly, a phenomena called “rarefaction waves”. Consequently, solutions to the Riemann problem can encompass both shock waves and rarefaction waves. This makes the Riemann problem a rich field of study, characterized by its diverse features that depend on a multitude of application domains. In our investigation of “wave-like” solutions, we seek to decompose the solutions of the Riemann problem into shock waves and rarefaction waves. This necessitates that the system of balance laws be hyperbolic within the domain of interest. However, over the entire domain of physical variables, the system may be entirely hyperbolic or be of elliptic-hyperbolic type. It is essential to possibly include elliptic regions in the phase space, since elliptic-hyperbolic systems of conservation laws find application in the modeling of phase transition problems. In such problems, the solution is required to cross a non-stable region sandwiched between two stable ones. The system maintains strict hyperbolicity in the stable regions, while being elliptic in nature in the non-stable region. This is one of the origin of the phase boundaries, or nonclassical shocks, under study in the present monograph. Owing to the simplified form of the initial data, the Riemann problem can be solved more efficiently than a general initial-value problem. This monograph is devoted to solving the Riemann problem within the context of continuum physics and, while we cannot cover all relevant models of balance laws, we do provide a comprehensive treatment of the most fundamental and important models. The significance of the Riemann problem extends beyond the sole construction of solutions from elementary waves such as shocks and rarefaction waves. It is also fundamental in understanding the general initial-value problem for nonlinear hyperbolic system. The Riemann problem also provides a key ingredient in order to design

1 Overview of this monograph

3

numerical methods for approximating solutions to the initial-value problem, such as the high-order Godunov method and its many variants. The methodology involves discretizing the initial data into constant states between adjacent mesh points, solving local Riemann problems at each grid point, and then integrating these local solutions to construct approximate solutions for the initial-value problem.

Outline of this monograph Specifically, in Chapter 2, we begin by deriving the governing equations for fluid flows from the fundamental laws of continuum physics. These equations encompass balance laws for mass, momentum, and energy of the unknown fields and are expressed as follows: ∂ρ + ∇ · (ρ u) = 0, ∂t ∂ (ρ u) + ∇ · (ρ uuT + pI) = ∇ · λ ∗ ∇ · uI + η ∇u + (∇u)T + ρ k, ∂t ∂ (ρ e) + ∇ · (κ ∇T ) + ρ (u · k), + ∇ · (ρ e + p)u = ∇ · u · λ ∗ ∇ · uI + η ∇u + (∇u)T ∂t

(1.0.4)

where I denotes the 3 × 3 identity matrix. In the system above, ρ represents the fluid density and u denotes the particle velocity, while p stands for the pressure, ε is the internal energy, e = ε + u|2 /2 is the total energy, T is the temperature, k is the mass body force, η is the shear viscosity coefficient, ηB = λ ∗ + (2/3)η is the bulk viscosity coefficient, and κ is the thermal conductivity coefficient. Additionally, we derive several models that will be further explored in subsequent chapters. We also review key concepts from thermodynamics and present various equations of state. In Chapter 3, we delve into the theory of nonlinear hyperbolic systems of balance laws. The system under consideration is derived from (1.0.4), with viscosity and heat conduction effects neglected. This can be expressed as a nonconservative system as follows:

∂u ∂u + A(u) = 0, x∈R I , t > 0. (1.0.5) ∂t ∂x Here, Ω is an open subset of R I n , and A = A(u) is an n × n matrix-valued map. We present the formulation of weak solutions, encompassing elements such as shock waves, contact discontinuities, rarefaction waves, characteristic fields, and more. As weak solutions containing shocks may not be unique, it becomes necessary to introduce an admissibility criterion. Accordingly, we discuss several types of admissibility criteria for shock waves, supplemented with illustrative examples. In particular, we present the concepts of classical and nonclassical shock waves. Chapters 4 and 5 focus on the Riemann problem as it pertains to fluid flows, governed by the following equations:

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1 Overview of this monograph

∂ ρ ∂ (ρ u) + = 0, ∂t ∂x ∂ (ρ u) ∂ (ρ u2 + p) + = 0, ∂t ∂x ∂ (ρ e) ∂ ((ρ e + p)u) + = 0. ∂t ∂x

(1.0.6)

In Chapter 4, we explore the case of an ideal fluid, while Chapter 5 broadens the scope to consider a general equation of state (EOS). The material presented in these chapters builds upon and extends existing theory, providing a review of the Riemann problem as it applies to gas dynamics equations. A range of EOSs, both convex and nonconvex, will be examined. We demonstrate that the Riemann problem for an isentropic fluid, as well as for an isentropic van der Waals fluid, admits a unique solution for large data. In Chapter 6 we treat the Riemann problem with nonclassical shock waves for the p-system arising, for instance, in elastodynamics. The governing equations are defined by ∂t v − ∂x σ (w) = 0, (1.0.7) ∂t w − ∂x v = 0. In this system, v and w > −1 represent the velocity and deformation gradient of the material, respectively. While the system is strictly hyperbolic, it is not genuinely nonlinear since the stress-strain function of typical interest admits an inflection point. • A classical Riemann solver can be implemented based on Wendroff’s approach, essentially allowing the shock speed to match the characteristic speed on one side of the shock, which corresponds to an equality in the more standard Lax shock inequalities. • However, more generally, the focus of this chapter is on nonclassical shocks which do not meet standard admissibility criteria, such as the Lax shock inequalities and the Liu entropy condition. Nonclassical shocks are undercompressive in nature, and were investigated systematically in material science by Abeyaratne, Knowles, and Truskinovsky and, mathematically, by LeFloch, Shearer, and their collaborators. We will combine shock satisfying Lax inequalities with nonclassical shocks that are selected by a so-called kinetic relation. • This latter notion is central throughout the following monograph and, as we will show it, play a fundamental role in solving the Riemann problem for several model of fluids and materials. Typically, we will construct a one-parameter family of solutions to the Riemann problem and next impose a kinetic condition (possibly together with a nucleation criterion) in order to finally select a unique solution. Interestingly, a specific choice of the kinetic function leads to the classical Riemann solver. Chapter 7 focuses on the Riemann problem for a hyperbolic-elliptic model of phase transition dynamics, which includes nonclassical shock waves. This model is a typical p-system as defined in equation (1.0.7), where the stress-strain curve

1 Overview of this monograph

5

can exhibit both increasing and decreasing intervals. This characteristic causes the system to behave hyperbolically in two regions that are separated by an elliptic region. Even when a kinetic relation is imposed, there may be instances of nonexistence. To address this, zero-speed shock waves are permitted to complete the wave curves. An intriguing phenomenon observed here is that the Riemann problem can potentially have two distinct solutions. Chapters 8 and 9 are dedicated to the Riemann problem for the model of fluid flow in a nozzle with a discontinuous cross-section. The main subject of study in these chapters are the nonconservative terms. Chapter 8 focuses on the case of ideal and isentropic fluids, whereas Chapter 9 examines the case of non-isentropic fluid flows. The governing equations read as follows:

∂t (aρ ) + ∂x (aρ u) = 0, ∂t (aρ u) + ∂x (a(ρ u2 + p)) = p(ρ )∂x a, ∂t (aρ e) + ∂x (au(ρ e + p)) = 0, x ∈ R I , t > 0.

(1.0.8)

Here, ρ , ε , T, S, and p denote the thermodynamical variables, that is, the density, internal energy, absolute temperature, entropy, and pressure, respectively, while u is the velocity, e = ε + u2 /2 the total energy, and a = a(x) > 0 the cross-sectional area of the nozzle. • The system described by these equations includes a nonconservative source term on its right-hand side, which is the product of a potentially discontinuous function and the spatial partial derivative of another discontinuous function. This nonconservative source term encapsulates the impact of the nozzle’s geometry on the fluid’s dynamics. The system can be reformulated into a nonconservative system of balance laws, and a broader definition of weak solutions, in terms of Dal Maso-LeFloch-Murat nonconservative products, may be applied. • It is important to observe that the system is not strictly hyperbolic, as the characteristic fields may overlap at certain points. In both isentropic and non-isentropic scenarios, solutions to the Riemann problem for large data are derived. A resonance phenomenon is observed, due to the presence of multiple waves associated with different characteristic families but all propagating at the same shock speed within a single solution. • Additionally, several qualitative properties of the entropy solutions are established. These include, on the one hand, a standard entropy inequality associated with the fluid equations, which takes a divergence form, and, on the other hand, a minimum entropy principle which demonstrates that entropy increases with respect to the time variable. In Chapter 10, we turn our attention to the Riemann problem for shallow water flows with discontinuous topography. The equations in this chapter again contains a source term, which introduces a significant challenge into the conventional shallow water model. In terms of practical applications, the model studied in this chapter is relevant in the area of water resource engineering. It is particularly useful in understanding phenomena such as dam breaking and open channel flows, among others.

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1 Overview of this monograph

Herein, we thereby seek to not only provide a mathematical framework to study the Riemann problem, but also to contribute to advance real-world engineering challenges. The governing equations are provided as follows:

∂t h + ∂x (hu) = 0, gh ∂t (hu) + ∂x h(u2 + ) + gh ∂x a = 0. 2

(1.0.9)

Here, h denotes the height of the water from the bottom to the surface, u is the fluid velocity, g is the gravity constant, and a = a(x) > 0 is the height of the topography bottom from a given level. The function a(x) is assumed to be piecewise constant of the form aL , x < 0, a(x) = aR , x > 0, where aL , aR are constants. In the presence of a discontinuous bottom, the nonconservative term within the momentum equation induces additional complexity in the dynamics of the problem and in the wave patterns. When solving the Riemann problem for large data we discover that the set of solutions reduces to a unique solution, for certain Riemann data but, often for a range of data, may contain up to three distinct solutions. Moreover, the resonance phenomenon reveals itself in that the Riemann problem can exhibit multiple waves propagating with identical shock speed. Chapter 11 continues to delve into the crucial nature of source terms in fluid evolution equations, and we then shift our focus toward the Riemann problem within the context of shallow water equations—this time with variable topography and horizontal temperature gradient included. The framework for our investigation is the well-known Ripa system, which was originally introduced to provide a mathematical model for ocean currents. It can be derived from the Saint-Venant system of shallow water equations, but with added horizontal water temperature fluctuations. This feature leads to a more comprehensive and realistic description. We work here with the following governing equations:

∂t h + ∂x (hu) = 0, g ∂t (hu) + ∂x hu2 + h2 θ = −ghθ ∂x a, 2 ∂t (hθ ) + ∂x (huθ ) = 0, x ∈ R I , t > 0.

(1.0.10)

Here, h, u and θ denote the water depth, the depth-averaged horizontal velocity, and the potential temperature field, respectively; g is the gravitational constant, and a is the bottom topography. The mathematical model (1.0.10) extends the framework considered in Chapter 10 and encompasses scenarios where the water’s behavior depends upon the temperature gradient.

1 Overview of this monograph

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The system under consideration is nonconservative and fails to be strictly hyperbolic. Among the four characteristic fields of this system, two exhibit linear degeneracy. The Ripa system exhibits an intriguing property that distinguishes it from the standard gas dynamics equations in a nozzle with variable cross-section. Specifically, across each contact discontinuity, denoted by λ = u, only the particle velocity remains continuous. This unique feature requires us to construct a novel type of composite wave curves. These curves bridge together waves associated with a nonlinear characteristic field and with the linearly degenerate field λ = u. The proposed approach is essential in order to determine the solution to the Riemann problem. We present all possible configurations of initial data and solutions for this rather intricate problem. Interestingly, the Riemann problem may admit up to three distinct solutions in certain cases, while we can still achieve uniqueness within specific regions. Chapter 12 tackles one of the most challenging models represented by nonconservative systems of balance laws: an example of multi-phase flow models. The complexity inherent in these models is due to the coexistence of distinct phases and the interplay among them. More specifically, Chapter 12 is dedicated to investigating a model of two-phase flows which is governed by six equations and is based on the balance of mass, momentum, and energy in each phase. The governing equations for this model are stated as follows:

∂t (α1 ρ1 ) + ∂x (α1 ρ1 u1 ) = 0, ∂t (α1 ρ1 u1 ) + ∂x (α1 (ρ1 u21 + p1 )) = P(U)∂x α1 , ∂t (α1 ρ1 e1 ) + ∂x (α1 u1 (ρ1 e1 + p1 )) = −P(U)∂t α1 , ∂t (α2 ρ2 ) + ∂x (α2 ρ2 u2 ) = 0,

(1.0.11)

∂t (α2 ρ2 u2 ) + ∂x (α2 (ρ2 u22 + p2 )) = −P(U)∂x α1 , ∂t (α2 ρ2 e2 ) + ∂x (α2 u2 (ρ2 e2 + p2 )) = P(U)∂t α1 . Here, αk , ρk , uk , pk , εk , Sk , Tk , ek = εk + u2k /2 (k = 1, 2) denote the volume fraction, density, velocity, pressure, internal energy, specific entropy, temperature, and total energy in the k-phase, respectively. The volume fractions satisfy the condition

α1 + α2 = 1.

(1.0.12)

Multi-phase flow models are represented by an extensive set of governing equations and often (but not always) exhibit complex properties due to the presence of nonconservative terms. The inherent complexity becomes more apparent in specific multiphase flow models which lack explicit expressions for their characteristic fields, and this significantly complicate the study of the Riemann problem. In Chapter 12, our primary focus is the Riemann problem for a model of twophase flows of the Baer-Nunziato type, a model originally designed for the dynamics of deflagration-to-detonation transitions in porous energetic materials. Unlike many multi-phase flow models, all characteristic fields of the BaerNunziato model can be expressed in explicit form. This helps us in our investigation

8

1 Overview of this monograph

of elementary waves and, by extension, the construction of the Riemann solutions. In particular, with this model, it becomes feasible to obtain explicit formulas for rarefaction waves. The system is then supplemented with the compaction dynamics equation. To handle shock waves in each phase, we introduce a phase decomposition approach and, in this setting, the solid contact waves act as a connecting link between the two phases. Our study leads us to the solutions of the Riemann problem in both subsonic and supersonic regions eventually via a numerical computation since a system of nonlinear algebraic equations must be solved. Furthermore, these equations pave the way for the development of efficient computational algorithms, and ultimately this chapter contributes to the development of computation-based Riemann solvers for particularly complex fluid flows.

In summary Our ambition with this work is to offer an accessible pathway to understanding the intricacies of a range of complex fluid flow models, including their derivations, fundamental characteristics, and the phenomena associated with shock waves and other elementary waves. We hope that this monograph enriches the theory of shock waves within the landscape of continuum physics. It is intended not only for those interested in constructing solutions to the Riemann problem, but also for a wide array of researchers across various scientific disciplines, ranging from applied mathematics to fluid dynamics. Consequently, we aspire for this book to serve as a valuable resource.

Chapter 2

Models arising in fluid and solid dynamics

We begin by introducing a list of models that describe complex flows in continuum physics, with the material being a gas, liquid, or solid. The governing equations of fluid flows are stated, consisting of balance laws for mass, momentum, and energy of the unknown fields. These equations express fundamental laws of continuum physics. Throughout this presentation, we discuss basic concepts of thermodynamics and provide examples of constitutive equations that describe the internal properties of materials. We present specific physical models for gases, liquids, and solids, including the system of fluid dynamics in Eulerian and Lagrangian coordinates. Additionally, we introduce a viscous-capillary model for fluid dynamics or elastodynamics, a model of fluid flow in a nozzle with variable cross-section, and the shallow water equations with variable topography in both two and one dimensions. It is important to note that the models discussed in this chapter and studied throughout this monograph can be written in either conservative or nonconservative form.

2.1 Derivation of the equations governing fluid flows 2.1.1 Material derivative A motivation for the concept of material derivatives can be based on the following example. The pressure acting on a moving particle at the position (x, y, z) in the three-dimensional space at the time t is a function p = p(x, y, z,t). Since the particle is moving, its position is a function of time, and we write

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 2

9

10

2 Models arising in fluid and solid dynamics

x = x(t),

y = y(t),

z = z(t).

Substituting these equations into the expression of the pressure p, we have p = p(x(t), y(t), z(t),t). Thus, the rate of change of the pressure on the moving particle with respect to time is given by the chain rule d p ∂ p ∂ p dx ∂ p dy ∂ p dz = + + + . dt ∂t ∂ x dt ∂ y dt ∂ z dt Here and throughout, the following notation for first-order partial derivatives is used

∂f = f x = ∂x f , ∂x together with the following notation for second-order partial derivatives:

∂2 f = fxx = ∂xx f , ∂ x2 ∂2 f ∂ ∂f = = ( fx )y = fxy = ∂yx f . ∂ y∂ x ∂ y ∂ x If the particle velocity is denoted by u = (u, v, w)T , then u=

dx , dt

v=

dy , dt

w=

dz . dt

Thus, the rate of change of the pressure can be computed as dp ∂ p ∂ p ∂p ∂p = + u(x, y, z,t) + v(x, y, z,t) + z(x, y, z,t). dt ∂t ∂x ∂y ∂z This quantity is distinct from the partial derivative with respect to time ∂ /∂ t, and we refer to the above rate of change as the material derivative of the pressure, which is denoted by Dp/Dt. So, we have

∂p Dp := + u · ∇p, Dt ∂t where ∇p = (∂ p/∂ x, ∂ p/∂ y, ∂ p/∂ z)T is the gradient of the function p with respect to the space variable. In general, let ϕ = ϕ (x, y, z,t) be a certain scalar field defined on a moving particle whose position at the time t is x = (x, y, z), where its coordinates are functions of time: x = x(t), y = y(t), z = z(t). Applying the chain rule, we see that the rate of change of ϕ with respect to time is

2.1 Derivation of the equations governing fluid flows

11

dϕ ∂ ϕ ∂ ϕ dx ∂ ϕ dy ∂ ϕ dz = + + + . dt ∂t ∂ x dt ∂ y dt ∂ z dt Now, let a scalar field ϕ be defined in a region, together with a vector field u = (u, v, w)T (x,t). Then, the value of ϕ at a moving point at the time t is ϕ (x(t),t), and the total rate of change of ϕ = ϕ (x(t),t) with respect to time at the position x(t) is d ∂ϕ ∂ϕ ∂ϕ ∂ϕ ∂ϕ +u +v +w = + u · ∇ϕ . ϕ (x(t),t) = dt ∂t ∂x ∂y ∂z ∂t This is the material derivative of ϕ : Dϕ ∂ϕ := + u · ∇ϕ , Dt ∂t

(2.1.1)

where ∇ϕ = (∂ ϕ /∂ x, ∂ ϕ /∂ y, ∂ ϕ /∂ z).

2.1.2 Reynolds’ transport theorem Reynolds’ transport theorem is established from a general transport theorem, which will play a key role in the derivation of the governing equations of fluid flows in the next subsections. A derivation of this theorem is as follows. Consider (for the sake of simplicity) a scalar-valued and differentiable function f (x,t), x ∈ G ⊂ R I 3 ,t ∈ I for some interval I ⊂ R I . Let G = G(t) ⊂ G ,t ∈ I be a time-dependent region with a smooth boundary denoted by S(t). Here, n = n(x,t) is the outward unit normal vector (understood as a row vector, say) and u = u(x,t) is a velocity field defined in the region G (as a column vector, say). For any arbitrary and fixed time t and for a sufficiently small time increment Δ t > 0, we consider the sets G1 = G(t) \ G(t + Δ t),

G2 = G(t + Δ t) \ G(t),

where “\” is the standard operator associated with the difference of two sets. In other words, G1 is a region left by the moving volume G, and G2 is the new region occupied by the volume G from the time t to the new time t + Δ t. Then, we have G(t + Δ t) ∪ G1 = G(t) ∪ G2 , G(t + Δ t) ∩ G1 = G(t) ∩ G2 = 0. /

(2.1.2)

See Figure 2.1. Therefore, from (2.1.2) and the additivity property of the integrals, we get G(t+Δ t)∪G1

and

f (x,t + Δ t)dV =

G(t)∪G2

f (x,t + Δ t)dV,

12

2 Models arising in fluid and solid dynamics

Fig. 2.1 A moving volume G(t) to the new volume G(t + Δ t)

G(t+Δ t)∪G1

G(t)∪G2

f (x,t + Δ t)dV =

f (x,t + Δ t)dV =

G(t)

G(t+Δ t)

f (x,t + Δ t)dV +

f (x,t + Δ t)dV +

G2

f (x,t + Δ t)dV,

G1

f (x,t + Δ t)dV.

Thus, we find G(t+Δ t)

f (x,t + Δ t)dV =

G(t)

f (x,t + Δ t)dV +

G2

f (x,t + Δ t)dV −

G1

f (x,t + Δ t)dV.

Now, the material derivative is the total rate of change with respect to time: d 1 f (x,t)dV = lim f (x,t + Δ t)dV − f (x,t)dV Δ t→0 Δ t dt G(t) G(t+Δ t) G(t) 1 = lim f (x,t + Δ t)dV − f (x,t)dV Δ t→0 Δ t G(t) G(t) 1 + lim f (x,t + Δ t)dV − f (x,t + Δ t)dV , Δ t→0 Δ t G2 G1 therefore d dt

G(t)

1 ( f (x,t + Δ t) − f (x,t)) dV Δt 1 f (x,t + Δ t)dV − f (x,t + Δ t)dV + lim Δ t→0 Δ t G2 G1 ∂ f (x,t) 1 dV + lim f (x,t + Δ t)dV − f (x,t + Δ t)dV . = Δ t→0 Δ t ∂t G(t) G2 G1

f (x,t)dV =

lim

G(t) Δ t→0

(2.1.3)

2.1 Derivation of the equations governing fluid flows

13

The second term of (2.1.3) consists of the limit of the difference of two spatial integrals which can be transformed into a surface integral in the limit as Δ t → 0, as follows. For small Δ t > 0, the volume of the region G1 and G2 can be approximated by the sums of the volumes |Vi j | of the “cylinders” Vi j whose “bases” are pieces Si j of the surface S(t), i = 1, 2, . . . , m and j = 1, 2, . . . , n, and the heights are Δ tu · n, that is, the scalar projections of the vectors Δ tu onto n. Observe that Si j , i = 1, 2, . . . , m, j = 1, 2, . . . , n, form a partition of the surface S(t), and Δ tu is an approximation of the displacement of the base between the time t and t + Δ t. This means that |Vi j | ≈ (Δ tuT · n)A(Si j ) on G2 , |Vi j | ≈ −(Δ tuT · n)A(Si j ) on G1 ,

i = 1, . . . , k, j = 1, . . . , l, i = k + 1, . . . , m, j = l + 1, . . . , n,

where A(Si j ) denotes the area of Si j , for some integers k, l. The approximation is expected to improve as Δ t becomes smaller. Moreover, by continuity we see that G2 ∪ G1 → S(t), as Δ t → 0. These approximations yield 1 Δ t→0 Δ t

A := lim

G2

1 = lim Δ t→0 Δ t

f (x,t + Δ t)dV − ⎛ lim

⎝

k,l,m,n→∞

1 lim Δ t→0 Δ t m,n→∞

∑

G1

f (x,t + Δ t)dV ⎞

f (x,t + Δ t)|Vi j | −

i=1,k, j=1,l

∑

= lim

∑

f (x,t + Δ t)|Vi j |⎠

i=k+1,m, j=l+1,n

f (x,t + Δ t)(Δ tuT · n)A(Si j ) .

i=1,m, j=1,n

Observe that we can write the dot product uT · n as uT · n = nu, where nu is the matrix product of the row vector n with the column vector u. Substituting uT ·n = nu into the last equation, we can therefore evaluate the second term of (2.1.3) as 1 T f (x,t + A = lim lim Δ t)( Δ tu · n)A(S ) ij ∑ Δ t→0 m,n→∞ Δt i=1,m, j=1,n = lim lim

Δ t→0 m,n→∞

= lim

Δ t→0 S(t)

∑

( f (x,t + Δ t)(nu)A(Si j ))

i=1,m, j=1,n

f (x,t + Δ t)(nu) dS =

S(t)

f (x,t)(nu) dS.

The identity (2.1.3) together with the last equation establishes the general transport theorem: d ∂f dV + fdV = (nu)fdS. (2.1.4) dt G(t) G(t) ∂ t S(t) On the other hand, the transport theorem can be derived more directly in the onedimensional case. Namely, f = f (x,t), a = a(t), b = b(t), x ∈ R I ,t > 0, being differentiable functions, we have

14

2 Models arising in fluid and solid dynamics

d dt

b(t) a(t)

f dx =

b(t) ∂f a(t)

∂t

dx + b (t) f (b(t),t) − a (t) f (a(t),t)

with, for instance, a (t) = da/dt. Consider next a body of fluid which consists of identical particles and is separated from the rest of the flow by a closed surface. When the fluid evolves, the volume and the surface of the body depend on time. Let V = V (t) be the region occupied by the body of fluid and let S(t) be the surface of the body at the time t. Then, u is the particle velocity, and the formula (2.1.4) above is referred to as Reynolds’ transport theorem, which thus reads ∂f d dV + fdV = (nu)fdS, (2.1.5) dt V (t) V (t) ∂ t S(t) where, by convention, n is a row vector and u, f are column vectors. A rigorous proof of Reynolds’ transport theorem is as follows. Take an arbitrary time t0 ∈ I and set the reference volume and its boundary as the reference boundary: V0 = V (t0 ),

S0 = S(t0 ) = boundary of V0 .

The motion of particles can be represented by a sufficiently regular (of class C1 , say) transformation x0 ∈ V0 → x = ϕ (x0 ,t). Then V (t) is the transformation map for V0 , that is, V (t) = ϕ (V0 ,t). We suppose that the Jacobian of the transformation is non-vanishing, i.e., ∂ xi (x0 ,t) J(x0 ,t) := det

= 0 for all x0 ∈ V0 . ∂ x0 j i, j=1,3 See Figure 2.2.

Fig. 2.2 Change of domain of integration by the transformation x = ϕ (x0 ,t)

Without loss of generality, we can assume that J(x0 ,t) > 0 for all x0 ∈ V0 . Then, by a change of variable in the integrals we find

2.1 Derivation of the equations governing fluid flows

V (t)

f(x,t)dV =

V0

f(ϕ (x0 ,t),t)J(x0 ,t)dV,

15

x0 ∈ V0 .

Thus, by taking the derivative with respect to time, the last equation gives us d dt

V (t)

f(x,t)dV = =

d dt

V0

V0

f(ϕ (x0 ,t),t)J(x0 ,t)dV

∂ (f(ϕ (x0 ,t),t)J(x0 ,t)) dV, ∂t

where the latter equation comes from the fact that the region V0 is independent of time. Using the product rule in the integrand of the last equation yields d ∂ ∂ [f(ϕ (x0 ,t),t)]J(x0 ,t) + f(ϕ (x0 ,t),t) [J(x0 ,t)] dV. f(x,t)dV = dt V (t) ∂t V0 ∂ t Now, by elementary algebraic calculations we can check that Df(x,t) ∂ ∂ f(x,t) [f(ϕ (x0 ,t),t)] = + ∇f(x,t) · u(x,t) = ∂t ∂t Dt and that

∂ J(x0 ,t) = J(x0 ,t)∇ · u(ϕ (x0 ,t),t) = J(x0 ,t)∇ · u(x,t). ∂t This implies that d dt

V (t)

∂ [f(ϕ (x0 ,t),t)]J(x0 ,t) + f(ϕ (x0 ,t),t)J(x0 ,t)∇ · u(x,t) dV V0 ) ∂ t ∂ f(ϕ (x0 ,t),t) + f(ϕ (x0 ,t),t)∇ · u(x,t) J(x0 ,t)dV = V0 ∂ t Df(x,t) + f(x,t)∇ · u(x,t) dV, = Dt V (t)

f(x,t)dV =

where the last equation is obtained by a backward transformation in the domain V (t). As usual, the outer product of the two column vectors ⎛ ⎞ ⎛ ⎞ v1 u1 u = ⎝u2 ⎠ , v = ⎝v2 ⎠ u3 v3 is defined as

⎞ ⎛ ⎞ ⎛ u1 u1 v1 u1 v2 u1 v3 uvT = ⎝u2 ⎠ (v1 , v2 , v3 ) = ⎝u2 v1 u2 v2 u2 v3 ⎠ . u3 u3 v1 u3 v2 u3 v3

The following identity holds for the outer product of two vector fields u and v: ∇ · (uvT ) = u(∇ · v) + ∇u · v.

16

2 Models arising in fluid and solid dynamics

Thus, if f and u are column vectors, we can write d ∂f ∂f + ∇f · u + f∇ · u dV = + ∇ · (fuT ) dV. f(x,t)dV = dt V (t) V (t) ∂ t V (t) ∂ t (2.1.6) Using the divergence theorem, from the equation (2.1.6) we obtain d dt

V (t)

f(x,t)dV =

∂f dV + V (t)) ∂ t

S(t)

(fuT ) · ndS.

Here, n = (n1 , n2 , n3 ) is a row vector and (fuT ) · n is a column vector whose i-component is the dot product of the ith row of the matrix (fuT ), that is, fi uT , with the row vector n (i = 1, 2, 3). It is easily checked that (fuT ) · n = (n · uT )f = (nu)f. The time derivative becomes d dt

V (t)

f(x,t)dV =

∂f dV + V (t)) ∂ t

S(t)

(nu)fdS,

which establishes (2.1.5), and also (2.1.4) where V (t) is replaced by G(t).

2.1.3 Conservation law of mass We denote by ρ = ρ (x, y, z,t) the density of the fluid flow under consideration. The mass m of the body of fluid is m=

V (t)

ρ (x, y, z,t)dV.

(2.1.7)

The principle of conservation of mass states that the mass of the body remains constant. This means that the material derivative of the mass function m vanishes identically, i.e., Dm = 0. (2.1.8) Dt From (2.1.5) it then follows that D Dm = Dt Dt

V

ρ dV =

V

∂ρ dV + ∂t

S

(n · ρ u) dS = 0.

Assume, for simplicity, that the volume occupied by the fluid body is fixed with a smooth boundary. Applying Gauss’ theorem, we have V

∇ · (ρ u)dV =

S

(n · ρ u) dS,

2.1 Derivation of the equations governing fluid flows

17

where ∇ · (.) = div(.) denotes the divergence of a vector field. Thus, the principle of conservation of mass is nothing but ∂ρ + ∇ · (ρ u) dV = 0. (2.1.9) ∂t V Since V is arbitrary, the equation (2.1.9) leads us to a differential form of the conservation of mass:

∂ρ + ∇ · (ρ u) = 0. ∂t

(2.1.10)

2.1.4 Balance law of momentum The balance law of momentum is obtained by stating that the rate of change of the momentum P of a body in an inertial frame is equal to the total acting force F on this body, that is, DP = F. (2.1.11) Dt The momentum of the body occupying a region V (t) is P=

V (t)

ρ udV.

(2.1.12)

The acting forces are classified into two types: body forces denoted by F1 and surface or contact forces denoted by F2 . An important body force is the gravitational force. The surface forces are exerted from the other bodies on the surface of the fluid under consideration. Let k = limΔ m→0 ΔΔFm1 be the mass body force, and let t = limΔ S→0 ΔΔFS2 be the stress vector. The total force F is thus F=

V

ρ k dV +

t dS.

(2.1.13)

S

Then, the balance law of momentum is expressed as D Dt

V

ρ udV =

V

ρ k dV +

t dS. S

Using (2.1.5), we can re-write the left-hand side of this equation as V

∂ (ρ u) dV + ∂t

S

(n · u)ρ udS =

V

ρ k dV +

t dS. S

It can be proven (see, for instance, [295]) that the stress tensor on the surface S can be expressed as t = n · T, for some matrix T. From these observations, the balance law of momentum thus reads

18

2 Models arising in fluid and solid dynamics

V

∂ (ρ u) dV + ∂t

S

ρ u(n · u) dS =

V

ρ k dV +

S

n · TdS.

(2.1.14)

We will now check that

ρ u(n · u) = n · (ρ uuT ),

(2.1.15)

where uuT is the usual matrix product of the column vector u = (u, v, w)T and the row vector uT = (u, v, w), that is, ⎞ ⎛ ⎞ ⎛ 2 u uv uw u uuT = ⎝ v ⎠ (u, v, w) = ⎝ vu v2 vw⎠ . w wu wv w2 Thus, letting n = (n1 , n2 , n3 ), we have ⎞ u2 uv uw n · (ρ uuT ) = ρ n · (uuT ) = ρ (n1 , n2 , n3 ) ⎝ vu v2 vw⎠ wu wv w2 ⎛

= ρ (n1 u2 + n2 vu + n3 wu, n1 uv + n2 v2 2 + n3 wv, n1 uw + n2 vw + n3 w2 ) = ρ (u(n1 u + n2 v + n3 w), v(n1 u + n2 v + n3 w), w(n1 u + n2 v + n3 w)) = ρ (n1 u + n2 v + n3 w)(u, v, w) = ρ u(n · u), which establishes (2.1.15). Substituting (2.1.15) into (2.1.14), we obtain the balance of momentum in the form V

∂ (ρ u) dV + ∂t

S

n · (ρ uuT ) dS =

V

ρ k dV +

S

n · TdS.

Without loss of generality, we can consider a fixed volume V with smooth surface S. Applying Gauss’ theorem, from the last equation we deduce that ∂ (ρ u) T + ∇ · (ρ uu ) − ρ k − ∇ · T dV = 0. (2.1.16) ∂t V Furthermore, the stress t in a fluid is generated by the fluid dynamical pressure p and the viscous stress τ , and T can be decomposed as T = −pI + τ .

(2.1.17)

Here, pI is isotropic and τ is the viscous stress tensor, with ⎞ ⎛ ⎛ ⎞ τ11 τ12 τ13 100 τ = ⎝τ21 τ22 τ23 ⎠ . I = ⎝0 1 0⎠ , τ31 τ32 τ33 001 Substituting (2.1.17) into (2.1.16), we arrive at the integral form of the balance law of momentum

2.1 Derivation of the equations governing fluid flows

V

19

∂ (ρ u) + ∇ · (ρ uuT + pI) − ρ k − ∇ · τ dV = 0. ∂t

(2.1.18)

Since the domain of integration V is arbitrary, (2.1.18) is equivalent to the following differential form of the balance law of momentum:

∂ (ρ u) + ∇ · (ρ uuT + pI) = ∇ · τ + ρ k. ∂t

(2.1.19)

The equation (2.1.19) can be expanded further in the form of the following CauchyPoisson law (see [295], pp. 77): T = (−p + λ ∗ ∇ · u)I + 2η E.

(2.1.20)

From (2.1.17) and (2.1.20) it follows that the viscous stress τ reads

τ = λ ∗ ∇ · uI + 2η E,

(2.1.21)

where λ ∗ and η are scalar functions of the thermodynamic state of the fluid, and E is the deformation tensor. The constant η is called the shear viscosity, while ηB = λ ∗ + 23 η is the bulk viscosity. Observe that in a Newtonian fluid the viscosity is independent of the velocity and the stress in the fluid. So, the viscous stress of a Newtonian fluid is linearly proportional to the local strain rate. The strain rate tensor E is given by 1 ∇u + (∇u)T , E= 2 where ∇u = (∇u, ∇v, ∇w)T . Thus, the viscous stress τ is given by

τ = λ ∗ ∇ · uI + η ∇u + (∇u)T .

(2.1.22)

In conclusion, the balance law of momentum (2.1.19) can be written in the form

∂ (ρ u) + ∇ · (ρ uuT + pI) = ∇ · λ ∗ ∇ · uI + η ∇u + (∇u)T + ρ k. ∂t

(2.1.23)

Observe that the vector equation (2.1.23) consists of a set of three scalar equations, where the divergence operator is taken on each row vector of the matrix ρ uuT , that is, ⎞ ⎛ ∇ · (ρ uu) ∇ · (ρ uuT ) = ⎝ ∇ · (ρ vu) ⎠ . ∇ · (ρ wu)

2.1.5 Balance law of energy According to the balance law of energy, also known as the first law of thermodynamics, the rate of change of a body’s total energy is equal to the sum of the power

20

2 Models arising in fluid and solid dynamics

exerted by all external forces and the rate of heat transfer into the body. This law is a statement of the conservation of energy and is fundamental to understanding energy transfer and transformation in physical systems. Let ε be the internal energy per unit mass so that the internal energy per unit volume is ρε . The internal energy EI of the fluid body is EI =

V (t)

ρε dV.

(2.1.24)

The kinetic energy of the fluid per unit mass is |u|2 /2, so that the kinetic energy per unit volume is ρ |u|2 /2. Thus, the kinetic energy of the body of the fluid is EK =

V (t)

ρ

u2 dV, 2

(2.1.25)

while the applying forces are the body and surface forces, as discussed earlier. The body force per unit volume is ρ k, and the power exerted by this body force per unit volume is u · ρ k. So, the power exerted by the body force is V (t)

u · ρ k dV.

On the other hand, the power exerted by the surface force t on S per unit area is u · t, so that the power exerted on the surface S is

u · t dS.

S(t)

Thus, the total power P developed by all applied forces reads P=

V (t)

u · ρ k dV +

S(t)

u · t dS.

(2.1.26)

Let us denote by q the heat flux vector, and −n · q the heat flux through the surface S. The rate of change of the heat transferred into the body is Q˙ = −

S(t)

n · q dS.

(2.1.27)

Observe that n is the outward normal vector to the surface S, so the heat flux vector q and the outward normal vector n point in opposite directions or, more precisely, they form an obtuse angle. The balance law of energy reads D ˙ (EI + EK ) = P + Q. Dt

(2.1.28)

The total energy per unit mass is the sum of the internal energy and the kinetic energy, defined as e = ε + |u|2 /2. We can re-write the equation of balance of energy (2.1.28) in terms of the total energy as follows:

2.1 Derivation of the equations governing fluid flows

D Dt

V (t)

ρ e dV =

V (t)

u · ρ k dV +

S(t)

21

u · t dS −

S(t)

n · q dS.

(2.1.29)

Applying (2.1.5) and substituting t = n · T into (2.1.29), we note that T is a symmetric tensor and V (t)

∂ (ρ e) dV + ∂t

S(t)

(n · (ρ e)u) dS =

V (t)

u · ρ k dV +

S(t)

n · (u · T) dS −

S(t)

n · q dS.

Re-arranging the terms, we arrive at

∂ (ρ e) − u · ρ k dV + (n · (ρ e)u) − n · (u · T) + n · q dS = 0. V (t) ∂ t S(t)

By considering a fixed domain V with a smooth surface S, the equation of balance of energy becomes ∂ V

∂t

(ρ e) − u · ρ k dV + n · ((ρ e)u − u · T + q) dS = 0.

(2.1.30)

S

Consequently, applying Gauss’ theorem, from (2.1.30) we deduce that ∂ (ρ e) − u · ρ k + ∇ · ((ρ e)u − u · T + q) dV = 0. ∂t V

(2.1.31)

Substituting the material derivative and the stress tensor T from (2.1.17) into (2.1.31) yields the integral form of balance law of energy ∂ (ρ e) + ∇ · ((ρ e + p)u − u · τ + q) − ρ (u · k) dV = 0. (2.1.32) ∂t V Since V is arbitrary, we also deduce from (2.1.28) the differential form of the balance law of energy

∂ (ρ e) + ∇ · (ρ e + p)u = ∇ · (u · τ ) − ∇ · q + ρ (u · k). ∂t

(2.1.33)

Suppose next that the heat flux has the form q = −κ ∇T, where κ denotes the thermal conductivity, and T is the temperature. Then, the equation for the balance of energy (2.1.33) can be written as

∂ (ρ e) + ∇ · [(ρ e + p)u] = ∇ · (u · τ ) + ∇ · (κ ∇T ) + ρ (u · k). ∂t Furthermore, substituting τ from (2.1.22) into the latter equation, we arrive at the following formulation of the balance law of energy:

22

2 Models arising in fluid and solid dynamics

∂ (ρ e) + ∇ · (κ ∇T ) + ρ (u · k). + ∇ · (ρ e + p)u = ∇ · u · λ ∗ ∇ · uI + η ∇u + (∇u)T ∂t

(2.1.34)

2.1.6 Governing equations and systems of conservation laws Summarizing our results above, we have obtained the following governing fluid dynamics equations: ∂ρ + ∇ · (ρ u) = 0, ∂t

∂ (ρ u) + ∇ · (ρ uuT + pI) = ∇ · λ ∗ ∇ · uI + η ∇u + (∇u)T + ρ k, ∂t

∂ (ρ e) + ∇ · (ρ e + p)u = ∇ · u · λ ∗ ∇ · uI + η ∇u + (∇u)T + ∇ · (κ ∇T ) + ρ (u · k). ∂t

(2.1.35)

Here, ρ denotes the density of the fluid, u its particle velocity, p is the pressure, ε its internal energy, e = ε + u|2 /2 its total energy, T its temperature, k its mass body force, η its shear viscosity coefficient, ηB = λ ∗ + (2/3)η its bulk viscosity coefficient, and κ its thermal conductivity coefficient. Recall that I is the 3 × 3 identity matrix. If we omit the viscosity, thermal conductivity, and external forces in (2.1.35), we find the three-dimensional inviscid gas dynamics equations in Eulerian coordinates

∂ρ + ∇ · (ρ u) = 0, ∂t ∂ (ρ u) + ∇ · (ρ uuT + pI) = 0, ∂t

∂ (ρ e) + ∇ · (ρ e + p)u = 0. ∂t

(2.1.36)

Clearly, the equations (2.1.36) take the form of a system of conservation laws

∂U + ∇ · F(U) = 0, ∂t

(2.1.37)

for an unknown function U = U(x,t) ∈ Ω ⊂ R I n with x ∈ R I d and t > 0. Indeed, it suffices to set ⎞ ⎛ ⎞ ⎛ ρu ρ I 5, U = ⎝ρ u⎠ ∈ R F(U) = ⎝ρ uuT + pI ⎠ . ρe (ρ e + p)u

2.2 Fundamental notions from thermodynamics

23

If G ⊂ R I 3 is a domain, then integrating the equation (2.1.37) over G and applying Gauss’ theorem, we get d dt

G

U(x,t)dx +

∂G

n · FdS = 0,

which means that the rate of change of the amount losses through the boundary ∂ G.

G U(x,t)dx

(2.1.38) in G is equal to the

2.2 Fundamental notions from thermodynamics 2.2.1 Equation of state and thermodynamic identity An equation of state of a fluid expresses a thermodynamic variable as a function of the other thermodynamic variables, and therefore takes one of the following form: p = p(ρ , ε ) = p(v, S),

ε = ε (ρ , S) = ε (v, T ),

T = T (ρ , S) = T (v, ε ). (2.2.1)

Here, ρ , ε , T, S, and p denote the thermodynamical variables, density, internal energy, absolute temperature, entropy, and the pressure, respectively, while v = 1/ρ denotes the specific volume. The thermodynamical variables are constrained by the thermodynamic identity (sometimes referred to as Gibbs’ relation; see [295]) d ε = T dS − pdv.

(2.2.2)

In view of thermodynamic identity, the pressure and temperature satisfy p=−

∂ ε (v, S) , ∂v

T=

∂ ε (v, S) . ∂S

The specific heat at constant volume Cv and the specific heat at constant pressure Cp are defined by Cv = T

∂ S(v, T ) , ∂T

Cp = T

∂ S(p, T ) , ∂T

while the isentropic compressibility coefficient KS and the isothermal compressibility coefficient KT are KS =

−1 ∂ v(p, S) , v ∂p

KT =

−1 ∂ v(p, T ) . v ∂p

The coefficient of thermal expansion β is defined as

β=

1 ∂ v(p, T ) . v ∂T

24

2 Models arising in fluid and solid dynamics

Finally, the above five quantities are related to each other by the equation KS β 2 vT Cv = 1− = . KT Cp KT Cp Another important relation involving T dS is derived as follows. Taking the differential of S = S(v, T ), we get dS =

∂ S(v, T ) ∂ S(v, T ) dv + dT. ∂v ∂T

Then, using the so-called Maxwell equation

∂ S(v, T ) ∂ p(v, T ) = , ∂v ∂T we obtain a second expression of the entropy T dS = Cv dT + T

∂ p(v, T ) dv. ∂T

(2.2.3)

Next, the adiabatic exponent γ and Gr¨uneisen coefficient Γ describe the variation of the pressure with respect to the density and entropy variables, via the equation vd p = −γ pdv + Γ T dS.

(2.2.4)

Thus, these exponents read

γ =−

1 v ∂ p(v, S) = , p ∂v pKS

Γ =v

∂ p(v, ε ) βv = . ∂ε Cv KT

(2.2.5)

One often imposes that the Gr¨uneisen coefficient is positive, i.e.,

Γ > 0,

U ∈ Ω,

(2.2.6)

which we assume throughout.

2.2.2 Helmholtz free energy The Helmholtz free energy F(v, T ) = ε (v, T ) − T S(v, T )

(2.2.7)

can be used in order to specify a choice of a complete equation of state. In fact, it follows from the thermodynamical identity (2.2.2) that

∂T ε (v, T ) = T ∂T S(v, T ).

2.2 Fundamental notions from thermodynamics

25

So, we can deduce from (2.2.7) that

∂T F(v, T ) = ∂T ε (v, T ) − S(v, T ) − T ∂T S(v, T ), or S = −∂T F(v, T ), and so

ε = F(v, T ) + T S(v, T ) = F(v, T ) − T ∂T F(v, T ). In addition, the thermodynamic identity (2.2.2) and the equation (2.2.7) yield p = −∂v F(v, T ).

2.2.3 The class of ideal fluids Ideal fluids correspond to the following choice of Helmholtz free energy: F(v, T ) = Cv T (1 − ln(T /T0 ) − (γ − 1) ln(v/v0 )) − S0 T,

(2.2.8)

where γ ,Cv , T0 , v0 , S0 are constant. We have S = −∂T F(v, T ) = Cv (ln(T /T0 ) + (γ − 1) ln(v/v0 )) + S0 , which yields the temperature as a function of the specific volume and the entropy v γ −1 S − S0 0 T = T0 exp . v Cv Furthermore, the internal energy reads ε = F + T S = Cv T and the pressure is given by Cv (γ − 1)T . p = −∂v F(v, T ) = v Observe that the pressure can be re-written as a function of the interval energy and the specific volume by (γ − 1)ε p= , v γ

p0 v 0 orasafunctionofthespecificvolumeandtheentropy: p = p(v, S) = γ 0 exp S−S . Cv v ∂ p(v,ε ) On the other hand, the Gr¨uneisen coefficient is Γ = v ∂ ε γ − 1 > 0.

2.2.4 The class of stiffened fluids Stiffened fluids are described by the equation p(v, ε ) = (γ − 1)(ε − ε∗ )/v − γ p∗ ,

(2.2.9)

26

2 Models arising in fluid and solid dynamics

where γ > 1 and ε∗ , p∗ are parameters. The stiffened gas equation (2.2.9) corresponds to the choice of Helmholtz free energy F(v, T ) = Cv T (1 − ln(T /T0 ) − (γ − 1) ln(v/v0 )) − S0 T + p∗ v + ε∗ ,

(2.2.10)

where T0 , v0 , S0 are constant; see for instance [242]. We find p(v, T ) = −∂v F(v, T ) = (γ − 1)Cv

T − p∗ , v

and S(v, T ) = −∂T F(v, T ) = Cv (ln(T /T0 ) + (γ − 1) ln(v/v0 )) + S0 . This yields T = T (v, S) = T0

v γ −1 0

v

exp

S − S0 Cv

,

and by substituting T = T (v, S) into p = p(v, T ) we express this equation of state in the form γ −1 (γ − 1)Cv T0 v0 S − S0 exp p = p(v, S) = − p∗ . vγ Cv ε) = γ − 1 > 0. Finally, the Gr¨uneisen coefficient is Γ = v ∂ p(v, ∂ε

2.2.5 The class of Hayes fluids Hayes fluids are defined by a more involved equation of state, namely Γ0 K0 p = p0 + ((v0 /v)N−1 − 1) + Γ0 (1 − v/v0 ) (v0 /v)N − 1 − N N −1 Γ0 (2.2.11) + (ε − ε0 +Cv T0Γ0 (1 − v/v0 ) − p0 (v0 − v)) , v0 where (v0 , ε0 , p0 , T0 ) is a reference state, and K0 , N, Γ0 are positive constants. This corresponds to the choice of free energy Γ0 F(v, T ) = ε0 + p0 (v0 − v) − S0 T +Cv (T − T0 ) 1 + (v0 − v) v0

K0 v0 (v0 /v)N−1 − (N − 1)(1 − v/v0 ) − 1 ; −Cv T ln(T /T0 ) + N(N − 1) (2.2.12) see again [242]. It is not difficult to check that

2.3 Physical models of particular interest

27

Γ0 Γ0 K0 N (v0 /v) − 1 +Cv T0 p(v, S) =p0 + (v0 − v) − 1 exp ((S − S0 )/Cv ) exp N v0 v0

and, moreover, Γ = (v/v0 )Γ0 > 0.

2.2.6 The class of Van der Waals fluids We now turn our attention to a typical nonconvex equation of state. A Van der Waals fluid, by definition, is governed by p=

a RT − , v − b v2

(2.2.13)

where a, b, R are positive constants. One can take Helmholtz free energy for Van der Waals EOS (2.2.13) a (v − b)T 3/2 − , (2.2.14) F(v, T ) = −RT 1 + ln c v where c is a parameter; see [297]. The entropy is expressed as 5 (v − b)T 3/2 S = −∂T F(v, T ) = R ln + , c 2 and this yields us the temperature as a function of (v, S): 2S 5 d − exp T (v, S) = , d = c2/3 . 3R 3 (v − b)2/3 Finally, substituting T from the last equation to (2.2.13), we obtain the pressure as a function of (v, S): 2S 5 Rd a − p(v, S) = exp − 2. 3R 3 v (v − b)5/3

2.3 Physical models of particular interest 2.3.1 Fluid equations in Eulerian coordinates The one-dimensional fluid dynamics equations in Eulerian coordinates is obtained from (2.1.36) by assuming a slab symmetry property of the flow. Precisely, we find

28

2 Models arising in fluid and solid dynamics

∂t ρ + ∂x (ρ u) = 0, ∂t (ρ u) + ∂x (ρ u2 + p) = 0, ∂t (ρ e) + ∂x (u(ρ e + p)) = 0,

(2.3.1)

where ρ , u, p, ε , T, S, and e = ε + u2 /2 represent the density, velocity, pressure, internal energy, temperature, entropy, and total energy, respectively. Moreover, by assuming the fluid flow is isentropic, we obtain the so-called isentropic model of fluid flows in Eulerian coordinates:

∂t ρ + ∂x (ρ u) = 0, ∂t (ρ u) + ∂x (ρ u2 + p) = 0.

(2.3.2)

2.3.2 Fluid equations in Lagrangian coordinates Given a sufficiently regular solution to (2.3.2), let y = y(x,t) be a solution to yx = ρ (x,t),

yt = −ρ u(x,t).

(2.3.3)

Thanks to the “consistency condition” ρt = −(ρ u)x , the equation (2.3.3) admits local solutions. Suppose that there is a one-to-one correspondence between x and y satisfying (2.3.3) in the form x = x(y,t). Then, given any function ϕ = ϕ (x,t), we write ϕ (y,t) = ϕ (x(y,t),t) and, for the sake of simplicity, we still write ϕ (y,t) = ϕ (y,t). It holds from (2.3.3) for any function ϕ that

∂t ϕ (x(y,t),t) + ∂x [ϕ u(x(y,t),t)] = ϕy yt + ϕt + (ϕ u)y yx = −ϕy ρ u + ϕt + (ϕ u)y ρ = −ϕy ρ u + ϕt + ϕy uρ + ρϕ uy = ϕt + ρϕ uy . (2.3.4) Using (2.3.4) for the choice of ϕ to be ρ , ρ u, and ρ e, we obtain from (2.3.2) the fluid dynamics equations in Lagrangian coordinates. Precisely, if we take ϕ = ρ , then from the conservation of mass in (2.3.2), we have 0 = ρt + ρ 2 uy . Let v be the specific volume, then ρt = −ρ 2 vt . From the last equation, we deduce 0 = −ρ 2 vt + ρ 2 uy . Canceling ρ 2 both sides and re-arranging terms, we can re-write the equation for conservation of mass in Lagrangian coordinates as vt − uy = 0.

(2.3.5)

2.3 Physical models of particular interest

29

For the equation of conservation of momentum, we take ϕ = ρ u. Then it holds from (2.3.4) that 0 = (ρ u)t + ρ 2 uuy + px . In view of px = py yx = py ρ and by the equation of conservation of mass (2.3.5) (ρ u)t = ρt u + ρ ut = −ρ 2 vt u + ρ ut = −ρ 2 uuy + ρ ut , the equation of conservation of momentum becomes 0 = −ρ 2 uuy + ρ ut + ρ 2 uuy + py ρ = ρ ut + py ρ . The equation of conservation of momentum in Lagrangian coordinates is thus ut + py = 0.

(2.3.6)

Third, taking ϕ = ρ e for the conservation of energy in (2.3.2) and applying (2.3.4) yield 0 = (ρ e)t + (ρ eu)x + (pu)x = (ρ e)t + ρ 2 euy + (pu)y yx = ρt e + ρ et + ρ 2 euy + (pu)y ρ = −ρ 2 vt e + ρ et + ρ 2 euy + (pu)y ρ . Using the equation of conservation of mass (2.3.5) in the last equation, we obtain 0 = −ρ 2 uy e + ρ et + ρ 2 euy + (pu)y ρ = ρ et + (pu)y ρ . Canceling ρ on both sides and changing notation E(y,t) = e(x(y,t),t) in the last equation, we can write the equation of conservation of energy in Lagrangian coordinates by (2.3.7) Et + (pu)y = 0. Summarizing our results (2.3.5)–(2.3.7) and still denoting the space variable by x instead of y for the sake of convenience, we have arrived at the fluid dynamics equations in Lagrangian coordinates: vt − ux = 0, ut + px = 0, Et + (pu)x = 0.

(2.3.8)

Assuming that the fluid is isentropic, we obtain the model of isentropic fluid flows in Lagrangian coordinates: vt − ux = 0, (2.3.9) ut + px = 0, which is referred to as the p-system.

30

2 Models arising in fluid and solid dynamics

2.3.3 Viscous-capillary models In some applications, it is suitable to consider the interfacial energy Ef =

λ (∂x v)2 , 2

(2.3.10)

where λ is a constant. Adding this term to Helmholtz free energy (2.2.7), we obtain the following modified free energy: F (v, T ) = F(v, T ) +

λ (∂x v)2 , 2

(2.3.11)

where μ is the viscosity and A > 0 is a constant. Therefore, the total stress in Lagrangian coordinates is T = −p + μ∂x u − λ ∂xx v. Then, the equation of balance of momentum becomes

∂t u + ∂x p = ∂x μ∂x u − λ ∂xx v .

(2.3.12)

(2.3.13)

The terms μ∂x u and λ ∂xx v represent the viscosity and capillarity effects, respectively. Using (2.3.12), we can write the equation for the balance of momentum as

∂t u = ∂x T.

(2.3.14)

The total energy is now given as the total of the internal, kinetic, and interfacial energies: u2 λ E = ε + + (∂x v)2 . (2.3.15) 2 2 In agreement with Felderhof postulate, we assume that the internal energy satisfies the balance of energy, that is,

εt = Π ∂x u + ∂x q,

Π = −p + μ∂x u,

(2.3.16)

where q is the heat flux; see for instance [131, 132]. The above equation is not in a divergence form; we find a divergence form for this equation by considering the total energy. First, note that Π = T + λ ∂xx v. Using the conservation of mass vt = ux , balance of momentum (2.3.14), the balance of energy (2.3.16), and the last identity, we obtain Et = εt + uut + λ vx vxt = Π ∂x u + ∂x q + uut + λ vx vxt = (T + λ vxx )ux + uut + λ vx vtx + qx = (Tux + uTx ) + λ (vxx ux + vx uxx ) + qx = (Tu)x + λ (vx ux )x + qx .

2.3 Physical models of particular interest

31

Using h = κ Tx and substituting T from (2.3.12), we re-write the equation for the balance of energy as

∂t E + ∂x (up) = ∂x (u (μ∂x u − λ ∂xx v)) + ∂x (λ ∂x u∂x v) + ∂x (κ∂x T ) . So, the viscous-capillary model of fluid flows is given by

∂t v − ∂x u = 0, ∂t u + ∂x p = ∂x (μ∂x u − λ ∂xx v) , ∂t E + ∂x (up) = ∂x (u (μ∂x u − λ ∂xx v)) + ∂x (λ ∂x u∂x v) + ∂x (κ∂x T ) .

(2.3.17)

In the isentropic case, the viscous-capillary model (2.3.15) reduces to vt − ux = 0, ∂t u + ∂x p = ∂x (μ∂x u − λ ∂xx v) .

(2.3.18)

The model (2.3.18) can be referred to as the viscous-capillary p-system. It is worth noting that traveling wave solutions to equation (2.3.18) exist and converge to shock waves as the viscosity and capillarity coefficients approach zero. These shock waves may or may not satisfy Lax shock inequalities, depending on whether the pressure function p = p(v) is nonconvex. An admissible shock that violates Lax shock inequalities is referred to as a nonclassical shock. Later in this text, we will explore solutions to the Riemann problem involving both classical and nonclassical shock waves.

2.3.4 Fluid flows in a nozzle with variable cross-section The following system describes the evolution of a fluid flow in a nozzle with variable cross-section:

∂t (aρ ) + ∂x (aρ u) = 0, ∂t (aρ u) + ∂x (a(ρ u2 + p)) = p∂x a, ∂t (aρ e) + ∂x (au(ρ e + p)) = 0,

(2.3.19) x∈R I , t > 0.

Here, the notation ρ , ε , and p stands for the thermodynamical variables: density, internal energy and the pressure, respectively; u is the velocity, and e = ε + u2 /2 is the total energy. The function a = a(x) > 0, x ∈ R I , is the cross-sectional area. The derivation of the model (2.3.19) can be justified as follows. Consider a “horizontal” nozzle placed in the x-direction. Each plane perpendicular to the x-axis intersects the nozzle at a point x on the x-axis. The intersection is called the cross-section at the point x. Suppose for simplicity that the area of cross-section is a smooth function a = a(x), x ∈ I ⊂ R I for an interval I; see Figure 2.3. Suppose also that the flow inside the nozzle has a symmetric property with respect to the z-direction so that its

32

2 Models arising in fluid and solid dynamics

Fig. 2.3 The cross-sectional area of the nozzle

dynamics inside the nozzle can be described by the two-dimensional Euler equations of fluid dynamics in the variable (x, y):

∂t ρ + ∂x (ρ u) + ∂y (ρ v) = 0, ∂t (ρ u) + ∂x (ρ u2 + p) + ∂y (ρ uv) = 0, ∂t (ρ v) + ∂x (ρ uv) + ∂y (ρ v2 + p) = 0, ∂t (ρ e) + ∂x (u(ρ e + p)) + ∂y (v(ρ e + p)) = 0,

(2.3.20) x, y ∈ R I ,t > 0.

We aim to simplify the two-dimensional Euler equations (2.3.20) by eliminating the variable y. First, integrating the first two equations of (2.3.20) with respect to y yields a(x) 0

(∂t ρ + ∂x (ρ u) + ∂y (ρ v)) dy = 0,

a(x) 0

∂t (ρ u) + ∂x (ρ u2 + p) + ∂y (ρ uv) dy = 0,

x∈R I ,t > 0,

that is a(x) a(x) ∂t ρ (x, y,t)dy + ∂x (ρ u)dy + ∂y (ρ v)dy = 0, 0 0 0 a(x) a(x) a(x) ∂t ρ u(x, y,t)dy + ∂x (ρ u2 + p)dy + ∂y (ρ uv)dy = 0.

0

a(x)

0

0

(2.3.21) The last integrals in each equation of (2.3.21) are equal to

2.3 Physical models of particular interest

a(x) 0

a(x) 0

33

∂y (ρ v)dy = ρ v(x, a(x),t) − ρ v(x, 0,t), (2.3.22)

∂y (ρ uv)dy = ρ uv(x, a(x),t) − ρ uv(x, 0,t).

Since particles at the boundary corresponding to y = a(x) move along the boundary, we have v(x, a(x),t) = u(x, a(x),t) tan α , where α is the angle between the tangent of the graph of a(x) and the x-axis. Using tan α = a (x), we obtain v(x, a(x),t) = a (x)u(x, a(x),t). Similarly, we have v(x, y = 0,t) = 0,

(2.3.23)

since the trajectories of the particles corresponding the boundary y = 0 are horizontally flat. It then follows from (2.3.22)–(2.3.23) that a(x) 0

a(x)

∂y (ρ v)dy = a (x)ρ u(x, a(x),t),

0

∂y (ρ uv)dy = a (x)ρ u2 (x, a(x),t).

Substituting the expressions in (2.3.24) into the last equations gives us a(x) a(x) ∂t ρ (x, y,t)dy + ∂x (ρ u)dy + a (x)ρ u(x, a(x),t) = 0, 0 0 a(x) a(x) ∂t ρ u(x, y,t)dy + ∂x (ρ u2 + p)dy + a (x)ρ u2 (x, a(x),t) = 0. 0

0

Combining the last two terms in each equation of the last system gives us a(x) ρ (x, y,t)dy + ∂x (ρ u)(x, y,t)dy = 0, 0 0 a(x) a(x) 2 ∂t (ρ u)(x, y,t)dy + ∂x (ρ u + p)(x, y,t)dy = a (x)p(x, a(x),t),

∂t

a(x)

0

0

(2.3.24) since a(x) ρ u(x, y,t)dy = ∂x (ρ u)(x, y,t)dy + a (x)ρ u(x, a(x),t), 0 0 a(x) a(x) ∂x ρ u2 (x, y,t)dy = ∂x (ρ u2 )(x, y,t)dy + a (x)ρ u2 (x, a(x),t), 0 0 a(x) a(x) ∂x p(x, y,t)dy = ∂x p(x, y,t)dy + a (x)p(x, a(x),t).

∂x

0

a(x)

0

34

2 Models arising in fluid and solid dynamics

Assume, in addition, that the u-component of the particle velocity and the pressure are constant along the y-direction. This means that uy (x, y,t) = 0,

py (x, y,t) = 0,

y ∈ (0, a(x)),

x∈R I,

t > 0.

Then, the first two equations of (2.3.19) from (2.3.24) for the y-average values ρ = ρ , u = u, p = p can be obtained. For example,

ρ (x,t) :=

1 a(x)

a(x) 0

ρ (x, y,t)dy,

x∈R I ,t > 0.

The third equation of (2.3.19) can also be similarly derived as the first equation. The derivation of the model (2.3.19) is thus complete.

2.3.5 Shallow water model for variable topography The shallow water equations are used to model free surface flow of incompressible fluid, such as water, over variable topography. This model is useful for simulating dam-breaking flows, river waves, and coastal flows. The shallow water equations with variable topography can be derived from the incompressible Navier-Stokes equations of fluid dynamics using a two-layer model of flows. Figure 2.4 provides an illustration. It is assumed that both u and v are independent of z, and the vertical component of the acceleration is negligible, that is, Dw/Dt = 0, where u = (u, v, w) denotes the velocity field.

Fig. 2.4 A two-layer model of water flow.

2.3 Physical models of particular interest

35

More precisely, consider an inviscid incompressible fluid in a constant gravitational field. Let us denote by u = (u, v, w) the velocity vector in three-dimensional coordinates (x, y, z)—the origin being within the fluid flow. The component z is the vertical coordinate, so the gravitational acceleration g is acting along the negative z-direction. Assume that the fluid density ρ is constant and that there is a gravitational force F ≈ −ρ gk, where k = (0, 0, 1) is the unique vector along the z-axis. Assume that the fluid is inviscid and incompressible. Then, we have

ρ = constant, and the law of conservation of mass becomes ∇ · u = 0. The law of balance of momentum in (2.1.35) can be simplified (by using the product rule for divergence) as follows: ∇ · (ρ uu) = ρ u(∇ · u) + u · (∇ρ u) = ρ u · (∇u), ∇ · (ρ vu) = ρ v(∇ · u) + u · (∇ρ v) = ρ u · (∇v), ∇ · (ρ wu) = ρ w(∇ · u) + u · (∇ρ w) = ρ u · (∇w). This enables us to write ∇ · (ρ uT u) = ρ (u · ∇)u. Expanding the terms and dividing both sides of the balance of momentum (2.1.35) by ρ , we obtain 1 ∂t u + (u · ∇)u + ∇p = −g k. ρ Therefore, the governing equations of an inviscid and incompressible fluid are ∇ · u = ∂x u + ∂y v + ∂z w = 0, 1 ∂t u + (u · ∇)u + ∇p = −g k. ρ

(2.3.25)

Suppose that the interface between the water and the air is described by the equation f (x, y, z,t) = 0. (2.3.26) The interface is characterized by the property that the fluid does not cross it, so (cf., for instance [328], p. 433) a particle at a specific and arbitrary point (x0 , y0 , z0 ) on the free surface and any given time t0 remains on the surface. Let (x(t), y(t), z(t)) denote the position of the particle at some time t, and set x(t0 ) = x0 , y(t0 ) = y0 , and z(t0 ) = z0 . The above argument tell us that, for all t near t0 , f (x(t), y(t), z(t),t) = 0. Taking the derivative of this equation, evaluated at t = t0 , gives us

∂t f + ∂x f

dx(t0 ) dy(t0 ) dz(t0 ) + ∂y f + ∂z f =0 dt dt dt

36

2 Models arising in fluid and solid dynamics

at the point (x0 , y0 , z0 ,t0 ). Since dx(t0 ) = u(x0 , y0 , z0 ,t0 ), dt

dy(t0 ) = v(x0 , y0 , z0 ,t0 ), dt

dz(t0 ) = w(x0 , y0 , z0 ,t0 ), dt

from the last equation we obtain (∂t f + u∂x f + v∂y f + w∂z f )(x0 , y0 , z0 ,t0 ) = 0. Since (x0 , y0 , z0 ,t0 ) is arbitrary, we conclude that the material derivative of the free surface vanishes identically: Df = ∂t f + u∂x f + v∂y f + w∂z f = 0. Dt

(2.3.27)

In addition, the normal velocity of the fluid on a solid boundary vanishes. Thus, suppose that the free surface and the fixed bottom are given by the following equations (respectively): f (x, y, z,t) = z − η (x, y,t) = 0,

z = −H(x, y).

(2.3.28)

Then, the following three boundary conditions must hold:

∂t η + u∂x η + v∂y η = w, p = p0 = air pressure on z = η (x, y,t), u∂x H + v∂y H + w = 0 on z = −H(x, y),

(2.3.29)

where p0 is constant. In view of these boundary conditions, we integrate the continuity equation in (2.3.25) with respect to z and obtain 0=

η −h

(∂x u + ∂y v + ∂z w) dy

= ∂t η + ∂x

η −h

udz + ∂y

η −h

dz = ∂t η + ∂x [u(η + H)] + ∂y [v(η + H)],

where the latter equation is obtained using the assumption that u and v are, both, independent of z. Setting h = η + H, and since H is independent of time so that ∂t H = 0, we arrive at the continuity equation in the form ∂t h + ∂x (hu) + ∂y (hv) = 0. (2.3.30) Next, we use the assumption that the vertical component of the acceleration is negligible, that is, Dw/Dt = 0, and substituting this into the third momentum equation in (2.3.25) we find

2.3 Physical models of particular interest

37

Dw 1 + ∂z p = −g, Dt ρ so that ∂z p = −ρ g. Integrating the latter equation with respect to z and using the second boundary condition in (2.3.29) gives us the hydrostatic pressure relation: p = p0 + ρ g(η − z). Consequently, we have

∂x p = ρ g∂x η ,

∂y p = ρ g∂y η .

Substituting the last equations into the first two momentum equations in (2.3.25), we obtain

∂t u + u∂x u + v∂y u + g∂x η = 0,

∂t v + u∂x v + v∂y v + g∂y η = 0.

(2.3.31)

Multiplying (2.3.30) by u, multiplying the first momentum equation in (2.3.31) by h, and finally adding up the resulting equations, we get u(∂t h + ∂x (hu) + ∂y (hv)) + h(∂t u + u∂x u + v∂y u) + gh∂x η = 0, gh ∂t (hu) + ∂x h(u2 + ) + ∂y (huv) − gh∂x H = 0. 2

or

Similarly, multiplying (2.3.30) by v, multiplying the second momentum equation in (2.3.31) by h, and then adding up the results, we get v(∂t h + ∂x (hu) + ∂y (hv)) + h(∂t v + u∂x v + v∂y v) + gh∂y η = 0,

or

∂t (hv) + ∂x (huv) + ∂y

gh h(v + ) − gh∂y H = 0. 2 2

Thus, the two equations of momentum become gh 2 ∂t (hu) + ∂x h(u + ) + ∂y (huv) − gh∂x H = 0, 2 gh 2 ∂t (hv) + ∂x (huv) + ∂y h(v + ) − gh∂y H = 0. 2

(2.3.32)

Now, let a = a(x, y), x, y ∈ R I , be the height of the topography bottom from a given level. Then we have a + H = constant and this yields

∂x a = −∂x H,

∂y a = −∂y H.

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2 Models arising in fluid and solid dynamics

Substituting the last equations into (2.3.32), from (2.3.30) we arrive at the model of two-dimensional shallow water equations with variable topography:

∂t h + ∂x (hu)+ ∂y (hv) = 0, gh ∂t (hu) + ∂x h(u2 + ) + ∂y (huv) + gh∂x a = 0, 2 gh 2 ∂t (hv) + ∂x (huv) + ∂y h(v + ) + gh∂y a = 0. 2

(2.3.33)

Here, h denotes the height of the water from the bottom to the surface, u and v are the two horizontal components of the fluid velocity, g represents the gravity constant, while a = a(x, y) (with x, y ∈ R I ) is the height of the topography bottom from a given level. Consider the situation where the variations of the unknowns in the y-axis are negligible so that v = 0 and all partial derivatives with respect to y vanish. From (2.3.33) we deduce the following one-dimensional shallow water equations with variable topography:

∂t h + ∂x (hu) = 0,

∂t (hu) + ∂x h(u2 + gh 2 ) + gh ∂x a = 0.

(2.3.34)

The Riemann problem for the shallow water equations with discontinuous topography (2.3.34) will be studied in Chapter 9.

2.3.6 A model of nonlinear elastodynamics with phase transitions The deformation of an elastic body can be described by a system similar to the above p-system in fluid dynamics. Under the change of variables (2.3.3), the velocity of deformation is v(y,t) = ∂t x, where x = x(y,t), and the deformation gradient reads w(y,t) = ∂y (x − y). The mapping x = x(y,t) is assumed to satisfy the principle of impenetrability of matter in the sense that ∂y x > 0, which yields us the condition w > −1. Furthermore, the commutation property for second-order partial derivatives gives us (x − y)yt = (x − y)ty , which is the so-called continuity equation: wt − vy = 0.

(2.3.35)

In a similar way as in the case of the p-system, but the pressure tensor −p being now replaced by the stress tensor σ , we postulate that the stress tensor coincides

2.3 Physical models of particular interest

39

with a prescribed stress-strain function of the form σ = σ (w). Thus, it is derived from the conservation of momentum (2.1.23), where the pressure −p in (2.1.17) is replaced by the stress tensor σ . Using a similar process to obtain the equation of conservation of momentum in Lagrangian coordinates from the equation in Eulerian coordinates, and noting that the velocity of the body is now denoted by v instead of u in the case of fluid dynamics equations, we write the equation of momentum as vt − σ (w)y = 0.

(2.3.36)

As in the case of the fluid dynamics equations in Lagrangian coordinates, when we concentrate on the governing equations in Lagrangian coordinates, and if there is no confusion, we denote the spatial variable by x instead of y. Thus, the governing equations for the model of nonlinear elastodynamics also read as follows: wt − vx = 0, vt − σ (w)x = 0.

(2.3.37)

If the stress is monotone and increasing with respect to the strain, we obtain a hyperbolic model. If there exists an interval in which the stress is decreasing with respect to the strain, and increasing elsewhere, we obtain an elliptic-hyperbolic model which is relevant for the description of a variety of phenomena in phase transition dynamics. A viscous-capillary version of this model (2.3.37) is given by wt − vx = 0,

∂t v − ∂x σ = ∂x (μ∂x v − λ ∂xx w) ,

(2.3.38)

in which μ and λ denote the so-called viscosity and capillarity coefficients, respectively. The derivation of (2.3.38) is similar to the one of the viscous-capillary p-system (2.3.18). It is interesting to note that the stress function is not convex as a function of the strain. Thus, the traveling waves of (2.3.38) will converge to a shock wave that may or may not satisfy Lax’s shock inequalities when the viscosity and capillarity go to zero. This justifies that both kinds of Lax shocks and non-Lax shocks may be called admissible for the vanishing viscosity-capillarity criterion. In Chapters 5 and 6 below, we will treat the Riemann problem for (2.3.37) when the solutions contain classical shocks satisfying Lax’s shock inequalities as well as nonclassical shocks violating these inequalities but satisfying a kinetic relation, as we call it.

40

2 Models arising in fluid and solid dynamics

2.4 Bibliographical notes For this chapter, we provide a brief selection of the most relevant papers. For additional references, please refer to the bibliography at the end of this monograph. Concerning the fluid dynamics modeling, we refer the reader to the textbooks by Courant and Friedrichs [105], Bowen [68], Spurk and Aksel [295], and Zucrow and Hoffman [334]. For questions related to thermodynamics, we refer to Swendsen [297], Menikoff and Plohr [243], and Menikoff [242]. In particular, [242] discusses the conditions satisfied by equations of state associated with classes of fluids and materials, especially the properties of the Helmholtz free energy. The equivalence between the Eulerian and Lagrangian equations of gas dynamics is standard for sufficiently regular solutions while, for weak solutions, this equivalence was established first by Wagner [324] even for solutions containing vacuum states. The derivation for the model (2.3.19) in the stationary regime is presented in Zucrow and Hoffman [334], while the derivation in the non-stationary regime can be found, for instance, in Kr¨oner and Thanh [202]. A two-dimensional simulation of debris flows in an erodible channel is studied by Armanini, Fraccarollo, and Rosatti [22], while two-phase flows through a duct with discontinuous cross-section are introduced in Brown, Martynov, and Mahgerefteh [71]. A derivation of the Saint-Venant system is found in [115], while the derivation of the two-dimensional shallow water system with variable topography (2.3.33) is discussed, for instance, in the textbook [314]. The Saint-Venant system with mass exchange in a shallow water flow is studied in [29, 32, 70].

Chapter 3

Nonlinear hyperbolic systems of balance laws

In this chapter, we introduce the fundamental concepts of the theory of systems of balance laws, which can be expressed in either conservative or nonconservative form. We provide several examples to illustrate these concepts. We define elementary waves, including shock waves, contact discontinuities, and rarefaction waves, and compare various admissibility criteria that can be applied at discontinuities. Additionally, we discuss the notion of classical and nonclassical shock waves and illustrate these concepts with a typical example: the isothermal model of van der Waals fluids.

3.1 Nonlinear hyperbolic systems of interest 3.1.1 Balance laws in conservative or nonconservative forms The physical models of interest can be formulated as a system in a nonconservative form ∂u ∂u + A(u) = 0, x∈R I , t > 0, (3.1.1) ∂t ∂x where Ω is an open subset of R I n and u = u(x,t) ∈ Ω ⊂ R I n is the unknown, while A = A(u) is a given n × n matrix-valued map. The formulation (3.1.1) covers numerous examples. On the other hand, a system of conservation laws, by definition, reads

∂ u ∂ f (u) + = 0, ∂t ∂x

x∈R I,

t > 0,

(3.1.2)

which clearly also takes the (nonconservative) form (3.1.1). Here, the unknown function is again denoted by u while the flux function f = f (u) is given and we use the notation

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 3

41

42

3 Nonlinear hyperbolic systems of balance laws

⎛

⎞

u1 ⎜ u2 ⎟ ⎟ u=⎜ ⎝ ··· ⎠, un

⎛

⎞ f1 (u) ⎜ f2 (u) ⎟ ⎟ f (u) = ⎜ ⎝ ··· ⎠. fn (u)

We refer to A(u) = D f (u) as the Jacobian matrix and we use the notation ⎛

⎞ ∂ f1 (u) ∂ f1 (u) ∂ f1 (u) ··· ⎜ ∂ u1 ∂ un ⎟ 2 ⎜ ∂ f (u) ∂ ∂f u(u) ∂ f2 (u) ⎟ ⎜ 2 ⎟ 2 ··· ⎜ ⎟ ⎜ ∂ u ∂ u ∂ u n ⎟. 1 2 D f (u) = ⎜ ⎟ . . . ⎜ ⎟ .. .. .. ··· ⎜ ⎟ ⎝ ∂ f (u) ∂ f (u) ∂ fn (u) ⎠ n n ··· ∂ u1 ∂ u2 ∂ un Many systems of physical interest can be decomposed into two parts, that is, conservative and nonconservative terms as in

∂ u ∂ f (u) ∂u + = B(u) , ∂t ∂x ∂x

x∈R I , t > 0,

(3.1.3)

where B(u) is an n × n matrix, and the product B(u) ∂∂ ux typically cannot be written as an exact derivative. For instance, the model of fluid flows in a nozzle with variable cross-section presented in Chapter 2, as well as the shallow water equations with variable topography, have the form (3.1.3). Obviously, (3.1.3) can be put in the form (3.1.1) by setting A(u) = D f (u) − B(u). We formulate the initial-value problem, or Cauchy problem, associated with (3.1.1) by prescribing an initial condition u(x, 0) = u0 (x),

x∈R I,

(3.1.4)

where u0 : R I → Ω is given. In particular, the Riemann problem associated with (3.1.1) corresponds to the simplest choice of (non-trivial) data, i.e., the piecewise constant function uL , x < 0, (3.1.5) u(x, 0) = uR , x > 0, where uL , uR are constant states in Ω .

3.1.2 Hyperbolicity The system (3.1.1) is said to be hyperbolic if, for every u ∈ Ω , the matrix A(u) admits n real eigenvalues, which we denote by

λ1 (u) ≤ λ2 (u) ≤ ... ≤ λn (u),

3.1 Nonlinear hyperbolic systems of interest

43

together with a complete set of (linearly independent) right-eigenvectors, denoted by r1 (u), r2 (u), ..., rn (u). Here, each vector ri (i = 1, 2, ..., n) is an eigenvector associated with the eigenvalue λi , that is, A(u) ri (u) = λi (u) ri (u), 1 ≤ i ≤ n.

We refer to the map u → λi (u), ri (u) as the ith-characteristic field of the system (3.1.1). If all the eigenvalues of the matrix (A(u) are distinct, that is, λ1 (u) < λ2 (u) < ... < λn (u), the system (3.1.1) is said to be strictly hyperbolic (and the existence of a full set of eigenvectors is immediate).

3.1.3 Examples of conservative models Example 3.1.1. The inviscid Burgers equation ut + (u2 /2)x = 0 is the prototype of a nonlinear scalar conservation law. Example 3.1.2. The p-system

∂t v − ∂x u = 0, ∂t u + ∂x p(v) = 0 governs isentropic fluid flows in Lagrangian coordinates, where v denotes the specific volume, u the velocity, and p the pressure of the fluid. This system is supplemented with an equation of state which determines the pressure p = p(v) as a function of the specific volume. The Jacobian matrix of this system in the variable (v, u) reads 0 −1 , A(v) = p (v) 0 where p (v) = d p/dv. The characteristic polynomial of the matrix A is q(λ ) = λ 2 + p(v). Hence, provided p (v) < 0, the matrix A(v) admits two real and distinct eigenvalues, i.e.,

λ1 (v) = − −p (v), λ2 (v) = −p (v), and the p-system is strictly hyperbolic. Depending on the behavior of the pressure function p = p(v), we may distinguish between different behaviors, as follows.

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3 Nonlinear hyperbolic systems of balance laws

Fig. 3.1 Monotone decreasing pressure for an isentropic ideal fluid: the p-system is strictly hyperbolic.

(a) First of all, consider the class of isentropic ideal fluids, corresponding to p(v) = vκγ for some constants κ > 0 and γ ≥ 1. We find p (v) =

−κγ < 0, vγ +1

v > 0,

and the system is strictly hyperbolic in the entire phase domain. See Figure 3.1. (b) Second, consider the isothermal model of van der Waals fluids, having the pressure function a RT − , p= v > b, T = T0 v − b v2 for some positive constants a, b, T0 . Since p (v) =

−RT 2a + , (v − b)2 v3

the hyperbolicity condition p (v) < 0 requires that T > g(v) :=

2a (v − b)2 , R v3

v > b.

A simple calculation gives us 2a −(v − b)(v − 3b) , R v4 3 a which yields us maxv>b g(v) = g(v = 3b) = 23 Rb and, therefore, g (v) =

dp

. 3 Rb

Therefore, the p-system is strictly hyperbolic in the domain v > b if and only if the temperature coefficient satisfies

3.1 Nonlinear hyperbolic systems of interest

T0 >

45

3 2 a . 3 Rb

Fig. 3.2 The non-monotone pressure function of an isentropic van der Waals fluid: The system is of hyperbolic-elliptic type.

On the other, in the interval 0 < T < 8a/27Rb, the equation p (v) = 0 admits exactly two roots, which we denote by α and β and satisfy b < α < 3b < β . We have p (v) < 0 for v ∈ (b, α ) ∪ (β , ∞), p (v) > 0

for v ∈ (α , β ),

and the p-system is thus strictly hyperbolic for v ∈ (b, α ) ∪ (β , ∞), but fails to be hyperbolic for v ∈ (α , β ). More precisely, when 0 < T < 8a/27Rb and v ∈ (α , β ), the characteristic equation associated with the Jacobian matrix admits two (purely imaginary, conjugate) complex roots, and the p-system is in fact an elliptic system in the region v ∈ (α , β ). Globally, the p-system with 0 < T < 8a/27Rb can therefore be referred to as a hyperbolic-elliptic model. See Figure 3.2 for an illustration. Example 3.1.3. The system of gas dynamics in Lagrangian coordinates reads

∂t v − ∂x u = 0, ∂t u + ∂x p = 0, ∂t e + ∂x (pu) = 0.

(3.1.6)

Here, ρ denotes the density of the fluid, v = 1/ρ its specific volume, p its pressure, u its velocity, while ε denotes the internal energy, S the specific entropy, T the absolute temperature, and e = ε + u2 /2 the total energy. The system (3.1.6) can be written in the following simpler, but nonconservative, form in the variables v, u, S:

∂t v − ∂x u = 0, ∂t u + ∂x p = 0, ∂t S = 0.

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3 Nonlinear hyperbolic systems of balance laws

The Jacobian matrix in these variables reads ⎛ ⎞ 0 −1 0 A(v, u, S) = ⎝∂v p 0 ∂S p⎠ . 0 0 0

Provided that ∂v p(v, S) < 0, the local sound speed c := v −∂v p(v, S) in the fluid is well-defined, and the matrix A(v, u, S) admits three real and distinct eigenvalues

λ1 (U) =

−c c < λ2 (U) = 0 < λ3 (U) = . v v

A corresponding basis of eigenvectors can be chosen to be ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ∂S p v v r1 (U) = ⎝c⎠ , r2 (U) = ⎝ 0 ⎠ , r3 (U) = ⎝−c⎠ . −∂v p 0 0 Example 3.1.4. The system of gas dynamics in Eulerian coordinates reads

∂t ρ + ∂x (ρ u) = 0, ∂t (ρ u) + ∂x (ρ u2 + p) = 0, ∂t (ρ e) + ∂x (u(ρ e + p)) = 0. In the variables U = (ρ , u, S)T ∈ Ω = R I + ×R I ×R I , this system can be written in the nonconservative form ∂t ρ + u∂x ρ + ρ∂x u = 0, 1 ∂t u + u∂x u + ∂x p = 0, ρ ∂t S + u∂x S = 0, with Jacobian matrix

⎞ u ρ 0 A(U) = ⎝∂ρ p/ρ u ∂S p/ρ ⎠ . 0 0 u ⎛

Provided ∂ρ p(ρ , S) > 0, this system admits three real and distinct eigenvalues, i.e.,

λ1 (U) = u − c,

λ2 (U) = u,

λ3 (U) = u + c,

where c = ∂ρ p(ρ , S) represents the sound speed in the fluid. A corresponding basis of eigenvectors can be chosen to be ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ρ ρ −∂S p(ρ , S) ⎠, 0 r3 (U) = ⎝ c ⎠ . r2 (U) = ⎝ r1 (U) = ⎝−c⎠ , ∂ρ p(ρ , S) 0 0

3.1 Nonlinear hyperbolic systems of interest

47

Since ∂ρ p(ρ , S) = −∂v p(v, S)/v2 , the fluid system in Eulerian coordinates is hyperbolic if and only if it is hyperbolic in Lagrangian coordinates.

3.1.4 Examples of nonconservative models Example 3.1.5. The one-dimensional shallow water equations with variable topography read ∂t h + ∂x (hu) = 0, h ∂t (hu) + ∂x h(u2 + g ) = −gh∂x a, 2 where h > 0 is the height of the water (measured from the bottom to the surface) and u is the velocity component. In addition, g > 0 is the gravity constant, and a = a(x) is the height of the bottom measured from a reference level. It is often convenient to supplement the two partial differential equations above with the trivial equation

∂t a = 0. In the variable U = (h, u, a)T ∈ Ω = R I + ×R I ×R I + , the full system of three equations reads also ∂t h + u∂x h + h∂x u = 0,

∂t u + g∂x h + u∂x u + g∂x a = 0, ∂t a = 0, which is a nonconservative system (while we started from a system of two balance laws). The later system has the nonconservative form ∂t U + A(U)∂xU = 0, where ⎛ ⎞ uu0 A(U) = ⎝g u g⎠ . 000 The matrix A(u) admits three real eigenvalues

λ1 (U) := u − gh, λ2 (U) := u + gh, together with the corresponding eigenvectors ⎞ ⎞ ⎛ ⎛ h √ √h r1 (U) := ⎝− gh⎠ , r2 (U) := ⎝ gh⎠ , 0 0

λ3 (U) := 0, ⎞ gh r3 (U) := ⎝ −gu ⎠ . u2 − gh ⎛

If u2 − gh = 0, the eigenvalues are distinct, so the above model of shallow water equations is strictly hyperbolic. However, when u2 − gh = 0, the eigenvalues coincide and so the shallow water system is not strictly hyperbolic. Precisely, the first and

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3 Nonlinear hyperbolic systems of balance laws

third characteristic eigenvalues and eigenvectors coincide, that is, (λ1 (U), r1 (U)) = (l3 (U), r3 (U)), on the surface

C+ := (h, u, a)| u = gh of the phase domain. Similarly, the second and third characteristic fields coincide, that is, (λ2 (U), r2 (U)) = (l3 (U), r3 (U)), on the surface

C− := (h, u, a)| u = − gh . These sets, referred to as critical surfaces, separate the (h, u, a)-space into three subdomains, in which the system is strictly hyperbolic: G1 := (h, u, a) ∈ R I + ×R I ×R I + / λ1 (U) > λ3 (U) , G2 := (h, u, a) ∈ R I + ×R I ×R I + / λ2 (U) > λ3 (U) > λ1 (U) , G3 := (h, u, a) ∈ R I + ×R I ×R I + / λ3 (U) > λ2 (U) . As will be explained later, it is natural to refer to G2 as the subcritical region, and G1 , G3 as supercritical regions. Example 3.1.6. The model of a fluid in a nozzle with variable cross-section is given by ∂t (aρ ) + ∂x (aρ u) = 0,

∂t (aρ u) + ∂x (a(ρ u2 + p)) = p(ρ )∂x a, ∂t (aρ e) + ∂x (au(ρ e + p)) = 0, x ∈ R I , t > 0, where ρ , ε , T, S, p, u, and e = ε + u2 /2 denote the density, internal energy, absolute temperature, entropy, pressure, velocity, and the total energy, respectively. The function a = a(x) > 0, x ∈ R I , represents the cross-sectional area. Often, one supplements the above model of a fluid in a nozzle with variable cross-section with the trivial equation ∂t a = 0, x ∈ R I , t > 0. For smooth solutions, the supplemented model of a fluid in a nozzle with variable cross-section becomes

ρu ∂x a = 0, a ∂ρ p(ρ , S) ∂S (ρ , S) ∂x ρ + u∂x u + ∂x S = 0, ∂t u + ρ ρ ∂t S + u∂x S = 0, ∂t a = 0. ∂t ρ + u∂x ρ + ρ∂x u +

This system has the nonconservative form ∂t U + A(U)∂xU = 0, where

3.1 Nonlinear hyperbolic systems of interest

⎛ ⎞ ρ ⎜u⎟ ⎟ U =⎜ ⎝S⎠ , a

⎛ u ⎜ ⎜ ∂ρ p(ρ , S) ⎜ A(U) = ⎜ ρ ⎜ ⎝ 0 0

49

ρ u 0 0

uρ ⎞ 0 a ⎟ ⎟ ∂S p(ρ , S) 0 ⎟ ⎟. ρ ⎟ u 0 ⎠ 0 0

The matrix A(U) admits four real eigenvalues, provided ∂ρ p(ρ , S) > 0: λ1 (U) = u − ∂ρ p(ρ , S) < λ2 (U) = u < λ3 (U) = u + ∂ρ p(ρ , S), λ4 (U) = 0. The eigenvalue λ4 (U) may coincide with any other eigenvalue. The hypersurfaces, referred to as the resonance surfaces,

Σ1 = {U : λ1 (U) = λ4 (U)}, Σ2 = {U : λ2 (U) = λ4 (U)}, Σ3 = {U : λ3 (U) = λ4 (U)}, divide the phase domain into four regions in which the system is strictly hyperbolic: G1 = {U : λ3 (U) < λ4 (U)}, G2 = {U : λ2 (U) < λ4 (U) < λ3 (U)}, G3 = {U : λ1 (U) < λ4 (U) < λ2 (U)}, G4 = {U : λ4 (U) < λ1 (U)}. Example 3.1.7. We now consider the following isentropic model of two-phase flows, which has applications in the modeling of deflagration-to-detonation transition in granular materials

∂t (αg ρg ) + ∂x (αg ρg ug ) = 0, ∂t (αg ρg ug ) + ∂x (αg (ρg u2g + pg )) = pg ∂x αg , ∂t (αs ρs ) + ∂x (αs ρs us ) = 0, ∂t (αs ρs us ) + ∂x (αs (ρs u2s + ps )) = −pg ∂x αg , ∂t ρs + ∂x (ρs us ) = 0, x ∈ R I ,t > 0. The first and the third equation of the above system describes the conservation of mass in each phase; the second and the four equation describes the balance of momentum in each phase; the latter equation is the so-called compaction dynamics equation. Throughout, we use the subscripts g and s to indicate the quantities in the gas phase and in the solid phase, respectively. The notation αk , ρk , uk , pk , k = g, s stands for the volume fraction, density, velocity, and pressure in the k-phase, k = g, s. The volume fractions satisfy αs + αg = 1. For smooth solutions, the above system of first-order differential equations is equivalent to the following system:

50

3 Nonlinear hyperbolic systems of balance laws

ρg (ug − us ) ∂x αg = 0, αg ∂t ug + hg (ρg )∂x ρg + ug ∂x ug = 0, ∂t ρs + us ∂x ρs + ρs ∂x us = 0, pg − ps ∂t us + hs (ρs )∂x ρs + us ∂x us + ∂x αg = 0, αs ρs ∂t αg + us ∂x αg = 0, x ∈ R I ,t > 0, ∂t ρg + ug ∂x ρg + ρg ∂x ug +

where hk is given by hk (ρ ) =

pk (ρ ) , ρ

k = s, g.

Thus, the system can be re-written as a system of balance laws in nonconservative form as ∂t U + A(U)∂xU = 0, where ⎛

⎞

ρg ⎜ ug ⎟ ⎜ ⎟ ⎟ U =⎜ ⎜ ρs ⎟ , ⎝ us ⎠ αg

⎛ ug

ρg

0

0

0

0

0

0

⎜ ⎜ ⎜hg (ρg ) ug 0 0 ⎜ A(U) = ⎜ 0 0 us ρs ⎜ ⎜ 0 hs (ρs ) us ⎝ 0

ρg (ug − us ) ⎞ ⎟ αg ⎟ ⎟ 0 ⎟ ⎟. 0 pg − ps ⎟ ⎟ ⎠ αs ρs us

The characteristic equation of the matrix A(U) is given by (us − λ )((ug − λ )2 − pg )((us − λ )2 − ps ) = 0, which admits five roots as

λ1 (U) = ug − pg < λ2 (U) = ug + pg , λ3 (U) = us − ps < λ4 (U) = us + ps < λ5 (U) = us . The eigenvalues λ1 and λ2 may coincide with any other three eigenvalues. Precisely, the six equations λi = λ j , i = 1, 2, j = 3, 4, 5, define six surfaces in the phase domain. Thus, the system is strictly hyperbolic, except on these surfaces. Example 3.1.8. The model of two-phase flows which arises from the modeling of deflagration-to-detonation transition in porous energetic materials is given by

3.2 Nonlinearity conditions and examples

51

∂t (αg ρg ) + ∂x (αg ρg ug ) = 0, ∂t (αg ρg ug ) + ∂x (αg (ρg u2g + pg )) = pg ∂x αg , ∂t (αg ρg eg ) + ∂x (αg ug (ρg eg + pg )) = pg us ∂x αg , ∂t (αs ρs ) + ∂x (αs ρs us ) = 0, ∂t (αs ρs us ) + ∂x (αs (ρs u2s + ps )) = pg ∂x αs , ∂t (αs ρs es ) + ∂x (αs us (ρs es + ps )) = pg us ∂x αs , ∂t ρs + ∂x (ρs us ) = 0, x ∈ R I ,t > 0. The latter equation of the above system is the compaction dynamics equation; the other six equations represent the balance of mass, momentum, and energy in each phase. Here, the subscripts g and s indicate the quantities in the gas phase and in the solid phase, respectively. The notation αk , ρk , uk , pk , εk , Sk , Tk , ek = εk +u2k /2, k = g, s stands for the volume fraction, density, velocity, pressure, internal energy, specific entropy, temperature, and the total energy in the k-phase, k = g, s, respectively. The volume fractions satisfy αs + αg = 1. The above system can also be written in the nonconservative form

∂t U + A(U)∂xU = 0, where the matrix A(U) has the following seven eigenvalues:

λ1 (U) = ug − cg < λ2 (U) = ug < λ3 (U) = ug + cg , λ4 (U) = us − cs < λ5 (U) = us < λ6 (U) = us + cs , λ7 (U) = us . Since λ5 ≡ λ7 , the system is not strictly hyperbolic in the entire phase domain. One can show that the system is hyperbolic in certain regions.

3.2 Nonlinearity conditions and examples 3.2.1 Nonlinearity conditions Let Ω ⊂ R I n be an open subset on which the system of balance laws

∂u ∂u + A(u) = 0, ∂t ∂x

x∈R I,

t > 0,

(3.2.1)

is strictly hyperbolic. The matrix A admits n real and distinct eigenvalues, denoted by λ1 (u) < λ2 (u) < ...λn (u). Each eigenvalue λi (u) is associated with a righteigenvector ri (u), i.e., (3.2.2) A(u)ri (u) = λi (u)ri (u), and a left-eigenvector li (u)T , i.e.,

52

3 Nonlinear hyperbolic systems of balance laws

li (u)T A(u) = λi (u)li (u)T .

(3.2.3)

In other words, li is an eigenvector of the transpose matrix AT corresponding to the eigenvalue λi , i = 1, 2, ..., n. Since the eigenvalues of AT are distinct, the corresponding eigenvectors of AT , which are the left-eigenvectors of A, are linearly independent. Furthermore, we can always choose the left-eigenvectors such that the following normalization conditions are fulfilled: 1, i = j, (3.2.4) li (u)r j (u) = δi j = 0, i = j, for u ∈ Ω , 1 ≤ i, j ≤ n. Definition 3.2.1. The characteristic field (λi , ri ) is said to be genuinely nonlinear on a set Ω if, for all u in Ω , Dλi (u) · ri (u) = 0,

(3.2.5)

while it is said to be linearly degenerate on a set Ω if, for all u in Ω , Dλi (u) · ri (u) = 0.

(3.2.6)

In many applications, a characteristic field may be genuinely nonlinear on certain sets, and linearly degenerate on other sets of the phase domains. In general, the hypersurface Dλi (.) · ri (.) = 0 divides the phase domain into disjoint sets such that in each of these sets the i-characteristic field is genuinely nonlinear, i = 1, 2, ..., n.

3.2.2 Examples Example 3.2.1. Consider the p-system in Example 3.1.2 and consider the set of v on which the condition p (v) < 0 holds. As seen in Example 3.1.2, the system is strictly hyperbolic, and the Jacobian matrix admits two real and distinct eigenvalues

λ1 (v) = − −p (v) < 0 < λ2 (v) = −p (v). The eigenvalues can be chosen as 1 r1 (v) = , −p (v)

r2 (v) =

1 . − −p (v)

We find p (v) , Dλ1 (v) · r1 (v) = 2 −p (v)

−p (v) Dλ2 (v) · r2 (v) = , 2 −p (v)

3.2 Nonlinearity conditions and examples

53

where p (v) = d 2 p/dv2 . Hence, the two characteristic fields of the p-system are genuinely nonlinear as long as p keeps a constant sign. Depending on the sign of d 2 p/dv2 , we may also have different classes of the p-systems as follows. (a) First, consider the case of an isentropic ideal fluid, where p(v) =

κ , vγ

for some positive constants κ and γ > 1. Then, a straightforward calculation gives us κγ (γ + 1) p (v) = > 0 for v > 0. vγ +2 Thus, in this case, the two characteristic fields of the p-system are genuinely nonlinear in the entire phase domain. (b) Second, consider an isothermal model of van der Waals fluids, where the pressure function p = p(v) is given by p=

a RT − 2, v−b v

v > b, T = T0 ,

for positive constants a, b, and T0 . A simple calculation gives us p (v) =

2RT 6a − 4, 3 (v − b) v

and the convexity condition p (v) > 0 gives T > h(v) :=

3a (v − b)3 , R v4

v > b.

A simple calculation gives us h (v) =

3a (v − b)2 (4b − v). Rv5

This yields max h(v) = h(v = 4b) = (3/4)4 v>b

a . Rb

If T > (3/4)4 a/Rb, then p (v) > 0, and so the two characteristic fields of the p-system are genuinely nonlinear. However, when 0 < T < (3/4)4 a/Rb, then the equation p (v) = 0 admits exactly two roots μ and ν such that b < μ < 4b < ν . Moreover p (v) > 0 p (v) < 0

for for

v ∈ (b, μ ) ∪ (ν , ∞), v ∈ (μ , ν ).

54

3 Nonlinear hyperbolic systems of balance laws

Fig. 3.3 The nonconvex pressure function of an isentropic van der Waals fluid: The characteristic fields fail to be genuinely nonlinear when v moves across the values μ and ν

The two characteristic fields of the p-system are genuinely nonlinear in each interval v ∈ (b, μ ), (μ , ν ) and (ν , ∞), but fail to be genuinely nonlinear when v moves across the values μ and ν . See Figure 3.3. Example 3.2.2. Consider the gas dynamics equations in Lagrangian coordinates in Example 3.1.3, where the equation of state is given in the form p = p(v, S). As seen from Example 3.1.3, providing that

∂v p(v, S) < 0 the Jacobian matrix of the system admits three real and distinct eigenvalues

λ1 (U) =

−c c < λ2 (U) = 0 < λ3 (U) = v v

together with a complete set of the corresponding eigenvectors ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ∂S p v v r1 (U) = ⎝c⎠ , r2 (U) = ⎝ 0 ⎠ , r3 (U) = ⎝−c⎠ . −∂v p 0 0 We have Dλ1 (U) · r1 (U) =

v2 ∂vv p(v, S), 2c

Dλ2 (U) · r2 (U) = 0,

Dλ3 (U) · r3 (U) = −

v2 ∂vv p(v, S). 2c

The second characteristic field is linearly degenerate. Moreover, under the condition

∂vv p(v, S) > 0, for all U, which means that the function v → p = p(v, S) is strictly convex for any fixed S, the first and the third characteristic fields are genuinely nonlinear.

3.3 Convex and nonconvex equations of state

55

Example 3.2.3. Consider the gas dynamics equations in Eulerian coordinates in Example 3.1.4, which are strictly hyperbolic under the assumption that

∂ρ p(ρ , S) = c2 > 0. Let us use the same hypotheses and notation as in Example 3.1.4. We find Dλ1 (U) · r1 (U) = −(c + ρ∂ρ c),

Dλ2 (U) · r2 (U) = 0,

Dλ3 (U) · r3 (U) = c + ρ∂ρ c.

The second characteristic field is linearly degenerate. Assume, in addition, that c + ρ∂ρ c > 0, for all U. Then, the first and the third characteristic fields are genuinely nonlinear. Since ∂vv p(v, S) = 2ρ 3 c(c + ρ∂ρ c), this result coincides with the one for gas dynamics equations in Lagrangian coordinates. The above examples show that the strict convexity of the function p = p(v, S) in the variable v for each fixed S reflects the genuine nonlinearity of the first and the third characteristic fields for the gas dynamics equations. Therefore, we sometimes refer to a convex EOS as the one where the function v → p = p(v, S) is convex for any fixed S.

3.3 Convex and nonconvex equations of state The governing equations of fluid flows in Eulerian or Lagrangian coordinates are strictly hyperbolic when ∂v p(v, S) < 0 and its first- and third-characteristic fields are genuinely nonlinear when ∂vv p(v, S) = 0. The domain of the state space Ω is defined to be the set of all states where the system is strictly hyperbolic, that is,

Ω := {U | ∂v p(v, S) < 0,

v > vinf ≥ 0,

S ≥ 0},

(3.3.1)

where vinf is a non-negative constant.

3.3.1 Fluid dynamics equations: ideal gas Consider a polytropic and ideal fluid, where the equation of state is in the form ε p = (γ −1) v . As seen in Chapter 2, the pressure can be expressed as a function of the specific volume and the entropy by γ p0 v0 S − S0 , p = p(v, S) = γ exp v Cv

56

3 Nonlinear hyperbolic systems of balance laws

where T0 , v0 , S0 are constant. This gives us

S∗ − S −γ −1 ∂v p(v, S) = −γ (γ − 1) exp < 0, v Cv for all v > 0, S ≥ 0. So, the domain of state space is Ω = {U|v > 0, Moreover, S∗ − S −γ −2 2 ∂vv p(v, S) = γ (γ − 1) exp > 0, v Cv

S ≥ 0}.

for all v > 0, S ≥ 0. So the first and the third characteristic fields are genuinely nonlinear on the whole domain. Hence, the equation of state for the polytropic and ideal fluid is a convex EOS.

3.3.2 Fluid dynamics equations: stiffened fluids As seen in Chapter 2, the pressure of a stiffened fluid can be expressed as a function of the specific volume and the entropy as

γ −1

p = p(v, S) =

(γ − 1)Cv T0 v0 vγ

∂v p(v, S) = −

γ (γ − 1)Cv T0 v0 vγ +1

exp

S − S0 Cv

− p∗ .

(3.3.2)

This gives us

γ −1

exp

S − S0 Cv

< 0,

for all v > 0, S ≥ 0. So, the domain of state space is Ω = {U|v > 0, Moreover, we have γ −1

γ (γ 2 − 1)Cv T0 v0 ∂vv p(v, S) = vγ +2

S − S0 exp Cv

S ≥ 0}.

> 0,

for all v > 0, S ≥ 0. So that the first and the third characteristic fields are genuinely nonlinear on the whole domain. Hence, (3.3.2) is a convex EOS.

3.3.3 Fluid dynamics equations: Hayes fluids As seen in Chapter 2, the pressure of a Hayes fluid can be expressed as a function of the specific volume and the entropy as

3.3 Convex and nonconvex equations of state

57

Γ0 Γ0 K0 N p(v, S) =p0 + (v0 − v) − 1 , exp ((S − S0 )/Cv ) exp (v0 /v) − 1 +Cv T0 N v0 v0

(3.3.3) where (v0 , ε0 , p0 , T0 ) is a reference state, and K0 , N, Γ0 are positive constants. Thus, we have 2 K0 vN0 Γ0 Γ0 ∂v p(v, S) = − N−1 −Cv T0 exp ((S − S0 )/Cv ) exp (v0 − v) < 0, v v0 v0 for all v > 0, S ≥ 0. Thus, the relevant domain is Ω = {U|v > 0, S ≥ 0}. Furthermore, we find 3 (N − 1)K0 vN0 Γ0 Γ0 ∂vv p(v, S) = +Cv T0 exp ((S − S0 )/Cv ) exp (v0 − v) > 0, vN−2 v0 v0 for all v > 0, S ≥ 0, and Γ0 > 0, N > 1. Thus, the first and the third characteristic fields are genuinely nonlinear. The equation of state (3.3.3) is therefore a convex EOS.

3.3.4 Fluid dynamics equations: van der Waals fluids

Fig. 3.4 van der Waals EOS: Graph of the pressure as a function of the specific volume at different values of the entropy

Recall from Chapter 2 that a van der Waals fluid is characterized by the equation of state a RT − . (3.3.4) p= v − b v2

58

3 Nonlinear hyperbolic systems of balance laws

Furthermore, it has been shown that the pressure can be expressed as a function of the specific volume and the entropy as 2S 5 Rd a p(v, S) = − exp − 2. 3R 3 v (v − b)5/3 See Figure 3.4. A straightforward calculation gives 2S 5 −5Rd 2a − ∂v p(v, S) = exp + 3. 3R 3 v 3(v − b)8/3

(3.3.5)

Recall that the system is strictly hyperbolic if and only if ∂v p(v, S) < 0, or equivalently, 6a 5 3R 8 ln(v − b) − 3 ln v + ln + S> . 2 3 5Rd 3 Thus, there is a curve in the (v, S)-plane, denoted by Ch , on which the system is hyperbolic, but not strictly hyperbolic: 6a 5 3R 8 ln(v − b) − 3 ln v + ln + Ch : S = Sh (v) := , v > b. (3.3.6) 2 3 5Rd 3 Let us investigate properties of the curve Ch . Taking the derivative of the function Sh in (3.3.6), we get 8 3 dSh (v) 3R = − , dv 2 3(v − b) v so that dSh (v) dv

dSh (v) dv

= 0 if and only if v = v@ := 9b. Moreover,

dSh (v) dv

> 0 for v < v@ and

< 0 for v > v@ . This means that Sh attains a maximum value at v = v@ . Thus, for any fixed S, if S > S@ := Sh (v@ ), then the system is strictly hyperbolic. If S < S@ , there are two values, denoted by v# = v# (S) < v# = v# (S) such that

∂v p(v, S) < 0 if and only if b < v < v# (S) or v > v# (S). The domain of state space is thus

Ω = {U | b < v < v# (S) or v > v# (S)}. The internal energy is defined as

ε = F +TS =

a 3R T− . 2 v

Substituting T from the latter equation to (3.3.4), we obtain

(3.3.7)

3.3 Convex and nonconvex equations of state

p(v, ε ) = We have

59

2 a a − 2. ε+ 3(v − b) v v

∂ p(v, ε ) 2 > 0, = ∂ε 3(v − b)

so that

∂ p(v, ε ) > 0. ∂ε Let us consider the convexity/concavity of the van der Waals EOS. A straightforward calculation gives us 2S 5 40Rd 6a − ∂vv p(v, S) = exp (3.3.8) − 4. 11/3 3R 3 v 9(v − b) Γ =v

Recall that the first and the third characteristic fields are genuinely nonlinear if and only if ∂vv (v, S) = 0. However, we see from (3.3.8) that ∂vv p(v, S) may vanish. In fact, the equation ∂vv p(v, S) = 0 determines a curve in the (v, S)-plane, denoted by Cn , on which the characteristic fields fail to be genuinely nonlinear: 27a 5 3R 11 ln(v − b) − 4 ln v + ln + Cn : S = Sn (v) := , v > b. (3.3.9) 2 3 20Rd 3 Let us investigate properties of the curve Cn . Taking the derivative of the function Sh in (3.3.9), we get 11 4 dSn (v) 3R = − , dv 2 3(v − b) v so that dSh (v) dv

dSn (v) dv

= 0 if and only if v = v@ := 12b. Moreover,

dSh (v) dv

> 0 for v < v@ and

< 0 for v > v@ . This means that Sn attains a maximum value at v = v@ . Thus, for any fixed S, if S > S@ := Sn (v@ ), then ∂vv p(v, S) > 0. If S < S@ , there are two values, denoted by v = v (S) < v = v (S) such that

∂vv p(v, S) = 0 if and only if v = v (S) or v = v (S).

(3.3.10)

The curve Cn divides the (v, S)-plane into two regions corresponding to the positive or negative sign of ∂vv p(v, S). The first and the third characteristic fields are genuinely nonlinear in each of these regions. A van der Waals fluid therefore has a nonconvex equation of state.

60

3 Nonlinear hyperbolic systems of balance laws

3.4 Weak solutions and elementary waves 3.4.1 Weak solutions of systems of conservation laws Smooth solutions of hyperbolic systems of conservation laws typically exist only for small times and become discontinuous at some finite time. Let u be a smooth solution of the system of conservation laws (3.1.2). Multiplying both sides of (3.1.2) I × [0, ∞), and by an arbitrary differentiable function ϕ with compact support in R then integrating gives us ∞ ∞ ∂ u ∂ f (u) + ϕ dxdt = 0. ∂x 0 −∞ ∂ t Applying Green’s formula, we see that the solution u satisfies the integral equation ∞ ∞ ∞ ∂ϕ ∂ϕ + f (u) u(x, 0)ϕ (x, 0)dx = 0. u dxdt + ∂t ∂x 0 −∞ −∞ It is interesting that this integral equation is well-defined for any measurable and locally bounded u. This suggests that one could use the latter equation to define a notion of weak solutions. As usual, for a given domain D ⊆ R I × [0, +∞) we denote by ϕ ∈ Cc∞ (D) the set of all infinitely differentiable functions with compact support on D. ∞ (R Definition 3.4.1. A function u ∈ Lloc I × [0, +∞), Ω ) is said to be a weak solution of the Cauchy problem (3.1.2) and (3.1.4) if ∞ ∞ ∞ ∂ϕ ∂ϕ + f (u) u0 (x)ϕ (0, x)dx = 0, (3.4.1) u dxdt + ∂t ∂x 0 −∞ −∞

I × [0, ∞)). for all ϕ ∈ Cc∞ (R Weak solutions are understood as follows. ∞ (R Definition 3.4.2. A function u ∈ Lloc I × [0, +∞), Ω ) is said to be a weak solution of the system of conservation laws (3.1.2) if ∞ ∞ ∂ϕ ∂ϕ + f (u) u dxdt = 0, (3.4.2) ∂t ∂x 0 −∞

I × (0, ∞)). for all ϕ ∈ Cc∞ (R Discontinuities in weak solutions propagate along certain curves. Let us consider an arbitrary weak solution which we assume to be piecewise smooth, say continuously differentiable except across a finite number of smooth curves in the (x,t)upper half plane t > 0. For such functions, algebraic equations involving the states

3.4 Weak solutions and elementary waves

61

on both sides of the discontinuity enable us to check whether the function is a weak solution. Let u be a function which is smooth everywhere in a domain D ⊂ R I × [0, ∞), except across a curve Σ , where u has a jump discontinuity. The limits of u on both sides of Σ can be defined by u± (x,t) = lim u((x,t) + ε n), ε →0±

t ∈ I,

(3.4.3)

where n = (n1 , n2 ) denotes a unit normal vector to Σ . The following theorem provides us with a useful tool to check whether a piecewise smooth function is a weak solution. Theorem 3.4.1. Let u : R I × [0, +∞) → Ω be a smooth function in (R I × [0, +∞)) \ Σ with a possible jump discontinuity across a curve Σ , and let u± be defined as in (3.4.3). Then, u is a weak solution of (3.1.2) if and only if the following two conditions are fulfilled: (i) (ii)

u satisfies pointwise (3.1.2) where it is differentiable; Across Σ , the following Rankine-Hugoniot relation holds: −n2 (u+ − u− ) + n1 ( f (u+ ) − f (u− )) = 0.

(3.4.4)

Proof. Consider a weak solution u : R I × [0, +∞) → Ω so that u has a jump discontiI × [0, +∞)) \ Σ . Let D nuity across a curve Σ in the (x,t)-plane and u is smooth in (R I × (0, +∞). Denote be a sufficiently small disk with its center on Σ such that D ⊂ R by D± the two open subdomains of D on both sides of Σ . It holds for any function ϕ ∈ Cc∞ (D) that 0= =

uϕt + f (u)ϕx dxdt

D D−

uϕt + f (u)ϕx dxdt +

D+

uϕt + f (u)ϕx dxdt.

Applying Green’s theorem in D+ and D− , assuming for simplicity that n point to the interior of D+ , and using the fact that u is a classical pointwise solution in each domain D± , we obtain

D+

uϕt + f (u)ϕx dxdt =

D+

=−

ut + f (u)x ϕ dxdt −

Σ ∩D

Σ ∩D

(n2 u+ + n1 f (u+ )ϕ ds

(n2 u+ + n1 f (u+ ))ϕ ds,

and D−

uϕt + f (u)ϕx dxdt = −

=

D−

Σ ∩D

ut + f (u)x ϕ dxdt +

(n2 u− + n1 f (u− ))ϕ ds.

Σ ∩D

(n2 u− + n1 f (u− ))ϕ ds

62

3 Nonlinear hyperbolic systems of balance laws

Subtracting these two equations side by side gives us

Σ ∩D

(n2 (u+ − u− ) + n1 ( f (u+ ) − f (u− )))ϕ dS = 0

for all functions ϕ ∈ Cc∞ (D). Since the latter equation holds for all functions ϕ ∈ Cc∞ (D), we conclude that n2 (u+ − u− ) + n1 ( f (u+ ) − f (u− )) = 0. Conversely, it is easy to see that if u is a piecewise smooth function that satisfies both conditions (i) and (ii), then u is a weak solution. We denote the jump of u across a discontinuity by [u] = u+ − u− and the jump of f (u) across a discontinuity by [ f (u)] = f (u+ ) − f (u− ). Let Σ be a piecewise smooth curve of the form

Σ = {(x,t) : x = x(t),

t ∈ I ⊂ (0, +∞)},

(3.4.5)

where x = x(t),t ∈ I, is a piecewise smooth function. Then a unit normal vector n of Σ is given by n = (1 + s2 )−1/2 (1, −s),

where s =

dx . dt

(3.4.6)

Then the Rankine-Hugoniot relation (3.4.4) becomes −s[u] + [ f (u)] = 0.

(3.4.7)

Example 3.4.1. Consider the p-system

∂t v − ∂x u = 0, ∂t u + ∂x p(v) = 0,

x∈R I ,t > 0,

where u, v > 0, and p denote the velocity, specific volume, and the pressure, respectively. Given two constant states (v− , u− ) and (v+ , u+ ), the following function (v− , u− ), x < st, (v, u)(x,t) = x > st, (v+ , u+ ), is a weak solution of the p-system if and only if the jump conditions −s(v+ − v− ) − (u+ − u− ) = 0, −s(u+ − u− ) + p(v+ ) − p(v− ) = 0 are satisfied. This is the case provided we have p(v+ ) − p(v− ) ≤0 v+ − v− −) . and then s2 = − p(vv++)−p(v −v−

3.4 Weak solutions and elementary waves

63

3.4.2 Weak solutions of nonconservative systems of balance laws Consider the following systems of balance laws in nonconservative form:

∂t U + A(U)∂xU = 0,

U ∈R I n.

(3.4.8)

Weak solutions of nonconservative systems of balance laws are taken from the space of functions of bounded variation and are understood in the sense of nonconservative products, as stated below. First, let us be given a family of Lipschitz paths φ : [0, 1] × R I n ×R I n→R I n , which satisfies the following conditions

φ (0;U,V ) = U φ (1;U,V ) = V, |∂s φ (s;U,V )| ≤ K|V −U|, |∂s φ (s;U1 ,V1 ) − ∂s φ (s;U2 ,V2 )| ≤ K(|V1 −V2 | + |U1 −U2 |),

(3.4.9)

for some constant K > 0, and for all s ∈ [0, 1],U,V,U1 ,U2 ,V1 ,V2 ∈ R I n . Let U : n [a, b] → R I be a function with bounded variation (or BV-function). Then, dU is a Borel measure which coincides with the distributional derivative of U, i.e., b a

U ϕ dx = −

b a

ϕ ∈ Cc∞ [a, b],

ϕ dU,

The notion of the nonconservative product of a locally Borel bounded function by a Borel measure is now stated, together with the notion of weak solutions for the system (3.4.8). Definition 3.4.3. Let φ : [0, 1]×R I n ×R I n→R I n be a family of Lipschitz paths satisfying (3.4.9), and let U = U(x), x ∈ [a, b] be a function with bounded variation. Then, the nonconservative product μ := g(U) · dU

g:R In

→R In

φ

of a locally Borel bounded function

by the vector-valued Borel measure dU is a real-valued bounded Borel measure μ with the following properties: (a) (b)

For any Borel set B such that U is continuous on B one has μ (B) = B g(U) dU. For any x0 ∈ [a, b], for any family of Lipschitz paths φ satisfying the conditions (3.4.9):

μ (x0 ) =

1 0

g(φ (s;U(x0 −),U(x0 +)))∂s φ (s;U(x0 −),U(x0 +)) ds.

Definition 3.4.4. A function U ∈ L∞ ∩ BVloc (R I ×R I +, R I n ) is called a weak solution of the system in nonconservative form Ut + A(U)Ux = 0, U = U(x,t) ∈ R In if the Borel measure ∂t U + A(U(.,t))∂xU(.,t) φ vanishes identically.

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3 Nonlinear hyperbolic systems of balance laws

3.4.3 Integral curves and rarefaction waves In this monograph, we consider hyperbolic systems that may contain genuinely nonlinear, linearly degenerate, and mixed-type characteristic fields. These mixed-type fields are genuinely nonlinear except on certain submanifolds where they are linearly degenerate. As such, the rarefaction waves we define below correspond to a characteristic field with arbitrary nonlinearity. We will consider the following hyperbolic system of balance laws in nonconservative form (3.1.1)

∂t U + A(U)∂xU = 0,

U ∈G⊂R I n.

(3.4.10)

Suppose that the system (3.4.10) is strictly hyperbolic with the eigenvalues

λ1 (u) < λ2 (u) < ... < λn (u). Let li , ri , i = 1, 2, ..., n be left- and right-eigenvectors of the matrix A(u), respectively. In the sequel, the left- and right-eigenvectors are chosen such that li · ri ≡ 1, and that Dλi (u) · ri (u) ≡ 1,

u ∈ G,

(3.4.11)

as long as the ith characteristic field is genuinely nonlinear in a domain G. Let us seek continuous piecewise smooth self-similar solutions of (3.4.10) of the form u(x,t) = v( xt ). We will show that v = v(ξ ), ξ = x/t is an integral curve of the characteristic field ri for some index i = 1, 2, ..., n. That means v is a solution of the differential equation v (ξ ) = ri (v(ξ )), for i = 1, 2, ..., n. Substituting u = v(x/t) into (3.4.10), one has 1 x x x x −( 2 )v ( ) + ( )A v( ) v ( ) = 0. t t t t t This yields

x ξ= . t Thus, either v (ξ ) = 0, or there is an index i ∈ {1, 2, ..., n} such that (A(v(ξ )) − ξ I)v (ξ ) = 0,

v (ξ ) = α (ξ )ri (v(ξ )),

λi (v(ξ )) = ξ .

(3.4.12)

Suppose that v (ξ ) = 0 in an interval. Then, since the eigenvalues are distinct, the index i does not depend on values ξ in this interval. Differentiating the second equation in (3.4.12), we have Dλi (v(ξ )) · v (ξ ) = 1. Then, using the first equation, we obtain α (ξ )Dλi (v(ξ )) · ri (v(ξ )) = 1. Let the i-characteristic field be genuinely nonlinear; using the normalization (3.4.11) we obtain α (ξ ) = 1. Thus, it follows from (3.4.12) that v is an integral curve of

3.4 Weak solutions and elementary waves

65

the field ri . The above argument indicates that if u = v(x/t) is a weak solution of (3.4.10), then either v (ξ ) = 0, or v (ξ ) = ri (v(ξ )),

λi (v(ξ )) = ξ .

Moreover, let us take two states u± ∈ Ω such that v(λi (u− )) = u− ,

v(λi (u+ )) = u+ .

Then, the above analysis shows that the function defined by ⎧ x ⎪ u− , ≤ λi (u− ), ⎪ ⎪ t ⎨ x x , λi (u− ) ≤ ≤ λi (u+ ), u(x,t) = v ⎪ t t ⎪ x ⎪ ⎩u+ , ≥ λi (u+ ) t

(3.4.13)

is a weak self-similar solution of (3.1.1) (Figure 3.5).

Fig. 3.5 An i-rarefaction wave in the (x,t)-plane

Definition 3.4.5. A self-similar solution of (3.1.1) of the form (3.4.13) is called an i-rarefaction wave connecting the left-hand state u− to the right-hand state u+ , i = 1, 2, ..., n. Let us now fix a left-hand state u− = u0 , and consider the set of right-hand states u+ = u that can be connected to u0 by an i-rarefaction wave of the form (3.4.13), for i = 1, 2, ..., n. The existence of rarefaction waves is given in the following theorem. Theorem 3.4.2. Assume that the ith characteristic field is genuinely nonlinear with the normalization (3.4.11). Given a left-hand state u0 ∈ Ω , there exists a curve Ri (u0 ) consisting of the right-hand states u ∈ Ω that can be connected to u0 on the right by an i-rarefaction wave. Moreover, there exists a parameterization of the curve Ri (u0 ) : s → Φi (s; u0 ), 0 ≤ s ≤ s0 such that

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3 Nonlinear hyperbolic systems of balance laws

Φi (s; u0 ) = u0 + sri (u0 ) +

s2 Dri (u0 ) · ri (u0 ) + O(s3 ). 2

(3.4.14)

Proof. Let v : ξ → v(ξ ) be the solution of the following initial-value problem for differential equations: v (ξ ) = ri (v(ξ )),

ξ > λi (u0 ),

v(λi (u0 )) = u0 .

Such a solution always locally exists. That is, a solution v exists for λi (u0 ) ≤ ξ ≤ λi (u0 ) + s0 for some positive s0 . We have d λi (v(ξ )) = Dλi (v(ξ )) · v (ξ ) = Dλi (v(ξ )) · ri (v(ξ )). dξ Using the normalization (3.4.11), we have

d λi (v(ξ )) dξ

= 1. Thus, the function v satisfies

λi (v(ξ )) − λi (v(λi (u0 ))) = ξ − λi (u0 ). This yields λi (v(ξ )) = ξ . We define a curve Ri (u0 ) by Ri (u0 ) = {v(ξ ) : λi (u0 ) ≤ ξ ≤ λi (u0 ) + s0 }. Then, the curve Ri (u0 ) consists of all states that can be connected to u0 by an i-rarefaction wave. Let us define

Φi (s; u0 ) := v(λi (u0 ) + s),

0 ≤ s ≤ s0 .

We find Φi (0; u0 ) = u0 and

Φi (0; u0 ) = v (λi (u0 )) = ri (v(λi (u0 ))) = ri (u0 ). Finally, since v (ξ ) = Dri (v(ξ )) · v (ξ ) = Dri (v(ξ )) · ri (v(ξ )), we get Φi ”(0; u0 ) = Dri (u0 ) · ri (u0 ), which establishes the expansion (3.4.14).

3.4.4 Shock waves and Hugoniot curves A shock wave of the general system of balance laws in nonconservative from (3.1.1) is a weak solution of the form u− , x < σ t, u(x,t) = (3.4.15) u+ , x > σ t,

3.4 Weak solutions and elementary waves

67

where the constants u− , u+ ∈ Ω are called the left-hand and right-hand states, respectively, and σ = σ (u− , u+ ) is the shock speed. In the rest of this section, we consider shock waves of systems of conservation laws (3.1.2). For the shock of the form (3.4.15), the Rankine-Hugoniot relation reads −σ (u+ − u− ) + ( f (u+ ) − f (u− )) = 0. See Figure 3.6. In the following we fix a left-hand state u− = u0 , and consider the set of right-hand states u+ = u that can be connected to u0 by a shock wave of the form (3.4.15).

Fig. 3.6 A shock wave in the (x,t)-plane

Definition 3.4.6. Given a left-hand state u0 ∈ Ω . The Rankine-Hugoniot set issuing from u0 of the system of conservation laws (3.1.2) is the set of all right-hand states I satisfying u ∈ Ω such that there exists a number σ (u0 , u) ∈ R −σ (u0 , u)(u − u0 ) + ( f (u) − f (u0 )) = 0. Thus, σ (u0 , u) is the corresponding shock speed. The existence of shock waves is given in the following theorem. Theorem 3.4.3. Given is a left-hand state u0 ∈ Ω . The Rankine-Hugoniot set issuing from u0 of the system of conservation laws (3.1.2) locally consists of n curves Hi (u0 ), 1 ≤ i ≤ n, called the i-Hugoniot curve. Moreover, for each i, there exists a parameterization of the curve Hi (u0 ) : s → Ψi (s; u0 ), |s| ≤ s1 such that s2 Dri (u0 ) · ri (u0 ) + O(s3 ). 2

(3.4.16)

s σ (u0 , Ψi (s; u0 )) = λi (u0 ) + Dλi (u0 ) · ri (u0 ) + O(s2 ). 2

(3.4.17)

Ψi (s; u0 ) = u0 + sri (u0 ) + The shock speed satisfies

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3 Nonlinear hyperbolic systems of balance laws

Proof. Since the matrix D f (u) is strictly hyperbolic, the average matrix A(u0 , u) :=

1 0

D f (θ u0 + (1 − θ )u)d θ

is also strictly hyperbolic. Let λ i (u0 , u), ri (u0 , u), l i (u0 , u) be the eigenvalue and the corresponding right- and left-eigenvectors of A(u0 , u), for i = 1, ..., n. By taking a normalization, we assume that l i (u0 , u) · ri (u0 , u) = 1,

i = 1, 2, .., n.

Then, the Rankine-Hugoniot relation for the shock connecting u and u0 can be written as (σ (u0 , u) − A)(u − u0 ) = 0. This implies that there is an index i such that σ (u0 , u) = λ i (u0 , u), and u − u0 is the i-eigenvector corresponding to the eigenvalue λ i (u0 , u). Therefore, we find l j (u0 , u) · (u − u0 ) = 0,

j = i, j = 1, ..., n.

(3.4.18)

The latter equations form a linear system of n − 1 equations for n unknowns u ∈ R I n. Let us consider the mapping u ∈ Ω → G(u) = l j (u0 , u) · (u − u0 ) ∈R I n−1 . j=i

Then, we have G(u0 ) = 0. Moreover, the fact that the left-eigenvectors l j (u0 ), j = 1, ..., n, j = i are linearly independent implies that the Jacobian matrix DG(u0 ) = (l j (u0 )) j=i has maximum rank, i.e., rank DG(u0 ) = n − 1. Applying the implicit function theorem, we see that there exists a smooth curve Hi (u0 ), called the ith Hugoniot curve, consisting of states u satisfying (3.4.18). This Hugoniot curve can be parameterized by ε → Ψi (ε ) with |ε | ≤ ε1 , for some positive constant ε1 such that

Ψi (0) = u0 .

(3.4.19)

Next, let us consider the coefficients in the expansion (3.4.16). Consider the state u and the shock speed λ along the Hugoniot curve Hi (u0 ) as functions of the parameter ε , i.e., u = v(ε ) = Ψi (ε ; u0 ),

λ (ε ) = λ i (u0 , Ψi (ε ; u0 )) = σ (u0 , Ψi (ε ; u0 )),

Differentiating with respect to ε the Rankine-Hugoniot equation −λ (ε )(v(ε ) − u0 ) + ( f (v(ε )) − f (u0 )) = 0

|ε | ≤ ε1 .

3.4 Weak solutions and elementary waves

yields

λ (ε )(v(ε ) − u0 ) = (A(v(ε )) − λ (ε ))v (ε ).

69

(3.4.20)

Letting ε → 0 in the latter equation, we have (λ (0) − A(u0 ))v (0) = 0. Thus, we have (λi (u0 ) − A(u0 ))v (0) = 0, that is, v (0) is an i-eigenvector of A(u0 ). By a suitable change of variables, we can assume that v (0) = ri (u0 ),

(3.4.21)

which leads to the second term of the expansion of the Hugoniot curve (3.4.16). Next, differentiating both sides with respect to s of (3.4.20) gives us

λ ”(ε )(v(ε ) − u0 ) + 2λ (ε )v (ε ) = (D2 f (v(ε )) · v (ε ))v (ε ) + (A(v(ε )) − λ (ε ))v”(ε ).

(3.4.22) Letting ε → 0 in (3.4.22) gives us

2λ (0)ri (u0 ) = (D2 f (u0 ) · ri (u0 )) + (A(v(0)) − λi (u0 ))v”(0). Besides, differentiating the equation Ari = λi ri yields us (D2 f · ri )ri = −(A − λi )Dri · ri + (∇λi · ri )ri . This implies that

(2λ (0)−∇λi ·ri (u0 ))ri (u0 ) = (A(v(0))− λi (u0 ))(v”(0)−Dri (u0 )ri (u0 )). (3.4.23) So, from (3.4.23) we find (A(v(0)) − λi (u0 ))(v”(0) − Dri (u0 )ri (u0 )) = 0. The latter equation means that v”(0) − Dri (u0 )ri (u0 ) is an i-eigenvector of A(v(0)). Thus, there exists a number θ such that v”(0) − Dri (u0 )ri (u0 ) = θ ri (u0 ), which yields v”(0) = Dri (u0 )ri (u0 ) + θ ri (u0 ).

(3.4.24)

By the change of the parameter s = ε + θ ε 2 /2, from (3.4.19), (3.4.19), and (3.4.24), we can write u along the Hugoniot curve Hi (u0 ) as

ε2 (Dri (u0 )ri (u0 ) + θ ri (u0 )) + O(ε 3 ) 2 s2 = u0 + sri (u0 ) + Dri (u0 )ri (u0 ) + O(s3 ) 2

u = u0 + ε ri (u0 ) +

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3 Nonlinear hyperbolic systems of balance laws

for |s| ≤ s1 , which establishes (3.4.16). For the shock speed, multiplying both sides of (3.4.23) with the left-eigenvector li (u0 ) gives us 1 λ (0) = ∇λi (u0 ) · ri (u0 ) 2 and this identity implies (3.4.17). This completes the proof of Theorem 3.4.3.

A shock wave associated with a linearly degenerate characteristic field is called a contact discontinuity. Theorem 3.4.4. Assume that the ith characteristic field is linearly degenerate, that is, ∇λi · ri ≡ 0. Then, the integral curve Ri (u0 ) coincides with the Hugoniot curve Hi (u0 ). Furthermore, the characteristic speed and the shock speed along this curve are constant and coincide. Proof. It holds along the integral curve s → v(s) = vi (s; u0 ) that

λk (v(s)) = ∇λk (v(s)) · rk (v(s)) = 0, which implies that the characteristic speed is constant. Next, let us consider the function h(s) = −λk (v(s))(v(s) − u0 ) + f (v(s)) − f (u0 ). Using v (s) = ri (v(s)), we obtain h (s) = −λk (v(s))v (s) + A(v(s))v (s) = 0. Since h(0) = 0, we find h(s) = 0 for all s. This shows that the Rankine-Hugoniot relation is satisfied along the integral curve Ri (u0 ), and the integral curve Ri (u0 ) coincides with the Hugoniot curve Hi (u0 ). Further, we find λk (v(s)) = λ k (u0 , v(s)).

Example 3.4.2. Consider the p-system

∂t v − ∂x u = 0, ∂t u + ∂x p(v) = 0,

x∈R I ,t > 0,

where u, v > 0, and p denote the velocity, specific volume, and the pressure, respectively. A discontinuity of the p-system connecting the left-hand state (v− , u− ) to the right-hand state (v+ , u+ ) propagating with speed s is a weak solution of the form (v− , u− ), x < st, (v, u)(x,t) = (v+ , u+ ), x > st. The Rankine-Hugoniot relations for the discontinuity

3.4 Weak solutions and elementary waves

71

−s(v+ − v− ) − (u+ − u− ) = 0, −s(u+ − u− ) + p(v+ ) − p(v− ) = 0 give us the shock speed s = s(v− , v+ ) = ∓ −

p(v+ ) − p(v− ) v+ − v−

which is well-defined if and only if p(v+ ) − p(v− ) ≤ 0. v+ − v− Here, the 1- and 2-shocks correspond to the minus and the plus sign, respectively. The Hugoniot set is thus composed of two Hugoniot curves H1 , H2 corresponding to s < 0 and s > 0:

H1,2 : u+ = u− ± −(p(v+ ) − p(v− ))(v+ − v− ). Finally, elementary waves of systems of balance laws consist of shock waves, rarefaction waves, and contact discontinuities.

3.4.5 Non-uniqueness of weak solutions Weak solutions of conservation laws are not unique, even in the scalar case. Indeed, consider a scalar conservation law

∂t u + ∂x f (u) = 0.

(3.4.25) 2

A typical counter-example can be described as follows. We take f (u) = u2 and consider the Riemann problem for the conservation law (3.4.25) with the initial data uL , x < 0, u0 (x) = uR , x > 0. It is not difficult to see that the following function uL + uR uL , x < st u(x,t) = , s := uR , x > st, 2 is a weak solution. Furthermore, there are many other weak solutions which are given by

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3 Nonlinear hyperbolic systems of balance laws

⎧ uL , x < s1t, ⎪ ⎪ ⎨ −a, s1t < x < 0, u(x,t) = ⎪ a, 0 < x < s2t, ⎪ ⎩ uR , x > s2t, where s1 :=

uL − a , 2

s2 :=

uR + a , 2

s ≥ max{uL , −uR }.

3.5 Admissibility criteria and Riemann problem 3.5.1 Entropy conditions To select the physical weak solution, an admissibility criterion is needed. In the scalar case, one may constrain any shock wave of (3.4.25) u− , x < σ t, u(x,t) = (3.5.1) u+ , x > σ t, where u− , u+ , and s are constants, to satisfy the following Oleinik criterion: f (v) − f (u− ) f (u+ ) − f (u− ) ≥ v − u− u+ − u−

u ∈ (u− , u+ ).

Geometric explanation of Oleinik’s criterion: The graph of the flux function f lies below (above) the line segment between the points (u− , f (u− )) and (u+ , f (u+ )) if u+ < u− (if u+ > u− , respectively). See Figure 3.7.

Fig. 3.7 Illustration of Oleinik’s criterion

The most standard admissibility criterion of the theory of hyperbolic systems of conservation laws is stated first for systems whose characteristic fields are either genuinely nonlinear or linearly degenerate.

3.5 Admissibility criteria and Riemann problem

73

Definition 3.5.1. A shock wave (3.5.1) is said to be a Lax shock if it satisfies the following Lax’s shock inequalities: there exists an index i ∈ {1, 2, ..., p} such that if the ith characteristic field is genuinely nonlinear then

λi (u+ ) < σ < λi+1 (u+ ),

λi−1 (u− ) < σ < λi (u− ),

(3.5.2)

and if the ith characteristic field is linearly degenerate, then

λi (u− ) = σ = λi (u+ ).

(3.5.3)

Using the parameterization as in Theorem 3.4.3, we define the curve of shock waves admissible for Lax’s shock inequalities Si (u0 ) to be the set of all states Ψi (s) ∈ Hi (u0 ) which can be connected to u0 by a Lax shock. Theorem 3.5.1. Assume that the ith characteristic field is genuinely nonlinear. Then, the shock curve Si (u0 ) consists of all states Ψ (s; u0 ) ∈ Hi (u0 ) corresponding to s ≥ 0, where Ψ (s; u0 ) is given as in Theorem 3.4.3. If the ith characteristic field is linearly degenerate, then Si (u0 ) = Hi (u0 ). Proof. Assume that the ith characteristic field is genuinely nonlinear. Set u(s) = Ψi (s; u0 ),

σ (s) = σ (u0 , Ψi (s; u0 )).

(3.5.4)

We have u(s) = u0 + sri (u0 ) + O(s2 ),

σ (s) = λi (u0 ) +

s + O(s2 ). 2

This yields

λi (u(s)) = λi (u0 ) + sDλi (u0 ) · ri (u0 ) + O(s2 ) s = λi (u0 ) + s + O(s2 ) = σ (s) + + O(s2 ). 2 Let us find out conditions such that λi (u(s)) < σ (s) < λi+1 (u(s)) λi−1 (u0 ) < σ (s) < λi (u0 ).

(3.5.5)

First, λi (u(s)) < σ (s) if and only if s < 0. Next, since λi (u0 ) < λi+1 (u0 ), and

σ (s) → λi (u0 ),

λi+1 (u(s)) → λi+1 (u0 ) as s → 0,

(3.5.6)

we have σ (s) < λi+1 (u0 ) for s small enough. Further, σ (s) < λi (u0 ) for sufficiently small s if and only if s < 0. Finally, since λi−1 (u0 ) < λi (u0 ), we have λi−1 (u0 ) < σ (s). This establishes the first conclusion of Theorem 3.5.1. The second conclusion of Theorem 3.5.1 can be deduced directly from (3.5.3). Besides, to deal with systems whose characteristic fields may not be genuinely nonlinear, one can use Liu’s entropy condition. Recall that Hi (u0 ) denotes the

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3 Nonlinear hyperbolic systems of balance laws

i-Hugoniot curve associated with the ith characteristic field, i = 1, 2, ..., n. Liu’s entropy condition for the shock of the form (3.4.15) is the requirement that

σ (u− , u) ≥ σ (u− , u+ ) for any u ∈ Hi (u− ) between u− and u+ ,

(3.5.7)

where σ (u− , u+ ) denotes the shock speed between u− and u+ . The strict version of Liu’s entropy condition is stated as follows.

σ (u− , u) > σ (u− , u+ ) for any u ∈ Hi (u− ) between u− and u+ , and u = u± . (3.5.8) A shock wave satisfying Liu’s entropy condition is called a classical shock. If the i-characteristic field is genuinely nonlinear, then Lax’s shock inequalities and Liu’s entropy condition are equivalent. However, this is no longer valid if the characteristic field is not entirely genuinely nonlinear.

3.5.2 Entropy inequality and the concept of nonclassical shocks Let U and F be smooth functions defined in a subset Ω of R I n and satisfy U (u) · f (u) = F (u),

u ∈ Ω.

(3.5.9)

Let u be a smooth solution of the system (3.1.2). Multiplying both sides of (3.1.2) by U (u), and using (3.5.9), we obtain

∂ U(u) ∂ F(u) + = 0. ∂t ∂x This leads us to define the concept of entropy as follows. Definition 3.5.2. Given is a subset Ω of R I n in which two functions U and F are defined and satisfy the equality (3.5.9). Then, the function U is called an entropy, the function F is called an entropy flux, and the pair (U, F) is called an entropy pair of the system (3.1.2). If the set Ω is a convex set and U is a convex function in Ω , then U is called a convex entropy, and the pair (U, F) is called a convex entropy pair. Example 3.5.1. Consider a scalar conservation law ut + f (u)x = 0 with u ∈ R I . For any smooth function U, from F (u) = U (u) f (u) we have F(u) = F(a) +

u a

U (v) f (v)dv.

Definition 3.5.3. A weak solution u of the system of conservation laws (3.1.2) is called an entropy solution corresponding to a given entropy pair (U, F) in Ω if the entropy condition ∂t U(u) + ∂x F(u) ≤ 0 (3.5.10)

3.5 Admissibility criteria and Riemann problem

75

is satisfied (in the sense of distributions). This means that the following inequality I × (0, +∞)), ϕ ≥ 0: holds for any function ϕ ∈ Cc∞ (R ∞ +∞ 0

−∞

U(u)ϕt + F(u)ϕx dxdt ≥ 0.

Consider a function u : R I × [0, +∞) → Ω . Suppose that u is smooth everywhere, except along a curve Σ . Suppose in addition that the curve Σ is a piecewise smooth curve given by Σ = {(x,t) : x = x(t), t ∈ I}, (3.5.11) where x = x(t),t ∈ I, for some interval I, is a piecewise smooth function. The leftand right limits u± of u on both sides of Σ are given by u± = lim u((x(t),t) + ε n), ε →0±

t ∈ I.

(3.5.12)

The following result provides us with a useful tool to check whether such a function u is an entropy solution. Theorem 3.5.2. Let u : R I × [0, +∞) → Ω be a smooth function in (R I × [0, +∞)) \ Σ , where Σ is a piecewise smooth curve given by (3.5.11), and let u± be defined as in (3.5.12). The function u is an entropy solution of (3.1.2) corresponding to the given entropy pair (U, F) if the following conditions are fulfilled: (a) (b)

u satisfies pointwise (3.1.2) everywhere it is differentiable; u satisfies across Σ : − s[u] + [ f (u)] = 0, − s[U(u)] + [F(u)] ≤ 0,

(3.5.13)

where s=

dx(t) , dt

[U(u)] = U(u+ ) −U(u− ),

[F(u)] = F(u+ ) − F(u− ).

The proof of Theorem 3.5.2 is similar to the one of Theorem 3.4.1 and is omitted. Example 3.5.2. Consider the p-system

∂t v − ∂x u = 0, ∂t u + ∂x p(v) = 0,

x∈R I , t > 0.

An entropy condition of the form (3.5.10) is given by

∂t U(u, v) + ∂x F(u, v) ≤ 0, U(u, v) :=

u2 + Σ (v), 2

(in the sense of distribution).

F(u, v) := u p(v), Σ (v) := −

v

p(w) dw, 0

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3 Nonlinear hyperbolic systems of balance laws

When a characteristic field is not entirely genuinely nonlinear, there exist shock waves in applications that violate Liu’s entropy condition. These shock waves are called nonclassical shocks. Definition 3.5.4. A shock wave satisfying the entropy inequality (3.5.10) for a given prescribed entropy pair (U, F), but violating Liu’s entropy condition is called a nonclassical shock.

3.5.3 Illustration of admissibility criteria: isothermal van der Waals fluids To better understand the admissibility criteria, let us consider the following example. Consider the p-system

∂t v − ∂x u = 0, ∂t u + ∂x p(v) = 0,

x∈R I ,t > 0,

where u, v > 0, and p denote the velocity, specific volume, and the pressure, respectively. The pressure is given by p=

a RT0 − , v − b v2

v > b,

where a, b, and T0 are given positive constants, and T0 satisfies (2/3)3 a (3/4)4 a < T0 < . Rb Rb As seen earlier, the p-system is then strictly hyperbolic, but fails to be genuinely nonlinear when v moves across μ and ν . First, we investigate properties of the pressure function. The tangent line to the graph of p = p(v) at v = μ intersects the graph at a point denoted by μ − . The tangent line to the graph of p = p(v) at v = ν intersects the graph at a point denoted by ν − . From any v ∈ [ν − , μ − ], there are two tangent lines to the graph of the pressure with the corresponding tangency points denoted by ϕ (v) ≤ ψ (v). This means that

p(v) − p ϕ (v) p(v) − p ψ (v) , p ψ (v) = . p ϕ (v) = v − ϕ (v) v − ψ (v) See Figure 3.8. It is easy to see that the values v and ψ (v) always lie on different sides with respect to ν , and the values v and ϕ (v) always lie on different sides with respect to μ , in the sense that

3.5 Admissibility criteria and Riemann problem

77

Fig. 3.8 The tangent functions ϕ and ψ

(ϕ (v) − μ )(v − μ ) < 0

for v = μ ,

ϕ (μ ) = μ ,

(ψ (v) − ν )(v − ν ) < 0 for v = ν ,

ψ (ν ) = ν .

There are also two points c < μ and d > ν such that

ψ (c) = d,

ϕ (d) = c.

The function ψ is increasing for v ∈ [ν − , c] and decreasing for v ∈ [c, μ − ]. The function ϕ is decreasing for v ∈ [ν − , d] and increasing for v ∈ [d, μ − ]. Moreover ϕ maps [ν − , μ − ] onto [c, ν ], while ψ maps [ν − , μ − ] onto [μ , d]. For all v ∈ (c, d), the tangent at the point with coordinate v intersects the graph of p at exactly two points, denoted by ϕ − (v) and ψ − (v), such that

ϕ − (v) ≤ ψ − (v). Let us recall that a shock wave of the p-system connecting a left-hand state (v− , u− ) to a right-hand state (v+ , u+ ) and propagating with speed s is a weak solution of the form x < st, (v− , u− ), (v, u)(x,t) = x > st. (v+ , u+ ),

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3 Nonlinear hyperbolic systems of balance laws

Definition 3.5.5. A shock wave is called a classical shock if it satisfies Liu’s entropy condition. A shock wave is called a nonclassical shock if it satisfies the entropy inequality

∂t U(u, v) + ∂x F(u, v) ≤ 0, U(u, v) :=

u2 + Σ (v), 2

F(u, v) := up(v),

Σ (v) := −

v

p(w)dw, 0

but violates Liu’s entropy condition. The following theorem can be checked geometrically by considering the graph of the pressure function. Theorem 3.5.3 (Classical shock waves). Fix an arbitrary left-hand state (v− , u− ), and consider the set of right-hand states (v+ , u+ ) attainable by an admissible 1-shock. A classical shock is characterized as follows: (a) (b) (c) (d)

If v− ∈ (0, c) ∪ (μ − , +∞), then v+ ∈ (0, v− ]. If v− ∈ [c, μ ], then v+ ∈ (0, v− ] ∪ [ϕ − (v− ), ψ (v− )]. If v− ∈ (μ .ν ), then v+ ∈ (0, ϕ − (v− )] ∪ [v− , ψ (v− )]. If v− ∈ [ν , μ − ], then v+ ∈ (0, ϕ − (ψ (v− ))] ∪ [ψ (v− ), v− ].

As indicated earlier, admissible shock waves satisfying the above entropy inequality are given by the following inequality: −s(U(u+ , v+ ) −U(u− , v− )) + (F(u+ , v+ ) − F(u− , v− )) ≤ 0. Substituting U(u, v) =

u2 + Σ (v), 2

F(u, v) = up(v),

Σ (v) = −

v

p(w)dw 0

into the latter inequality, and by a straightforward calculation we deduce that the entropy inequality is equivalent to the condition p(v+ ) + p(v− ) (v+ − v− ) ≤ 0. E(v− , v+ ) := −s(v− , v+ ) Σ (v+ ) − Σ (v− ) + 2 The function E is referred to as the entropy dissipation. We have just seen that the entropy inequality is equivalent to the fact that the entropy dissipation is nonpositive. Since s < 0 for 1-shocks and s > 0 for 2-shocks, the entropy inequality now becomes D(v− , v+ ) :=

v− v+

p(w)dw +

p(v+ ) + p(v− ) (v+ − v− ), 2

D(v− , v+ ) ≤ 0 for 1-shocks, D(v− , v+ ) ≥ 0 for 2-shocks.

(3.5.14)

3.5 Admissibility criteria and Riemann problem

79

Let us fix a left-hand state (v− , u− ), and let us determine the set of all right-hand states (v+ , u+ ) that can be connected to (v− , u− ) by an admissible 1-shock, which satisfies the entropy inequality (3.5.14). This can be done by investigating D(v− , v+ ) as a function of v+ . Since Dv+ (v− , v+ ) = (p(v− ) − p(v+ ) − p (v+ )(v− − v+ ))/2, we conclude that Dv+ (v− , ϕ (v− )) = Dv+ (v− , ψ (v− )) = 0, Dv+ (v− , v+ ) < 0,

ϕ (v− ) < v+ < ψ (v− ),

Dv+ (v− , v+ ) > 0,

v+ < ϕ (v− ),

v > ψ (v− )).

The entropy dissipation D(v− , v+ ) thus attains a local maximum at v+ = ϕ (v− ) and a local minimum at v+ = ψ (v− ). To determine the sign of the entropy dissipation, we need to know the sign of the local maximum P(v− ) := D(v− , ϕ (v− )),

(3.5.15)

and the sign of the local minimum Q(v− ) := D(v− , ψ (v− )).

(3.5.16)

Fundamental properties of the function P(v− ) defined by (3.5.15) are given below. Lemma 3.5.1. The function P = P(v) defined by (3.5.15) for v ∈ (b− , a− ) is increasing in the interval (μ , d) and decreasing in the intervals (b, μ ) and (d, +∞). Moreover, there exists exactly one value f ∈ (d, a− ) such that P(μ ) = P( f ) = 0, P(v) < 0 if v > f , P(v) > 0,

if v < f , v = μ .

Proof. The derivative of P is given by d d D(v, ϕ (v)) = Dv− (v, ϕ (v)) + Dv+ (v, ϕ (v)) ϕ (v) dv dv −1 = Dv− (v, ϕ (v)) = (p(ϕ (v)) − p(v) − p (v)(ϕ (v) − v)) 2 (v − ϕ (v)) p(ϕ (v)) − p(v) = − p (v) . 2 ϕ (v) − v

P (v) =

Therefore, we have

(3.5.17)

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3 Nonlinear hyperbolic systems of balance laws

P (v) > 0 for P (v) < 0 for

μ < v < d, v < μ , v > d.

This establishes the monotonicity property of the function v → P(v), v > b. Since P(μ ) = 0 and P (v) > 0 for v ∈ (μ , d), we find P(d) > 0. Moreover, we get P(a− ) =

a− μ

p(y)dy −

p(μ ) + p(a− ) − (a − μ ). 2

Observe that the tangent line to the graph of the pressure at (μ , p(μ )) is above the graph of the pressure in the interval (μ , a− ). So, the area bounded by this tangent line, the lines v = μ and v = a− , and the v-axis is given by p(μ ) + p(a− ) − (a − μ ) > 2

a− μ

p(y)dy,

where the right-hand side is the area under the curve p = p(v), above the v-axis in the same interval. This means that P(a− ) < 0. Since P(d)P(b− ) < 0 and that the function P is strictly decreasing for v > d, there is exactly one value f ∈ (d, a− ) such that P( f ) = 0. The monotonicity property of P shown above implies that P(v) < 0 if v > f , P(v) > 0, if v < f , v = μ ,

P(μ ) = 0,

which establishes (3.5.17). Arguing similarly, we can obtain the properties of the function Q in (3.5.16) as in the following lemma. Lemma 3.5.2. The function Q = Q(v) defined by (3.5.16) for v ∈ (b− , a− ) is increasing in the interval (c, ν ) and decreasing in the intervals (0, c) and (ν , +∞). Moreover, there exists exactly one value e ∈ (d, a− ) such that Q(ν ) = Q(e) = 0, Q(v) > 0 if v < e, Q(v) < 0,

if v > e, v = ν .

Lemmas 3.5.1 and 3.5.2 yield the following theorem, which describes the set of admissible shock waves satisfying the entropy inequality. Theorem 3.5.4. Given an arbitrary fixed v− . If v− ∈ (0, ν − ) ∪ (μ − , +∞), the entropy dissipation D(v− , v+ ) is a decreasing function of v+ > 0. If v− ∈ [ν − , μ − ], the function v+ → D(v− , v+ ) is increasing in the intervals 0, ϕ (v− ) and

ψ (v− ), +∞ , decreasing in the interval ϕ (v− ), ψ (v− ) , and has a local maximum P(v− ) at ϕ (v− ) and a local minimum Q(v− ) at ψ (v− ). The functions

3.5 Admissibility criteria and Riemann problem

81

P(v), Q(v), v > 0 are characterized by Lemmas 3.5.1 and 3.5.2. The following conclusions on admissible shock waves hold: (a)

If v− ≤ e or v− ≥ f , then (3.5.14) selects the set v+ ≤ v− .

(b)

If v− ∈ (e, μ ], then the entropy dissipation D admits three roots: v− and two other roots denoted by ϕ∞ (v− ) < ψ∞ (v− ); see Figure 3.9. The entropy condition (3.5.14) selects the following set:

Fig. 3.9 The boundary functions ϕ∞ and ψ∞ for v ∈ (e, μ ]

v+ ∈ (b, v− ] ∪ [ϕ∞ (v− ), ψ∞ (v− )]. (c)

If v− ∈ (μ , ν ), then the entropy dissipation D admits three roots: v− and two other roots denoted by ϕ∞ (v− ) < ψ∞ (v− ); see Figure 3.10. The entropy condition (3.5.14) selects the following set:

Fig. 3.10 The boundary functions ϕ∞ and ψ∞ for v ∈ (μ , ν )

82

3 Nonlinear hyperbolic systems of balance laws

v+ ∈ (b, ϕ∞ (v− )] ∪ [v− , ψ∞ (v− )]. (d)

If v− ∈ [ν , f ), then the entropy dissipation D admits three roots: v− and two other roots denoted by ϕ∞ (v− ) < ψ∞ (v− ); see Figure 3.11. The entropy condition (3.5.14) selects the following set: v+ ∈ (b, ϕ∞ (v− )] ∪ [ψ∞ (v− ), v− ].

Fig. 3.11 The boundary functions ϕ∞ and ψ∞ for v ∈ [ν , f )

The following theorem characterizes the sets of nonclassical shock waves. Theorem 3.5.5 (Nonclassical admissible shock waves). Fix an arbitrary left-hand state (v− , u− ), and consider the set of right-hand states (v+ , u+ ) attainable by an admissible 1-shock. A nonclassical shock is characterized as follows: (a) (b)

If v− ∈ (e, c], then

v+ ∈ [ϕ∞ (v− ), ψ∞ (v− )].

If v− ∈ (c, μ ], then v+ ∈ [ϕ∞ (v− ), ϕ − (v− )) ∪ (ψ (v− ), ψ∞ (v− )].

(c)

If v− ∈ (μ .ν ), then v+ ∈ (ϕ − (v− ), ϕ∞ (v− )] ∪ (ψ (v− ), ψ∞ (v− )].

(d)

If v− ∈ [ν , f ), then v+ ∈ (ϕ − (ψ (v− )), ϕ∞ (v− )] ∪ [ψ∞ (v− ), ψ (v− )).

(e)

If v− ∈ [ f , μ − ), then v+ ∈ (ϕ − (ψ (v− )), ψ (v− )).

3.5 Admissibility criteria and Riemann problem

83

3.5.4 Entropy condition for nonconservative systems Consider the hyperbolic system in nonconservative form

∂t U + A(U) ∂xU = 0.

(3.5.18)

The entropy condition for the nonconservative system of balance laws (3.5.18) is also understood in the sense of nonconservative product and has the form ∂t U (U) + DU U (U) A(U(.,t))∂xU(.,t) ≤ 0, φ

where φ is a given Lipschitz family of paths, and U is a convex function satisfying DU2 U (U) A(U) = A(U)T DU2 U (U).

3.5.5 The Riemann problem Let us now consider the Riemann problem for the following system of conservation laws: ∂u ∂ + f (u) = 0, x ∈ R I ,t > 0, (3.5.19) ∂t ∂x subject to the initial condition uL , x < 0, u(x, 0) = (3.5.20) uR , x > 0, where uL , uR are constant states in an open and connected domain Ω ⊂ R I n. If the ith characteristic field is genuinely nonlinear, we use the normalization (3.4.11). Given an arbitrary u0 ∈ Ω , let Φi (s; u0 ) and Ψi (s; u0 ) be defined as in Theorem 3.4.2 and Theorem 3.4.3, respectively. Then, let us consider the function Ψi (s; u0 ), s < 0, (3.5.21) χi (s; u0 ) = Φi (s; u0 ), s ≥ 0. By definition, the set {χi (s; u0 ),

s small enough}

(3.5.22)

consists of all right-hand states u that can be connected to the given left-hand state u0 by either an i-Lax shock for s < 0, or an i-rarefaction wave for s ≥ 0. If the ith characteristic field is linearly degenerate, we set

χi (s; u0 ) = Ψi (s; u0 )

(3.5.23)

84

3 Nonlinear hyperbolic systems of balance laws

which consists of all states u that can be connected to u0 by an i-contact discontinuity. Theorem 3.5.6. Assume that for each i = 1, 2, ..., n, the ith characteristic field is either genuinely nonlinear, or linearly degenerate. Then for every uL ∈ Ω , there exists a neighborhood G of uL in Ω such that whenever uR ∈ G, the Riemann problem for (3.1.2) admits a unique weak solution which constitutes from at most n + 1 states separated by rarefaction wave, Lax shocks, and contact discontinuities. Proof. Let uL ∈ Ω . Consider the mapping

χ : s = (s1 , s2 , ..., s p ) → χ (s) = χ p (s p ; χ p−1 (s p−1 ; ...; χ1 (s1 ; uL )...)) which is defined in a neighborhood of 0 ∈ R I n and takes values in Ω ⊂ R I p . This means that the left-hand state uL is connected on the right with χ1 (s1 ; uL ) = u1 by a 1-wave, then the state u1 is connected to u2 = χ2 (s2 ; u1 ) on the right by a 2-wave,..., and finally the state un−1 is connected on the right with un = χ p (sn ; un−1 ) by an n-wave. We need to find out what the range of the mapping χ is. Thus, we solve the equation χ (s) = uR . (3.5.24) Since χ is a twice differentiable mapping with

χ (0) = χn (0; χn−1 (0; ...; χ1 (0; uL )...)) = uL , and, moreover, we find

χi (si ; u) = u + si ri (u) + O(s2i ), thus, we have

χ2 (s2 ; χ1 (s1 ; uL )) = χ2 (s2 ; ul + s1 r1 (uL ) + O(s21 )) = uL + s1 r1 (uL ) + O(s21 ) + s2 r2 (uL + s1 r1 (uL ) + O(s21 )) + O(s22 ) = uL + s1 r1 (uL ) + s2 r2 (uL ) + O(s21 + s22 ). Using an induction argument, we get n

χ (s) = uL + ∑ si ri (uL ) + O(|s|2 ). i=1

This means that the derivative Dχ (0) of χ at the origin is given by n

Dχ (0) · η = ∑ ηi ri (uL ) for all η = (η1 , η2 , ..., ηn )T ∈ R I n. i=1

Since the vectors ri (uL ), 1 ≤ i ≤ n are linearly independent, the matrix Dχ (0) is invertible. By the inverse function theorem, there exists a neighborhood D of uL in Ω

3.6 Bibliographical notes

85

such that for any uR ∈ D, the equation (3.5.24) admits a unique solution s ∈ R I n . This gives us a solution u consisting of (n + 1) states u0 = uL , u1 , ..., un = uR separated from each other by the i-waves and satisfy Lax’s shock inequalities.

3.6 Bibliographical notes We provide here a brief selection of the most relevant papers and refer the reader to the bibliography at the end of this monograph for additional references. The mathematical theory of nonlinear hyperbolic systems of conservation laws was pioneered by Lax [203–205] (Riemann problem) and Glimm [138] (Cauchy problem). We do not review the many formulations of the entropy condition but refer the reader to Oleinik [255] (scalar equations) and Liu [232, 233] (systems), and the references cited therein. For the initial-value problem for nonlinear hyperbolic systems in one space dimension, we refer to the textbooks by Bressan [69], Dafermos [110], Holden and Risebro [166], LeFloch [211], and Smoller [294]. For the mathematical theory of multi-dimensional compressible fluids, we refer to Majda [240]. For the study of the dynamics of phase transitions for a model of material science, we refer to Abayaratne-Knowles [2, 4] and Truskinovsky [318], as well as Slemrod [290] and Fan and Slemrod [130, 131]. The theory of nonclassical shocks for hyperbolic systems is developed in the textbook by LeFloch [211] and the review papers [210, 212]. For the existence and uniqueness of nonclassical solutions, we refer to Amadori-Baiti-LeFloch-Piccoli [13] and Baiti-LeFloch-Piccoli [34, 35]. The Riemann problem was treated by Shearer et al. [163] and Hayes-LeFloch [160, 162], as well as in [223–226]. The relation between a kinetic function and the equivalent equation of a numerical scheme is investigated in LeFloch-Mohammadian [213]. Boutin et al. [62] introduced a converging scheme for nonclassical shocks. See also Chalons and LeFloch [89] and the more recent works [82–86]. Various aspects of the theory of nonlinear hyperbolic systems are discussed in Isaacson-Temple [174, 175], Marchesin-Paes Leme [241], Goatin-LeFloch[141], Hong-Temple [169, 170], LeFloch-Thanh [227–229], Kr¨oner-LeFloch-Thanh [201], Thanh [301], and Andrianov-Warnecke [20, 21]. The report [99] overviews developments in phase transition dynamics. The theory of nonconservative hyperbolic systems was pioneered by LeFloch [207, 208], in which the nonconservative reformulation of balance laws in Examples 3.1.5 and 3.1.6 was proposed. Dal Maso-LeFloch-Murat [114] introduced the concept of nonconservative products in the framework of functions with bounded variation; their definition arose as a generalization of Volpert’s product [322]. The existence theory for the Riemann problem associated with nonconservative hyperbolic systems is established in [114] and LeFloch-Tzavaras [221]. Kinetic relations were also introduced for nonconservative systems by Berthon-Coquel-LeFloch [51].

Chapter 4

Riemann problem for ideal fluid

4.1 Introduction In this chapter we consider the Riemann problem for the gas dynamics equations in the Eulerian coordinates (x ∈ R I ,t > 0)

∂ ρ ∂ (ρ u) + = 0, ∂t ∂x ∂ (ρ u) ∂ (ρ u2 + p) + = 0, ∂t ∂x ∂ (ρ e) ∂ ((ρ e + p)u) + = 0, ∂t ∂x in both isentropic and non-isentropic cases, assuming that the fluid is polytropic and ideal. Throughout, the notation ρ , p, T, ε , and S stands for the five thermodynamic variables: density, pressure, temperature, internal energy, and specific entropy, respectively. Moreover, v = 1/ρ , u, and e = ε + u2 /2 stand for the specific volume, the particle velocity, and the total energy, respectively. For an ideal fluid, the quantities pv and ε —as functions of (v, T )—depend only on T . In a polytropic fluid, the adiabatic exponent is constant. The choice of the polytropic ideal fluid leads us to a simpler mathematical setting and corresponds to the equation of state p = (γ − 1) ρε for some constant γ > 1.

4.2 Riemann problem for isentropic ideal flows 4.2.1 Hyperbolicity and genuine nonlinearity We begin our study with isentropic fluids, with general equation of state p = p(ρ ) and treat the equations in Euler coordinates (t ≥ 0, x ∈ R I) © Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 4

87

88

4 Riemann problem for ideal fluid

∂t ρ + ∂x (ρ u) = 0,

(4.2.1)

∂t (ρ u) + ∂x (ρ u2 + p) = 0,

whose unknowns are the density ρ and the velocity u. The equations (4.2.1) describe the conservation of mass and momentum, respectively. For smooth solutions (ρ , u), the first equation can be written as ρt + uρx + ρ ux = 0, while the second equation in (4.2.1) is equivalent to p (ρ ) u(ρt + (ρ u)x ) + ρ ut + ρx + uux = 0, ρ so the conservation of momentum can be replaced by ut + u ux +

p (ρ ) ρx = 0. ρ

The Jacobian matrix of this latter system reads u ρ , A(ρ , u) = p (ρ ) u ρ and admits two distinct eigenvalues λ1 (ρ , u) = u − p (ρ ) < λ2 (v) = u + p (ρ ), provided ρ > 0, (4.2.2) p (ρ ) > 0, which we always assume from now. For example, the condition (4.2.2) holds if the flow is isentropic and ideal. That is, the pressure law p = κρ γ , where κ > 0 is a constant. Corresponding eigenvectors can be chosen to be ρ ρ r1 (ρ , u) = ( ρ , u) = , r , 2 − p (ρ ) p (ρ )

so that Dλ1 (ρ , u) =

−p (ρ ) √ 2 p (ρ )

1

,

Dλ2 (ρ , u) =

2

p (ρ ) √

p (ρ )

1

,

which yields Dλ1 (ρ , u)T r1 (ρ , u) = −

p (ρ )ρ + 2p (ρ ) = −Dλ2 (ρ , u)T r2 (ρ , u). 2 p (ρ )

Thus, the two characteristic fields are genuinely nonlinear provided

ρ p (ρ ) + 2 p (ρ ) > 0,

ρ > 0,

which, for example, is satisfied by the isentropic ideal fluids.

(4.2.3)

4.2 Riemann problem for isentropic ideal flows

89

Since v = 1/ρ , we have dp dp = −ρ 2 , dv dρ 2 d2 p d p dp 3 = ρ ρ 2 +2 . dv2 dρ dρ Thus, the strictly hyperbolic condition (4.2.2) is equivalent to d2 p

dp dv

< 0 and the gen-

uinely nonlinear condition (4.2.3) is equivalent to dv2 > 0. In the rest of this section, we assume that the hyperbolicity condition (4.2.2) as well as the genuine nonlinearity condition (4.2.3) holds.

4.2.2 Rarefaction waves We describe rarefaction waves to the isentropic Euler equations. It is convenient to introduce the following normalization of the eigenvectors: r¯k (ρ , u) := rk /Dλk (ρ , u)T · rk (ρ , u),

Dλk (ρ , u) · r¯k (ρ , u) ≡ 1,

k = 1, 2.

Recall that a k-rarefaction wave (for k = 1, 2) connecting a left-hand state ρL , uL to a right-hand state ρR , uR is a continuous self-similar solution ⎧ x ⎪ (ρL , uL ) ≤ λk (ρL , uL ), ⎪ ⎪ ⎨ x t x , λk (wL ) ≤ ≤ λk (wR ), (ρ , u)(x,t) = V (4.2.4) ⎪ t t ⎪ x ⎪ ⎩ (ρR , uR ), ≥ λk (wR ), t where V = (ρ , u) is determined by solving the ordinary differential equations V (ξ ) = r¯k (V (ξ )),

ξ > λk (ρL , uL ), V (λk (ρL , uL )) = (ρL , uL ).

Observe that

2ρ p (ρ ) dρ < 0, = ∓ dξ p (ρ )ρ + 2p (ρ ) > 0,

(4.2.5)

k = 1, k = 2,

thanks to (4.2.3). Thus, we can use ρ as a parameter of the integral curve of (4.2.5). It follows from (4.2.5) that the integral curve corresponding to the first characteristic field is given by − p (ρ ) du = , u|ρ =ρL = uL . dρ ρ

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4 Riemann problem for ideal fluid

After integration, we arrive at the forward rarefaction curve, denoted by R1F (ρL , uL ), associated with the first characteristic family consisting of all right-hand states (ρR , uR ) that can be connected to (ρL , uL ) on the left by a 1-rarefaction wave: ρ p (z) F R1 (ρL , uL ) : u = ϕ1 (ρ ) := uL − dz, ρ ≤ ρL . (4.2.6) z ρL It is easy to check that the function [0, ρL ] ρ → ϕ1 (ρ ) is strictly decreasing. Similarly, the backward rarefaction curves, denoted by R2B (ρR , uR ), associated with the second characteristic family consist of all left-hand states (ρL , uL ) that can be connected to a given state (ρR , uR ) by a 2-rarefaction wave: ρ p (z) B R2 (ρR , uR ) : u = ϕ2 (ρ ) := uR + dz, ρ ≤ ρR . (4.2.7) z ρR It is easy to see that the function [0, ρR ] ρ → ϕ2 (ρ ) is strictly increasing.

4.2.3 Shock waves We next construct shock wave solutions to the isentropic ideal fluid equations. The Rankine-Hugoniot relations for a shock wave connecting (ρ0 , u0 ) to (ρ , u) for the system (4.2.1) read − σ (ρ − ρ0 ) + (ρ u − ρ0 u0 ) = 0, − σ (ρ u − ρ0 u0 ) + (ρ u2 + p) − (ρ0 u20 + p0 ) = 0. Eliminating σ , we obtain (ρ u − ρ0 u0 )2 = (ρ − ρ0 )(ρ u2 + p − ρ0 u20 − p0 ), which yields

ρ 2 u2 − 2ρ uρ0 u0 + ρ02 u20 = ρ 2 u2 + ρ02 u20 + ρ (p − p0 ) − ρρ0 u20 − ρ0 ρ u2 − ρ0 (p − p0 ). Canceling terms in the last equation, after re-arranging terms, we get (p − p0 )(ρ − ρ0 ) = (u − u0 )2 ρ0 ρ , which yields the Hugoniot set H (w0 ) :

u = u0 ±

(p − p0 )(ρ − ρ0 ) . ρ0 ρ

Recall from Chapter 3 that the k-Hugoniot curve is tangent to the vector rk at the point (ρ0 , u0 ), k = 1, 2. Along the 1-Hugoniot curve we have

4.2 Riemann problem for isentropic ideal flows

91

− p (ρ0 ) u − u0 = + O(ε ). ρ − ρ0 ρ0 This means that

>0 u − u0 ρ0 .

The 1-Hugoniot curve is therefore determined by ⎧ ⎪ (p − p0 )(ρ − ρ0 ) ⎪ ⎪ , ρ > ρ0 , ⎪ ⎨− ρ0 ρ H1 (ρ0 , u0 ) : u = u0 + ⎪ (p − p0 )(ρ − ρ0 ) ⎪ ⎪ ⎪ , ρ < ρ0 . ⎩ ρ0 ρ Similarly, the 2-Hugoniot curve is determined by ⎧ ⎪ (p − p0 )(ρ − ρ0 ) ⎪ ⎪ , ρ > ρ0 , ⎪ ⎨ ρ0 ρ H2 (ρ0 , u0 ) : u = u0 + ⎪ (p − p0 )(ρ − ρ0 ) ⎪ ⎪ ⎪ , ρ < ρ0 . ⎩− ρ0 ρ In the following we postulate that any shock wave must satisfy Lax’s shock inequalities as the admissibility criterion. Since λ1 < 0 < λ2 , it is easy to see that Lax’s shock inequalities for an admissible shock connecting a left-hand state w0 with a right-hand state w become

λk (w) < σk < λk (w0 ),

k = 1, 2.

Since p > 0, the function p is strictly increasing. Thus, Lax’s shock inequalities for 1-shocks are equivalent to the condition ρ > ρ0 . Thus, the 1-shock curve consisting of all right-hand states w = (ρ , u) that can be connected to a left-hand state wL = (ρL , uL ) by an admissible shock wave is given by (p − pL )(ρ − ρL ) , ρ > ρL . (4.2.8) S1F (wL ) : u = ϕ1 (ρ ) := uL − ρL ρ It is easy to check that the function (ρL , +∞) → ϕ1 (ρ ) defined by (4.2.8) is strictly decreasing. Similarly, the backward 2-shock curve consisting of all left-hand states w = (ρ , u) that can be connected to a right-hand state wR = (ρR , uR ) by an admissible shock wave satisfying Lax’s shock inequalities is given by (p − pR )(ρ − ρR ) B , ρ > ρR . (4.2.9) S2 (wR ) : u = ϕ2 (ρ ) := uR + ρR ρ

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4 Riemann problem for ideal fluid

It is easy to check that the function (ρR , +∞) → ϕ2 (ρ ) defined by (4.2.9) is strictly increasing. The function ρ → ϕ2 (ρ ) defined by (4.2.7) and (4.2.9) is thus strictly increasing.

4.2.4 Wave curves and Riemann problem Let w = (ρ , u). Define the forward 1-wave curve issuing from wL by W1F (wL ) = R1F (wL ) ∪ S1F (wL ) :

ρ → u = ϕ1 (ρ ),

(4.2.10)

and the backward 2-wave curve issuing from wR by W2B (wR ) = R2B (wR ) ∪ S2B (wR ) :

ρ → u = ϕ2 (ρ ).

(4.2.11)

From the above argument, we can see that the function ρ → ϕ1 (ρ ) in (4.2.10) is strictly decreasing, and the function ρ → ϕ2 (ρ ) in (4.2.11) is strictly increasing. Consider the asymptotic behavior of the wave curves and set

ϕ1 (0+) := lim ϕ1 (ρ ) ≤ +∞, ρ →0+

(4.2.12)

ϕ2 (0+) := lim ϕ2 (ρ ) ≥ −∞. ρ →0+

For example, consider an isentropic ideal gas where the pressure is given by p = κρ γ ,

κ > 0, 1 < γ < 5/3.

One has

ϕ1 (0+) = uL +

√ 2 κγ (γ −1)/2 ρ , γ −1 L

ϕ2 (0+) = uR −

√ 2 κγ (γ −1)/2 ρ . γ −1 R

Moreover, we find lim ϕ1 (ρ ) = −∞,

ρ →+∞

lim ϕ2 (ρ ) = +∞,

ρ →+∞

(4.2.13)

provided lim p(ρ ) = +∞.

ρ →+∞

Therefore, whenever

ϕ1 (0+) > ϕ2 (0+), W1F (wL )

the two wave curves and W2B (wR ) intersect at a unique point. This intersection point determines a unique Riemann solution. Theorem 4.2.1. (Riemann problem for isentropic ideal fluids) Let w = (ρ , u). Assume that

4.2 Riemann problem for isentropic ideal flows

p (ρ ) > 0,

93

ρ p (ρ ) + 2 p (ρ ) > 0,

(ρ > 0).

In addition, assume that the function ϕ1 defined by (4.2.6) and (4.2.8), and the function ϕ2 defined by (4.2.7) and (4.2.9) satisfy

ϕ1 (0) > ϕ2 (0),

lim p(ρ ) = +∞.

ρ →+∞

Then, the two wave curves W1F (wL ) and W2B (wR ) intersect at a unique point: W1F (wL ) ∩ W2B (wR ) = {w∗ = (ρ∗ , u∗ )}.

(4.2.14)

Accordingly, the Riemann problem for (4.2.1) admits a unique solution which can be described as follows:

Fig. 4.1 Riemann solution determined by the intersection of the two wave curves W1F (wL ) and W2B (wR )

(i) If ρ∗ > ρL and ρ∗ > ρR , the Riemann solution consists of a 1-shock from ρL , uL to ρ∗ , u∗ , followed by a 2-shock connecting to ρR , uR . (Cf. the upper-left illustration in Figure 4.1.) (ii) If ρ∗ > ρL and ρ∗ ≤ ρR , the Riemann solution consists of a 1-shock from ρL , uL to ρ∗ , u∗ , followed by a 2-rarefaction wave connecting to ρR , uR . (Cf. the upperright illustration in Figure 4.1.) (iii) If ρ∗ ≤ ρL and ρ∗ > ρR , the Riemann solution consists of a 1-rarefaction wave from ρL , uL to ρ∗ , u∗ , followed by a 2-shock to ρR , uR . (Cf. the lower-left illustration in Figure 4.1.)

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4 Riemann problem for ideal fluid

(iv) If ρ∗ ≤ ρL and ρ∗ ≤ ρR , the Riemann solution consists of a 1-rarefaction wave from ρL , uL to ρ∗ , u∗ , followed by a 2-rarefaction wave to ρR , uR . (Cf. the lowerright illustration in Figure 4.1.)

4.3 Riemann problem for polytropic ideal fluids In this section we consider the Riemann problem for the gas dynamics equations in the Euler coordinates ∂ ρ ∂ (ρ u) + = 0, ∂t ∂x ∂ (ρ u) ∂ (ρ u2 + p) (4.3.1) + = 0, ∂t ∂x ∂ (ρ e) ∂ ((ρ e + p)u) + = 0, ∂t ∂x where the fluid is assumed to be polytropic and ideal. It is not difficult to check that the pressure of a polytropic and ideal fluid can be given as a function of the density and the entropy: S∗ − S p = p(ρ , S) = (γ − 1) exp ργ , (4.3.2) Cv where γ > 1,Cv > 0, and S∗ are constants.

4.3.1 Basic properties The system (4.3.1) for any smooth solution U(x,t) := (ρ (x,t), u(x,t), S(x,t)) can be re-written as ∂t ρ + u∂x ρ + ρ∂x u = 0, 1 ∂t u + u∂x u + ∂x p = 0, ρ ∂t S + u∂x S = 0, whose Jacobian matrix is given by ⎛

u ρ ⎜ ∂ρ p A(U) = ⎜ ⎝ ρ u 0 0

⎞ 0 ∂S p ⎟ ⎟. ρ ⎠ u

The characteristic polynomial of the Jacobian matrix A(U) is given by (u − λ )[(u − λ )2 − ∂ρ p(ρ , S)] = 0. Since

S∗ − S ∂ρ p(ρ , S) = γ (γ − 1) exp ρ γ −1 > 0, Cv

(4.3.3)

4.3 Riemann problem for polytropic ideal fluids

95

the matrix A(U) admits three real and distinct eigenvalues λ1 = u − c < λ2 = u < λ3 = u + c, where c denotes the sound speed S∗ − S c = c(ρ , S) = ∂ρ p(ρ , S) = γ (γ − 1) exp ρ (γ −1)/2 . (4.3.4) 2Cv Thus, the system (4.3.1) is strictly hyperbolic. Let us choose the corresponding eigenvectors by ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ρ ρ −∂S p(ρ , S) ⎠ , r3 = ⎝ c ⎠ . 0 r1 = ⎝−c⎠ , r2 = ⎝ ∂ρ p(ρ , S) 0 0 Then, we have ⎞ −cρ (ρ , S) ⎠, 1 Dλ1 = ⎝ −cS (ρ , S) ⎛

⎛ ⎞ 0 Dλ2 = ⎝1⎠ , 0

⎞ cρ (ρ , S) Dλ3 = ⎝ 1 ⎠ . cS (ρ , S) ⎛

This gives us Dλ1 · r1 = −(c + ρ∂ρ c(ρ , S)) < 0,

Dλ2 · r2 = 0,

Dλ3 · r3 = c + ρ∂ρ c(ρ , S) > 0.

Therefore, the second characteristic field is linearly degenerate, while the first and the third characteristic fields are genuinely nonlinear.

4.3.2 Rarefaction waves Recall that an i-rarefaction wave is a continuous piecewise smooth self-similar solution of the form U(x,t) = V (ξ ), ξ = x/t, which satisfies the ordinary differential equation ri (V (ξ )) , i = 1, 3, V (ξ ) = Dλi · ri (V (ξ )) V (ξ0 ) = U0 . The conditions hold (for i = 1) provided dρ ρ =− , dξ c + ρ cρ du c = , dξ c + ρ cρ S (ξ ) = 0, ξ > ξ0 , (ρ , u, p)(ξ0 ) = (ρ0 , u0 , p0 ),

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4 Riemann problem for ideal fluid

which clearly imply that the entropy S is constant. Moreover, we have

ρ pρ dp dρ = pρ =− < 0, dξ dξ c + ρ cρ so that the variable p can be used as a parameter along the integral curves (of the first characteristic field). Indeed, it follows from (4.3.2) that the density can be written as a function of the pressure and the entropy as

ρ = ρ (p, S) =

p γ −1

1/γ

S − S∗ exp γ Cv

.

(4.3.5)

This yields us du c =− dp ρ∂ρ p(ρ , S) 1 =− =− ρ ∂ρ (ρ , S)

∂ p ρ (p, S0 ) . ρ (p, S0 )

Thus, the integral curve associated with the first characteristic field issuing from U0 is given (in the (p, u)-plane) by p ∂ p ρ ( p, ˜ S0 ) u = u0 − d p, ˜ ρ ( p, ˜ S0 ) p0 where ρ = ρ (p, S) is given by (4.3.5). Moreover, since d p/d ξ < 0, the following inequality holds: p ≤ p0 along the curve of 1-rarefaction waves. We conclude that the forward rarefaction curve R1F (U0 ) consisting of all righthand states U connected to a given left-hand state U0 by a 1-rarefaction wave is R1F (U0 ) :

S = S0 ,

ρ = ρ (p, S0 ), u = u0 −

p ∂ p ρ ( p, ˜ S0 ) p0

ρ ( p, ˜ S0 )

d p, ˜

2γ 1/2 S0 − S∗ (γ −1)/2γ exp ), = u0 − (p(γ −1)/2γ − p0 2γ Cv (γ − 1)1−1/2γ

p ≤ p0 .

(4.3.6)

Arguing similarly, we obtain the backward rarefaction curve R3B (U0 ) consisting of all left-hand states U connected to a given right-hand state U0 by a 3-rarefaction wave as

4.3 Riemann problem for polytropic ideal fluids

R3B (U0 ) :

97

S = S0 ,

ρ = ρ (p, S0 ), u = u0 + = u0 +

p ∂ p ρ ( p, ˜ S0 )

ρ ( p, ˜ S0 )

p0

d p, ˜

S0 − S∗ 2γ 1/2 (γ −1)/2γ exp ), (p(γ −1)/2γ − p0 2γ Cv (γ − 1)1−1/2γ

p ≤ p0 .

(4.3.7)

4.3.3 Hugoniot curves Across a jump discontinuity between two given states U± = (ρ± , ρ± u± , ρ± e± ) propagating with velocity σ , the Rankine-Hugoniot relations read −σ [ρ ] + [ρ u] = 0, −σ [ρ u] + [ρ u2 + p] = 0, −σ [ρ e] + [u(ρ e + p)] = 0.

(4.3.8)

Here [A] = A+ − A− is the jump in a quantity A, where A± is the limit value of A on the right and on the left of the discontinuity, respectively. Let A¯ denote the average A¯ = (A+ + A− )/2. It follows from the first equation in (4.3.8) that m = ρ− (u− − σ ) = ρ+ (u+ − σ )

(4.3.9)

is constant. The quantity m can be regarded as the mass flux across the discontinuity. There are two types of non-trivial solutions corresponding to the two cases m = 0 and m = 0. Consider first the case m = 0. Then [u] = 0,

[p] = 0,

and the specific volume can have an arbitrary jump. This solution is a 2-contact discontinuity which propagates with the particle velocity:

σ = u+ = u− . Second, consider the case m = 0. Lemma 4.3.1. Let m = ρ− (u− − σ ) = ρ+ (u+ − σ ) = 0.

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4 Riemann problem for ideal fluid

For non-trivial solutions, the Rankine-Hugoniot relations (4.3.8) are equivalent to [u] −[p] = , [v] [u] −[p] m=± , [v] m=

(4.3.10)

1 [ε ] + (p+ + p− )[v] = 0, 2 where the positive sign corresponds to 1-shocks, and the negative sign corresponds to the 3-shocks. Proof. First, it follows from (4.3.9) that mv± = u± − σ , which gives m[v] = [u].

(4.3.11)

Next, the second equation in (4.3.8) can be expressed as 0 = −σ [ρ u] + [ρ u2 + p] = [ρ u2 − σ ρ u + p] = [ρ u(u − σ ) + p] = [mu + p], or m[u] + [p] = 0.

(4.3.12)

Next, we claim that for non-trivial solutions, [u] = 0,

[p] = 0,

[v] = 0.

(4.3.13)

Indeed, if [u] = 0, then (4.3.11) yields [v] = 0, and (4.3.12) yields [p] = 0, so that this is a trivial solution. If [p] = 0, then since m = 0, the equation (4.3.12) gives [u] = 0 and so the equation (4.3.11) gives [v] = 0, therefore the solution is trivial. If [v] = 0, then (4.3.11) gives [u] = 0 and so (4.3.12) gives [p] = 0 so that the solution is trivial, too. In view of (4.3.13), we obtain the equations in the first two lines of (4.3.10). Next, we consider the third equation in (4.3.8). One has 0 = −σ [ρ e] + [u(ρ e + p)] = [−σ ρ e + u(ρ e + p)] = [ρ (u − σ )e + up] = [me + up], where the last equation is obtained by applying (4.3.9). Let us continue our computations from the last equation

4.3 Riemann problem for polytropic ideal fluids

99

0 = m[e] + [up] = m[e] + u+ p+ − u− p+ + u− p+ − u− p− = m[e] + p+ [u] + u− [p]. Substituting [u] = m[v] from (4.3.11) and [p] = −m[u] from (4.3.12) into the last equation, one gets 0 = m[e] + m[v]p+ − mu− [u]. Dividing the last equation by m = 0, we have 0 = [e] + p+ [v] − u− [u]. Substituting for the total energy into the last equation yields u2+ − u2− + p+ [v] − u− [u] 2 u+ + u− + p+ [v] − u− [u], = [ε ] + [u] 2

0 = [ε + u2 /2] + p+ [v] − u− [u] = [ε ] +

or 0 = [ε ] + [u]

u+ + u− [u]2 − u− + p+ [v] = [ε ] + + p+ [v]. 2 2

(4.3.14)

On the other hand, from (4.3.11) and (4.3.12) we get [u]2 = −[p][v].

(4.3.15)

Substituting [u]2 from (4.3.15) into (4.3.14), one obtains 0 = [ε ] −

[p][v] p+ + p− + p+ [v] = [ε ] + [v], 2 2

which gives us the third equation in (4.3.10). Furthermore, by the standard theory of hyperbolic systems of conservation laws (see Chapter 3), the k-Hugoniot curve is tangent to the k-eigenvectors at the center. Thus, one can determine the sign of m by using any parameterization of the Hugoniot curves determined by (4.3.10). Since v = v− − sv− + O(s2 ), and

u = u− − sc− + O(s2 )

for 1-shocks,

u = u− + sc− + O(s )

for 3-shocks,

2

one gets m=

u − u− = v − v−

c− ρ− + O(s), for 1-shocks, −c− ρ− + O(s), for 3-shocks.

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4 Riemann problem for ideal fluid

Therefore, m > 0 for 1-shocks and m < 0 for 3-shocks. The proof of Lemma 4.3.1 is complete. As seen by Lemma 4.3.1, the set of all the final states that can be connected to a given state U0 by a shock wave satisfy the equation

ε − ε0 +

p + p0 (v − v0 ) = 0 2

(4.3.16)

which is called the Hugoniot equation. Substituting ε = pv/(γ − 1) into (4.3.16), we obtain p p0 p + p0 p + p0 pv − p0 v0 p + p0 + (v − v0 ) = v + + 0= − v0 . γ −1 2 γ −1 2 γ −1 2 The last equation yields v − v0 = v(p) =

2v0 (p0 − p) , (γ + 1)p + (γ − 1)p0

(4.3.17)

i.e., the specific volume along the Hugoniot curves can be given as a function of the pressure. This also implies that the Hugoniot curves can be parameterized by the pressure. Indeed, the second equation in (4.3.1) and (4.3.17) give the mass flux across the discontinuity m by ⎧ (γ + 1)p + (γ − 1)p0 ⎪ ⎪ ⎨ , for 1-shocks, 2v0 m = m(p) = (γ + 1)p + (γ − 1)p0 ⎪ ⎪ ⎩− , for 3-shocks. 2v0 And so, the velocity along the Hugoniot curves is given by ⎧ 2v0 ⎪ ⎪ −(p − p0 ) , for 1-shocks, ⎨ p − p0 ( γ + 1)p + (γ − 1)p0 = u(p) = u−u0 = − 2v0 ⎪ m ⎪ , for 3-shocks. ⎩(p − p0 ) (γ + 1)p + (γ − 1)p0

4.3.4 Admissible shock waves To determine the physically realizable solutions, admissibility criteria are imposed. As seen earlier, (−ρ S, −ρ Su) is a convex entropy pair, where −ρ S is a strictly convex function of the conservative variables (ρ , ρ u, ρ e). We say that a weak solution is admissible if it satisfies the entropy condition (in the weak sense) (−ρ S)t + (−ρ Su)x ≤ 0,

4.3 Riemann problem for polytropic ideal fluids

101

which is equivalent for a shock wave with shock speed σ to the following condition: −σ [ρ S] + [ρ Su] ≥ 0.

(4.3.18)

The inequality (4.3.18) can be re-written as 0 ≤ −σ [ρ S] + [ρ Su] = [ρ (u − σ )S] = m[S]. Let us recall from Lemma 4.3.1 that m > 0 corresponds to a 1-shock wave, and m < 0 corresponds to a 3-shock wave. Thus, the admissibility condition (4.3.18) becomes S ≥ S0 for 1-shocks, (4.3.19) S ≤ S0 for 3-shocks. Let us investigate the Hugoniot curves to see which parts of the Hugoniot curves are admissible. First, it follows from (4.3.17) that along the Hugoniot curves −4γ v0 p0 dv = . d p ((γ + 1)p + (γ − 1)p0 )2

(4.3.20)

Second, differentiate the Hugoniot equation (4.3.16) to get dε +

v − v0 p + p0 dv + d p = 0. 2 2

Substituting d ε = T dS − pdv from the thermodynamic identity to the last equation, we have p + p0 v − v0 T dS − pdv + dv + d p = 0, 2 2 or p − p0 v − v0 T dS = dv − d p. 2 2 It follows from the last equation and (4.3.17) that T

dS p − p0 dv v − v0 = − dp 2 dp 2 2v0 p − p0 dv + = . 2 d p (γ + 1)p + (γ − 1)p0

Substituting dv/d p from (4.3.20) to the last equation, and simplifying the terms, we obtain (γ + 1)v0 (p − p0 )2 dS = > 0, T d p (γ + 1)p + (γ − 1)p0 for p = p0 . This means that the entropy is increasing with respect to the pressure along the Hugoniot curves. The admissibility criterion (4.3.19) yields

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4 Riemann problem for ideal fluid

p ≥ p0 , p ≤ p0 ,

for 1-shocks, for 3-shocks.

(4.3.21)

Moreover, it follows from (4.3.20) that dv 0. First, let us consider the wave curves in the (p, u)-plane. The forward wave curve W1F (UL ) consisting of all right-hand states that can be connected to UL on the left by either a 1-shock or a 1-rarefaction wave is given by W1F (UL ) = R1F (UL ) ∪ S1F (UL ). It follows from (4.3.6) and (4.3.22) that the curve W1F (UL ) can be parameterized by the pressure as ⎧ 2v0 ⎪ ⎨(p − p0 ) , p > p0 , (γ + u = ϕ1 (U0 ; p) = u0 − 1)p + (γ − 1)p0 1/2 ⎪ (γ −1)/2γ ∗ ⎩ 2γ (p(γ −1)/2γ − p0 exp S20γ−S ), p ≤ p0 . Cv (γ −1)1−1/2γ (4.3.24) The backward wave curve W3B (UR ) consisting of all left-hand states that can be connected to UR on the right by either a 3-shock or a 3-rarefaction wave is given by W3B (UR ) = R3B (UR ) ∪ S3B (UR ). It follows from (4.3.7) and (4.3.23) that the curve W3B (UR ) can be parameterized by the pressure as

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4 Riemann problem for ideal fluid

u = ϕ3 (U0 ; p) ⎧ 2v0 ⎪ ⎨(p − p0 ) , p > p0 , (γ + = u0 + 1)p + (γ − 1)p0 1/2 ⎪ (γ −1)/2γ ∗ ⎩ 2γ (p(γ −1)/2γ − p0 exp S20γ−S ), Cv (γ −1)1−1/2γ

(4.3.25) p ≤ p0 .

It is not difficult to check that the function p → ϕ1 (p) is continuous and strictly decreasing, and the function p → ϕ3 (p) is continuous and strictly increasing for p ≥ 0. To find the intersection of these two wave curves, we need to determine their u-intercepts. The u-intercept u∗ of the wave curve W1F (UL ) is given by (Figure 4.2) pL ρ p (p, SL ) ∗ u = ϕ1 (UL ; 0) = uL + dp (4.3.26) ρ (p, SL ) 0 The u-intercept u∗ of the wave curve W3B (UR ) is given by pR ρ p (p, SR ) u∗ = ϕ3 (UR ; 0) = uR − d p. ρ (p, SR ) 0 Whenever

(4.3.27)

u∗ > u∗ ,

the wave curves W1F (UL ) and W3B (UR ) intersect at a unique point in the (p, u)-plane, denoted by UM = (pm , um ). Precisely, the pressure value p = pm across the 2-contact discontinuity is the root of the nonlinear algebraic equation

ϕ1 (UL ; p) − ϕ3 (UR ; p) = 0,

Fig. 4.2 The wave curves W1F (UL ) and W3B (UR )

4.3 Riemann problem for polytropic ideal fluids

105

where ϕ1 (UL ; p) and ϕ3 (UR ; p) are given by (4.3.24) and (4.3.25), respectively. And then the velocity value across the 2-contact discontinuity is given by um = ϕ1 (UL ; pm ) = ϕ3 (UR ; pm ). The point UM determines the 2-contact discontinuity, since u and p do not change across any 2-contact. In fact, the contact discontinuity separates the two states, denoted by UI = (ρI , um , pm ) and UII = (ρII , um , pm ). Observe that the projections of these two states onto the (p, u)-plane coincide with the point UM . Then, the specific volume value vI on the left-hand side of the 2-contact discontinuity, and therefore the density ρI = 1/vI , is determined by (4.3.6) or (4.3.17) as ρ (pm , SL ), if pm > pL , 1 = vI = 2vL (pL −pm ) ρI vL + (γ +1)p , if pm ≤ pL , m +(γ −1)pL where ρ = ρ (p, S) is given as an equation of state. Similarly, the specific volume value vII on the right-hand side of the 2-contact discontinuity, and therefore the density ρII = 1/vII , is determined by (4.3.6) or (4.3.17) as ρ (pm , SR ), if pm > pR , 1 = vII = 2vR (pR −pm ) ρII vR + (γ +1)p , if pm ≤ pR . m +(γ −1)pR Thus, the Riemann solution is defined as follows. First, the solution begins with a 1-wave (a shock if pm > pL , or a 1-rarefaction wave if pm ≤ pL ) from UL to UI . Second, the solution continues with a 2-contact discontinuity from UI to UII . Finally, the solution arrives at UR from UII by a 3-wave (a shock if pm > pR , or a 1-rarefaction wave if pm ≤ pR ).

Notation (i) Wk (Ui ,U j ) (Sk (Ui ,U j ), Rk (Ui ,U j )) denotes the k-wave (k-shock, k-rarefaction wave, respectively) connecting the left-hand state Ui to the right-hand state U j . (ii) Wm (Ui ,U j ) Wn (U j ,Uk ) means that an m-wave from the left-hand state Ui to the right-hand state U j is followed by an n-wave from the left-hand state U j to the right-hand state Uk .

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4 Riemann problem for ideal fluid

We summarize the above argument in the following theorem.

Fig. 4.3 A Riemann solution of the type S1 (UL ,UI ) W2 (UI ,UII ) S3 (UII ,UR )

Fig. 4.4 A Riemann solution of the type R1 (UL ,UI ) W2 (UI ,UII ) R3 (UII ,UR )

Theorem 4.3.2 (Riemann problem for polytropic ideal fluids). Consider a polytropic ideal fluid. The Riemann problem for the Euler equations (4.3.1) admits a unique solution if and only if u∗ > u∗ , where u∗ and u∗ are defined by (4.3.26) and (4.3.27), respectively. In this case, the Riemann solution is defined as follows: (i) If pm > pL and pm > pR , then the solution is given by S1 (UL ,UI ) W2 (UI ,UII ) S3 (UII ,UR ), (see Figure 4.3). (ii) If pm > pL and pm ≤ pR , then the solution is given by S1 (UL ,UI ) W2 (UI ,UII ) R3 (UII ,UR ), (see Figure 4.5).

4.3 Riemann problem for polytropic ideal fluids

Fig. 4.5 A Riemann solution of the type S1 (UL ,UI ) W2 (UI ,UII ) R3 (UII ,UR )

(iii) If pm ≤ pL and pm > pR , then the solution is given by R1 (UL ,UI ) W2 (UI ,UII ) S3 (UII ,UR ), (see Figure 4.6).

Fig. 4.6 A Riemann solution of the type R1 (UL ,UI ) W2 (UI ,UII ) S3 (UII ,UR )

(iv) If pm ≤ pL and pm ≤ pR , then the solution is given by R1 (UL ,UI ) W2 (UI ,UII ) R3 (UII ,UR ), (see Figure 4.4).

107

108

4 Riemann problem for ideal fluid

4.4 Bibliographical notes For this chapter, we provide a brief selection of the most relevant papers. For additional references, please refer to the bibliography at the end of this monograph. An overview of the structure of the p-system can be found in Young [330]. For the selfsimilar viscosity approach to the Riemann problem in isentropic gas dynamics, we refer to Kim [197], which is based on Tzavaras [319]. The existence and uniqueness of solutions to the generalized Riemann problem for isentropic flows is established by Reigstad [267]. For the Riemann problem for convex equations of state, we refer to Smoller [294] and Toro [314]. Global BV solutions for the p-system with frictional damping were studied by Dafermos-Pan [112]. A wave-front tracking algorithm for the equations of isentropic gas dynamics was studied by Asakura [24].

Chapter 5

Compressible fluids governed by a general equation of state

5.1 Introduction This chapter focuses on the Riemann problem for fluids with a general equation of state (EOS), covering both convex and nonconvex EOSs. We begin with the simple case of isentropic van der Waals fluids, for which the Riemann problem admits a unique solution for large data. We then present non-isentropic fluids with a general EOS, relying solely on the requirement that the Gr¨uneisen coefficient is positive. Examples of EOSs that satisfy this assumption will be provided. The curves of rarefaction waves can be parameterized by either specific volume or pressure. Choosing a specific volume as the parameterization would make the presentation consistent, as shock curves will also be parameterized by a specific volume. However, choosing pressure as the parameterization would be useful when solving the Riemann problem, as one may want to project the wave curves onto the (p, u)-plane. Hugoniot and shock curves can be parameterized by specific volume, provided that the Gr¨uneisen coefficient is positive. Examples of convex and nonconvex EOSs are given. Hugoniot and shock curves can be defined at any state that locally reflects the convexity or concavity of the EOS. The behavior of Hugoniot curves issuing from a given point depends on the convexity of the EOS near that point: the pressure function may be strictly convex or concave near the point or may have an inflection point at it. In the latter case, it is interesting to note that when the pressure function changes from a convex to a concave shape, the behavior of shock curves differs from when it changes from a concave to a convex shape. The Riemann problem is addressed, distinguishing between convex and nonconvex EOSs when describing wave curves. In the former case, wave curves behave like those of an ideal gas. In the latter case, composite waves are available to extend wave curves. As a result, a Riemann solution for a nonconvex EOS may contain up to seven elementary waves.

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 5

109

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5 Compressible fluids governed by a general equation of state

5.2 Riemann problem for isentropic van der Waals fluids 5.2.1 Preliminaries Consider the isentropic model of van der Waals fluids

∂t ρ + ∂x (ρ u) = 0, ∂t (ρ u) + ∂x (ρ u2 + p) = 0,

(5.2.1)

where ρ , v = 1/ρ , u are the density, specific volume, and the velocity, respectively. The pressure p is given by a RT0 − 2, v−b v

p=

v > b,

for some positive constants R, T0 , a and b. A straightforward calculation shows that the system (5.2.1) for smooth solutions U = (v, u)T can be re-written as

∂t v + u∂x v − v∂x u = 0, ∂t u + vp (v)∂x v + u∂x u = 0, where p (v) = d p/dv. The Jacobian matrix of the last system is given by u −v , A(U) = vp (v) u which admits two distinct eigenvalues λ (v, u) = u − v −p (v) < λ2 (v) = u + 1 v −p (v), if and only if p (v) < 0. As seen from Chapter 3 (in the Lagrange coordinates), it is easy to check that dp

3 2 a . 3 Rb

Thus, the system is strictly hyperbolic if and only if T0 >

3 2 a . 3 Rb

The corresponding eigenvectors can be chosen to be 1 −1 r1 (v, u) = , r2 (v, u) = . −p (v) −p (v)

5.2 Riemann problem for isentropic van der Waals fluids

111

We have ⎛ ⎞ vp (v) (v) + − −p Dλ1 (v, u) = ⎝ 2 −p (v) ⎠ , 1

⎛ ⎞ vp (v) (v) − −p Dλ2 (v, u) = ⎝ 2 −p (v) ⎠ . 1

Then one has vp (v) = −Dλ2 (v, u)T r2 (v, u). Dλ1 (v, u)T r1 (v, u) = 2 −p (v) This implies that the two characteristic fields are genuinely nonlinear if and only if p (v) > 0. As shown in Chapter 2, we find 4 3 a . T0 > 4 Rb

p (v) > 0 iff

The case p (v) > 0 has been considered earlier in Chapter 4. So, in the following we will consider the case where the system is strictly hyperbolic, but the two characteristic fields are not genuinely nonlinear. This means that the equation of state is given by a RT0 − 2 , v > b, p= v−b v 4 3 (5.2.2) 3 2 a a < T0 < . 3 Rb 4 Rb In this case, as seen in Chapter 3 (in the Lagrange coordinates), there exist exactly two values μ and ν such that p (μ ) = p (ν ) = 0,

p (v) > 0, p (v) < 0,

for for

b < μ < 4b < ν ,

v ∈ (b, μ ) ∪ (ν , ∞), v ∈ (μ , ν ).

(5.2.3)

We emphasize that the two characteristic fields of the p-system are genuinely nonlinear in each interval v ∈ (b, μ ), (μ , ν ) and (ν , ∞), but fail to be genuinely nonlinear at the planes {v = μ } and {v = ν }.

5.2.2 Rarefaction waves For v = μ , ν , we make a normalization by setting

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5 Compressible fluids governed by a general equation of state

⎛ ⎞ 2 −p (v) ⎜ vp (v) ⎟ 1 ⎟ r¯1 := r1 = ⎜ ⎝ −2p (v) ⎠ , Dλ1 · r1 vp (v)

⎛ ⎞ 2 −p (v) ⎜ vp (v) ⎟ 1 ⎟ r2 = ⎜ r¯2 := ⎝ 2p (v) ⎠ . Dλ2 · r2 vp (v)

A k-rarefaction wave (for k = 1, 2) connecting a left-hand state vL , uL to a right-hand state vR , uR is a continuous self-similar solution ⎧ x ⎪ (vL , uL ) ≤ λk (vL , uL ), ⎪ ⎪ t ⎨ x x , λk (wL ) ≤ ≤ λk (wR ), (v, u)(x,t) = V ⎪ t t ⎪ ⎪ ⎩ρR , uR , x ≥ λk (wR ), t where V = (v, u) is an integral curve of r¯k , k = 1, 2 issuing from (vL , uL ). Consider in general the integral curves of r¯k , k = 1, 2 dV (ξ ) = r¯k (V (ξ )). dξ We have

2 −p (v) dv = 0, = dξ vp (v)

so that v can be used as a parameter of the integral curve. We have du = ± −p (v). dv Thus, the integral curves issuing from a given state U0 = (v0 , u0 ) are given by O1 (U0 ) :

u = u0 +

O2 (U0 ) :

u = u0 −

v v 0v v0

−p (y)dy, −p (y)dy.

Using the fact that the characteristic speed must be increasing through a rarefaction fan, a 1-rarefaction wave from a left-hand state UL to a right-hand state UR satisfies UR ∈ O1 (UL ) and vR > vL , if p (v) > 0, v ∈ [vL , vR ], vR < vL , if p (v) < 0, v ∈ [vL , vR ]. Similarly, a 2-rarefaction wave from a left-hand state UL to a right-hand state UR satisfies UR ∈ O1 (UL ) and vR > vL , if p (v) < 0, v ∈ [vL , vR ], vR < vL , if p (v) > 0, v ∈ [vL , vR ].

5.2 Riemann problem for isentropic van der Waals fluids

113

5.2.3 Shock waves A shock wave connecting two states (ρ0 , u0 ) and (ρ , u) for the system (5.2.1) satisfies the Rankine-Hogoniot relations − σ (ρ − ρ0 ) + (ρ u − ρ0 u0 ) = 0, − σ (ρ u − ρ0 u0 ) + (ρ u2 + p) − (ρ0 u20 + p0 ) = 0. Eliminating σ in the last system of equations, substituting v = 1/ρ , after simplifying, we obtain 1 1 1 1 − , (p − p0 ) = (u − u0 )2 v v0 v0 v which yields the Hugoniot set H (w0 ) :

u = u0 ±

(p − p0 )(v0 − v).

Recall from Chapter 3 that the k-Hugoniot curve is tangent to the vector rk at the point (v0 , u0 ), k = 1, 2. Precisely, we find along the 1-Hugoniot curve u − u0 = −p (v0 ) + O(ε ). v − v0 This means that

> 0 for u − u0 < 0 for

v > v0 , v < v0 .

The 1-Hugoniot curve is therefore determined by − (p − p0 )(v − v0 ), v < v0 , H1 (v0 , u0 ) : u = u0 + (p − p0 )(v − v0 ), v > v0 . Similarly, the 2-Hugoniot curve is determined by (p − p0 )(v − v0 ), v < v0 , H2 (v0 , u0 ) : u = u0 + − (p − p0 )(v − v0 ), v > v0 . Shock waves will be a constraint to the Liu entropy condition. Consider a shock wave connecting a left-hand state U0 = (u0 , v0 ) to a right-hand state U1 = (u1 , v1 ) and propagating with a shock speed σ (U0 ,U1 ). Since p(v1 ) − p(v0 ) p(v1 ) − p(v0 ) σ1 (U0 ,U1 ) = − − < 0, σ2 (U0 ,U1 ) = − > 0, v1 − v0 v1 − v0 which depend only on the v-component, we can suppress the u-argument from our notation, and so we denote the shock speeds by σ (v0 , v1 ), σ1 (v0 , v1 ), and σ2 (v0 , v1 ).

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5 Compressible fluids governed by a general equation of state

The shock satisfies the Liu entropy condition iff

σ (v0 , v) ≥ σ (v0 , v1 )

for all v between v0 and v1 .

Geometrically, the Liu condition for a 1-shock is equivalent to p(v) − p(v0 ) p(v1 ) − p(v0 ) ≥ v − v0 v1 − v0

for all v between v0 and v1 .

This means that for all v between v0 and v1 , the graph of p is below (above) the line connecting v0 to v1 when v1 < v0 (resp. v1 > v0 ). A similar interpretation can be made for 2-shocks. Unlike the case of a convex EOS, where the 1-shock set contains only one curve, the 1-shock set in van der Waals fluids can be made up of several disjoint curves. To describe the set of shock waves satisfying the Liu entropy condition, we recall some concepts and notation introduced earlier in Chapter 3. The tangent lines to the graph of p = p(v) at μ and ν cut the graph at a point denoted by μ − and ν − , respectively. From any v ∈ [ν − , μ − ], we can draw two tangent lines to the graph of the pressure p = p(v) at ϕ (v) ≤ ψ (v), i.e., p(v) − p ϕ (v) , p ϕ (v) = v − ϕ (v) p(v) − p ψ (v) p ψ (v) = . v − ψ (v) See Figure 5.1. Let c < μ and d > ν be the values such that

Fig. 5.1 The pressure function and the tangent functions ϕ and ψ

5.2 Riemann problem for isentropic van der Waals fluids

ψ (c) = d,

115

ϕ (d) = c.

The tangent line to the graph of the pressure function at any v ∈ (c, d) cuts the graph of p at exactly two values ϕ − (v) ≤ ψ − (v). Admissible shock waves satisfying Liu’s entropy condition are given in the following lemma. Lemma 5.2.1 (Admissible shock waves). Given a left-hand state (u0 , v0 ), the set of right-hand states (u1 , v1 ) attainable by a 1-shock satisfying the Liu entropy condition corresponds to the following parts on the 1-Hugoniot curves: (i) (ii) (iii) (iv)

If v0 ∈ (b, c) ∪ (μ − , +∞), then v1 ∈ (b, v0 ]. If v0 ∈ [c, μ ], then v1 ∈ (b, v0 ] ∪ [ϕ − (v0 ), ψ (v0 )]. If v0 ∈ (μ , ν ), then v1 ∈ (ν , ϕ − (v0 )] ∪ [v0 , ψ (v0 )]. If v0 ∈ [ν , μ − ], then v1 ∈ (b, ϕ − (ψ (v0 ))] ∪ [ψ (v0 ), v0 ].

Similar results hold for backward 2-shock waves.

5.2.4 Composite waves and Riemann problem In this subsection we first describe the forward 1-wave curve W1F (vL , uL ) consisting of all right-hand states (v, u) that can be arrived at by a combination of Liu admissible shocks and rarefaction waves. A similar construction can be made for the backward 2-wave curve W2B (vR , uR ) consisting of all left-hand states (v, u) that can be arrived at by a combination of of Liu admissible shocks and rarefaction waves. First, consider the case vL < c. All the states (v, u) having v ∈ (b, vL ) can be arrived at by a single Liu admissible 1-shock. All of the points (v, u) with v ∈ (vL , μ ] can be arrived at by a single 1-rarefaction. For v ∈ [μ , d], we have ϕ (v) ∈ [c, μ ], and so the solution begins with a 1-rarefaction wave from vL to ϕ (v), followed by a 1-shock from ϕ (v) to v. If v > d, the solution begins with a 1-rarefaction wave from vL to c, followed by a 1-shock from c to d, and then followed by a 1-rarefaction wave from d to v. Second, consider the case vL ∈ [c, μ ]. For v < vL , the Riemann solution is a single Liu admissible 1-shock. For v ∈ (vL , μ ], the solution is a single 1-rarefaction wave. For v ∈ [μ , ϕ − (vL )], we have ϕ (v) ∈ [vL , μ ], and so the Riemann solution begins with a 1-rarefaction wave from vL to ϕ (v), followed by a 1-shock from ϕ (v) to v. For v ∈ (ϕ − (vL ), ψ (vL ], the solution is a single 1-shock. For v > ψ (vL ), the solution begins with a 1-shock from vL to ψ (vL ), followed by a 1-rarefaction wave connecting ψ (vL ) to v. Third, consider the case vL ∈ (μ , ν ). For v ∈ (0, ϕ − (vL )] ∪ [vL , ψ (vL )], the solution is a single 1-shock. For v ∈ [μ , vL ], the solution is a single 1-rarefaction wave. For each v ∈ (ϕ − (vL ), μ ), we set v∗ = ϕ (v), and so the Riemann solution begins with a 1-rarefaction wave from vL to v∗ , followed by a 1-shock from v∗ to v. For

116

5 Compressible fluids governed by a general equation of state

each v > ψ (vL ), the Riemann solution begins with a 1-shock from vL to ψ (vL ), followed by a 1-rarefaction wave from ψ (vL ) to v. Fourth, consider the case vL ∈ [ν , μ − ]. For v ∈ (0, ϕ − (ψ (vL ))] ∪ [ψ (vL ), vL ], the Riemann solution is a single 1-shock. For v ∈ [vL , +∞), the solution is a single 1-rarefaction wave. For v ∈ [μ , ψ (vL )), the Riemann solution begins with a 1-shock from vL to ψ (vL ), followed by a 1-rarefaction from ψ (vL ) to v. For v ∈ (ϕ − (ψ (vL )), μ ), the solution begins with a 1-shock from vL to ψ (vL ), followed by a 1-rarefaction from ψ (vL ) to ϕ (v), and finally arrived at v by a shock wave. Finally, consider the case vL ∈ (μ − , +∞). For v < vL , the Riemann solution is simply a 1-shock. For v ≥ vL , the solution is a 1-rarefaction wave. It is not difficult to see that the forward 1-wave curve W1F (vL , uL ), being parameterized by v, is strictly increasing and covers the whole range of values u ∈ (−∞, +∞). Similarly, the backward 2-wave curve W2B (vR , uR ), being parameterized by v, is strictly decreasing and covers the whole range of values u ∈ (−∞, +∞). These two wave curves therefore intersect exactly once. This unique intersection point leads to a unique Riemann solution. Thus, we arrive at the following main result in this section. Theorem 5.2.1. The Riemann problem for the isentropic van der Waals fluid (5.2.2) admits a unique solution in the class of piecewise smooth self-similar functions made of rarefaction fans and shock waves satisfying the Liu entropy criterion.

5.3 General EOS: rarefaction curves 5.3.1 Parameterization by the specific volume Recall from Chapter 3 that the gas dynamics equations in the Eulerian coordinates

∂ ρ ∂ (ρ u) + = 0, ∂t ∂x ∂ (ρ u) ∂ (ρ u2 + p) + = 0, ∂t ∂x ∂ (ρ e) ∂ ((ρ e + p)u) + = 0, ∂t ∂x for smooth solutions can be re-written in the matrix form as

∂t U + A(U)∂xU = 0,

(5.3.1)

5.3 General EOS: rarefaction curves

117

where the unknown U and the Jacobian matrix A(U) are given by ⎛ ⎞ ⎛ ⎞ u ρ 0 ρ ⎜ ∂ρ p(ρ , S) ∂S p(ρ , S) ⎟ ⎟. U = ⎝ u ⎠ , A(U) = ⎜ u ⎝ ⎠ ρ ρ S 0 0 u

(5.3.2)

The characteristic polynomial of A(U) is given by (u − λ )[(u − λ )2 − ∂ρ p(ρ , S)] = 0. Since ∂v p(v, S) < 0, we have c2 = ∂ρ p(ρ , S) = −v2 ∂v p(v, S) > 0.

(5.3.3)

Thus, the Jacobian matrix A(U) admits three real and distinct eigenvalues

λ1 = u − c < λ2 = u < λ3 = u + c, where c denotes the sound speed c = c(ρ , S) =

∂ρ p(ρ , S).

(5.3.4)

In the following, we choose the corresponding eigenvectors of the Jacobian matrix A(U) in (5.3.2) as ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ρ ρ −∂S p(ρ , S) ⎠ , r3 = ⎝ c ⎠ . 0 r1 = ⎝−c⎠ , r2 = ⎝ ∂ρ p(ρ , S) 0 0 This choice gives us Dλ1 · r1 = −(c + ρ∂ρ c),

Dλ2 · r2 = 0,

Dλ3 · r3 = c + ρ∂ρ c.

Thus, shock waves associated with the second characteristic field are just contact discontinuities. A contact discontinuity of this kind is specified in the following as a 2-contact discontinuity, or a 2-contact in short. Moreover, observe that 2 2c ∂vv p(v, S) = ∂ρρ p(ρ , S) + ∂ρ p(ρ , S) = (ρ cρ + c). ρ ρ This yields

ρ∂vv p(v, S) ρ∂vv p(v, S) , Dλ3 · r3 = . 2c 2c Thus, the first and the third characteristic fields for a nonconvex equation of state could fail to be genuinely nonlinear if ∂vv p(v, S) vanishes. Dλ1 · r1 = −

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5 Compressible fluids governed by a general equation of state

By definition, an i-rarefaction wave is a continuous piecewise smooth selfsimilar solution of the form U(x,t) = V (ξ ), ξ = x/t. As discussed in Chapter 3, an i-rarefaction wave starting from a given state U0 is the solution of the following initial value problem for ordinary differential equations ri (V (ξ )) , Dλi · ri (V (ξ )) V (ξ0 ) = U0 , ξ > ξ0 , V (ξ ) =

i = 1, 3,

where (.) = d/d ξ . Assume in the following that U0 is a point where ∂vv p(v0 , S0 ) = 0. So, it holds that c + ρ∂ρ c = 2cv∂vv p(v, S) = 0 in a neighborhood of U0 . Therefore, one can define rarefaction waves of (5.3.1) starting from U0 . Indeed, the last differential equations become for i = 1, for instance

ρ dρ , =− dξ c + ρ∂ρ c du c , = dξ c + ρ∂ρ c S (ξ ) = 0,

(5.3.5)

ξ > ξ0 .

The last equation in (5.3.5) implies that the entropy S is constant across a rarefaction wave: S ≡ S0 . By (5.3.3) we have dv dv d ρ = dξ dρ dξ −1 (−ρ ) = 2 ρ c + ρ∂ρ c

2v3 −∂v p(v, S) 1 = = 0, = ρ (c + ρ∂ρ c) ∂vv p(v, S)

provided that ∂vv p(v0 , S0 ) = 0. Therefore, the integral curve can also be parameterized by specific volume v: du d ρ du = dv d ρ dv −c −1 = ρ v2 c = = −∂v p(v, S). v

5.3 General EOS: rarefaction curves

119

Integrating the last equation gives us the integral curve O1 (U0 ) defined by O1 (U0 ) :

u = u0 +

v v0

−∂v p(ν , S)d ν .

(5.3.6)

Suppose now that ∂vv p(v0 , S0 ) > 0. Then, 2v3 −∂v p(v, S) dv > 0, = dξ ∂vv p(v, S) in a neighborhood of U0 . This yields v ≥ v0 . on the integral curve O1 (U0 ). Thus, as long as ∂vv p(v, S) does not change sign along the integral curve O1 (U0 ) defined by (5.3.6) from U0 to U ∈ O1 (U0 ), the curve of 1-rarefaction waves of (5.3.1) can be defined. Precisely, the curve of 1-rarefaction waves R1F (U0 ) consisting of all right-hand states U connected to a given left-hand state U0 by a 1-rarefaction wave can be determined by R1F (U0 ) : where v ≥ v0 , v ≥ v0 ,

if if

v

u = ϕ1 (v) := u0 +

v0

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

−∂v p(ν , S)d ν ,

to U to U

along along

(5.3.7)

O1 (U0 ), O1 (U0 ),

where O1 (U0 ) is defined by (5.3.6). Similarly, we can define the integral curve O3 (U0 ) issuing from U0 by O3 (U0 ) :

u = u0 −

v v0

−∂v p(ν , S)d ν .

(5.3.8)

The curve of 3-rarefaction waves R3B (U0 ) consisting of all right-hand states U connected to a given left-hand state U0 by a 3-rarefaction wave can be determined by R3B (U0 ) : where v ≤ v0 , v ≥ v0 ,

if if

u = ϕ3 (v) := u0 −

v

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

v0

−∂v p(ν , S)d ν .

to U to U

along along

O3 (U0 ), O3 (U0 ).

(5.3.9)

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5 Compressible fluids governed by a general equation of state

5.3.2 Parameterization by the pressure In solving the Riemann problem for the gas dynamics equations as seen later, wave curves will be projected into the (p, u)-plane. So, it is convenient to find a parameterization of the integral curve in terms of the pressure p. This can be done as follows. First, we observe that

ρ∂ρ p(ρ , S) dρ dp = 0, = ∂ρ p(ρ , S) =− dξ dξ c + ρ∂ρ c so that the variable p can be used as a parameter along the integral curves (of the first characteristic field). This means that u can be defined in terms of p as ∂ p ρ (p, S0 ) du c 1 =− =− . =− dp ρ∂ρ p ρ (p, S0 ) ρ ∂ρ p Here,

ρ = ρ (p, S0 ) is the inverse function of p = p(ρ , S0 ). Thus, 1-rarefaction waves trace out a subset of the integral curve O1 (U0 ) issuing from U0 , which is defined (in the (p, u)-plane) by p ∂ p ρ ( p, ˜ S0 ) d p. ˜ (5.3.10) O1 (U0 ) : u = u0 − ρ ( p, ˜ S0 ) p0 Suppose now that ∂vv p(v, S) > 0. Then, since d p/d ξ < 0, and so the 1-rarefaction waves starting from U0 correspond to the part p ≤ p0 on the integral curve O1 (U0 ). Similarly, if ∂vv p(v, S) < 0, then since d p/d ξ > 0, and so the 1-rarefaction waves starting from U0 correspond to the part p ≥ p0 on the integral curve O1 (U0 ). Thus, as long as ∂vv p(v, S) does not change sign along the integral curve O1 (U0 ) defined by (5.3.10) from U0 to U ∈ O1 (U0 ), the curve of 1-rarefaction waves of (5.3.1) can be defined. Precisely, the curve of 1-rarefaction waves R1F (U0 ) consisting of all right-hand states U connected to a given left-hand state U0 by a 1-rarefaction wave can be determined by p ∂ p ρ ( p, ˜ S0 ) d p, ˜ (5.3.11) R1F (U0 ) : u = φ1 (p) = u0 − ρ ( p, ˜ S0 ) p0 where

5.4 General EOS: shock curves

p ≤ p0 , p ≥ p0 ,

if if

121

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

to U to U

along along

O1 (U0 ), O1 (U0 ),

where O1 (U0 ) is defined by (5.3.10). Similarly, we can define the integral curve O3 (U0 ) issuing from U0 by p ρ p ( p, ˜ S0 ) O3 (U0 ) : u = u0 + d p. ˜ (5.3.12) ρ ( p, ˜ S0 ) p0 The curve of 3-rarefaction waves R3B (U0 ) consisting of all right-hand states U connected to a given left-hand state U0 by a 3-rarefaction wave can be determined by p ∂ p ρ ( p, ˜ S0 ) B R3 (U0 ) : u = φ3 (p) = u0 + d p, ˜ (5.3.13) ρ ( p, ˜ S0 ) p0 where

p ≥ p0 , p ≤ p0 ,

if if

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

to U to U

along along

O3 (U0 ), O3 (U0 ),

where O3 (U0 ) is defined by (5.3.12).

5.4 General EOS: shock curves 5.4.1 Hugoniot curves In this subsection, we fix an arbitrary U0 , and we investigate the Hugoniot curves Hi (U0 ), i = 1, 2, 3, centered at U0 . A jump discontinuity for (5.2.1) between two states U0 and U propagating with velocity σ satisfies the Rankine-Hugoniot relations −σ [ρ ] + [ρ u] = 0, −σ [ρ u] + [ρ u2 + p] = 0, −σ [ρ e] + [u(ρ e + p)] = 0,

(5.4.1)

where [ρ ] = ρ − ρ0 , etc. The first equation of (5.4.1) yields m = ρ0 (u0 − σ ) = ρ (u − σ ). First, if m = 0, then

[u] = 0,

[p] = 0,

and so the solution is a 2-contact discontinuity which speed

(5.4.2)

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5 Compressible fluids governed by a general equation of state

σ = u = u0 . This means that the Hugoniot curve H2 (U0 ) associated with the second (linearly degenerate) characteristic field can be parameterized simply as v = v for any v. Second, if m = 0, then as seen in the previous chapter, for non-trivial solutions, the Rankine-Hugoniot relations (5.4.1) are equivalent to the relations u − u0 p − p0 0 = m = =− , v − v0 u − u0 −(p − p0 ) m=± , v − v0

(5.4.3)

together with the Hugoniot equation 1 Φ (v, p) = ε − ε0 + (p + p0 )(v − v0 ) = 0, 2

(5.4.4)

where ε = ε (v, p). Observe that in the second equation of (5.4.3), the positive sign corresponds to 1-shocks, and the negative sign corresponds to the 3-shocks. The function Φ of the variable (v, p) in (5.4.4) is referred to as the Hugoniot function. The equations (5.4.3) and (5.4.4) define two Hugoniot curves Hi (U0 ), i = 1, 3, associated with the first and the third characteristic fields. These two curves are sometimes referred to as the shock curves, to distinguish with the curve of contact discontinuities H2 (U0 ). The projection of the shock curves onto the space of the thermodynamic variables is called the Hugoniot locus. The Hugoniot locus is thus the zero set of the Hugoniot function, i.e., Φ (v, p) = 0. Theorem 5.4.1 (Hugoniot curves for general fluids). Assume that the Gr¨uneisen coefficient Γ is positive, the Hugoniot equation (5.4.4) defines the pressure as a function of the specific volume. Precisely, there exists δ > 0 and a function (v0 − δ , v0 + δ ) v → p = p(U0 ; v) such that

1 ε (v, p(U0 ; v)) − ε0 + (p(U0 ; v) + p0 )(v − v0 ) = 0. 2 Consequently, the Hugoniot curves Hi (U0 ), i = 1, 3, can always be locally parameterized as functions of the specific volume for U sufficiently closed to U0 . Precisely, there exists δ > 0 such that the Hugoniot curve H1 (U0 ) can be locally parameterized as

5.4 General EOS: shock curves

H1 (U0 ) :

H3 (U0 ) :

123

(v0 − δ , v0 + δ ) v → U(v) = (v, u, p), ⎧ ⎨ p = p(U0 ; v), where −(p(U0 ; v) − p0 ) ⎩u − u0 = u(U0 ; v) := (v − v0 ) , v − v0 (v0 − δ , v0 + δ ) v → U(v) = (v, u, p), p = p(U0 ; v), where u − u0 = −u(U0 ; v). (5.4.5)

Proof. Let ε = ε (p, v). Differentiating the Hugoniot function 1 Φ (v, p) = ε (v, p) − ε0 + (p + p0 )(v − v0 ), 2 with respect to p, and since Γ > 0, one gets 1 ∂ p Φ (v, p) = ∂ p ε (v, p) + (v − v0 ) 2 1 1 + (v − v0 ) = ∂ε p(ε , v) 2 v 1 = + (v − v0 ) > 0. Γ 2 This gives

∂ p Φ (v0 , p0 ) =

v0 > 0. Γ (v0 , p0 )

Thus, it follows from the implicit function theorem that there exists a constant δ > 0 such that the equation Φ (v, p) = 0 determines a function p = p(U0 ; v) for v ∈ (v0 − δ , v0 + δ ). The parameterization of the Hugoniot curve H1 (U0 ) in (5.4.5) then follows from (5.4.3), since −(p − p0 ) m= , v − v0 and the parameterization of the Hugoniot curve H3 (U0 ) in (5.4.5) follows from (5.4.3), since −(p − p0 ) m=− . v − v0 The proof of Theorem 5.4.1 is complete. The above argument implies that all the three Hugoniot curves can always be locally parameterized as a function of the specific volume.

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5 Compressible fluids governed by a general equation of state

Example 5.4.1. Consider the stiffened gas dynamics equation, where

ε=

(p + γ p∗ )v + ε∗ . γ −1

So the Hugoniot equation (5.4.4) becomes

Φ (v, p) =

(p + γ p∗ )v 1 + ε∗ − ε0 + (p + p0 )(v − v0 ) = 0. γ −1 2

The last equation yields p(v/(γ − 1) + (v − v0 )/2) = −[γ p∗ v/(γ − 1) + ε∗ − ε0 + p0 (v − v0 )/2]. Substituting

ε0 =

(p0 + γ p∗ )v0 + ε∗ γ −1

into the last equation, after re-arranging terms, we obtain p((γ + 1)v − (γ − 1)v0 ) = −[(2γ p∗ (v − v0 ) + p0 ((γ − 1)v − (γ + 1)v0 )]. Substituting 2γ (v − v0 ) = ((γ + 1)v − (γ − 1)v0 ) + ((γ − 1)v − (γ + 1)v0 ) into the right-hand side of the last equation, and re-arranging terms, we get ((γ + 1)v − (γ − 1)v0 )(p + p∗ ) = −((γ − 1)v − (γ + 1)v0 )(p0 + p∗ ). Thus, the pressure can be parameterized by the specific volume as p = p(U0 ; v) = −p∗ −

((γ − 1)v − (γ + 1)v0 )(p0 + p∗ ) . (γ + 1)v − (γ − 1)v0

In order for p is always positive, we have p = −p∗ − This yields v# :=

((γ − 1)v − (γ + 1)v0 )(p0 + p∗ ) > 0. (γ + 1)v − (γ − 1)v0

γ −1 (γ + 1)p0 + 2γ p∗ v0 < v < v# := . γ +1 (γ − 1)p0 + 2γ p∗

We can verify that the interval v# < v < v# contains v0 . Thus, p(U0 ; v) = −p∗ −

((γ − 1)v − (γ + 1)v0 )(p0 + p∗ ) , (γ + 1)v − (γ − 1)v0

v# < v < v# .

(5.4.6)

5.4 General EOS: shock curves

125

Moreover, the pressure is extended from 0 to infinity in this interval: lim p(U0 ; v) = 0,

v→v# −

and

lim p(U0 ; v) = ∞.

v→v# +

Example 5.4.2. Consider the van der Waals equation of state presented in Chapter 2. We have which also yields the internal energy of the form ε = ε (v, p):

ε = ε (v, p) =

a a 3 p + 2 (v − b) − . 2 v v

So the Hugoniot equation (5.4.4) becomes a a 3 1 p + 2 (v − b) − − ε0 + (p + p0 )(v − v0 ) = 0. 2 v v 2

Φ (v, p) =

A straightforward calculation yields p = p(U0 ; v) = where

−p0 v3 + (2ε0 + p0 v0 )v2 − av + 3ab , v2 (4v − (3b + v0 ))

3 ε0 = ε (v0 , p0 ) = 2

a p0 + 2 v0

(v0 − b) −

a . v0

Set Q(v) = −p0 v3 + (2ε0 + p0 v0 )v2 − av + 3ab. Then, we find Q(v0 ) = 3p0 v20 (v0 − b) > 0. Since the pressure is positive at least in a neighborhood of v0 , the denominator in the expression of p = p(U0 ; v) must be positive v2 (4v − (3b + v0 )) > 0, so that

3b + v0 > v0 . 4 Let v# be the maximum value of (3b + v0 )/4 and the largest root of Q(v) less than v0 . This maximum value exists by setting the root is equal to minus infinity if there is no such a root. And let v# be the smallest root of Q(v) larger than v0 . This root always exists, since Q(v0 ) > 0 and Q(v) → −∞ as v → ∞. Thus, the pressure can be parameterized by the specific volume as v>

p = p(U0 ; v) =

−p0 v3 + (2ε0 + p0 v0 )v2 − av + 3ab , v2 (4v − (3b + v0 ))

v# < v < v# .

(5.4.7)

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5 Compressible fluids governed by a general equation of state

5.4.2 Admissibility criteria States on the Hugoniot curves are not all admissible, since physically relevant shock waves must satisfy certain admissibility criteria, and therefore, only certain parts of the Hugoniot curves should be kept, for instance, in the construction of the Riemann problem (see below). The most common admissibility criterion is to require that a shock wave must satisfy the (single) entropy inequality

∂t (−ρ S) + ∂x (−ρ uS) ≤ 0,

(5.4.8)

in the sense of distributions. Let us fix an arbitrary left-hand state U0 and consider the right-hand states U that can be connected to U0 by a shock satisfying the entropy condition (5.4.8). As seen from Chapter 3, the entropy inequality (5.4.8) for this shock wave becomes −σ [−ρ S] + [−ρ uS] ≤ 0, where σ is the shock speed, [−ρ S] = (−ρ S) − (−ρ0 S0 ), etc. The last inequality yields 0 ≤ [ρ (u − σ )S] = m[S], where the equality comes from (5.4.2). Since m > 0 for 1-shocks and m < 0 for 3-shocks, the last inequality deduces S ≥ S0

for 1-shocks,

S ≤ S0

for 3-shocks.

(5.4.9)

Consequently, the entropy conditions (5.4.9) will select admissible part(s) on the Hugoniot curves. The following theorem provides us with a local description of the curves of admissible shock waves. Theorem 5.4.2 (Shock curves for general fluids). Assume that the Gr¨uneisen coefficient Γ is positive. Let the Hugoniot curves issuing from a given left-hand state U0 be parameterized by v ∈ [v0 − δ , v0 + δ ] as in Theorem 5.4.1. There exists a positive constant δ1 ≤ δ such that the following conclusions hold: (i) If ∂vv p(v0 , S0 ) > 0, then the parts on the Hugoniot curves v ∈ [v0 − δ1 , v0 ] for 1-shocks, v ∈ [v0 , v0 + δ1 ] for 3-shocks,

(5.4.10)

satisfy the entropy conditions (5.4.9). (ii) If ∂vv p(v0 , S0 ) < 0, then the parts on the Hugoniot curves v ∈ [v0 , v0 + δ1 ] for 1-shocks, v ∈ [v0 − δ1 , v0 ] for 3-shocks, satisfy the entropy conditions (5.4.9).

(5.4.11)

5.4 General EOS: shock curves

127

Proof. As seen from Theorem 5.4.1, the Hugoniot locus can be parameterized by v at least for U near U0 . For simplicity, we omit U0 in the function p = p(U0 ; v) in (5.4.5). This means that S = S(v), p = p(v), for v ∈ (v0 − δ , v0 + δ ) for some δ > 0. First, we will show that dS(v0 ) = 0, dv

d 2 S(v0 ) = 0, dv2

d 3 S(v0 ) −1 = ∂vv p(v0 , S0 ). dv3 2T0

(5.4.12)

Indeed, writing ε = ε (v, S), and differentiating the identity 1 ε (v, S(v)) − ε0 + (p(v) + p0 )(v − v0 ) = 0, 2 we get 1 εv (v, S(v)) + εS (v, S(v))S (v) + (p (v)(v − v0 ) + p(v) + p0 ) = 0, 2 where the prime ”’” stands for d/dv. Using the thermodynamic identity obtained from the last equation 1 −p(v, S(v)) + T (v, S(v))S (v) + (p (v)(v − v0 ) + p(v) + p0 ) = 0. 2 Observe that p(v, S(v)) is the pressure along the Hugoniot locus, so that p(v, S(v)) = p(v) and T (v, S(v)) = T (v). Hence, the last equation yields or 1 T (v)S (v) = − (p (v)(v − v0 ) − p(v) + p0 ). 2

(5.4.13)

Setting v = v0 in (5.4.13) gives us S (v0 ) = 0, which establishes the first equality in (5.4.12). Next, differentiating (5.4.13) with respect to v gives us 1 T (v)S (v) + T (v)S (v) = − (p”(v)(v − v0 )). 2

(5.4.14)

Setting v = v0 in (5.4.14) gives us, since S (v0 ) = 0, S”(v0 ) = 0, which establishes the second equality in (5.4.12). Next, differentiating (5.4.14) with respect to v again, we have 1 T (v)S (v) + 2T (v)S”(v) + T (v)S (v) = − (p (v)(v − v0 ) + p”(v)). (5.4.15) 2

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5 Compressible fluids governed by a general equation of state

Setting v = v0 in (5.4.15) gives us 1 T0 S (v0 ) = − (p”(v0 )). 2

(5.4.16)

Let us show that p”(v0 ) = ∂vv p(v0 , S0 ). Observe that it holds locally along the Hugoniot locus p(v) = p(v, S(v)), so that p (v) = pv (v, S(v)) + pS (v, S(v))S (v), p (v) = ∂vv p(v, S(v)) + 2∂vS p(v, S(v))S (v) + ∂SS p(v, S(v))(S (v))2 .

(5.4.17)

Setting v = v0 in (5.4.17), we have p”(v0 ) = ∂vv p(v0 , S(v0 )).

(5.4.18)

From (5.4.16) and (5.4.18), we deduce the third equality in (5.4.12). Finally, it follows from (5.4.12) that S(v) − S0 =

−1 ∂vv p(v0 , S0 )(v − v0 )3 + O((v − v0 )4 ), 3!2T0

so that the conclusions in (5.4.10) and (5.4.11) follow directly from the conditions (5.4.9). The proof of Theorem 5.4.2 is complete. Example 5.4.3. Consider the stiffened gas equation of state presented in Chapter 2. We find γ −1 S = S(v, p) = S0 +Cv ln(p + p∗ ) + γ ln v − ln((γ − 1)Cv T0 v0 ) . As shown in Example 5.4.1, along the Hugoniot curve, the pressure can be parameterized by the specific volume as in (5.4.6). Substituting p = p(U0 ; v) from (5.4.6) into S = S(v, p), we obtain ((γ + 1)v0 − (γ − 1)v) p0 + p∗ S(v) = S(U0 ; v) = S0 +Cv ln . + γ ln v + ln γ −1 (γ + 1)v − (γ − 1)v0 (γ − 1)Cv T0 v 0

Taking the derivative of S(v) with respect to v, we have 1−γ γ +1 γ − + S (v) = Cv . ((γ + 1)v0 − (γ − 1)v) (γ + 1)v − (γ − 1)v0 v A straightforward calculation gives us

5.4 General EOS: shock curves

S (v) = −γ (γ 2 − 1)(v − v0 )2 < 0,

129

v = v0 .

Thus, S is strictly decreasing along the Hugoniot curve. The admissible parts of the Hugoniot curves (5.4.5) are therefore corresponding to p(U0 ; v) = −p∗ −

((γ − 1)v − (γ + 1)v0 )(p0 + p∗ ) , (γ + 1)v − (γ − 1)v0

v# < v ≤ v0 ,

((γ − 1)v − (γ + 1)v0 )(p0 + p∗ ) , (γ + 1)v − (γ − 1)v0

v0 ≤ v < v# ,

for 1-shocks, and p(U0 ; v) = −p∗ − for 3-shocks Remark. The description of an admissible shock set of nonconvex equations of state along the Hugoniot curves is much more complicated. Besides, the admissible criterion (5.4.8) may not be sufficient to select a unique solution for fluids with a nonconvex equation of state. This topic has left open questions for further studies. In a neighborhood of a state where the equation of state resembles a strictly convex or strictly concave function, one can see that Lax’s shock inequalities and Liu’s entropy condition are equivalent to the increase of entropy. Lemma 5.4.1. Assume that the Gr¨uneisen coefficient Γ is positive, and consider the Hugoniot curves issuing from a given left-hand state U0 being parameterized by v ∈ [v0 − δ , v0 + δ ] as in Theorem 5.4.1. There exists a positive constant δ2 ≤ δ such that the following conclusions hold. (i) If ∂vv p(v0 , S0 ) > 0, then the 1-shock speed σ1 (U0 ,U(v)), v ∈ [v0 − δ2 , v0 + δ2 ], is strictly increasing along the Hugoniot curve H1 (U0 ), and the 3-shock speed σ3 (U0 ,U(v)), v ∈ [v0 − δ2 , v0 + δ2 ], is strictly decreasing along the Hugoniot curve H3 (U0 ); (ii) If ∂vv p(v0 , S0 ) < 0, then the 1-shock speed σ1 (U0 ,U(v)), v ∈ [v0 − δ2 , v0 + δ2 ], is strictly decreasing along the Hugoniot curve H1 (U0 ), and the 3-shock speed σ3 (U0 ,U(v)), v ∈ [v0 − δ2 , v0 + δ2 ], is strictly increasing along the Hugoniot curve H3 (U0 ). Proof. From (5.4.2) and (5.4.3), we obtain −(p − p0 ) −(p − p0 ) σ1 (U0 ,U) = u0 − v0 , σ3 (U0 ,U) = u0 + v0 . v − v0 v − v0 It follows from Theorem 5.4.1 that p = p(v) is given as a function of v. Thus, −(p(v) − p0 ) := σ1 (v). σ1 (U0 ,U) = u0 − v0 v − v0

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5 Compressible fluids governed by a general equation of state

A straightforward calculation yields 1 d σ1 (v) v − v0 p0 − p(v) − p (v)(v0 − v) . = − 2 dv 2(v − v0 ) p − p0

(5.4.19)

In the proof of Theorem 5.4.2, we have shown that p”(v0 ) = ∂vv p(v0 , S0 ). This means that p” remains the same sign as ∂vv p(v0 , S0 ) in a neighborhood of v0 . First, if ∂vv p(v0 , S0 ) > 0, then there exists a positive constant δ2 ≤ δ , where δ as in Theorem 5.4.1, such that p”(v) > 0, v ∈ [v0 − δ2 , v0 + δ2 ]. This implies that the function [v0 − δ2 , v0 + δ2 ] v → p(v), is strictly convex. This implies that p0 − p(v) − p (v)(v0 − v) > 0,

v ∈ [v0 − δ2 , v0 + δ2 ], v = v0 ,

so that (5.4.19) gives d σ1 (v) > 0, dv

v ∈ [v0 − δ2 , v0 + δ2 ], v = v0 .

The last inequality shows that the shock speed σ1 is strictly increasing along the Hugoniot curve H1 (U0 ) for v ∈ [v0 − δ2 , v0 + δ2 ]. Similar argument can be made for other cases. If the function p = p(v, S0 ) changes the concavity across v0 , then the entropy inequality (5.4.8) locally takes one Hugoniot curve in both directions, while it takes only a trivial solution on the Hugoniot curve. Precisely, this result is stated in the following theorem. Theorem 5.4.3 (Shock curves for general fluids near an extremum). Assume that the Gr¨uneisen coefficient Γ is positive. Given a state U0 where ∂vv p(v0 , S0 ) = 0. Let the Hugoniot curves issuing from a given left-hand state U0 be parameterized by v ∈ [v0 − δ , v0 + δ ] as in Theorem 5.4.1. There exists a positive constant δ1 ≤ δ such that the following conclusions hold: (i) If ∂vvv p(v0 , S0 ) < 0, then the entropy has S0 as its strict local minimum value at v0 , i.e. S(v) > S0 , v ∈ [v0 − δ1 , v0 + δ1 ], v = v0 . Consequently, admissible parts on the portion v ∈ [v0 − δ1 , v0 + δ1 ] of the Hugoniot curve H1 (U0 ) contain the whole portion, while admissible parts on the portion v ∈ [v0 − δ1 , v0 + δ1 ] of the Hugoniot curve H3 (U0 ) contains only the trivial state {v = v0 }. (ii) If ∂vvv p(v0 , S0 ) > 0, then the entropy has S0 as its strict local maximum value at v0 , i.e.

5.4 General EOS: shock curves

131

S(v) < S0 ,

v ∈ [v0 − δ1 , v0 + δ1 ], v = v0 .

Consequently, admissible parts on the portion v ∈ [v0 − δ1 , v0 + δ1 ] of the Hugoniot curve H3 (U0 ) contain the whole portion, while admissible parts on the portion v ∈ [v0 − δ1 , v0 + δ1 ] of the Hugoniot curve H1 (U0 ) contains only the trivial state {v = v0 }. Proof. It follows from (5.4.12), (5.4.18) and the assumption ∂vv p(v0 , S0 ) = 0 that S (v0 ) = S (v0 ) = S (v0 ) = 0.

(5.4.20)

Now, differentiating (5.4.17) with respect to v, we obtain p (v) = ∂vvv p(v, S(v)) + ∂vvS (v, S(v))S (v) + 2[∂vvS p(v, S(v))S (v) + 2∂vSS p(v, S(v))]S (v) + 2∂vS p(v, S(v))S (v) + [∂vSS p(v, S(v)) + ∂SSS p(v, S(v))S (v)](S (v))2 + 2∂SS (v, S(v))S (v)S (v). Setting v = v0 in the last equality, and using the fact that S (v0 ) = 0, we get p (v0 ) = ∂vvv p(v0 , S0 ).

(5.4.21)

On the other hand, differentiating (5.4.15) with respect to v, we have T (v)S (v) + T (v)S(4) (v) + 2T ”(v)S”(v) + 2T (v)S (v) + T (v)S (v) + T (v)S (v) 1 = − (p(4) (v)(v − v0 ) + 2p (v)). 2

Substituting v = v0 into the last equality, using and (5.4.21), we obtain T0 S(4) (v0 ) = −∂vvv p(v0 , S0 ). The last equality and (5.4.20) imply that it holds along the Hugoniot curves that S = S0 −

1 ∂vvv p(v0 , S0 )(v − v0 )4 + O((v − v0 )5 ), 4!T0

which establishes the conclusions of the theorem. Lemma 5.4.2. Assume that the Gr¨uneisen coefficient Γ is positive. Given a state U0 where ∂vv p(v0 , S0 ) = 0. Consider the Hugoniot curves issuing from a given lefthand state U0 being parameterized by v ∈ [v0 − δ , v0 + δ ] as in Theorem 5.4.1. There exists a positive constant δ2 ≤ δ such that the following conclusions hold. (i) If ∂vvv p(v0 , S0 ) < 0, then the 1-shock speed σ1 (U0 ,U(v)), v ∈ [v0 − δ2 , v0 + δ2 ], is strictly increasing for v ≤ v0 and strictly decreasing for v ≥ v0 along the Hugoniot curve H1 (U0 );

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5 Compressible fluids governed by a general equation of state

(ii) If ∂vvv p(v0 , S0 ) > 0, then the 3-shock speed σ3 (U0 ,U(v)), v ∈ [v0 − δ2 , v0 + δ2 ], is strictly decreasing for v ≤ v0 and strictly increasing for v ≥ v0 along the Hugoniot curve H3 (U0 ). Proof. Let us first prove (i). From (5.4.19) d σ1 (v)/dv has the same sign as q(v) := p0 − p(v) − p (v)(v0 − v).

(5.4.22)

It follows from (5.4.21) and the assumption ∂vvv p(v0 , S0 ) < 0 that p (v0 ) < 0 so that there exists a positive constant δ2 ≤ δ such that p (v) < 0,

v ∈ [v0 − δ2 , v0 + δ2 ].

From (5.4.18) and the assumption ∂vv p(v0 , S0 ) = 0 we have p (v0 ) = 0. Thus,

p”(v) > 0, p”(v) < 0,

v ∈ [v0 − δ2 , v0 ), v ∈ (v0 , v0 + δ2 ].

This shows that q(v) > 0 for v ∈ [v0 − δ2 , v0 ) and q(v) < 0 for v ∈ (v0 , v0 + δ2 ]. So does d σ1 (v)/dv. This completes the proof of Lemma 5.4.2. From Lemmas 5.4.1 and 5.4.2, we obtain the following result. Proposition 5.4.1. Assume that the Gr¨uneisen coefficient Γ is positive. If either ∂vv p(v0 , S0 ) = 0, or ∂vv p(v0 , S0 ) = 0 and ∂vvv p(v0 , S0 ) = 0, then there exists a neighborhood of U0 in which Lax’s shock inequalities, Liu’s entropy condition, and the entropy conditions (5.4.8) are equivalent. It is important to note that if the equation of state is nonconvex, the increase of the entropy does not guarantee the uniqueness of Riemann solutions for large initial data. In such cases, a kinetic relation should be used as an additional admissibility criterion. In the limit case, a kinetic relation reduces to the usual Liu entropy condition.

5.5 Wave curves and Riemann problem

133

5.5 Wave curves and Riemann problem 5.5.1 Wave curves We have examined the basic scale-invariant solutions that are elementary waves, including rarefaction waves, shock waves, and contact discontinuities. Waves associated with the same characteristic field are referred to as waves in the same family and form a wave curve. Let W1F (U0 ) be the forward 1-wave curve consisting of all right-hand state U that can be connected to a given left-hand state U0 by an admissible 1-shock wave, or a 1-rarefaction wave, or a combination of these waves, called composite waves. Let W3B (U0 ) be the backward 3-wave curve consisting of all lefthand state U that can be connected to a given right-hand state U0 by an admissible 3-shock wave, or a 3-rarefaction wave, or a combination of these waves. Let W2 (U0 ) denote the 2-wave curve which consists of contact discontinuities associated with the second characteristic field. For a fluid with a convex equation of state of the form p = p(v, S), the each of the wave curves W1F (U0 ) and W3B (U0 ) is the union of the curve of admissible shock waves on one side with respect to U0 and the curve of rarefaction waves on the other side of U0 , as long as these curves are well-defined. These wave curves were investigated before. For a fluid with a nonconvex equation of state, wave curves exhibit a more complicated structure. Apart from shock waves and rarefaction waves, composite waves are also available. Clearly, the wave curves still depend on the admissibility criterion. Assume, for definitiveness, that the Liu entropy condition is chosen as the admissibility criterion. Composite waves will help extend the wave curves in the following manner. For simplicity, we take a van der Waals fluid to illustrate the case. First, let us consider the case where the EOS is strictly convex near a given state U0 , and changes the convexity in the interval v < v0 . This means that

∂vv p(v0 , S0 ) > 0, and ∂vv p(v, S0 ), as a function of v, changes sign in the interval v < v0 . The wave curve W1F (U0 ) contains the curve of admissible shock waves S1F (U0 ), v < v0 , and the curve of 1-rarefaction waves R1 (U0 ), v ≥ v0 . In the case of a van der Waals fluid, the curve rarefaction waves can be extended for v ∈ [v0 , +∞). Moreover, the wave curve W1F (U0 ) contains also composite waves which consist of a shock wave from U0 to some state U1 ∈ S1F (U0 ) at which

∂vv p(v1 , S1 ) < 0,

and

σ1 (U0 ,U1 ) = λ1 (U1 ),

followed by a rarefaction wave from U1 to U2 ∈ R1F (U1 ). This configuration may be continued until a critical state where the pressure p = p(v, S) changes the concavity. Precisely, let v (S) be a state where ∂vv p(v, S) vanishes and ∂vv p(v, S) > 0 for v < v (S). Then, the solution can be continued with a 1-shock wave from U2 to a state

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5 Compressible fluids governed by a general equation of state

U3 ∈ S1F (U2 ), v3 < v , at which

∂vv p(v3 , S3 ) > 0,

and

σ1 (U2 ,U3 ) = λ1 (U2 ).

See Figure 5.2, where the figure also shows the graph of the pressure as a function of the specific volume for different values of the specific entropy at the states that separate elementary waves.

Fig. 5.2 A possible Riemann solution that consists of shock wave and composite waves in the same family: A shock wave from U0 to U1 , followed by a rarefaction wave from U1 to U2 , followed by another shock wave from U2 to U3 .

Second, let us consider the case where the EOS is strictly convex near a given state U0 , and changes the convexity in the interval v > v0 . This means that

∂vv p(v0 , S0 ) > 0, and ∂vv p(v, S0 ), as a function of v, changes sign in the interval v > v0 . The wave curve W1F (U0 ) contains the curve of admissible shock waves S1F (U0 ), v < v0 , and the curve of 1-rarefaction waves R1F (U0 ), v ≥ v0 . The part of shock waves can be extended beyond v < v0 as long as the Hugoniot curve H1 (U0 ) is well-defined. The part of rarefaction waves can be extended until a state v (S) at which the pressure p = p(v, S) changes the concavity, i.e., ∂vv p(v (S), S) = 0. If ∂vv p(v, S) < 0 for v > v (S), composite waves can be formed. For example, a rarefaction wave from U0 to some state U1 ∈ R1F (U0 ), v0 < v1 < v (S) can be followed by an admissible shock wave from U1 to a state U2 ∈ S1F (U1 ) at which

∂vv p(v2 , S2 ) < 0,

and

σ1 (U1 ,U2 ) = λ1 (U1 ).

5.5 Wave curves and Riemann problem

135

See Figure 5.3.

Fig. 5.3 A Riemann solution that consists of shock wave and composite waves in the same family: A rarefaction wave from U0 to U1 , followed by a shock wave from U1 to U2 .

Third, consider the case where the EOS is strictly concave near a given state U0 , and changes the convexity in each interval v > v0 and v < v0 . This means that

∂vv p(v0 , S0 ) < 0, and ∂vv p(v, S0 ), as a function of v, changes sign in each interval v > v0 and v < v0 . In the direction v > v0 , the wave curve W1F (U0 ) would contain the curve of admissible shock waves S1F (U0 ), v > v0 until a first state U1 := φ (U0 ) on H1 (U0 ), where

σ1 (U0 ,U1 ) = λ1 (U1 ),

∂vv p(v1 , S1 ) > 0.

The curve W1F (U0 ) can be extended beyond v > v1 by composite waves: A shock wave from U0 to U1 followed by a rarefaction wave from U1 to U2 with v2 ≥ v1 . See Figure 5.4 (left). In the direction v ≤ v0 , the curve W1F (U0 ) can contain the curve of 1-rarefaction waves R1F (U0 ), v ≤ v0 until a value v∗ where

∂vv p(v∗ , S∗ = S0 ) = 0. The curve W1F (U0 ) can be extended for v < v∗ by composite waves: A rarefaction wave from U0 to some state U1 , followed by a shock wave from U1 to some state U2 , where σ1 (U1 ,U2 ) = λ1 (U1 ), ∂vv p(v2 , S2 ) > 0.

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5 Compressible fluids governed by a general equation of state

See Figure 5.4 (right).

Fig. 5.4 A shock wave followed by a rarefaction wave (left), and a rarefaction wave followed by a shock wave (right).

Fourth, let us consider a critical case where

∂vv p(v0 , S0 ) = 0, and that the EOS changes from convexity to concavity. This means that ∂vv p(v, S0 ) is positive for v < v0 and negative in the interval (v0 , v∗ ). The wave curve W1F (U0 ) would contain shock waves on both sides of U0 near U0 . Precisely, S1F (U0 ) can be parameterized by v, where v can take values in both intervals v ≤ v0 and v > v0 . In the part v > v0 , S1F (U0 ) may reach a state U1 , where

σ1 (U0 ,U1 ) = λ1 (U1 ),

∂vv p(v1 , S1 ) > 0.

The curve W1F (U0 ) can be extended beyond v > v1 by composite waves: A shock wave from U0 to U1 followed by a rarefaction wave from U1 to U2 with v2 ≥ v1 . Finally, consider another critical case where

∂vv p(v0 , S0 ) = 0, and that the EOS changes from concavity to convexity. This means that ∂vv p(v, S0 ) is positive for v > v0 and negative in the interval (v∗ , v0 ). The curve W1F (U0 ) contain the curve of rarefaction wave R1F (U0 ) for v ≥ v0 . The curve of rarefaction waves R1F (U0 ) can also be defined for v < v0 until a value v∗ where

∂vv p(v∗ , S0 ) = 0. The curve W1F (U0 ) can be extended beyond v < v∗ using composite waves: A rarefaction wave from U0 to a state U1 , followed by a shock wave from U1 to some state U2 , where v2 < v∗ and

σ1 (U1 ,U2 ) = λ1 (U1 ),

∂vv p(v2 , S2 ) > 0.

5.5 Wave curves and Riemann problem

137

A similar argument can be made for the backward wave curve W3B (U0 ) consisting of all admissible 3-shocks, 3-rarefaction waves, and combinations of these waves.

5.5.2 Riemann problem In general, a Riemann solution of the gas dynamics equations consists of 1-waves (left-facing waves) separated from 3-waves (right-facing waves) by a 2-contact discontinuity. Since the particle velocity and the pressure remain constant across the 2-contact, the projections of the states on both sides of the 2-contact on the (p, u)plane coincide. This simplifies the construction of the Riemann solution by projecting the wave curves onto the (p, u)-plane. Precisely, let us choose (ρ , u, p) to be the variables to characterize the state of the fluid. Consider the Riemann problem for the gas dynamics equations with the initial data UL = (ρL , uL , pL ), x < 0, (ρ , u, p)(x, 0) = UR = (ρR , uR , pR ), x > 0. By the general theory of hyperbolic systems of conservation laws, we can establish the existence of the Riemann problem at least locally: The left-hand state (ρL , uL , pL ) is connected with the right-hand state (ρR , uR , pR ) by one or more 1-waves (1-shock wave, or 1-rarefaction wave, or composite 1-waves) followed by a 2-contact, and then followed by one or more 3-waves. Since u and p do not change across any 2-contact, the contact discontinuity separates the two states UI = (ρI , um , pm ) and UII = (ρII , um , pm ). The state UI ∈ W1F (UL ), where W1F (UL ) is the forward 1-wave curve consists of all right-hand state that can be connected to UL by either a 1-shock or a 1-rarefaction wave, or composite solutions combining these two kinds of waves in the case the characteristic field fails to be genuinely nonlinear. Similarly, the state UII ∈ W3B (UR ), where W3B (UR ) is the backward 3-wave curve consisting of all left-hand state that can be connected to UR by either a 3-shock or a 3-rarefaction wave, or composite solutions combining these two kinds of waves in the case the characteristic field fails to be genuinely nonlinear. The projections of UI and UII onto the (u, p)-plane coincide and has the coordinates (um , pm ). Thus, the intersection of the two wave curves W1F (UL ) and W3B (UR ) on the (u, p)-plane can determine the Riemann solution. Finally, we emphasize that a Riemann solution for a convex EOS may have at most three waves. However, it is interesting to note that a Riemann solution for a van der Waals fluid may have up to seven elementary waves, since it may contain up to three elementary waves in each family, together with a 2-contact wave, see Figure 5.5. Observe that in the case of a polytropic ideal gas, both wave curves W1F (UL ) and B W3 (UR ) can be parameterized by p. In fact, the velocity along W1F (UL ) can be given as a decreasing function of p and varies to −∞ as p → ∞, and the velocity along W3B (UR ) can be given as a increasing function of p and varies to ∞ as p → ∞ in the

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5 Compressible fluids governed by a general equation of state

Fig. 5.5 In a van der Waals fluid, a Riemann solution may contain up to seven elementary waves.

(p, u)-plane. In the (p, u)-plane, the u-intercept of W1F (UL ) is u∗ := uL + γ2c−1L , where

R c is given by (5.3.3), and the u-intercept of W3B (UR ) is u∗ := uR − γ2c−1 . Whenever ∗ u > u∗ , these two wave curves intersect at a unique point, which determines a unique solution of the Riemann problem. If u∗ ≤ u∗ , a vacuum occurs in the solution. We thus arrive at the following theorem.

Theorem 5.5.1. (a) For a polytropic ideal gas, the Riemann problem for the gas dynamics equations (5.3.1) in the Eulerian coordinates has a unique solution if and only if 2cR 2cL < uL + , uR − γ −1 γ −1 where c is given by (5.3.3). Otherwise, a vacuum occurs in the solution. (b) For a general fluid, each intersection point UM of the two wave curves W1F (UL ) and W3B (UR ) on the (u, p)-plane can determine a Riemann solution: the solution can be constructed from UL to UM following the wave curve W1F (UL ), then a 2contact at UM keeping u and p the same, followed by 3-waves from the right-hand state of this 2-contact to UR .

5.6 Bibliographical notes We provide a brief selection of the most relevant papers. For additional references, please refer to the bibliography at the end of this monograph. Composite waves for materials with nonconvex equations of state were constructed first by Wendroff [325] for the isentropic system, and by Wendroff [326] for general flows. Later on, Liu [232, 233] introduced an entropy condition for general systems in which the characteristic fields need not be genuinely nonlinear.

5.6 Bibliographical notes

139

The existence and uniqueness of solutions to the Riemann problem in gas dynamics with convex equations of state was studied by Smith [293]; see also Smoller [294]. The gas dynamics equations were studied in Liu [234], Roe [271], and Holden-Risebro-Sande [167] among many others. Wave-front tracking algorithms for the equations of non-isentropic gas dynamics were analyzed by Asakura [25] and Asakura-Corli [27]. The Riemann problem for a nonconvex EOS in gas dynamics was also considered by Hattori [152, 153], Shearer-Yang [289], Fan [126], Menikoff-Plohr [243], LeFloch-Thanh [223], and Thanh [302]. The construction of wave curves for nonconvex EOS were given explicitly by LeFloch-Thanh [226], as well as in Dahmen-M¨uller-Voss [113], and M¨uller-Voss[251]. The Riemann problem for a particle-fluid system was investigated in Aguillon [10] while the problem for a multi-pressure Euler system was studied by Chalons-Coquel [85]. The generalized Riemann problems for compressible fluid flows were considered by Glimm-Marshall-Plohr [139], Bourgeade-LeFloch-Raviart [67], Ben-Artzi and Falcovitz [46, 47], Ben Artzi-Falcovitz [48], and Qian-Li-Wang [264]. Reactive flows were studied by Chorin [95], Levy [231], Majda [239], Corli-Fan [104], and Fan-Lin [129]. The Riemann problem in magneto-gasdynamics was investigated by MischaikowHattori [246], Myong-Roe [252, 253], and Liu-Sun [238]. For admissibility criteria based on traveling waves we refer to Slemrod [290, 291], Bedjaoui-ChalonsCoquel-LeFloch [40], and Bedjaoui-LeFloch [44].

Chapter 6

Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

6.1 Introduction This chapter serves as an example of how a kinetic relation can be used to implement Lax’s shock inequalities. Specifically, we consider a model from elastodynamics, given by a p-system consisting of conservation laws for mass and momentum. In the conservation of momentum, the stress is given as an increasing and concaveconvex function of the strain, making the system strictly hyperbolic with two characteristic fields that are not genuinely nonlinear. Lax’s shock inequalities can only handle limited initial data, but a global construction for a Riemann solver can be achieved by following Wendroff’s idea, where the shock speed is allowed to coincide with the characteristic speed at the state(s) of the shock. This construction satisfies the Liu entropy condition. However, the elastodynamics model considered in this chapter possesses shock waves that violate standard admissibility criteria such as Lax’s shock inequalities and the Liu entropy condition. These shocks were observed in material science by Abeyaratne and Knowles. LeFloch and his collaborators introduced the concept of nonclassical shocks for these shocks, relying on a kinetic relation. Lax’s shock inequalities can still be used, and when they fail at a critical point, a kinetic relation is supplemented around that point to overcome the obstacle. Both Lax and non-Lax shocks are involved in a Riemann solver with kinetics. By definition, kinetic relations provide a one-parameter family of choices for determining Riemann solvers in the range of kinetic functions. A specific choice of kinetic relation will give the classical Riemann solver. Nonclassical shocks can be obtained as the limit of smooth solutions of a diffusive-dispersive system when viscosity and capillarity are suitably added to the system. Diffusive and dispersive models are convenient for Riemann solvers with kinetics; they can provide traveling waves that approximate both Lax and non-Lax shocks.

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 6

141

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

6.2 Background 6.2.1 Strict hyperbolicity and non-genuine nonlinearity The model in nonlinear elastodynamics involves the following system of conservation laws: ∂t v − ∂x σ (w) = 0, (6.2.1) ∂t w − ∂x v = 0, where v, w > −1 represent the velocity and the deformation gradient of the material, respectively. In fact, the system (6.2.1) is a p-system. The stress function σ = σ (w) in (6.2.1) is assumed to be twice differentiable and satisfy the following conditions:

σ (w) > 0, w > −1, wσ (w) > 0 for w = 0, lim σ (w) = −∞, lim σ (w) = +∞,

(6.2.2)

w→+∞

w→−1

where (.) = d/dw and (.)” = d 2 /dw2 . This implies that the stress function σ is always increasing, and is concave upward for w > 0 and concave downward for −1 < w < 0, and it has an inflection point on the graph at w = 0, see Figure 6.1.

Fig. 6.1 The stress function

The system (6.2.1) can be re-written as

∂t v − σ (w)∂x w = 0, ∂t w − ∂x v = 0,

6.2 Background

143

so that, for the unknown vector function u = (v, w), the Jacobian matrix of the system (6.2.1) is given by 0 −σ (w) A(u) = . −1 0 The characteristic polynomial of the matrix A(u) is given by |A(u) − λ I| = λ 2 − σ (w), where I is the 2×2 identity matrix. We can see that A(u) depends only on the second component w. Under the assumptions (6.2.1), σ (w) > 0 for all w > −1. Thus, the matrix A(u) admits two real and distinct eigenvalues which depend only on w, and are, therefore, denoted by λ1 (w) := − σ (w) < 0 < σ (w) := λ2 (w). The model (6.2.1) is thus strictly hyperbolic. The right eigenvectors can be chosen as −λ1 (w) σ (w) r1 (w) := , = 1 1

and r2 (w) :=

−λ2 (w) 1

− σ (w) = . 1

Under the assumptions (6.2.2), we have −σ (w) =0 iff w = 0, Dλ1 (w) · r1 (w) = 2 σ (w) −σ (w) Dλ2 (w) · r2 (w) = = 0 iff w = 0. 2 σ (w) Thus, one can see that the two characteristic fields of the system (6.2.1) are genuinely nonlinear for w = 0, but fail to be genuinely nonlinear when w = 0.

6.2.2 Hugoniot curves Consider a shock wave u = u(x,t) of (6.2.1) connecting a given left-hand state u0 = (v0 , w0 ) to a right-hand state u = (v, w) and propagating with the speed s u0 for x < st, u(x,t) = u for x > st. The Rankine-Hugoniot relations for this shock wave are given by s(v − v0 ) + σ (w) − σ (w0 ) = 0,

s(w − w0 ) + v − v0 = 0.

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

Eliminating v from the last equations, we obtain the shock speed s2 (u0 , u) =

σ (w) − σ (w0 ) , w − w0

and the Hugoniot set (v − v0 )2 + (σ (w) − σ (w0 ))(w − w0 ) = 0. Using the fact that the i-Hugoniot curve is tangent to the eigenvector ri (u0 ) and the shock speed along the i-Hugoniot curve approximates the characteristic speed near u0 , one can obtain the i-Hugoniot curves and the corresponding shock speed along each curve. Precisely, the 1-Hugoniot curve H1 (u0 ) issuing from a given left-hand state u0 = (v0 , w0 ) consisting of all right-hand states u = (v, w) that can be connected to u0 by a 1-shock wave is given by σ (w1 ) − σ (w0 ) H1 (u0 ) : v = v0 + (w − w0 ), (6.2.3) w − w0 and the corresponding 1-shock speed is given by σ (w) − σ (w0 ) s = s1 (w0 , w) = − . w − w0 The 2-Hugoniot curve H2 (u0 ) issuing from a given left-hand state u0 = (v0 , w0 ) consisting of all right-hand states u = (v, w) that can be connected to u0 by a 2-shock wave is given by σ (w1 ) − σ (w0 ) H2 (u0 ) : v = v0 − (w − w0 ), (6.2.4) w − w0 and the corresponding 2-shock speed is given by σ (w) − σ (w0 ) s = s2 (w0 , w) = . w − w0

6.2.3 Lax’s shock inequalities and Lax shocks Next, let us discuss shock waves satisfying Lax’s shock inequalities: for a shock wave between a left-hand state u0 and a right-hand state u

λi (u0 ) > s(w0 , w) > λi (u),

i = 1, 2.

As before, such a shock wave is called a Lax shock.

(6.2.5)

6.2 Background

145

Lax’s shock inequalities (6.2.5) mean that

σ (w) − σ (w0 ) < σ (w), w − w0 σ (w) − σ (w0 ) σ (w0 ) > s2 = > σ (w), w − w0 σ (w0 ) < s2 =

for 1-shocks, (6.2.6) for 2-shocks.

Consider a 2-shock and w0 > 0, for instance. Under the assumptions (6.2.2), the last inequalities in (6.2.6) will select w such that w0 > w > ϕ (w0 ), where ϕ (w0 ) < 0 is the unique value such that

σ (ϕ (w0 )) =

σ (w0 ) − σ (ϕ (w0 )) . w0 − ϕ (w0 )

(6.2.7)

This means that the tangent line to the graph drawing from (w0 , σ (w0 )) coincides with the secant line between the two points (w0 , σ (w0 )) and (ϕ (w0 ), σ (ϕ (w0 ))), see Figure 6.2. This also holds for the case w0 < 0, where ϕ (w0 ) > 0 and satisfies (6.2.7), and Lax’s shock inequalities select the interval w0 < w < ϕ (w0 ). We extend the definition of ϕ (w0 ) when w0 = 0 to be ϕ (0) = 0.

Fig. 6.2 The tangent function to the graph of the stress function

Thus, a tangent function ϕ (w), w > −1 is well-defined: it satisfies (6.2.7) for any w = w0 , and wϕ (w) < 0, w = 0, ϕ (0) = 0. Let us investigate the monotonicity of the function ϕ (w) defined by

σ (ϕ (w)) =

σ (w) − σ (ϕ (w)) , w − ϕ (w)

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

where (.) = d/dw. That is

σ (ϕ (w)) − σ (w) + σ (ϕ (w))(w − ϕ (w)) = 0. Differentiating the last equation yields σ (ϕ (w))

d ϕ (w) d ϕ (w) d ϕ (w) − σ (w) + σ ”(ϕ (w)) (w − ϕ (w)) + σ (ϕ (w)) 1 − = 0. dw dw dw

Simplifying the last equality, we obtain

σ (w) − σ (ϕ (w)) d ϕ (w) < 0, = dw (w − ϕ (w))(σ ”(ϕ (w)))

w = 0,

since the numerator is positive and the denominator is negative. This means that the function ϕ is decreasing. Moreover, it is not difficult to check that under the assumptions (6.2.2), the function w → ϕ (w), w > −1 is onto. Next, we consider the admissibility conditions (6.2.6) for 1-shocks. In this case, since the inequalities are reversed, we should proceed “conversely” as follows. Consider first the case w0 > 0. The tangent line to the graph of the function σ from any value w0 > 0 intersects the graph at a unique point corresponding to w = ϕ − (w0 ). The value ϕ − (w0 ) satisfies w0 · ϕ − (w0 ) < 0,

σ (ϕ − (w0 )) =

σ (ϕ − (w0 )) − σ (w0 ) . ϕ − (w0 ) − w0

(6.2.8)

Lax’s shock inequalities (6.2.6) and therefore (6.2.5) for 1-shocks are equivalent to w > w0

or

w < ϕ − (w0 ).

This argument stills holds for w0 < 0, so that ϕ − (w0 ) is well-defined by (6.2.8) for w0 < 0, and we extend the definition of ϕ − (w0 ) when w0 = 0 by ϕ − (0) = 0. We can check that

ϕ − (ϕ (w)) = w,

ϕ (ϕ − (w)) = w,

w > −1.

This means that the function ϕ − is the inverse of the function ϕ − . So, the function ϕ − is strictly decreasing for w > −1. We can now summarize the above argument concerning admissible shocks satisfying Lax’s shock inequalities as follows. The set of right-hand states u that can be connected to a given left-hand state u0 by a 1-Lax shock is given by {(v, w) ∈ H1 (u0 ) : S1,Lax (u0 ) = {(v, w) ∈ H1 (u0 ) :

w < ϕ − (w0 ) w > ϕ − (w0 )

or or

w > w0 } w < w0 }

for for

w0 ≥ 0, (6.2.9) w0 < 0.

6.2 Background

147

Similarly, the set of right-hand states u that can be connected to a given left-hand state u0 by a 2-Lax shock is given by {(v, w) ∈ H2 (u0 ) : ϕ (w0 ) < w < w0 } for w0 ≥ 0, S2,Lax (u0 ) = for w0 < 0. {(v, w) ∈ H2 (u0 ) : ϕ (w0 ) > w > w0 } (6.2.10)

6.2.4 Integral curves and rarefaction waves Now, consider the integral curves associated with each characteristic field of the system (6.2.1) dw dv (ξ ) = ± σ (w(ξ )), = 1, dξ dξ where (.) = d/dw. Since dw/d ξ = 0, one can use w as a parameterization of the integral curves. Further, the integral curve associated with the first characteristic field issuing from a given state u0 = (v0 , w0 ) is given by v = v0 +

w w0

σ (z)dz,

w > −1.

The integral curve associated with the second characteristic field issuing from a given state u0 = (v0 , w0 ) is given by v = v0 −

w w0

σ (z)dz,

w > −1.

Using the condition that the characteristic speed λi must be increasing through a rarefaction fan i = 1, 2, we can define the curves of rarefaction waves as follows. The curve of 1-rarefaction waves R1F (u0 ) issuing from a given left-hand state u0 consisting of all right-hand state u = (v, w) that can be connected to u0 by a 1rarefaction wave is given by w 0 ≤ w ≤ w0 , if w0 ≥ 0, F σ (z)dz, (6.2.11) R1 (u0 ) : v = v0 + if w0 < 0, 0 ≥ w ≥ w0 , w0 The curve of 2-rarefaction waves R2B (u0 ) issuing from a given left-hand state u0 consisting of all right-hand state u = (v, w) that can be connected to u0 by a 2rarefaction wave is given by w w ≥ w0 , if w0 ≥ 0, B σ (z)dz, R2 (u0 ) : v = v0 − (6.2.12) if w0 < 0. w ≤ w0 , w0

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

6.3 Admissible shocks satisfying a single entropy inequality 6.3.1 The entropy inequality Lax’s shock inequalities are quite strict for selecting shock waves of the model (6.2.1)-(6.2.2). In this section, we will require a weaker admissibility criterion for weak solutions of (6.2.1) to satisfy the following single entropy inequality

∂t U(u) + ∂x F(u) ≤ 0, v2 + Σ (w), w2 Σ (w) := σ (z)dz.

U(v, w) :=

F(v, w) = −vσ (w),

(6.3.1)

0

Under the assumptions (6.2.2), the entropy pair (U, F) is convex, since σ is increasing. Definition 6.3.1. An admissible shock wave is the one satisfying the entropy inequality (6.3.1); a non-Lax shock is an admissible shock which violates Lax’s shock inequalities (6.2.5). Consider a shock wave of (6.2.1) connecting a left-hand state (v0 , w0 ) to some right-hand state (v, w) with a shock speed s. from the general theory of hyperbolic systems in Chapter 3, we know that the entropy inequality (6.3.1) for this discontinuity is equivalent to the condition −s(U(u) −U(u0 )) + F(u) − F(u0 ) ≤ 0. It holds that + Σ (w) − Σ (w0 ) − vσ (w) + v0 σ (w0 ) w2 σ (w) + σ (w0 ) (w − w0 ) , = −s σ (z)dz − 2 w0

−s(U(u) −U(u0 )) + F(u) − F(u0 ) = −s

v2 − v2 0

where s is the shock speed. So, the condition (6.3.1) is equivalent to the condition w σ (w) + σ (w0 ) E(w0 , w) = −s (w − w0 ) ≤ 0. σ (z)dz − (6.3.2) 2 w0

6.3 Admissible shocks satisfying a single entropy inequality

149

6.3.2 Sets of admissible shock waves Let us fix a left-hand state u0 , and look for the set of all right-hand states u that can be connected to u0 by a shock wave satisfying the entropy inequality (6.3.1). By writing E(w0 , w) = −sG(w0 , w), where G(w0 , w) :=

w w0

σ (z)dz −

σ (w) + σ (w0 ) (w − w0 ) := g(w), 2

w > −1,

(6.3.3)

we can investigate the inequality (6.3.2) more easily since the shock speed in each characteristic field does not change sign. For 1-shocks, since s < 0, the entropy inequality (6.3.2) becomes g(w) ≤ 0,

or

w ∈ g−1 ((−∞, 0]).

(6.3.4)

Similarly, for 2-shocks, since s > 0, the entropy inequality (6.3.2) becomes g(w) ≥ 0,

or

w ∈ g−1 ([0, ∞)).

(6.3.5)

Thus, admissible 1-shocks correspond to the non-positive values of G, and admissible 2-shocks correspond to the non-negative values of G, where G is defined by (6.3.3). These values will be obtained in the following by investigating the properties of the function G. We have dg(w) 1

= σ (w) − σ (w0 ) − σ (w)(w − w0 ) , dw 2 dh(w) d 2 g(w) −1 = = σ (w)(w − w0 ). dw dw2 2

h(w) :=

w > −1,

First, assume that w0 > 0. Using the assumptions (6.2.2), we can see that ⎧ ⎪ ⎨> 0 for 0 < w < w0 , dh(w) −1 = σ (w)(w − w0 ) < 0 for w < 0 or w > w0 , ⎪ dw 2 ⎩ = 0 for w = 0, w = w0 , h(ϕ (w0 ) = h(w0 ) = 0. This implies that 0

for w > ϕ (w0 ), w = w0 , for w < ϕ (w0 ),

(6.3.6)

which means that the function G is increasing for w < ϕ (w0 ) and decreasing for w > ϕ (w0 ). Further, we have

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

g(w0 ) = 0,

g(ϕ (w0 )) > 0,

g(ϕ − (w0 )) < 0.

(6.3.7)

From (6.3.6) and (6.3.7), there exists a unique value, denoted by ϕ∞ (w0 ) between the two values (ϕ − (w0 ) and w0 ϕ (w0 )) such that g(ϕ∞ (w0 )) = 0.

(6.3.8)

Thus, g−1 ([0, ∞)) = [ϕ∞ (w0 ), w0 ],

g−1 ((−∞, 0]) = (−1, ϕ∞ (w0 )] ∪ [w0 , ∞).

Similarly, for w0 ≤ 0, there exists a unique value, denoted by ϕ∞ (w0 ) between the two values (ϕ − (w0 ) and w0 ϕ (w0 )) such that g(ϕ∞ (w0 )) = 0, and g−1 ([0, ∞)) = [w0 , ϕ∞ (w0 )],

g−1 ((−∞, 0]) = (−1, w0 ] ∪ [ϕ∞ (w0 ), ∞).

Fig. 6.3 The zero function ϕ∞ of the entropy dissipation

The function w → ϕ∞ (w), w > −1 is strictly decreasing, and by the skewsymmetry of G, we have G(ϕ∞ (w), w) = −G(w, ϕ∞ (w)) = 0,

G(ϕ∞ (w), ϕ∞ (ϕ∞ (w))) = 0.

This implies that the function ϕ∞ is its own inverse

ϕ∞ (ϕ∞ (w)) = w

for all w > −1.

6.3 Admissible shocks satisfying a single entropy inequality

151

Thus, right-hand states u = (v, w) that can be reached by shock waves satisfying the entropy inequality (6.3.1) are characterized as follows (Figure 6.3). For 1-shocks:

w∈

For 2-shocks:

w∈

( − 1, ϕ∞ (w0 )] ∪ [w0 , ∞) (−1, w0 ] ∪ [ϕ∞ (w0 ), ∞) [ϕ∞ (w0 ), w0 ], [w0 , ϕ∞ (w0 )],

for for

for for

w0 ≥ 0, w0 < 0.

w0 ≥ 0, w0 < 0.

(6.3.9)

6.3.3 Non-Lax shocks and composite waves Moreover, the right-hand states u = (v, w) that can be reached by an admissible nonLax shock are given by − [ϕ (w0 ), ϕ∞ (w0 )] for w0 ≥ 0, For 1-shocks: w ∈ for w0 < 0. [ϕ∞ (w0 ), ϕ − (w0 )] (6.3.10) [ϕ∞ (w0 ), ϕ (w0 )], for w0 ≥ 0, For 2-shocks: w ∈ for w0 < 0. [ϕ (w0 ), ϕ∞ (w0 )],

Notation We will use the following notation: (i) Wi (u1 , u2 ): An i-wave connecting a state u1 and a state u2 , where W = S: admissible shock, W = SL : Lax shock, W = SN : non-Lax shock, W = R: rarefaction wave; (ii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the left-hand state u1 to the right-hand state u2 is followed by a j-wave from the left-hand state u2 to the right-hand state u3 ; (iii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the right-hand state u1 to the left-hand state u2 is preceded by a j-wave from the right-hand state u2 to the left-hand state u3 . Composite waves which contain several waves associated with the same characteristic field can be made. Let us now introduce a mapping as follows. For any pair (w1 , w2 ), the secant line passing through (w1 , σ (w1 )) and (w2 , σ (w2 )) cuts the graph of the stress function at a third point corresponding to w = w3 := ϕ (w1 , w2 ). This means that the slopes between w1 , w2 and w2 , w3 are the same

σ (w2 ) − σ (w1 ) σ (ϕ (w1 , w2 )) − σ (w2 ) = , w2 − w1 ϕ (w1 , w2 ) − w2

w1 , w2 > −1.

(6.3.11)

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

Moreover, we set

ϕ∞ (w) := ϕ (w, ϕ∞ (w)),

(6.3.12)

ϕ∞

where is defined by (6.3.8). Multiple waves in the same characteristic field can be composed together in a Riemann solution. To see this, let us consider 1-waves. A similar argument can be made for 2-waves. First, let us investigate the case of rarefaction wave-shock. Actually, we can claim that a 1-rarefaction wave from a left-hand state u0 = (v0 , w0 ) to a right-hand state u1 = (v1 , w1 ) can be followed by a 1-shock wave from the left-hand state u1 to a right-hand state u2 = (v2 , w2 ): R1 (u0 , u1 ) S1 (u1 , u2 ),

for w2 between ϕ − (w1 ) and ϕ∞ (w1 ),

(6.3.13)

see Figure 6.4. Indeed, assume for simplicity that w0 > 0, since a similar argument can be applied when w0 ≤ 0. Since u1 ∈ R1F (u0 ), 0 ≤ w1 ≤ w0 . For the admissible shock between u1 and u2 , it follows from (6.3.9) that

Fig. 6.4 A composition of a rarefaction wave and a shock wave

w2 ≤ ϕ∞ (w1 ). Furthermore, the constraint that the shock speed s1 (w1 , w2 ) is greater than the maximum speed of the rarefaction wave −c(w1 ) gives −c(w1 , w2 ) ≥ −c(w1 ). This yields or, as seen earlier,

σ (w1 )(w1 − w2 ) − σ (w1 ) + σ (w2 ) ≥ 0,

6.3 Admissible shocks satisfying a single entropy inequality

153

w1 ≥ ϕ (w2 ). Since the function ϕ − is strictly decreasing, the last inequality yields w2 ≥ ϕ − (w1 ), as claimed above. Composite waves of two shocks can also be made: a 1-shock from a left-hand state u0 = (v0 , w0 ) to a right-hand state u1 = (v1 , w1 ) can be followed by another 1-shock from the left-hand state u1 to a right-hand state u2 = (v2 , w2 ) S1 (u0 , u1 ) S1 (u1 , u2 ), whenever

w1 ≥ w0 and ϕ (w1 , w0 ) ≤ w2 ≤ ϕ∞ (w1 ), w1 ≤ w0 and ϕ (w1 , w0 ) ≥ w2 ≥ ϕ∞ (w1 ),

(6.3.14)

for w0 ≥ 0 for w0 < 0,

see Figure 6.5. Indeed, assume for simplicity that w0 > 0, since the case of the proof for w0 ≤ 0 is similar. It follows from (6.3.9) that the state u1 ∈ H1 (u0 ) with w1 ≥ w0 corresponds an admissible shock. It is easy to see that this shock is a Lax shock. The state u2 ∈ H1 (u1 ) with w2 ≤ ϕ∞ (w1 ) corresponds to an admissible shock. Observe that the second shock is non-Lax if w2 > ϕ − (w1 ). The condition that the speed of the second shock must be greater than the one of the first shock s1 (w0 , w1 ) ≤ s1 (w1 , w2 ), yields

Fig. 6.5 A composition of two shock waves: a shock from w0 to w1 , followed by another shock from w1 to w2 .

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

w2 ≥ ϕ (w1 , w0 ), as claimed earlier.

6.3.4 Two-parameter sets of waves In this subsection, we can construct two-parameter sets of waves corresponding to the same characteristic field, rather than wave curves. It is sufficient to do that for 1waves since a similar argument can be made for 2-waves. Assume first that wL > 0. For any given left-hand state uL = (vL , wL ), the set of 1-waves consisting of all righthand states uR = (vR , wR ) that can be connected to uL by 1-shocks, 1-rarefaction waves, and composite waves of 1-shocks and 1-rarefaction waves is described as follows. First, if wR > wL , the solution consists of a Lax shock S1L (uL , uR ). Second, if 0 ≤ wR < wL , the solution consists of a rarefaction wave R1 (uL , uR ) connecting uL to uR ∈ R1F (uL ). Third, if ϕ∞ (wL ) ≤ wR < 0, there is a one-parameter family of admissible solutions described as follows. The solution can begin with a rarefaction wave from uL to some intermediate state uM ∈ R1F (uL ),

ϕ (wR ) ≤ wM ≤ ϕ∞ (wR ),

followed by a non-Lax shock from uM and uR ∈ H1 (uM ). That is the solution is given by R1 (uL , uM ) S1N (uM , uR ). Fourth, if ϕ − (wL ) < wR < ϕ∞ (wL ), there is also a one-parameter family of admissible solutions described as follows. The solution can begin with either a rarefaction wave from uL to some state uM uM ∈ R1F (uL ),

ϕ (wR ) ≤ wM < wL ,

or a Lax shock from uL to some state uM uM ∈ H1 (uL ),

wL < wM ≤ ϕ∞ (wR ),

followed by a non-Lax shock from uM to uR ∈ H1 (uM ). Thus, the solution is either R1 (uL , uM ) S1N (uM , uR ), or S1L (uL , uM ) S1N (uM , uR ). Finally, if wR < ϕ − (wL ), there is also a one-parameter family of admissible solutions described as follows. The solution can begin with a Lax shock from uL to some state uM

6.4 General Riemann solver based on a kinetic relation

uM ∈ H1 (uL ),

155

wM ≥ ϕ (wR , wL ),

followed by a non-Lax shock from uM to uR ∈ H1 (uM ), so that the solution is S1L (uL , uM ) S1N (uM , uR ). For wL ≤ 0, the construction of solutions is similar. We can also construct a twoparameter family set of 2-waves.

6.4 General Riemann solver based on a kinetic relation 6.4.1 Kinetic relation imposed from left-to-right of the shock A kinetic function for the model (6.2.1) is a smooth function which takes the values in the set of admissible non-Lax shocks given by (6.3.10). Precisely, a kinetic function ϕ for 1-shocks satisfies the following constraints: − [ϕ (w), ϕ∞ (w)], for w ≥ 0, (6.4.1) ϕ (w) ∈ [ϕ∞ (w), ϕ − (w)], for w < 0. A kinetic function φ for 2-shocks satisfies the following constraints: [ϕ∞ (w), ϕ (w)], for w ≥ 0, φ (w) ∈ [ϕ (w), ϕ∞ (w)], for w < 0.

(6.4.2)

Since the upper and lower limits of the range of the kinetic functions are strictly decreasing functions, it is natural to require in the following that both kinetic functions ϕ and φ in (6.4.2) are strictly decreasing for w > −1. Therefore, the kinetic functions are invertible. Let ϕ −1 and φ −1 denote the inverse functions of the kinetic functions ϕ and φ , respectively. In this section, a kinetic relation is a relation that is imposed from left-to-right. That is, a kinetic relation requires that the right-hand state be attained by using the kinetic function at the left-hand state. Precisely, any non-Lax shock between a lefthand state u− = (v− , w− ) and a right-hand state u+ = (v+ , w+ ) is required to satisfy the following kinetic relation: w+ = ϕ (w− )

for non-Lax 1-shocks,

w+ = φ (w− )

for non-Lax 2-shocks.

(6.4.3)

Furthermore, to be deterministic, the following nucleation criterion is imposed: (C)

Non-Lax shocks are preferred whenever available.

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

6.4.2 Construction of wave curves Given a left-hand state u0 = (v0 , w0 ), the forward i-wave curve Wi F (u0 ) is defined as the set of all right-hand states u = (v, w) that can be connected to u0 using single waves or composite waves from Lax shocks, admissible non-Lax shocks, and rarefaction waves associated with the ith characteristic field, i = 1, 2. Given a righthand state u0 = (v0 , w0 ), the backward i-wave curve Wi B (u0 ) is defined as the set of all left-hand states u = (v, w) that can be connected to u0 using single waves or composite waves from Lax shocks, admissible non-Lax shocks, and rarefaction waves associated with the ith characteristic field, i = 1, 2.

Fig. 6.6 The solution is a 1-rarefaction wave followed by a non-Lax 1-shock

Next, we will show that the wave curves can be parameterized by the form v = v(w), w > −1. Forward 1-wave curve W1F (u0 ) First, consider the forward 1-wave curve associated with the first characteristic field W1F (u0 ). We will describe the wave curve only for w0 > 0 since the argument is similar for w0 ≤ 0. For w > w0 , the solution is a Lax shock from u0 to u S1L (u0 , u1 ). For 0 ≤ w < w0 , the solution is a rarefaction wave from u0 to u R1 (u0 , u1 ).

6.4 General Riemann solver based on a kinetic relation

157

For ϕ (w0 ) ≤ w < 0, the solution is a composite wave that consists of a rarefaction wave from u0 to u1 = (v1 , w1 = ϕ −1 (w)) ∈ R1F (u0 ) followed by a non-Lax shock from u1 to u R1 (u0 , u1 ) S1N (u1 , u). See Figure 6.6. For w < ϕ (w0 ), the solution can be either a Lax shock from u0 to u1 = (v1 , w1 = ϕ −1 (w)), followed by a non-Lax shock from u1 to u, if w0 ϕ −1 (w) > w0 ϕ (w0 , w), or a single Lax shock, otherwise S1L (u0 , u1 ) S1N (u1 , u), S1L (u0 , u),

if

if w0 ϕ −1 (w) > w0 ϕ (w0 , w),

w0 ϕ −1 (w) ≤ w0 ϕ (w0 , w),

see Figure 6.7.

Fig. 6.7 The solution is a Lax 1-shock followed by a non-Lax 1-shock

Forward 2-wave curve W2F (u0 ) Second, the forward 2-wave curve W2F (u0 ) associated with the second characteristic field is given as follows. Consider only the case w0 > 0, since the argument is similar for w0 ≤ 0. For w > w0 , the solution is a rarefaction wave from u0 to u ∈ R2 (u0 ) R2 (u0 , u). For ϕ (w, w0 ) ≤ w < w0 , the solution is a Lax shock from u0 to u ∈ H2 (u0 ) S2L (u0 , u).

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

Fig. 6.8 The solution is a non-Lax 2-shock preceded by a Lax 2-shock

Third, if φ (w0 ) ≤ w < ϕ (w, w0 ), the solution is a composite wave consisting of a non-Lax shock from u0 to u1 = (v1 , w1 = φ (w0 )) ∈ H2 (u0 ), followed by a Lax shock from u1 to u ∈ H2 (u1 ) S2N (u0 , u1 ) S2L (u1 , u), see Figure 6.8. For w < φ (w0 ), the solution consists of a non-Lax shock from u0 to u1 = (v1 , w1 = φ (w0 )) ∈ H2 (u0 ), followed by a rarefaction wave from u1 to u ∈ H2 (u1 ) S2N (u0 , u1 ) R2 (u1 , u), see Figure 6.9. Backward 2-wave curve W2B (u0 ) The backward 2-wave curve W2B (uR ) will be used in solving the Riemann problem in the next section. Under the condition (C), W2B (u0 ) is uniquely defined as follows. Consider only the case w0 ≥ 0, since the argument for case w0 ≤ 0 is similar. For w > w0 , the solution is a Lax shock from u0 to u S2L (u0 , u). For 0 ≤ w < w0 , the solution is a rarefaction wave from u0 to u R2 (u0 , u). For φ −1 (w0 ) ≤ w < 0, the solution is a composite wave that consists of a rarefaction wave from u0 to u1 = (v1 , w1 = φ (w)), preceded by a non-Lax shock from u1 to u

6.4 General Riemann solver based on a kinetic relation

159

Fig. 6.9 The solution is a non-Lax 2-shock preceded by a 2-rarefaction wave

Fig. 6.10 The solution is a 2-rarefaction wave preceded by a non-Lax 2-shock

R2 (u0 , u1 ) S2N (u1 , u), see Figure 6.10. For w < ϕ (w0 ), the solution can be either a Lax shock from u0 to u1 = (v1 , w1 = ϕ −1 (w)), followed by a non-Lax shock from u1 to u, if w0 ϕ −1 (w) > w0 ϕ (w0 , w), or a single Lax shock, otherwise S2L (u0 , u1 ) S2N (u1 , u), S2L (u0 , u), see Figure 6.11.

if

if w0 ϕ −1 (w) > w0 ϕ (w0 , w),

w0 ϕ −1 (w) ≤ w0 ϕ (w0 , w),

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

Fig. 6.11 The solution is a 2-shock preceded by a non-Lax 2-shock

6.4.3 General Riemann solvers The following theorem is the main result in this section. Theorem 6.4.1 (General Riemann solvers by kinetic relations). The Riemann problem for the elastodynamic model (6.2.1)-(6.2.2) admits a unique solution made of Lax shocks, non-Lax shocks satisfying the kinetic relation (6.4.3), and rarefaction waves. Proof Without loss of generality, assume that wL > 0. First, we need to show that the forward 1-wave curve W1F (uL ) can be parameterized in the form w → v = v(w), w > −1, where v is a strictly increasing function. This is evident for w ≥ 0, since the wave curve coincides with S1,Lax (uL ) on the part w > wL and with R1F (uL ) on the part 0 ≤ w ≤ wL , along which v is decreasing as a function of w. Moreover, we have lim v(w) = ∞.

w→∞

Next, we consider w ∈ (ϕ (wL ), 0). The solution is a 1-rarefaction wave from wL to w1 = ϕ −1 (w) followed by a non-Lax shock from w1 to w. Thus, v = v(w1 ) + c(w1 , w)(w − w1 ), v(w1 ) = vL +

w1

c(z)dz,

wL

where c(w, w1 ) =

w1 = ϕ −1 (w),

σ (w) − σ (w1 ) , w − w1

6.4 General Riemann solver based on a kinetic relation

and c(w) =

161

σ (w).

The last equations yield −d(w )

2 dv 1 1 = c(w1 , w) − c(w1 ) + c2 (w) + c2 (w1 , w) > 0. dw 2c(w1 , w) dw The last inequality shows that the wave curve is also strictly increasing in the interval (ϕ (wL ), 0). Next, let us study the pattern of the wave curve corresponding to w < ϕ (wL ). We have to distinguish between the two cases (a)

ϕ −1 (w) > ϕ (wL , w),

(b)

ϕ −1 (w) ≤ ϕ (wL , w).

Consider the case (a). It is easy to check that in this case the speed of the Lax 1-shock between wL and w1 = ϕ −1 (w) is smaller than non-Lax 1-shock between w1 = ϕ −1 (w) and w, i.e., s1 (wL , w1 ) ≤ s1 (w1 , w). This means that, in view of the nucleation criterion (C), the solution is a composition of a Lax shock from wL to w1 = ϕ −1 (w) followed by a non-Lax shock from w1 to w. Consider the case (b). The solution is merely a single Lax shock from wL to w. By continuity, the set Ω := w ∈ (−1, ϕ (wL ))/ϕ −1 (w) − ϕ (wL , w) > 0 is an open set, and thus can be expressed as a union of open intervals. At the endpoint of each of these intervals, the speeds of the Lax shock and non-Lax shock coincide. This yields the continuity of the wave curve in the entire interval. As shown above, in each interval of the complement set (−1, ϕ (wL )) \ Ω , the wave curve coincides with S1,Lax (uL ) and so the conclusion follows. Consider now an interval (w , w ) ⊂ Ω . It holds by the construction above that v = v(w1 ) + c(w1 , w)(w − w1 ), v(w1 ) = vL + c(wL , w1 )(w1 − wL ). It holds that c2 (w1 ) −d(w1 ) dv = (c(w1 , wL ) − c(w1 , w)) −1 dw 2dw c(w1 , wL )c(w1 , w) +

c2 (w) + 1 . 2c(w1 , w)

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

Since a Lax shock from wL to w1 is followed by the non-Lax shock from w1 to w, the shock speed of the later one must be larger than the one of the earlier: s1 (wL , w1 ) ≤ s1 (w1 , w). Moreover, we can see that w1 ≥ wL > ϕ (w). Thus, we get the following estimates

σ (w1 ) − σ (wL ) ≤ σ (w1 ), w1 − wL

and

σ (w1 ) − σ (w) ≤ σ (w1 ), w1 − w

The last inequality yields c2 (w1 ) − 1 ≥ 0. c(w1 , wL )c(w1 , w) This therefore gives us dv > 0, dw and the conclusion of the monotonicity of the curve is established. Now, it is an easy matter to check that lim v = −∞.

w→−1

Arguing similarly, we can prove that the backward 2-wave curve W2B (uR ) is a strictly decreasing function w → v = v(w; uR ), w > −1, and lim v(w; uR ) = ∞,

w→−1

lim v(w; uR ) = −∞.

w→∞

The above argument shows that the forward 1-wave curve W1F (uL ) and the backward 2-wave curve W2B (uR ) always intersect at a unique point uM = (vM , wM ). Thus, the Riemann problem possesses a unique solution. The proof of Theorem 6.4.1 is completed.

6.4.4 Classical Riemann solver based on Wendroff’s construction The classical construction of a Riemann solver for the elastodynamics model (6.2.1)-(6.2.2) was first proposed by Wendroff. The construction for Riemann solvers with kinetics in the previous section actually covers the classical Riemann solver with the choice ϕ = ϕ − , for 1-shocks,

φ = ϕ ,

for 2-shocks.

Precisely, we can describe the wave curves as follows.

6.4 General Riemann solver based on a kinetic relation

163

Forward 1-wave curve W1F (u0 ) Consider the wave curve W1F (u0 ). Assume that w0 > 0, since the case w0 ≤ 0 is similar. The forward 1-wave curve W1F (u0 ), for any given left-hand state u0 , can be parameterized by w → v = v(u0 ; w), w > −1, as follows. First, for either w > w0 , or w < ϕ − (w0 ), the solution is a Lax shock, thus the solution is S1L (u0 , u). Second, for 0 ≤ w ≤ w0 , the solution is a rarefaction wave R1 (u0 , u). Third, for ϕ − (w0 ) ≤ w < 0, the solution is a rarefaction wave from u0 = (v0 , w0 ) to u1 = (v1 , w1 = ϕ (w)), followed by a shock wave from u1 to u = (v, w) R1 (u0 , u1 ) S1N (u1 , u). The forward 1-wave curve W1F (u0 ) given as a function w → v = vF1 (u0 ; w), w > −1 is strictly increasing and satisfies lim v(u0 ; w) = −∞,

w→−1

lim v(u0 ; w) = ∞.

w→∞

Forward 2-wave curve W2F (u0 ) Next, let us describe the forward 2-wave curve W2F (u0 ). The forward 2-wave curve W2F (u0 ), for any given left-hand state u0 , can be parameterized by w → v = v(u0 ; w), w > −1, as follows. First, for w ≥ w0 , the solution is a single rarefaction wave R2 (u0 , u). Second, for ϕ (w0 ) < w < w0 , the solution is a single Lax shock S2L (u0 , u). Third, for w < ϕ (w0 ), the solution consists of a non-Lax shock from u0 = (v0 , w0 ) to u1 = (v1 , w1 = ϕ (w0 )), followed by a rarefaction wave from u1 to u = (v, w) S2N (u0 , u1 ) R2 (u1 , u). Backward 2-wave curve W2B (u0 ) and solving Riemann problem To solve the Riemann problem, we will use the backward 2-wave curve W2B (u0 ), which can be constructed as follows. We will describe the construction for w0 > 0, since the case w0 ≤ 0 is similar. This wave curve W2B (u0 ), for any given right-hand state u0 , can be given in the form w → v = vB2 (u0 ; w), w > −1, as follows. First, for either w > w0 , or w < ϕ − (w0 ), the solution is a Lax shock, thus the solution is S2L (u0 , u).

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6 Nonclassical Riemann solver with prescribed kinetics—The hyperbolic regime

Second, for 0 ≤ w ≤ w0 , the solution is a rarefaction wave R2 (u0 , u). Third, for ϕ − (w0 ) ≤ w < 0, the solution is a rarefaction wave from u0 = (v0 , w0 ) to u1 = (v1 , w1 = ϕ (w)), preceded by a shock wave from u1 to u = (v, w) R2 (u0 , u1 ) S2N (u1 , u). It is not difficult to check that the function w → v = vB2 (u0 ; w), w > −1 is strictly decreasing and satisfies lim v(u0 ; w) = ∞,

w→−1

lim v(u0 ; w) = −∞.

w→∞

The above argument shows that the forward 1-wave curve W1F (uL ) and the backward 2-wave curve W2B (uL ) always intersect at a unique point uM . Accordingly, the Riemann problem for (6.2.1)-(6.2.2) admits a unique solution. Theorem 6.4.2 (Classical Riemann solver). The Riemann problem for the elastodynamic model (6.2.1)-(6.2.2) admits a unique solution made of admissible shocks and rarefaction waves. Remark 6.4.1. In this chapter, a kinetic relation is imposed and defined from the left-hand to the right-hand sides of the shock. In the next chapter, we will see that the kinetic relation can also be imposed from the right-hand to the left-hand sides.

6.5 Bibliographical notes We list only a short selection of the most relevant papers for this chapter and also refer the reader to the bibliography at the end of this monograph. Shock waves that violate Lax’s shock inequalities arise in material science, as observed in Abayaratne-Knowles [2, 4]. A mathematical formulation was proposed in HayesLeFloch [160, 162] and LeFloch [210, 211]. The elastodynamics model in this chapter was studied in LeFloch-Thanh [224]. Kinetic relations for undercompressive shocks were studied by Shearer-Schecter [287, 288], Schecter-Shearer [280], LeFloch [212], Schulze-Shearer [282], Bertozzi-M¨unch-Shearer [53, 54], M¨unchBertozzi [248], Asakura [23], LeFloch-Thanh [223, 225, 226], and Cox-Kluwick [106]. The analysis of diffusive-dispersive admissibility criteria was pioneered by Slemrod [290, 291], and by Fan [124, 126]. Traveling waves associated with nonclassical shocks in diffusive-dispersive models are studied in Bedjaoui-LeFloch [41– 45], Bedjaoui-Chalons-Coquel-LeFloch [40], and Bertozzi-Shearer [55]. The mathematical theory of self-similar boundary layers for nonlinear hyperbolic systems with viscosity and capillarity is developed in Choudhury-Joseph-LeFloch [96]. Self-

6.5 Bibliographical notes

165

similar approximations and boundary layers were studied by Joseph-LeFloch [181– 184]. Coupling techniques for nonlinear hyperbolic equations were investigated by Boutin-Coquel-LeFloch [63–65]. Numerical schemes for nonclassical shock waves and undercompressive shocks were constructed by Hayes-LeFloch [161], Chalons-LeFloch [87–89], LeFlochRohde, [218], Chalons [82, 83], Zhong-Hou-LeFloch [333], and Chalons-EngelRohde [86]. Schemes with well-controlled dissipation were studied by LeFlochMishra [214].

Chapter 7

Nonclassical Riemann solver with prescribed kinetics—The hyperbolic-elliptic regime

7.1 Introduction This chapter deals with a hyperbolic-elliptic model of phase transition dynamics. The model is a typical p-system, where the stress can be decreased as a function of the strain in a certain interval. Consequently, this causes the system to become elliptic in the corresponding region. The elliptic region separates the phase domain into two regions, called the phases, in which the system is strictly hyperbolic. The system admits undercompressive subsonic phase boundaries to be characterized via a kinetic relation. The dynamics of phase boundaries in solids undergoing phase transformations play an important role in many applications of material science. It is interesting to note that the Riemann problem may admit more than one solution. In fact, there can be two distinct solutions.

7.2 Background 7.2.1 Elliptic-hyperbolic model Precisely, we consider in this chapter the Riemann problem for the following model of phase transitions: ∂t v − ∂x σ (w) = 0, (7.2.1) ∂t w − ∂x v = 0, where v = v(x,t) is the velocity, and w = w(x,t) > −1 is the deformation gradient, and the stress σ = σ (w) is a twice differentiable function of w, which satisfies the following assumptions:

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 7

167

168

7

Nonclassical Riemann solver with prescribed kinetics . . .

σ (0) < 0, w σ (w) > 0 for w = 0, lim σ (w) = +∞, lim σ (w) = +∞,

(7.2.2)

w→+∞

w→−1

where (.) = d/dw and (.)” = d 2 /dw2 . Note that under the conditions (7.2.2), there exist constants a, b ∈ (−1, +∞), a < 0 < b, such that

σ (w) > 0 σ (w) < 0

if

− 1 < w < a,

if

a < w < b.

w > b,

(7.2.3)

That is, the function σ is increasing in each interval −1 < w < a and w > b, and is decreasing in the interval a < w < b. The function σ attains a (unique) local maximum value at w = a and a (unique) local minimum value at w = b; it is concave upward for w > 0 and concave downward for −1 < w < 0, and admits an inflection point at w = 0, see Figure 7.1.

Fig. 7.1 The stress-strain function for the elliptic-hyperbolic model (7.2.1)-(7.2.2)

Let us investigate the properties of the model (7.2.1)-(7.2.2). The system (7.2.1) can be re-written as ∂t v − σ (w)∂x w = 0, ∂t w − ∂x v = 0, so that, the Jacobian matrix of the system (7.2.1) for the unknown vector function u = (v, w) is given by 0 −σ (w) A(u) = . −1 0 The characteristic polynomial of the matrix A(u) is given by |A(u) − λ I| = λ 2 − σ (w), where I is the 2 × 2 identity matrix. It follows from (7.2.3) that the matrix A(u) admits two real and distinct eigenvalues

7.2 Background

169

λ1 (w) := − σ (w) < 0 < σ (w) := λ2 (w),

(7.2.4)

in the regions −1 < w < a and w > b; and it admits two purely complex eigenvalues in the region a < w < b. This implies that the model (7.2.1)-(7.2.2) is strictly hyperbolic for −1 < w < a, w > b, but elliptic for a < w < b. For this reason, the model (7.2.1)-(7.2.2) is referred to as an elliptic-hyperbolic model. The corresponding eigenvectors of (7.2.1)-(7.2.3) in the strictly hyperbolic regions can be chosen as −λ1 (w) σ (w) r1 (w) := = , 1 1

and r2 (w) :=

−λ2 (w) 1

=

− σ (w) . 1

Under the assumptions (7.2.2), we have −σ (w) Dλ1 (w) · r1 (w) = =0 2 σ (w)

iff w = 0,

−σ (w) Dλ2 (w) · r2 (w) = = 0 iff 2 σ (w)

w = 0.

The two characteristic fields of the model (7.2.1)-(7.2.2) are genuinely nonlinear for w = 0, but fail to be genuinely nonlinear when w = 0. Thus, the two characteristic fields of (7.2.1)-(7.2.2) are genuinely nonlinear in the hyperbolic regions. However, as seen later, the shock speed may not be defined even in strictly hyperbolic regions, when the two states of the jump belong to different sides with respect to the elliptic region. In constructing wave curves, and therefore, solutions to the Riemann problem, we restrict our discussions to values in the hyperbolic regions only. That is, we consider only the model (7.2.1)-(7.2.2) for −1 < w ≤ a

or

w ≥ b.

The region corresponding to −1 < w < a will be referred to as Phase I, and the region corresponding to w > b will be referred to as Phase II, see Figure 7.2.

7.2.2 Hugoniot curves and integral curves First, let us consider Hugoniot curves. A shock wave of (7.2.1) connecting a lefthand state u0 = (v0 , w0 ) to a right-hand state u1 = (v1 , w1 ) and propagating with the I is a weak solution of the form speed s = s(u0 , u1 ) ∈ R u0 for x < st, u(x,t) = u1 for x > st, The Rankine-Hugoniot relations for this shock are given by

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Fig. 7.2 The two phases in the (w, z)-plane

s (v1 − v0 ) + σ (w1 ) − σ (w0 ) = 0,

s (w1 − w0 ) + v1 − v0 = 0,

(7.2.5)

which yields the shock speed s as s = s(u0 , u1 ) = −

σ (w1 ) − σ (w0 ) v1 − v0 =− . v1 − v0 w1 − w0

Therefore, whenever (σ (w1 ) − σ (w0 ))/(w1 − w0 ) ≥ 0, the shock speed σ (w1 ) − σ (w0 ) s = ∓c(w0 , w1 ) := ∓ w1 − w0

(7.2.6)

is well-defined. Since the shock speed by (7.2.6) is independent of v0 and v1 , we can simply write s = s(w0 , w1 ). In (7.2.6), the 1- and 2-shocks correspond to the ∓ signs, respectively. We also define the sound speed to be c(w) := σ (w). The Hugoniot curves of (7.2.1) under the assumptions (7.2.3) in the hyperbolic regions issuing from a given state u0 = (v0 , w0 ) can be determined as follows. Eliminating s in the Rankine-Hugoniot relations (7.2.5), we obtain the Hugoniot set issuing from u0 , which consists of all the states u such that (v − v0 )2 + (σ (w) − σ (w0 ))(w − w0 ) = 0. Using the fact that the Hugoniot curve H1 (u0 ) corresponding to the first characteristic field is tangent to the eigenvector r1 (u0 ) at u0 , we obtain H1 (u0 ) := (v, w) / v − v0 = c(w0 , w) (w − w0 ) , (7.2.7)

7.2 Background

171

and that the Hugoniot curve H2 (u0 ) corresponding to the second characteristic field is tangent to the eigenvector r2 (u0 ) at u0 , we obtain H2 (u0 ) := (v, w) / v − v0 = −c(w0 , w) (w − w0 ) , (7.2.8) where c is defined by (7.2.6). As before, we rely on Lax’s shock inequalities for selecting shock waves between a left-hand state u0 and a right-hand state u1 in strictly hyperbolic regions, corresponding to Phases I and II:

λi (u0 ) > s(w0 , w1 ) > λi (u1 ),

i = 1, 2.

(7.2.9)

Observe that the simpler form (7.2.9) of Lax’s shock inequalities is equivalent to the original form introduced in Chapter 3 since the two eigenvalues have opposite signs in strictly hyperbolic regions. The conditions (7.2.9) can be expressed as

σ (w1 ) − σ (w0 ) < σ (w1 ), w1 − w0 σ (w1 ) − σ (w0 ) σ (w0 ) > > σ (w1 ), w1 − w0 σ (w0 )

−1 there exists a unique line that passes through the point with coordinates (w, σ (w)) and is tangent to the graph at a point ϕ (w), σ (ϕ (w) with ϕ (w) = w. These values satisfy the equation

σ (w) − σ ϕ (w) for w >= 0. (7.2.12) σ ϕ (w) = w − ϕ (w)

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The tangent line to the graph at w = 0 is the one drawing from it to the graph, so we set ϕ (0) = 0. Note that wϕ (w) < 0, w = 0. It is not difficult to see that under the hypotheses (7.2.2), the function ϕ : (−1, +∞) → (−1, +∞) is strictly decreasing and onto. The inverse function of ϕ is denoted by ϕ − , see Figure 7.3. We can check that

Fig. 7.3 The tangent functions ϕ and ϕ − to the graph of the stress function

w ϕ − (w) ≤ w ϕ (w) ≤ 0 and

for all w ∈ (−1, +∞),

ϕ − (0) = 0.

Furthermore, it will be useful to consider points on the graph of the stress functions having the same stress value. In view of (7.2.2)-(7.2.3), there exist unique points a− and b− satisfying b− < a < 0 < b < a− and

σ (a) = σ (a− )

and

σ (b) = σ (b− ).

(7.2.13)

Any horizontal line z = z0 for σ (a) > z0 > σ (b) will cut the graph of σ at three distinct points: each point on each Phase I and Phase II, and another point in the elliptic region a < w < b. The shock connecting the two points on such the same horizontal line has a zero-shock speed, see Figure 7.4. It will be helpful to define a “zero-shock speed” function as follows. For each point w ∈ [b− , a] ∪ [b, a− ], there corresponds a unique point ϕ0 (w) ∈ [b− , a] ∪ [b, a− ] such that wϕ0 (w) < 0, σ (w) = σ (ϕ0 (w)). This means that the horizontal line through the point (w, σ (w)) in a phase cuts the graph of the stress function at another point (ϕ0 (w), σ (w)) in the other phase, see Figure 7.4. At the limiting cases, one can see that

7.2 Background

173

Fig. 7.4 The zero-speed shock between w and ϕ0 (w)

ϕ0 (a) = a− ,

ϕ0 (b) = b− .

This determines a bijection

ϕ0 (w) : [b− , a] ∪ [b, a− ] → [b− , a] ∪ [b, a− ], σ (ϕ0 (w)) = σ (w), wϕ0 (w) < 0.

(7.2.14)

7.2.4 Lax shocks and rarefaction waves Relying on the geometry, we can now determine the parts of Lax shocks as follows. The set of all right-hand states that can be connected to a given left-hand state uL by a Lax shock associated with the first characteristic field corresponds to the following two disjoint curves:

-If wL > 0 : S1F (uL ) = u = (v, w) ∈ H1 (uL ) : w < ϕ − (wL ) or w > wL ,

-If wL < 0 : S1F (uL ) = u = (v, w) ∈ H1 (uL ) : w > ϕ − (wL ) or w < wL . (7.2.15) The set of all left-hand states that can be connected to a given right-hand state uR by a Lax shock associated with the second characteristic field corresponds to the following two disjoint curves:

-If wR > 0 : S2B (uR ) = u = (v, w) ∈ H2 (uR ) : w < ϕ − (wR ) or w > wR ,

-If wR < 0 : S2B (uR ) = u = (v, w) ∈ H2 (uR ) : w > ϕ − (wR ) or w < wR . (7.2.16) That the wave speed must be strictly increasing inside a rarefaction fan yields the parts for rarefaction waves as follows. The forward 1-rarefaction waves connecting

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a given left-hand state uL to a right-hand state u = (v, w) cover the following section of the 1-wave curve:

-If wL ≥ b : R1F (uL ) = (v, w) ∈ O1 (uL ) : b ≤ w ≤ wL ,

(7.2.17) -If wL ≤ a : R1F (uL ) = (v, w) ∈ O1 (uL ) : wL ≤ w ≤ a . The backward 2-rarefaction waves connecting a given right-hand state uR to a left-hand state u = (v, w) cover the following section of the 2-wave curve:

-If wR ≥ b : R2B (uR ) = (v, w) ∈ O2 (uR ) : b ≤ w ≤ wR ,

(7.2.18) -If wR ≤ a : R2B (uR ) = (v, w) ∈ O2 (uR ) : wR ≤ w ≤ a .

7.3 Admissible shocks satisfying an entropy inequality 7.3.1 The entropy inequality In the following, we constrain the weak solutions to satisfy the entropy inequality

∂t U(u) + ∂x F(u) ≤ 0, v2 U(v, w) := + 2

w 0

σ (z) dz,

F(v, w) = −vσ (w).

(7.3.1)

Observe that (U, F) is no longer a convex entropy pair. Definition 7.3.1. (i) An admissible non-Lax shock is a shock which satisfies the entropy condition (7.3.1), but violates Lax’s shock inequalities (7.2.9); (ii) Furthermore, if both left-hand state u0 and right-hand state u1 of an i-admissible non-Lax shock belong to the different phases, it is calleda subsonic phase boundary, denoted by Pisub (u0 , u1 ), i = 1, 2; (iii) A subsonic phase boundary Pisub (u0 , u1 ) with zero-shock speed s(u0 , u1 ) = 0 is called a stationary phase boundary from u0 to u1 , and is denoted by Z(u0 , u1 ), i = 1, 2.

Notation We will use the following notation: (i) Wi (u1 , u2 ): An i-wave connecting a state u1 and a state u2 , where W = S: admissible shock, W = SL : Lax shock, W = SN : non-Lax shock, W = R: rarefaction

7.3 Admissible shocks satisfying an entropy inequality

175

super

wave W = Pi (u0 , u1 ): supersonic phase boundary, W = Pisub (u0 , u1 ): subsonic phase boundary, i = 1, 2; (ii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the left-hand state u1 to the right-hand state u2 is followed by a j-wave from the left-hand state u2 to the right-hand state u3 ; (iii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the right-hand state u1 to the left-hand state u2 is preceded by a j-wave from the right-hand state u2 to the left-hand state u3 . Let us now investigate the properties of the entropy inequality (7.3.1). First, from the general framework of hyperbolic systems of conservation laws, we can see that the entropy inequality (7.3.1) is equivalent to the condition −s[U(u)] + [F(u)] ≤ 0, or E(u0 ; u1 ) ≤ 0, where E(u0 ; u1 ), called the entropy dissipation, is given by E(u0 ; u1 ) := −s

v2 − v2 1

0

2

+

w1 w0

σ (w) dw − v1 σ (w1 ) + v0 σ (w0 ).

The Rankine-Hugoniot relations (7.2.5) lead us to the simpler expression E(v0 , w0 ; v1 , w1 ) = −s

w1

w0

σ (w) dw −

σ (w1 ) + σ (w0 ) (w1 − w0 ) , 2

so that E is independent of v0 and v1 . In the following, we simply write E = E(w0 , w1 ). Thus, to fix a left-hand state u0 , we need to determine all the values u1 such that

σ (w1 ) + σ (w0 ) (w1 − w0 ) 2 w0 = −s(w0 , w1 ) G(w0 , w1 ) ≤ 0,

E(w0 , w1 ) = −s

w1

σ (w) dw −

(7.3.2)

where G(w0 , w1 ) :=

w1 w0

σ (w) dw −

σ (w1 ) + σ (w0 ) (w1 − w0 ). 2

(7.3.3)

It is not difficult to check that G is the area limited by the graph of σ and the line connecting the points with coordinates w0 and w1 . is As seen in the previous chapter, the function G(w0 , w1 ) defined by (7.3.3)

strictly increasing in −1, ϕ (w0 ) and strictly decreasing in ϕ (w0 ), +∞ . Moreover, we find

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∂w1 G(w0 , .) > 0 ∂w1 G(w0 , .) < 0

G w0 , w0 = 0,

G w0 , ϕ (w0 ) > 0,

in the interval −1, ϕ (w0 ) ,

in the interval ϕ (w0 ), +∞ ,

G w0 , ϕ − (w0 ) < 0.

Therefore, there exists a value ϕ∞ (w0 ) satisfying G(w0 , ϕ∞ (w0 )) = 0,

w0 ϕ∞ (w0 ) ∈ w0 ϕ − (w0 ), w0 ϕ (w0 ) .

(7.3.4)

Moreover, the function ϕ∞ : (−1, +∞) → (−1, +∞) is strictly decreasing (as are both functions ϕ and ϕ − ) d ϕ∞ (w) < 0, dw

for all w ∈ (−1, +∞).

The function ϕ∞ defined in (7.3.4) is of class C1 (by the implicit function theorem), strictly decreasing, and is its own inverse:

ϕ∞ (ϕ∞ (w)) = w,

w ∈ (−1, +∞).

The last property is a direct consequence of the skew-symmetry of G G(ϕ∞ (w), w) = −G(w, ϕ∞ (w)) = 0, and of

G(ϕ∞ (w), ϕ∞ (ϕ∞ (w))) = 0.

Since the shock speed may vanish, E may have three roots instead of two as in the hyperbolic model of the previous chapter. Since s2 =

σ (w1 ) − σ (w0 ) ≥ 0, w1 − w0

the value w1 has to fall out of the open interval between ϕ0 (w0 ) and w0 . On the other hand, as seen above, the value w1 = ϕ∞ (w0 ) is a root of G and it can be a root of E if it falls outside the open interval between ϕ0 (w0 ) and w0 . Thus, we need to investigate when ϕ∞ and ϕ0 coincide, i.e., we find w such that

ϕ∞ (w) = ϕ0 (w).

(7.3.5)

In the following, we will show that (7.3.5) admits exactly two roots w = c ∈ (b, a− ), and w = ϕ0 (c) ∈ (b− , a). Indeed, assume first that w > 0. As seen earlier, the function G(w0 , w1 ) vanishes exactly when w1 = w0 or w1 = ϕ∞ (w0 ). Hence, it is sufficient for our purpose to check that the equation G(w, ϕ0 (w)) = 0 has exactly one root w = c in the interval (b, a− ). Set

7.3 Admissible shocks satisfying an entropy inequality

g(w) := G(w, ϕ0 (w)) =

w ϕ0 (w)

σ (τ ) d τ − σ (w) (w − ϕ0 (w)),

177

w ∈ [b, a− ].

Clearly, the function g is strictly increasing, and we have g(b) < 0 and g(a− ) > 0. Consequently, there exists a unique value w = c ∈ (b, a− ) such that G(c, ϕ0 (c)) = g(c) = 0. Then, we have ϕ0 (c) = ϕ∞ (c). By symmetry, w = ϕ0 (c) ∈ (b− , a) is also a root of (7.3.5). The case for w ≤ 0 is similarly argued. Moreover, by the monotonicity properties of the function ϕ∞ − ϕ0 on each of the intervals [b− , a] and [b, a− ], we also have ϕ∞ (w) = ϕ0 (w), for w = c ∈ (b, a− ), w = ϕ0 (c),

ϕ∞ (w) < ϕ0 (w),

for w ∈ (ϕ0 (c), a] ∪ [c, a− ),

ϕ∞ (w) > ϕ0 (w),

for w ∈ [b− , ϕ0 (c)] ∪ [a, c).

7.3.2 Properties of entropy dissipation and the set of admissible shocks Properties of the entropy dissipation E are given in the following theorem. We state the results for 1-shocks only, since the results can similarly be made for 2-shocks. Theorem 7.3.1. Consider the entropy dissipation function E = E(w0 , w1 ) associated with 1-shock waves and restrict attention to values w0 ≥ b. (i) If E(w the entropy dissipation 0 , w1 ) is strictly increasing in

w0 ∈ [b, c), −1, ϕ0 (w0 ) , strictly decreasing in b, +∞ , and vanishes at exactly two points, i.e., E(w0 , ϕ0 (w0 )) = E(w0 , w0 ) = 0. (ii) If w0 ∈ [c, a− ], the entropy dissipation E(w0 , w1 ) is strictly increasing in −1, θ (w0 ) , strictly decreasing in [θ (w0 ), ϕ0 (w0 )] and in b, +∞ , and vanishes at exactly three points with, in particular, E(w0 , θ (w0 )) > E(w0 , ϕ0 (w0 )) = E(w0 , ϕ∞ (w0 )) = E(w0 , w0 ) = 0.

(iii) If w0 > a− , the entropy dissipation E(w0 , w1 ) is strictly increasing in −1, ϕ∞ (w0 ) , strictly decreasing in b, +∞ , and vanishes at exactly two points, i.e., E(w0 , ϕ∞ (w0 )) = E(w0 , w0 ) = 0. For w0 ≤ a, the same properties hold provided we replace a, a− , b and c with b, b− , a, and ϕ0 (c), respectively.

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Proof. We already know that w1 = w0 , ϕ0 (w0 ), or ϕ∞ (w0 ) are the roots of E. However, only values remaining in the intervals of interest are relevant. The following arguments focus on deriving the monotonicity of E. First of all, a simple calculation yields ∂ E(w0 , w1 ) 1 w1 − w0 1 = P(w0 , w1 ) Q(w0 , w1 ), (7.3.6) ∂ w1 2 σ (w1 ) − σ (w0 ) (w − w0 )2 where

P(w0 , w1 ) := σ (w1 ) − σ (w0 ) − σ (w1 ) (w1 − w0 )

and Q(w0 , w1 ) := 2

w0 w1

σ (w)dw + (3σ (w1 ) − σ (w0 ))(w1 − w0 ).

Therefore, the sign of E depends on the signs of P and Q, which we now investigate. For the function P, we find

∂P (w0 , w1 ) = −σ (w1 ) (w1 − w0 ) > 0 ∂ w1

iff

0 < w1 < w0 .

Clearly, from the definition of ϕ (w0 ), we get P(w0 , ϕ (w0 )) = P(w0 , w0 ) = 0. Hence, the previous inequality gives us P(w0 , w1 ) < 0

for all w1 > ϕ (w0 ),

w1 = w0 ,

P(w0 , w1 ) > 0

for all w1 < ϕ (w0 ),

where ϕ (w0 ) < 0.

(7.3.7)

For the function Q, we have ∂Q σ (w1 ) − σ (w0 ) > 0 if and only if w1 > w0 . (w0 , w1 ) = (w1 −w0 ) 3 σ (w1 )+ ∂ w1 w1 − w0 (7.3.8) In the following, we study the entropy dissipation E in each interval: w1 ∈ [b, +∞) and w1 ∈ (−1, a]. First, we consider the case w1 ∈ [b, +∞). In view of (7.3.8), the function Q achieves a strict minimum in this interval at the point w0 , i.e., Q(w0 , w1 ) > Q(w0 , w0 ) = 0

for all w1 = w0 .

In view of (7.3.7) and the latter inequality, (7.3.6) yields

∂ E(w0 , w1 ) 0

for all w1 = ϕ0 (w0 ).

By (7.3.7) and the above inequality, (7.3.6) gives

∂ E(w0 , w1 ) >0 ∂ w1

for all w1 = ϕ0 (w0 ),

which establishes that E is strictly increasing. Third, consider w1 ∈ (−1, a] and w0 ∈ [c, a− ]. The entropy dissipation E is welldefined only if w1 ∈ (−1, ϕ0 (w0 )]. Then we have ϕ (w0 ) ∈ [a, 0). It is easy to see that ϕ∞ (w0 ) ≤ ϕ0 (w0 ). And from (7.3.8), it follows that min

w1 ∈(−1,ϕ0 (w0 )]

Q(w0 , w1 ) = Q(w0 , ϕ0 (w0 )) = G(w0 ) < 0.

(7.3.10)

One can easily verify that Q(w0 , w1 ) → +∞

as w1 → −1.

(7.3.11)

Thus, from (7.3.8), (7.3.10), and (7.3.11), we deduce that there exists a unique point, depending on w0 and denoted by θ (w0 ) ∈ (−1, ϕ0 (w0 )) such that Q(w0 , w1 ) = 0 Q(w0 , w1 ) > 0

w1 = θ (w0 ), for all w1 ∈ (−1, θ (w0 )),

Q(w0 , w1 ) < 0

for all w1 ∈ (θ (w0 ), ϕ0 (w0 )).

(7.3.12)

Combining (7.3.7) and (7.3.12) proves the monotonicity of E. Finally, consider w1 ∈ (−1, a] and w0 > a− . In this case, the function Q is strictly decreasing in the interval (−1, a], and thus achieves a strict minimum value at w1 = a, i.e., Q(w0 , w1 ) > Q(w0 , a) for all w1 = w0 . One needs to investigate the sign of Q(w0 , a). It is not difficult to check that the function w → Q(w, a) is strictly increasing in the interval w ∈ [a− , +∞), that Q(w, a) → +∞ as w → +∞, and that Q(a− , a) = 2

a

a−

σ (w)dw − σ (a)(a− − a) < 0.

Thus, there exists a unique root w = d ∈ (a− , +∞) of Q(w, a)

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Q(d, a) = 0. From this observation, we see that, if w0 ∈ [a− , d), then Q(w0 , a) < 0, and therefore, there exists a unique value, still denoted by θ (w0 ) such that Q(w0 , w1 ) = 0

for w1 = θ (w0 ),

Q(w0 , w1 ) > 0 Q(w0 , w1 ) < 0

for all w1 ∈ (−1, θ (w0 )), for all w1 ∈ (θ (w0 ), ϕ0 (w0 )).

(7.3.13)

From (7.3.7) and (7.3.13), one obtains

∂ E(w0 , w1 ) >0 ∂ w1 ∂ E(w0 , w1 ) ≤0 ∂ w1

for all w1 ∈ min{ϕ (w0 ), θ (w0 )}, max{ϕ (w0 ), θ (w0 )} , elsewhere.

This yields the monotonicity property of E, as claimed. Theorem 7.3.1 provides us with a complete description of the sign of the entropy dissipation, where the two functions ϕ∞ and ϕ0 play a central role, as they correspond to the (non-trivial) roots of E. However, the points corresponding to w ∈ (ϕ0 (w0 ), ϕ∞ (w0 )) cannot be reached, since the associated shock speed is not defined. So, it seems natural to introduce the following (continuous) function which determines the attainable region: ϕ∞ (w) if w ∈ (−1, ϕ0 (c)] ∪ [c, +∞), ϕ∞,0 (w) = (7.3.14) ϕ0 (w) if w ∈ (ϕ0 (c), a] ∪ [b, c). It follows from Theorem 7.3.1 that right-hand states u = (v, w) that can be reached by a 1-shock from a given left-hand state u0 = (v0 , w0 ) are given by − If w0 ≥ 0 :

w ≤ ϕ∞,0 (w0 ),

w ≥ w0 ,

or

w1 = ϕ0 (w0 ),

− If w0 < 0 :

w ≥ ϕ∞,0 (w0 ),

w ≤ w0 ,

or

w1 = ϕ0 (w0 ).

(7.3.15)

Similarly, left-hand states u = (v, w) that can be reached by a 2-shock from a given right-hand state u0 = (v0 , w0 ) are given by − If w0 ≥ 0 :

w ≤ ϕ∞,0 (w0 ),

w ≥ w0 ,

or

w1 = ϕ0 (w0 ),

− If w0 < 0 :

w ≥ ϕ∞,0 (w0 ),

w ≤ w0 ,

or

w1 = ϕ0 (w0 ).

(7.3.16)

7.3 Admissible shocks satisfying an entropy inequality

181

7.3.3 Subsonic phase boundaries and composite waves Furthermore, subsonic phase boundaries can be characterized as follows. The righthand states u = (v, w) that can be reached by a subsonic 1-phase boundary from a given left-hand state u0 = (v0 , w0 ) are given by − If w0 ≥ 0 : − If w0 < 0 :

ϕ − (w0 ) ≤ w ≤ ϕ∞,0 (w0 ),

ϕ

−

(w0 ) ≥ w ≥ ϕ∞,0 (w0 ),

w ≥ w0 ,

or

w1 = ϕ0 (w0 ),

w ≤ w0 ,

or

w1 = ϕ0 (w0 ). (7.3.17)

Similarly, the left-hand states u = (v, w) that can be reached by a subsonic 2-phase boundary from a given right-hand state u0 = (v0 , w0 ) are given by − If w0 ≥ 0 : − If w0 < 0 :

ϕ − (w0 ) ≤ w ≤ ϕ∞,0 (w0 ),

ϕ

−

(w0 ) ≥ w ≥ ϕ∞,0 (w0 ),

w ≥ w0 ,

or

w1 = ϕ0 (w0 ),

w ≤ w0 ,

or

w1 = ϕ0 (w0 ). (7.3.18)

It will be convenient to introduce the continuous mapping ϕ : (w∗ , w) ∈ (−1, +∞)2 → ϕ (w∗ , w) ∈ (−1, +∞) defined for w∗ = w, ϕ (w), ϕ − (w) by

σ (w) − σ (w∗ ) σ (ϕ (w∗ , w)) − σ (w∗ ) = , w − w∗ ϕ (w∗ , w) − w∗

(7.3.19)

and extended by continuity, see Figure 7.5. This means that the points corresponding to w, w∗ , and ϕ (w∗ , w) on the graph of f belong to the same straight line. Shock waves between any two of the three states corresponding to these w-values would have the same speed.

Fig. 7.5 The function ϕ : The shock speeds s(w, ϕ (w∗ , w)) and s(ϕ (w∗ , w), w∗ ) are the same

Let us consider composite waves by combining different waves in the same characteristic field. We claim that a rarefaction wave connecting u0 to some u1 =

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(v1 , w1 ) ∈ O1 (u0 ), with of course 0 ≤ w1 w0 ≤ w20 , can be followed with a shock wave connecting u1 to some point u2 = (v2 , w2 ) ∈ H1 (u1 ), provided we have the following inequalities: w1 ϕ − (w1 ) ≤ w1 w2 ≤ w1 ϕ∞,0 (w1 ),

(7.3.20)

or else, when w1 ∈ [b, a− ], the second wave can be a stationary phase boundary. Indeed, we restrict attention to the case w0 ≥ b. The condition (7.3.20) re-writes as ϕ − (w1 ) ≤ w2 ≤ ϕ∞,0 (w1 ). (7.3.21) On one hand, the assumption u2 ∈ H1 (u1 ) with w2 ≤ 0 yields −1 < w2 ≤ ϕ∞,0 (w1 ).

(7.3.22)

On the other hand, the constrain that the shock from (v1 , w1 ) to (v2 , w2 ) must follow the rarefaction wave connecting u0 to u1 implies that the characteristic speeds in the rarefaction fan be smaller than the shock speed s1 (w1 , w2 ). In other words, we must have −c(w1 , w2 ) ≥ −c(w1 ) or, equivalently,

σ (w1 ) (w1 − w2 ) − σ (w1 ) + σ (w2 ) ≥ 0.

Thanks to of (7.3.12), the last inequality holds true provided w1 ≥ ϕ (w2 ). Since the function ϕ − is decreasing, we get w2 ≥ ϕ − (w1 ).

(7.3.23)

From (7.3.22) and (7.3.23) we obtain (7.3.21). Furthermore, a Lax shock connecting the left-hand state u0 to u1 = (v1 , w1 ) ∈ H1 (u0 ), with w0 w1 ≥ w20 , can be followed with a subsonic 1-phase boundaries connecting to some u2 = (v2 , w2 ) ∈ H1 (u1 ), provided w0 ϕ (w1 , w0 ) < w0 w2 < w0 ϕ∞,0 (w1 ).

(7.3.24)

Actually, assume for simplicity that w0 ≥ b, so we have w1 ≥ w0 . The inequalities (7.3.24) become ϕ (w1 , w0 ) ≤ w2 ≤ ϕ∞,0 (w1 ). On one hand, in view of (7.3.15), the condition u2 ∈ H1 (u1 ) imposes w2 ≤ ϕ∞,0 (w1 ).

On the other hand, in order for the subsonic phase boundary to follow the Lax shock, we need s1 (w0 , w1 ) ≤ s1 (w1 , w2 ), that is w2 ≥ ϕ (w1 , w0 ).

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7.3.4 Two-parameter sets of waves We can now construct a two-parameter set of waves associated with the same characteristic field. The set of all right-hand states u = (v, w) that can be attained from a given u0 = (v0 , w) by combining Lax shocks, rarefaction fans, subsonic (including stationary) phase boundaries, and supersonic phase boundaries of the 1-wave family are described as follows. First, if w > w0 , the solution is a single Lax shock S1L (u0 , u). Second, if w ∈ [b, w0 ), the solution consists of a rarefaction wave from u1 to u: R1 (u0 , u1 ). Third, if w ∈ [ϕ0 (c), a], then the solution begins either with a Lax shock if wL < ϕ0 (w), or a rarefaction wave, if wL ≥ ϕ0 (w) from u0 to u1 = (v1 , w1 = ϕ0 (w)), followed by a stationary phase boundary from u1 to u: S1L (u0 , u1 ) Z(u1 , u), R1 (u0 , u1 ) Z(u1 , u),

if wL < ϕ0 (w), if wL ≥ ϕ0 (w).

Fourth, if ϕ − (w0 ) ≤ w < ϕ0 (c), then there exists a one-parameter family of admissible solutions. The solution can begin with either a rarefaction wave from u0 to some intermediate state u1 = (v1 , w1 ), w1 ≤ w0 , or a Lax shock from u0 to u1 , w1 > w0 , (w)] followed by a subsonic phase boundary from u1 to u for either w1 ∈ [ϕ0 (w), ϕ∞,0 − − if w ≥ b , or w1 ∈ [ϕ (w), ϕ∞,0 (w)] if w ≥ b : R1 (u0 , u1 ) P1sub (u1 , u)or

S1L (u0 , u1 ) P1sub (u1 , u).

Finally, if w < ϕ − (w0 ), there exists also a one-parameter family of admissible solutions. The solution can begin with a Lax shock from u0 to some state u1 = (w )], followed by a subsonic phase boundary from (v1 , w1 ), w1 ∈ (ϕ (w, w0 ), ϕ∞,0 0 u1 to u: S1L (u0 , u1 ) P1sub (u1 , u). The solution can be a single supersonic phase boundary from u0 to u: super

P1

(u1 , u).

Similarly, for any given right-hand state u0 , we can always construct a twoparameter set of 2-waves consisting of left-hand states u that can be attained from u0 by 2-waves. Precisely, we can describe such a construction as follows. As usual, we will construct the wave set only for the case w0 > 0, since the case w0 ≤ 0 can be done similarly. First, if w > w0 , the solution is a single Lax shock:

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S2L (u0 , 0). Second, if b ≤ w ≤ w0 , the solution is a single rarefaction wave from u0 to u: R2 (u0 , u). Third, if w ∈ [ϕ0 (c), a], then the solution begins either with a Lax shock if wL < ϕ0 (w), or a rarefaction wave, if wL ≥ ϕ0 (w) from u0 to u1 = (v1 , w1 = ϕ0 (w)), preceded by a stationary phase boundary from u1 to u: S2L (u0 , u1 ) Z(u1 , u), R2 (u0 , u1 ) Z(u1 , u),

if wL < ϕ0 (w), if wL ≥ ϕ0 (w).

Fourth, if ϕ − (w0 ) ≤ w < ϕ0 (c), then there exists a one-parameter family of admissible solutions. The solution can begin with either a rarefaction wave from u0 to some intermediate state u1 = (v1 , w1 ), w1 ≤ w0 , or a Lax shock from u0 to u1 , w1 > w0 , followed by a subsonic phase boundary from u1 to u for either w1 ∈ (w)] if w ≥ b− , or w ∈ [ϕ (w), ϕ (w)] if w ≥ b− : [ϕ0 (w), ϕ∞,0 1 ∞,0 R2 (u0 , u1 ) P2sub (u1 , u)or

S2L (u0 , u1 ) P2sub (u1 , u).

Finally, if w < ϕ − (w0 ), there exists also a one-parameter family of admissible solutions. The solution can begin with a Lax shock from u0 to some state u1 = (w )], preceded by a subsonic phase boundary from (v1 , w1 ), w1 ∈ (ϕ (w, w0 ), ϕ∞,0 0 u1 to u: S2L (u0 , u1 ) P2sub (u1 , u). The solution can be a single supersonic phase boundary from u0 to u: super

P2

(u1 , u).

7.4 Riemann solver based on a kinetic relation 7.4.1 Kinetic relation imposed from back side to front side of the shock Let ϕ ,i : (−1, +∞) → (−1, +∞) be locally Lipschitz continuous and strictly decreasing kinetic functions satisfying w ϕ∞ (w) ≤ w ϕ ,i (w) ≤ w ϕ (w),

w ∈ (−1, ∞),

i = 1, 2.

(7.4.1)

Their inverse functions, denoted by ϕ −,i : (−1, ∞) → (−1, ∞), i = 1, 2, are strictly decreasing functions as well. It follows from (7.4.1) that

7.4 Riemann solver based on a kinetic relation

w ϕ − (w) ≤ w ϕ −,i (w) ≤ w ϕ∞ (w),

185

w ∈ (−1, ∞).

(7.4.2)

We now formulate the following kinetic relation for subsonic phase boundaries: The state in the front side of the subsonic phase boundary can be attained by the kinetic function at the value on its back side. Precisely, any subsonic 1-phase boundary connecting a left-hand state (v0 , w0 ) to some right-hand state (v1 , w1 ) should satisfy (7.4.3) w0 = ϕ ,1 (w1 ). Similarly, for subsonic 2-phase boundaries, we require that any subsonic 2-phase boundary connecting a left-hand state (v0 , w0 ) to some right-hand state (v1 , w1 ) should satisfy (7.4.4) w1 = ϕ ,2 (w0 ). We postulate that subsonic phase boundaries are preferred by imposing the following condition: (C) The Riemann solution always uses non-stationary subsonic phase boundaries whenever available. Since the inverses of the kinetic functions ϕ −,i , i = 1, 2, are decreasing, the zeroshock speed function ϕ0 is increasing on the interval [b− , a], and by (7.4.2)

ϕ0 (b− ) = b = ϕ (b− ) ≤ ϕ∞ (b− ) ≤ ϕ −,i (b− ), ϕ0 (a) = a− = ϕ − (a) ≥ ϕ −,i (a), the continuity implies that there exists a unique value, denoted by αi in [b− , a] such that ϕ −,i (αi ) = ϕ0 (αi ), i = 1, 2. (7.4.5) Similarly, since the inverses of the kinetic functions ϕ −,i , i = 1, 2, are decreasing, the zero-shock speed function ϕ0 is increasing on the interval [b, a− ], and

ϕ0 (a− ) = a = ϕ (a− ) ≥ ϕ∞ (a− ) ≥ ϕ −,i (a− ), ϕ0 (b) = b− = ϕ − (b) ≤ ϕ −,i (b), the continuity implies that there exists a unique value, denoted by βi in [b, a− ] such that ϕ −,i (βi ) = ϕ0 (βi ), i = 1, 2. (7.4.6)

7.4.2 Construction of wave curves Under the condition (C), we can now construct wave curves. We will indicate that all the wave curves can be parameterized by the form w → v = v(w), w ∈ (−1, a] ∪ [b, ∞), as follows, see Figure 7.6.

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Forward 1-wave curve W1F (u0 ) Let us denote by W1F (u0 ) the forward 1-wave curve consisting of all right-hand states u = (v, w) that can be reached by using single waves or composite waves associated with the first characteristic field: Lax shocks, rarefaction waves, supersonic phase boundaries, subsonic phase boundaries, and stationary phase boundaries. We will describe the wave curve only for w0 > 0 since the argument for the case w0 ≤ 0 is similar. The forward 1-wave curve W1F (u0 ), for any given left-hand state u0 , can be parameterized by w → v = v1 (u0 ; w), w ∈ (−1, a] ∪ [b, ∞), as follows, see Figure 7.6. First, if w ≥ w0 , then the solution is a Lax shock from uL to u: S1L (u0 , u). Second, if w ∈ [b, w0 ), the solution is a rarefaction wave connecting u0 to u: R1 (u0 , u). Third, if w ∈ [α1 , a], then the solution consists of either a Lax shock if ϕ0 (w) > w0 , or a rarefaction if ϕ0 (w) ≤ w0 , from u0 to u1 = (v1 , w1 = ϕ0 (w)), followed by a stationary phase boundary from u1 to u: S1L (u0 , u1 ) Z(u1 , u), R1 (u0 , u1 ) Z(u1 , u),

if ϕ0 (w) > w0 , if ϕ0 (w) ≤ w0 .

Fourth, if w ∈ [ϕ −,1 (w0 ), α1 ), the solution consists of a rarefaction wave from u0 to u1 = (v1 , w1 = ϕ ,1 (w0 )), followed by a subsonic phase boundary from u1 to u: R1 (u0 , u1 ) P1sub (u1 , u). Fifth, if w ∈ (ϕ − (w0 ), ϕ −,1 (w0 )), then the solution is a Lax shock from u0 to u1 = (v1 , w1 = ϕ ,1 (w0 )), followed by a subsonic phase boundary from u1 to u: S1L (u0 , u1 ) P1sub (u1 , u). Finally, if w ≤ ϕ − (w0 ), then the solution can begin either with a Lax shock from u0 to u1 = (v1 , w1 = ϕ ,1 (w)), followed by a subsonic phase boundary from u1 to u, if ϕ ,1 (w) > ϕ (w0 , w), or a supersonic phase boundary from u0 directly to u, otherwise S1L (u0 , u1 ) P1sub (u1 , u), super

P1

(u1 , u),

if ϕ ,1 (w) > ϕ (w0 , w), if ϕ ,1 (w) ≤ ϕ (w0 , w).

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Fig. 7.6 The forward curve of 1-waves W1F (u0 )

Backward 2-wave curve W2B (u0 ) Let us denote by W2B (u0 ) the backward 2-wave curve consisting of all left-hand states u = (v, w) that can be reached by using single waves or composite waves associated with the second characteristic field: Lax shocks, rarefaction waves, supersonic phase boundaries, subsonic phase boundaries, and stationary phase boundaries. Let us describe the wave curve only for w0 > 0, since the argument for the case w0 ≤ 0 is similar. We are going to show that the backward 2-wave curve W2B (u0 ), for any given right-hand state u0 , can be parameterized by w → v = v2 (u0 ; w), w ∈ (−1, a] ∪ [b, ∞), see Figure 7.7. First, if w ≥ w0 , then the solution is a Lax shock from uL to u: S2L (u0 , u). Second, if w ∈ [b, w0 ), the solution is a rarefaction wave connecting u0 to u: R2 (u0 , u). Third, if w ∈ [α2 , a], then the solution consists of either a Lax shock if ϕ0 (w) > w0 , or a rarefaction if ϕ0 (w) ≤ w0 , from u0 to u1 = (v1 , w1 = ϕ0 (w)), followed by a stationary phase boundary from u1 to u: S2L (u0 , u1 ) Z(u1 , u), R2 (u0 , u1 ) Z(u1 , u),

if ϕ0 (w) > w0 , if ϕ0 (w) ≤ w0 .

Fourth, if w ∈ [ϕ −,1 (w0 ), α2 ), the solution consists of a rarefaction wave from u0 to u1 = (v1 , w1 = ϕ ,1 (w0 )), preceded by a subsonic phase boundary from u1 to u: R2 (u0 , u1 ) P2sub (u1 , u).

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Fifth, if w ∈ (ϕ − (w0 ), ϕ −,1 (w0 )), then the solution is a Lax shock from u0 to u1 = (v1 , w1 = ϕ ,1 (w0 )), preceded by a subsonic phase boundary from u1 to u: S2L (u0 , u1 ) P2sub (u1 , u). Finally, if w ≤ ϕ − (w0 ), then the solution can begin either with a Lax shock from u0 to u1 = (v1 , w1 = ϕ ,1 (w)), preceded by a subsonic phase boundary from u1 to u, if ϕ ,1 (w) > ϕ (w0 , w), or a supersonic phase boundary from u0 directly to u, otherwise: S2L (u0 , u1 ) P2sub (u1 , u), super

P2

if ϕ ,1 (w) > ϕ (w0 , w), if ϕ ,1 (w) ≤ ϕ (w0 , w).

(u1 , u),

Fig. 7.7 The backward curve of 2-waves W2B (u0 )

The following theorem provides us with properties of the wave curves constructed above. Theorem 7.4.1. The forward 1-wave curve W1F (u0 ) can be parameterized as v = v1 (u0 , w), w ∈ (−1, a] ∪ [b, ∞), which is a continuous, strictly increasing function and satisfies lim v1 (u0 , w) = ∞. lim v1 (u0 , w) = −∞, w→∞

w→−1

W2B (u0 )

The backward 2-wave curve can be parameterized as v = v2 (u0 , w), w ∈ (−1, a] ∪ [b, ∞), which is a continuous, strictly decreasing function and satisfies lim v2 (u0 , w) = ∞,

w→−1

lim v2 (u0 , w) = −∞.

w→∞

Moreover, we have v1 (u0 , w) = v1 (u0 , ϕ0 (w))

for all w ∈ [α1 , a] ∪ [b, β1 ],

v2 (u0 , w) = v2 (u0 , ϕ0 (w))

for all w ∈ [α2 , a] ∪ [b, β2 ],

where αi , βi , i = 1, 2, are defined by (7.4.5) and (7.4.6), respectively.

(7.4.7)

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189

Proof. It is sufficient to prove these results for the 1-wave curve W1F (u0 ) with wL ≥ b, as the other cases can be done similarly. If w ≥ b, the solution is a Lax shock if w > w0 , and a rarefaction wave otherwise. From the formulas (7.2.7) and (7.2.10), one can see that v1 (w), w ≥ b, is increasing, and lim v1 (w) = ∞.

w→∞

Consider now α1 ≤ w ≤ a. The solution consists of either a rarefaction wave if w0 > ϕ0 (w), or a Lax shock otherwise, followed with a stationary phase boundary (w) = ϕ (w) in from ϕ0 (w) to w. By the monotonicity property of the function ϕ∞,0 0 this interval, the wave curve is strictly increasing as well. Observe that the value of v at w coincides with the one of v at ϕ0 (w). This is due to the fact that we use here stationary jumps. Therefore, (7.4.7) holds true. If w ∈ (ϕ −,1 (w0 ), α1 ), the solution consists of a rarefaction wave from w0 to ,1 ϕ (w), followed by a subsonic phase boundary connecting ϕ ,1 (w) to w. It follows from (7.2.7) and (7.2.10) that v − v(ϕ ,1 (w)) = c(ϕ ,1 (w), w) (w − ϕ ,1 (w)), v(ϕ ,1 (w)) − v0 =

ϕ ,1 (w)

(7.4.8) c(z) dz.

w0

Then, we find dv θ (w) = dw ,1 σ (ϕ (w)) − σ (w) 2 ϕ ,1 (w) − w 2 σ (ϕ ,1 (w)) − σ (w) d ϕ ,1 ,1 (w) ( σ (ϕ (w)) − θ (w) := − dw ϕ ,1 (w) − w + σ (w) +

σ (ϕ ,1 (w)) − σ (w) > 0. ϕ ,1 (w) − w

This yields the monotonicity property of the wave curve. If w ≤ ϕ −,1 (w0 ), the solution is either a composite of a Lax shock connecting u0 to u1 = (v1 , w1 = ϕ ,1 (w)) followed by a subsonic phase boundary if

ξ (w) := ϕ ,1 (w) − ϕ (w0 , w) > 0, or a single Lax shock, otherwise. Since the function ξ is continuous, the set N C := w ∈ (−1, ϕ ,1 (w0 )) / ϕ ,1 (w) − ϕ (w0 , w) > 0

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is an open set, and is therefore a (countable) union of open intervals. At an endpoint, say w , of such an interval (w , w ) ⊂ N C, the speeds of the Lax shock and subsonic phase boundaries tend to the speed of the Lax shock connecting w0 to w . Therefore, the wave curve is (at least) continuous. In each interval of the set (−1, ∞) \ N C, which corresponds to Lax shock waves, the wave curve is clearly strictly increasing. Consider now some w ∈ (w , w ) ⊂ N C. From the above construction, we have v − v(ϕ ,1 (w)) = c(ϕ ,1 (w), w)(w − ϕ ,1 (w)), v(ϕ ,1 (w)) − v0 = c(w0 , ϕ ,1 (w))(ϕ ,1 (w) − w0 ).

(7.4.9)

Then, (7.4.9) yields

θ1 θ2 σ (w) + 1 dv = + , dw 2 σ (ϕ ,1 (w)) − σ (w) 2 ϕ ,1 (w) − w σ (ϕ ,1 (w)) − σ (w0 ) σ (ϕ ,1 (w)) − σ (w) θ1 := − , ϕ ,1 (w) − w0 ϕ ,1 (w) − w θ2 := 2

(7.4.10)

σ (ϕ ,1 (w)) − 1. σ (ϕ ,1 (w)) − σ (w0 ) σ (ϕ ,1 (w)) − σ (w) ϕ ,1 (w) − w0 ϕ ,1 (w) − w

Using the condition that the shock speed is increasing, one obtains σ (ϕ ,1 (w)) − σ (w0 ) σ (ϕ ,1 (w)) − σ (w) − ≤ − . ϕ ,1 (w) − w0 ϕ ,1 (w) − w By definition, ϕ ,1 (w) ≥ w0 > ϕ (w). Then it follows from the inequality (7.3.12) that σ (ϕ ,1 (w)) − σ (w0 ) ≤ σ (ϕ ,1 (w)), ϕ ,1 (w) − w0 (7.4.11) σ (ϕ ,1 (w)) − σ (w) ,1 ≤ σ (ϕ (w)). ϕ ,1 (w) − w From (7.4.10)–(7.4.11), we deduce dv > 0, dw which means that the wave curve can be parameterized by (w, v(w)), where the velocity v = v(w) is a strictly increasing function of the gradient w. Besides, it follows from our construction that v → −∞ as w → −1. The proof of Theorem 7.4.1 is completed.

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7.4.3 Riemann solver based on a kinetic relation A nonempty intersection of the forward 1-wave curve W1F (u0 ) and the backward 2-wave curve W2F (uR ) will lead to the existence of Riemann solutions. From the previous section, we can see that they always intersect. By convention, we name W1F (u0 ) and W2F (uR ) as “curves” in an extended sense, since the domain of them is not an entire interval. Each of them consists of two disjoint curves in each phase in the usual sense. The two curves of W1F (u0 ) can be parameterized where v is a decreasing function of w. The two curves of W2B (u0 ) can be parameterized where v is a increasing function of w. Thus, the forward 1-wave curve W1F (u0 ) and the backward 2-wave curve W2F (uR ) intersect at most once in each phase. Therefore, these two wave curves always intersect and they intersect at most twice in the phase domain. This implies that the Rieman problem always admits one or two solutions. Moreover, it is interesting that even when the intersection of W1F (u0 ) and W2F (uR ) has two different points, if they are corresponding to a stationary wave, the Riemann solution is unique. In this case, if w = wm is the w-component of an intersection point in one phase, then w = ϕ0 (wm ) is the w-component of the intersection point in the other phase; the Riemann solution between wm and ϕ0 (wm ) is just a stationary wave, see Figure 7.10. From Theorem 7.4.1, we can obtain the following main result of this chapter.

Fig. 7.8 The wave curves intersect once giving a unique Riemann solution

Theorem 7.4.2 (General Riemann solvers by kinetic relations). Consider the Riemann problem for the model of phase transition (7.2.1)-(7.2.2) with the Riemann data uL , uR . The forward 1-wave curve W1F (uL ) and the backward 2-wave curve W2F (uR ) always intersect: the intersection consists of either a single point which leads to a unique solution of the Riemann problem, see Figure 7.8, or two points which give two Riemann solutions, see Figure 7.9. Moreover, if the two intersection points are the two states of a stationary phase boundary, then the two Riemann solutions coincide, see Figure 7.10.

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Fig. 7.9 The wave curves intersect twice resulting two distinct solutions

Fig. 7.10 The wave curves intersect twice corresponding to a unique solution

Under certain conditions, we can obtain the uniqueness of the Riemann problem as follows, by imposing additional conditions on the kinetic functions. Corollary 7.1 Suppose that the kinetic functions (7.4.1) satisfy also the restriction

ϕ ,i (a− ) = a

and

ϕ ,i (b− ) = b,

i = 1, 2.

Under condition (C), the Riemann problem (7.2.1)-(7.2.2) admits a unique solution made of Lax shocks, rarefaction fans, supersonic phase boundaries, and stationary phase boundaries, as well as subsonic phase boundaries satisfying the kinetic relations (7.4.3)-(7.4.4).

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193

7.4.4 Classical Riemann solver based on stationary phase boundaries The construction for Riemann solvers with kinetics in the previous section actually covers the classical Riemann solver by choosing the kinetic functions

ϕ ,1 = ϕ − , ϕ

,2

for 1-shocks,

=ϕ ,

for 2-shocks.

(7.4.12)

We will point out that the forward 1-wave curve W1F (u0 ) can be parameterized by v = vF1 (u0 ; w), w ∈ (−1, a] ∪ [b, ∞). Let us consider only for the case w0 ≥ b, since the argument is similar for the case −1 < w0 ≤ a. If −1 < w < ϕ − (w0 ), then the solution is a supersonic phase boundary from u0 to u: super

P1

(u0 , u).

If w > w0 , then the solution is a Lax shock from u0 to u = (v, w): S1L (u0 , u). If b ≤ w ≤ w0 , the solution is a 1-rarefaction wave: R1 (u0 , u). If w ∈ (ϕ − (w0 ), b− ), then the solution is a rarefaction wave from u0 to u1 = (v1 , w1 = ϕ (w)), followed by a supersonic phase boundary from u1 to u: super

R1 (u0 , u1 ) P1

(u1 , u).

Finally, if w ∈ [b− , a], the solution is a rarefaction wave if w0 ≥ ϕ0 (w), or a Lax shock, otherwise, from u0 to u1 = (v1 , w1 = ϕ0 (w)), followed by a stationary phase boundary: R1 (u0 , u1 ) Z(u1 , u), if w0 ≥ ϕ0 (w), S1L (u0 , u1 ) Z(u1 , u),

if b ≤ w0 < ϕ0 (w).

Next, we can also show that the backward 2-wave curve W2B (u0 ) can be parameterized by v = vB2 (u0 ; w), w ∈ (−1, a] ∪ [b, ∞). Again, we can assume that w0 ≥ b, since the argument is similar when −1 < w0 ≤ a. If −1 < w < ϕ − (w0 ), then the solution is a supersonic phase boundary from u0 to u: super

P2

(u0 , u).

If w > w0 , then the solution is a Lax shock from u0 to u = (v, w): S2L (u0 , u).

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If b ≤ w ≤ w0 , the solution is a 2-rarefaction wave: R2 (u0 , u). If w ∈ (ϕ − (w0 ), b− ), then the solution is a rarefaction wave from u0 to u1 = (v1 , w1 = ϕ (w)), followed by a supersonic phase boundary from u1 to u: super

R2 (u0 , u1 ) P2

(u1 , u).

Finally, if w ∈ [b− , a], the solution is a rarefaction wave if w0 ≥ ϕ0 (w), or a Lax shock, otherwise, from u0 to u1 = (v1 , w1 = ϕ0 (w)), followed by a stationary phase boundary: R2 (u0 , u1 ) Z(u1 , u), if w0 ≥ ϕ0 (w), S2L (u0 , u1 ) Z(u1 , u),

if b ≤ w0 < ϕ0 (w).

From Theorem 7.4.2, we can deduce the following result. Theorem 7.4.3 (Classical Riemann solver based on stationary phase boundaries). Consider the Riemann problem for the model of phase transition (7.2.1)(7.2.2) with the Riemann data data uL , uR . The forward 1-wave curve W1F (uL ) and the backward 2-wave curve W2F (uR ) corresponding to the kinetic functions (7.4.12) intersect either at a unique point, or two points corresponding to the two states of a stationary phase boundary. Consequently, the Riemann problem for (7.2.1)-(7.2.2) under the condition (C) with the choice of kinetic functions (7.4.12) admits a unique solution.

7.5 Bibliographical notes We list only a short selection of the most relevant papers for this chapter, and also refer the reader to the bibliography at the end of this monograph. For results on phase transitions in fluids and material science based on the system (7.2.1) and related models, see Abeyaratne-Knowles [2, 3] and Hayes-LeFloch [160, 162]. The hyperbolic-elliptic model (7.2.1)-(7.2.2) was treated in [225], as a generalization of works by Abeyaratne and Knowles (piecewise linear law) and Shearer and Yang [289] (cubic law). The Riemann problem for phase transition problems was also considered by Hattori-Mischaikow [158, 159], Fan-Slemrod [130], Hattori [154– 157], Asakura [23], and Chen-Hattori [91, 92], Admissibility criteria for systems of mixed type were considered first by Shearer [285] and [286], as well as Keyfitz [187], Ames-Keyfitz [18], and Keyfitz [188, 189].

Chapter 8

Compressible fluids in a nozzle with discontinuous cross-section: Isentropic flows

8.1 Introduction In this chapter, we consider the Riemann problem for the model of an isentropic fluid in a nozzle with a discontinuous cross-sectional area. The modeling for fluid flows in a nozzle with variable cross-section is given in Chapter 2. For simplicity, we assume that the fluid is isentropic and ideal. Unlike the models considered in previous chapters, this model contains a source term involving the space partial derivative in nonconservative form in the equation for the balance of momentum. This nonconservative source term represents the influence of the nozzle’s geometry on the fluid’s dynamics. However, the model can be written as a nonconservative system of balance laws, allowing for a general definition of weak solutions in terms of nonconservative products. Studying such a system may provide better understanding of more complicated nonconservative models of multi-phase flows. The model under study is given as a system consisting of an equation for conservation of mass and an equation for balance of momentum:

∂t (aρ ) + ∂x (aρ u) = 0, ∂t (aρ u) + ∂x (a(ρ u2 + p)) = p∂x a,

x∈R I , t > 0.

(8.1.1)

Here, ρ , u, and p stand for the density, the particle velocity, and the pressure of the fluid under consideration, respectively. The function a = a(x) is the cross-sectional area. In the following, we will consider the function a = a(x), x ∈ R I , to be a piecewise constant function: aL , x < 0, a(x) = aR , x > 0, where aL , aR are constants. The system (8.1.1) has a nonconservative term p∂x a on the right-hand side, making it challenging for the study. To investigate the system (8.1.1), we formally supplement it with the trivial equation © Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 8

195

196

8 Compressible fluids in a nozzle with discontinuous . . .

∂t a = 0.

(8.1.2)

Then, the supplemented system (8.1.1)–(8.1.2) can be written as a system of balance laws in nonconservative form

∂t U + A(U)∂xU = 0,

(8.1.3)

where U = (ρ , u, a)T , and A(U), as seen later, is a matrix with a complete set of eigenvectors. So, the model can be studied in the framework of systems of balance laws in nonconservative form. As we will see in the following sections, the system may not be strictly hyperbolic. Characteristic fields may coincide on certain surfaces and the order of characteristic speeds may change from one region to another. The initial-value problem is not well-posed, as it may not have a unique solution. Specifically, the Riemann problem may have multiple solutions in one region but none in another. A significant feature of this model is the appearance of stationary contact discontinuities. Unlike the usual contact discontinuities of hyperbolic systems of conservation laws, which can be obtained using Rankine-Hugoniot relations, stationary contact discontinuities are demonstrated to be weak solutions by taking the limit of stationary smooth solutions. However, they are shown to be one type of elementary wave, as their curve coincides with the integral curve associated with a linearly degenerate characteristic field. Stationary contact discontinuities represent the action of the nonconservative source term in fluid flow. Two-parameter sets of waves composed of a wave associated with a genuinely nonlinear characteristic field and a stationary wave can be constructed. As a result, the Riemann problem may admit up to a one-parameter set of solutions in certain regions. To select a unique physical solution, an additional admissibility criterion for stationary contact discontinuities is imposed. The Riemann problem may then have a unique solution or admit up to two or three different solutions. It is interesting to observe the resonant phenomenon that occurs in the latter case, where three different waves with the same speed can coexist in a solution. As usual, solutions to the Riemann problem are understood to be composed of a finite number of elementary waves, each associated with one characteristic field. An elementary wave can be either an admissible shock wave or a rarefaction wave associated with a nonlinear characteristic field or a contact discontinuity associated with a linearly degenerate characteristic field. Each Riemann solution corresponds to a set of algebraic equations that can be used for computational purposes.

8.2 Basic properties

197

8.2 Basic properties 8.2.1 Model reduced to a system in nonconservative form Consider an unknown smooth solution (ρ , u, a)T , which can be seen as a primitive variable, of the system (8.1.1)–(8.1.2). The first equation of (8.1.1) can be written in terms of the primitive variable (ρ , u, a)T as follows. Let us expand the first equation of (8.1.1) by ∂t aρ + a∂t ρ + au∂x ρ + aρ∂x u + ρ u∂x a = 0. Using the equation (8.1.2), and dividing both sides of the last equation by a = 0, we get ρu ∂t ρ + u∂x ρ + ρ∂x u + ∂x a = 0. a Next, let us expand the second equation in (8.1.1) as

∂t (aρ )u + aρ∂t u + aρ u∂x u + ∂x (aρ u)u + p∂x a + a∂x p = p∂x a. Canceling the term p∂x a on both sides, and using the first equation of (8.1.1) so that the first and the fourth terms cancel each other, we obtain from the last equation aρ∂t u + aρ u∂x u + a∂x p = 0. Dividing both sides of the last equation by aρ = 0 gives us

∂t u + u∂x u + or

∂t u +

∂x p = 0, ρ

p (ρ ) ∂x ρ + u∂x u = 0, ρ

where, and in the sequel, we use the notation (.) = d/d ρ . Thus, the system (8.1.1)-(8.1.2) for the primitive variable U = (ρ , u, a)T can be written as a system of balance laws in nonconservative form

∂t U + A(U)∂xU = 0, where

⎛

⎞ u ρ ρ u/a ⎜ p (ρ ) ⎟ ⎟ A(U) = ⎜ ⎝ ρ u 0 ⎠. 0 0 0

(8.2.1)

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8 Compressible fluids in a nozzle with discontinuous . . .

8.2.2 A non-strictly hyperbolic system Let us investigate the characteristic fields of the system (8.2.1). The characteristic equation of the matrix A(U) of (8.2.1) is given by det(A − λ I) = λ ((u − λ )2 − p (ρ )) = 0, where (.) = d/d ρ . Thus, providing that p (ρ ) > 0, the matrix A admits the following three real eigenvalues:

λ1 (U) = u − c,

λ2 (U) = u + c,

where c = c(ρ ) =

λ3 (U) = 0,

(8.2.2)

p (ρ ).

The corresponding right-eigenvectors can be chosen as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ρu ρ ρ r1 (U) = ⎝−c⎠ , r2 (U) = ⎝ c ⎠ , r3 (U) = ⎝ −c2 ⎠ . 2 0 0 a( cu − u)

(8.2.3)

Thus, the system (8.2.1) is hyperbolic as long as the eigenvectors are linearly independent. The eigenvalues in (8.2.2) can coincide on a certain curve. In the next sections, it will be useful to consider the projection of the phase domain onto the (ρ , u)-plane. We will define some typical sets as follows: C + := {(ρ , u)| u = c}, C − := {(ρ , u)| u = −c}.

(8.2.4)

So, λ1 = λ3 = 0 on C + , and λ2 = λ3 = 0 on C − . The system (8.1.1)–(8.1.2) is therefore not strictly hyperbolic. The curves C ± are called sonic curves. We also define the regions in which the system is strictly hyperbolic G1 = {(ρ , u)| u > p (ρ )}, G2 = {(ρ , u)| |u| < p (ρ )}, G+ 2 = {(ρ , u) ∈ G2 | u > 0}, G− 2

= {(ρ , u) ∈ G2 | u < 0}, G3 = {(ρ , u)| u < − p (ρ )}. Let us introduce some terminologies. A state is said to be subsonic if

(8.2.5)

8.2 Basic properties

199

|u| < c =

p (ρ ),

a state is said to be supersonic if |u| > c, and a state is said to be sonic if |u| = c. Thus, the subsonic set is G2 , called the subsonic region, and the supersonic set is G1 ∪ G3 , called the supersonic region. See Figure 8.1. We observe that the system (8.2.1) is strictly hyperbolic in each of these subsonic or supersonic regions, where

λ1 > 0 = λ3 , in G1 , λ1 < λ3 = 0 < λ2 , in G2 , λ2 < λ3 = 0, in G3 .

Fig. 8.1 Supersonic, subsonic regions, and the sonic curves

Next, we consider the genuine nonlinearity and linear degeneracy of the characteristic fields. Assume from now on that p”(ρ ) =

d 2 p(ρ ) > 0. dρ 2

(8.2.6)

Then, the 1- and the 2-characteristic fields are genuinely nonlinear, since −Dλ1 · r1 = Dλ2 · r2 =

ρ p (ρ ) + 2p (ρ ) > 0. 2 p (ρ )

(8.2.7)

200

8 Compressible fluids in a nozzle with discontinuous . . .

Since Dλ3 · r3 = 0,

(8.2.8)

the third characteristic field is linearly degenerate.

8.2.3 Rarefaction waves First, let us consider rarefaction waves, which are smooth self-similar solutions U(x,t) = (ρ , u, a)(ξ ), ξ = x/t of the system (8.2.1). Remember that in Chapter 3 we derive the equations for rarefaction waves of hyperbolic systems of conservation laws starting from its nonconservative form. Thus, this method is still valid for hyperbolic nonconservative systems. Indeed, substituting U(x,t) = V (x/t) into (8.2.1) yields us 1 x x x x −( 2 )V ( ) + ( )A V ( ) V ) = 0, t t t t t or x (A(U(ξ )) − ξ I)U (ξ ) = 0, ξ = . (8.2.9) t This means that if U (ξ ) = 0, then there is an index i ∈ {1, 2} such that V (ξ ) = α (ξ )ri (V (ξ )),

λi (V (ξ )) = ξ ,

(8.2.10)

for some function α (ξ ). Differentiating the second equation of (8.2.10) with respect to ξ gives us Dλi (V (ξ )) ·V (ξ ) = 1. So, multiplying the first equation of (8.2.10) by Dλi (U(ξ )) and using the last equation, we have α (ξ )Dλi (V (ξ )) · ri (V (ξ )) = 1. This yields us

α (ξ ) =

1 . Dλi (V (ξ )) · ri (V (ξ ))

(8.2.11)

Substitute α (ξ ) from (8.2.11) into (8.2.10) to get V (ξ ) =

1 ri (V (ξ )), Dλi (V (ξ )) · ri (V (ξ ))

λi (V (ξ )) = ξ ,

i = 1, 2.

(8.2.12)

Now, let U± be on the integral curve of the first equation in (8.2.12) such that V (λi (U− )) = U− ,

V (λi (U+ )) = U+ .

Then, the rarefaction wave of (8.2.1) connecting the left-hand state U− and the righthand state U+ given by

8.2 Basic properties

201

⎧ x ⎪ U− , ≤ λi (U− ), ⎪ ⎪ t ⎨ x x , λi (u− ) ≤ ≤ λi (U+ ), U(x,t) = V ⎪ t t ⎪ x ⎪ ⎩U+ , ≥ λi (U+ ) t

(8.2.13)

is a weak self-similar solution of (8.1.1). From the above argument, we consider the integral curves associated with the first characteristic field: dU r1 (U) , = dξ Dλ1 · r1 (U)

U(ξ0 ) = U0 .

(8.2.14)

From (8.2.3), (8.2.7), and (8.2.14), we see that the integral curve passing through U0 = (ρ0 , u0 , a0 ) satisfies 2ρ p (ρ ) dρ , = − dξ ρ p (ρ ) + 2p (ρ ) du 2p (ρ ) (8.2.15) , = dξ ρ p (ρ ) + 2p (ρ ) da = 0. dξ The third equation of (8.2.15) indicates that the cross-section a remains constant along the integral curve, and therefore a is constant through any 1-rarefaction fan. Due to the condition (8.2.7), from the first equation of (8.2.15) we see that dd ρξ = 0 along the integral curve. Therefore, ρ can be used as a parameterization of the integral curve. So, dividing the second equation by the first equation of (8.2.15) side by side, we get p (ρ ) du =− . (8.2.16) dρ ρ Integrating (8.2.16), we obtain the integral curve O1 (U0 ) passing through U0 associated with the first characteristic field as ρ p (ρ ) O1 (U0 ) : u = u0 − dρ . (8.2.17) ρ ρ0 Using the fact that the characteristic speed λ1 (U) must be increasing through any 1-rarefaction wave, from (8.2.17) we can obtain the forward curve of 1-rarefaction waves R1F (U0 ) consisting of all right-hand states that can be connected to a given left-hand state U0 by a 1-rarefaction wave as ρ p (ρ ) F F R1 (U0 ) : u = ω1 (U0 ; ρ ) = u0 − d ρ ρ ≤ ρ0 . (8.2.18) ρ ρ0

202

8 Compressible fluids in a nozzle with discontinuous . . .

That the characteristic speed λ1 (U) must be increasing through any 1-rarefaction wave also yields the backward curve of 1-rarefaction waves R1B (U0 ) consisting of all left-hand states that can be connected to a given right-hand state U0 by a 1rarefaction wave as ρ p (y) B B dy, ρ ≥ ρ0 . R1 (U0 ) : u = ω1 (U0 ; ρ ) = u0 − (8.2.19) y ρ0 Next, let us consider rarefaction waves associated with the second characteristic field. The integral curves associated with the second characteristic field are given by dU r2 (U) , = dξ Dλ2 · r2 (U) U(ξ0 ) = U0 ,

(8.2.20)

where U0 is a given state. From (8.2.3) and (8.2.7), we see that the system (8.2.20) yields us ρ dρ (ρ p (ρ ) + 2p (ρ )), = dξ 2 p (ρ ) p (ρ ) du (8.2.21) (ρ p (ρ ) + 2p (ρ )), = dξ 2 p (ρ ) da = 0, dξ for the integral curve passing through U0 = (ρ0 , u0 , a0 ). The third equation of (8.2.21) indicates that the cross-section a remains constant along the integral curve, and therefore a is constant through any 2-rarefaction fan. From (8.2.7) and from the first equation of (8.2.21) we see that dd ρξ = 0 along the integral curve. Therefore, ρ can be used as a parameterization of the integral curve, and from the first two equations of (8.2.21), we obtain p (ρ ) du = . (8.2.22) dρ ρ Integrating (8.2.22), we obtain the integral curve O2 (U0 ) passing through U0 associated with the second characteristic field as ρ p (y) O2 (U0 ) : u = u0 + dy. (8.2.23) y ρ0 Using the fact that the characteristic speed λ2 (U) must be increasing through any 2-rarefaction wave, from (8.2.23) we can obtain the forward curve of 3-rarefaction waves R2F (U0 ) consisting of all right-hand states that can be connected to a given left-hand state U0 by a 3-rarefaction wave as

8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves

R2F (U0 ) :

u = ω2F (U0 ; ρ ) = u0 +

ρ ρ0

p (y) dy, y

ρ ≥ ρ0 .

203

(8.2.24)

And, that the characteristic speed λ2 (U) must be increasing through any 3-rarefaction wave from (8.2.23) yields us the backward curve of 3-rarefaction waves R2B (U0 ) consisting of all left-hand states that can be connected to a given right-hand state U0 by a 3-rarefaction wave as ρ p (y) B B dy, ρ ≤ ρ0 . R2 (U0 ) : u = ω2 (U0 ; ρ ) = u0 + (8.2.25) y ρ0

8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves Let us now consider shock waves. Given a shock wave of the form U0 , x < λ t, U(x,t) = U1 , x > λ t,

(8.3.1)

where U0 ,U1 are the left-hand and right-hand states, respectively, and λ is the shock speed. Let us start with the Rankine-Hugoniot relation associated with the third equation in (8.1.1), that is, −λ [a] = 0, where [a] := a1 − a0 is the jump of the quantity a. The last equation implies that shock waves of (8.2.1) can be classified into two distinct families. (i) Shock waves across which the cross-sectional area a remains constant: a1 = a0 . (ii) Shock waves with zero propagation speed λ = 0.

8.3.1 Shock waves with constant cross-section Consider Case (i) when a remains constant across the shock, a = a0 . This means that ∂x a = 0 across the shock, and thus the left- and right-hand states of the shock are given by the usual Rankine-Hugoniot relations − λ [ρ ] + [ρ u] = 0, − λ [ρ u] + [ρ u2 + p(ρ )] = 0,

(8.3.2)

where [ρ ] := ρ+ − ρ+ , etc. Thus, sometimes in this subsection we may denote a state by its first two components U = (ρ , u) for simplicity, with the understanding that we are referring to the state U = (ρ , u, a0 ).

204

8 Compressible fluids in a nozzle with discontinuous . . .

Fix a given left-hand state U0 ; we are now looking for the set of all right-hand states U = U1 that satisfy the Rankine-Hugoniot relations (8.3.2). Eliminating λ , we obtain from these equations the Hugoniot set as (p − p0 )(ρ − ρ0 ) . H (U0 ) : u = u0 ± ρ0 ρ Using the fact that the 1-Hugoniot curve H1 (U0 ) is tangent to the vector r1 (U0 ) at the point U0 , we see that the 1-Hugoniot curve is given by (p − p0 )(ρ − ρ0 ) . H1 (U0 ) : u = u0 − ρ0 ρ And, that the 3-Hugoniot curve H2 (U0 ) is tangent to the vector r2 (U0 ) at the point U0 yields us the 3-Hugoniot curve as (p − p0 )(ρ − ρ0 ) . H2 (U0 ) : u = u0 + ρ0 ρ Along Hugoniot curves, Lax shock inequalities will be imposed to select admissible shock waves. Recall that a shock wave connecting a given left-hand state U0 to a right-hand state U will be admissible if it satisfies the Lax shock inequalities

λi (U) < λ¯ i (U,U0 ) < λi (U0 ),

i = 1, 3,

(8.3.3)

where λ¯ i (U,U0 ) denotes the shock speed. Assume from now on for simplicity that p”(ρ ) > 0.

(8.3.4)

This implies that the first and the second characteristic fields are genuinely nonlinear. Then, the Lax shock inequalities for 1-shocks are equivalent to the condition

ρ > ρ0 . Thus, the forward 1-shock curve consisting of all right-hand states U = (ρ , u) that can be connected to a given left-hand state U0 = (ρ0 , u0 ) by a Lax shock wave is given by (p − p0 )(ρ − ρ0 ) , ρ > ρ0 . (8.3.5) S1F (U0 ) : u = ω1F (U0 ; ρ ) := u0 − ρ0 ρ Similarly, the forward 3-shock curve consisting of all right-hand states U = (ρ , u) that can be connected to a given left-hand state U0 = (ρ0 , u0 ) by a Lax shock is given by

8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves

S2F (U0 ) :

u = ω2F (U0 ; ρ ) := u0 +

(p − p0 )(ρ − ρ0 ) , ρ0 ρ

205

ρ < ρ0 .

(8.3.6)

Backward constructions of shock curves will also be useful later in this chapter. The backward 1-shock curve consisting of all left-hand states U = (ρ , u) that can be connected to a given right-hand state U0 = (ρ0 , u0 ) by a Lax shock wave is given by (p − p0 )(ρ − ρ0 ) B B S1 (U0 ) : u = ω1 (U0 ; ρ ) := u0 − , ρ < ρ0 . (8.3.7) ρ0 ρ The backward 3-shock curve consisting of all left-hand states U = (ρ , u) that can be connected to a given right-hand state U0 = (ρ0 , u0 ) by a Lax shock is given by (p − p0 )(ρ − ρ0 ) S2B (U0 ) : u = ω2B (U0 ; ρ ) := u0 + , ρ > ρ0 . (8.3.8) ρ0 ρ For example, for an isentropic ideal fluid, the forward 1-shock curves S1F (U0 ) and the backward 3-shock curve S2F (U0 ) are given by

1 1 γ 1/2 S1F (U0 ) : u = ω1F (U0 ; ρ ) := u0 − κ ( − )(ρ γ − ρ0 ) , ρ0 ρ

1 1 γ 1/2 S2F (U0 ) : u = ω2F (U0 ; ρ ) := u0 + κ ( − )(ρ γ − ρ0 ) , ρ0 ρ

ρ > ρ0 , (8.3.9)

ρ < ρ0 .

We can now define the 1-wave forward curve W1F (U0 ), and the 3-wave backward curve W2B (U0 ) issuing from a given state U0 by Wi F (U0 ) := SiF (U0 ) ∪ RiF (U0 ), Wi B (U0 ) := SiB (U0 ) ∪ RiB (U0 ),

i = 1, 2.

(8.3.10)

It is not difficult to see that the functions ω1F (U0 ; ρ ) and ω1B (U0 ; ρ ), ρ > 0 are strictly decreasing, and the functions ω2F (U0 ; ρ ) and ω2B (U0 ; ρ ), ρ > 0 are strictly increasing.

8.3.2 Shock waves with zero speed: stationary contact waves Let us now consider Case (ii), where the shock speed vanishes. Since the solution is independent of time, it is natural to search for a solution to be the limit of a sequence of time-independent smooth solutions of (8.1.1)–(8.1.2). For this purpose we first find a divergence form of the system. As shown above, the second equation of (8.1.1) can be transformed to the equation

206

8 Compressible fluids in a nozzle with discontinuous . . .

∂t u + u∂x u +

∂x p = 0, ρ

which can be put in the divergence form

∂t u + ∂x ( where h(ρ ) =

u2 + h(ρ )) = 0, 2

1 d p(ρ ) dρ . ρ dρ

For example, for an isentropic ideal fluid with an equation of state p(ρ ) = κρ γ ,

κ > 0, γ > 1,

γ −1 . we can take h(ρ ) = γκγ −1 ρ Thus, the system (8.1.1)–(8.1.2) for the unknown smooth function U = (ρ , u, a)T can be written in the divergence form as

∂t (aρ ) + ∂x (aρ u) = 0, ∂t u + ∂x (

u2 + h(ρ )) = 0, 2

∂t a = 0. Consequently, smooth stationary solutions of the system (8.1.1)–(8.1.2) satisfy (aρ u) = 0, (

u2 + h(ρ )) = 0, 2

(8.3.11)

where (.) = d/dx. The integral curve of (8.3.11) passing through each point (ρ0 , u0 , a0 ) can be parameterized by u, say u → (ρ (u), u, a(u)). Let u → u± ,

ρ± = ρ (u± ),

a± = a(u± ).

The states (ρ± , u± , a± ) then satisfy the jump relations [aρ u] = 0,

[

u2 + h(ρ )] = 0. 2

(8.3.12)

Any jump of the system (8.1.1)–(8.1.2) that satisfies the jump relations (8.3.12) is a weak solution in the sense of nonconservative products. The equation (8.3.12) defines a curve issuing from any given state U0 , denoted by W3 (U0 ). The curve W3 (U0 ) is the set of all states U that can be connected to U0 by

8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves

207

a stationary jump, which is a weak solution of the system (8.1.1)–(8.1.2). We will show that this kind of weak solutions are associated with the 3-characteristic field. This is given by the following lemma. Lemma 8.3.1. The curve of stationary jumps defined by (8.3.12) coincides with the integral curve associated with the 3-characteristic field (λ3 (U), r3 (U)). Proof. Since the curve defined by (8.3.12) coincides with the integral curve (8.3.11), we will show that this integral curve coincides with the integral curve associated with the 3-characteristic field. Let U = (ρ , u, a)T . Consider the integral curve associated with the 3-characteristic field, which is given by ⎛ ⎞ ρu dU(ξ ) = r3 (U(ξ )) = ⎝ −p (ρ ) ⎠ , dξ a( p (uρ ) − u) where (.) = d/d ρ ; see (8.2.3). This implies along this curve that du h (ρ ) p (ρ ) =− , =− dρ ρu u 2

where (.) = d/d ρ . This yields us udu + h (ρ )d ρ = 0, or d( u2 + h(ρ )) = 0, which is the second equation of (8.3.11). Furthermore, it holds along the integral curve associated with the 3-characteristic field that p (ρ ) 1 da =a − dρ ρ u2 ρ h (ρ ) 1 1 1 du =a − + = −a , u2 ρ ρ u dρ which yields us −ρ uda = aud ρ + aρ du or d(aρ u) = 0. This establishes the first equation in (8.3.11). The proof of Lemma 8.3.1 is complete. In view of Lemma 8.3.1, the 3-contact discontinuities of the system are given by (8.3.12). Fix an arbitrary state U0 ; it holds from (8.3.12) that the curve W3 (U0 ) can be parameterized by ρ :

1/2 . W3 (U0 ) : u = ω3 (U0 ; ρ ) := sgn(u0 ) u20 − 2(h(ρ ) − h(ρ0 ))

(8.3.13)

For simplicity, from now on we assume that the fluid is isentropic and ideal. Then, the 2-wave curve is given by

2κγ γ −1 γ −1 1/2 (ρ − ρ0 ) . W3 (U0 ) : u = ω3 (U0 , ρ ) := sgn(u0 ) u20 − γ −1 A state U = (ρ , u, a) can be connected to U0 = (ρ0 , u0 , a0 ) by a stationary wave if and only if

208

8 Compressible fluids in a nozzle with discontinuous . . .

u = ω3 (U0 ; ρ ),

(8.3.14) a0 u0 ρ0 = 0. a For a given state U0 , we can use the cross-section a as a parameter in the system (8.3.14) to resolve for the density ρ . We will see below that depending on the value of a, the equations (8.3.14) may admit no root, one root, or two roots. If u0 = 0, then the equations (8.3.14) determine three points (γ −1)/2 ). (ρ0 , 0), (0, ± (2κγ )/(γ − 1)ρ0

Φ (U0 ; ρ , a) := ω3 (U0 ; ρ )ρ −

Assume u0 = 0. The function ρ → ω3 (U0 , ρ ) is defined provided the expression under the square root is non-negative: u20 −

2κγ γ −1 γ −1 (ρ − ρ0 ) ≥ 0, γ −1

which requires that

ρ ≤ ρ¯ (U0 ) :=

γ −1 2κγ

γ −1

u20 + ρ0

1 γ −1

.

(8.3.15)

In the interval [0, ρ¯ (U0 )], the function ρ → ω3 (U0 , ρ ) is decreasing for u0 > 0 and increasing for u0 < 0. On the other hand, γ −1

2κγ 2 γ −1 − ρ γ −1 ∂ Φ (U0 ; ρ , a) u0 − γ −1 (ρ 0 ) − κγρ =

. ∂ρ γ −1 1/2 κγ u20 − γ2−1 (ρ γ −1 − ρ0 )

(8.3.16)

Assume, for simplicity, that u0 > 0. The last expression means that

∂ Φ (U0 ; ρ , a) > 0, ∂ρ ∂ Φ (U0 ; ρ , a) < 0, ∂ρ where

ρmax (ρ0 , u0 ) :=

ρ < ρmax (ρ0 , u0 ), (8.3.17)

ρ > ρmax (ρ0 , u0 ),

1 γ −1 2 γ −1 γ −1 u20 + ρ0 . κγ (γ + 1) γ +1

(8.3.18)

The function ρ → Φ (U0 ; ρ , a) takes negative values at the endpoints, i.e.,

Φ (U0 ; ρ = 0, a) < 0,

Φ (U0 ; ρ = ρ¯ (U0 ), a) < 0,

where ρ¯ is defined by (8.3.15). Thus, this function admits a root if and only if the maximum value on the interval [0, ρ¯ (U0 )] is non-negative. This is equivalent to the condition

8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves

a ≥ amin (U0 ) :=

a0 ρ0 |u0 | . γ +1 √ 2 κγρmax (ρ0 , u0 )

209

(8.3.19)

For u0 < 0, similar properties hold. We are in a position to state the following basic results in this subsection. Lemma 8.3.2 (Existence of stationary waves). Given U0 , a stationary shock issuing from U0 and connecting to some state U = (ρ , u, a) exists if and only if a ≥ amin (U0 ). More precisely, we have (i) If a < amin (U0 ), there are no stationary shocks. (ii) If a > amin (U0 ), then there are exactly two values ϕ1 (U0 , a) < ρmax (U0 ) < ϕ2 (U0 , a) such that

Φ (U0 ; ϕ1 (U0 , a), a) = Φ (U0 ; ϕ2 (U0 , a), a) = 0. Accordingly, along the curve W3 (U0 ), there are two distinct points that can be attained from U0 using a stationary shock. (iii) If a = amin (U0 ), then on the curve W3 (U0 ) there is a unique point that can be attained from U0 using a stationary shock. The following lemma provides us with comparisons of quantities defined above. Lemma 8.3.3. a) The following results hold

ρmax (ρ0 , u0 ) < ρ0 , ρmax (ρ0 , u0 ) > ρ0 , ρmax (ρ0 , u0 ) = ρ0 ,

(ρ0 , u0 ) ∈ G2 , (ρ0 , u0 ) ∈ G1 ∪ G3 , (ρ0 , u0 ) ∈ C± .

b) The state (ϕ1 (U0 , a), ω3 (U0 , ϕ1 (U0 , a))) belongs to G1 if u0 > 0; it belongs to G3 if u0 < 0, while the state (ϕ2 (U0 , a), ω3 (U0 , ϕ2 (U0 , a))) always belongs to G2 . Moreover, it holds for u0 = 0 that (ρmax (U0 , a), ω3 (U0 , ρmax (U0 , a))) ∈ C +

if

u0 > 0,

−

if

u0 < 0.

(ρmax (U0 , a), ω3 (U0 , ρmax (U0 , a))) ∈ C

(8.3.20)

In addition, the following estimates hold: (i) If a > a0 , then (ii) If a < a0 , then

ϕ1 (U0 , a) < ρ0 < ϕ2 (U0 , a). ρ0 < ϕ1 (U0 , a) for ρ0 > ϕ2 (U0 , a) for

U0 ∈ G1 ∪ G3 , U0 ∈ G2 .

(8.3.21)

(8.3.22)

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8 Compressible fluids in a nozzle with discontinuous . . .

c)

amin (U, a) < a,

(ρ , u) ∈ Gi , i = 1, 2, 3,

amin (U, a) = a, amin (U, a) = 0,

(ρ , u) ∈ C± , ρ = 0 or u = 0.

(8.3.23)

Proof. It is easy to check a) directly from the definition of ρmax in (8.3.18). Let us prove b) and c). Assume for simplicity that u0 > 0, since the case u0 < 0 can be similarly argued. Let us define the function g(ρ ) = ω22 (U0 , ρ ) − κγρ γ −1 2κγ γ −1 γ −1 (ρ = u20 − − ρ0 ) − κγρ γ −1 . γ −1 Then, a straightforward calculation gives us g(ρmax (U0 )) = 0, which establishes (8.3.20). On the other hand, since dg(ρ ) = −(γ + 1)κγρ γ −2 < 0, dρ and that ϕ1 (U0 , a) < ρmax (U0 , a) < ϕ2 (U0 , a) we find g(ϕ1 (U0 , a)) > g(ρmax (U0 )) = 0 > g(ϕ2 (U0 , a)). The last two inequalities establish the first statement of b). Next, we have

Φ (U0 ; ρ0 , a) = ρ0 u0 (1 − a0 /a) > 0 iff a > a0 . This gives (8.3.21) when a > a0 . Moreover, the last inequality also implies that ρ0 is located outside of the interval [ϕ1 (U0 , a), ϕ2 (U0 , a)] when a < a0 . It follows from (8.3.16) that γ −1

∂ Φ (U0 ; ρ0 , a) u20 − κγρ0 = ∂ρ u0

< 0 iff U0 ∈ G2 ,

which, together with the earlier observation, implies (8.3.22). Since amin (U0 ) < a0 if and only if γ +1 √ κγρ ∗ 2 > ρ0 |u0 |,

or l(m) := γ −1

where m := ρ0

1−γ 2 2 γ −1 m − (κγ ) γ +1 m γ +1 + > 0, γ +1 κγ (γ + 1)

/u20 , then, we see that l(1/κγ ) = 0,

(8.3.24)

8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves

211

which shows that the second equation in (8.3.23) also holds, since (ρ0 , u0 ) ∈ C± for m = 1/κγ . Furthermore, we have 1−γ dl(m) 2 = (1 − (κγ m) γ +1 ), dm γ +1

which is positive for m > 1/κγ and negative for m < κγ . This together with (8.3.24) establishes the first statement in (8.3.23). The third statement in (8.3.23) is straightforward. This completes the proof of Lemma 8.3.3. Proposition 8.3.1. i) Given a left-hand state U0 , the 1-shock speed λ¯ 1 (U0 ,U), (for ρ > ρ0 ) may change sign along the (forward) 1-shock curve S1F (U0 ) iff U0 ∈ G1 . More precisely, if U0 ∈ G1 , then λ¯ 1 (U0 ,U) vanishes once at some point U˜ 0 = (ρ˜ 0 , u˜0 ) ∈ S1F (U0 ) ∩ G+ 2 such that

λ¯ 1 (U0 , U˜ 0 ) = 0, λ¯ 1 (U0 ,U) > 0, ρ ∈ (ρ0 , ρ˜ 0 ), λ¯ 1 (U0 ,U) < 0, ρ ∈ (ρ˜ 0 , +∞).

(8.3.25)

If U0 ∈ G2 ∪ G3 then λ¯ 1 (U0 ,U) is always negative:

λ¯ 1 (U0 ,U) < 0,

U ∈ S1F (U0 ).

(8.3.26)

ii) Given a right-hand state U0 , the 3-shock speed λ¯ 2 (U0 ,U) (for ρ > ρ0 ) may change sign along the (backward) 3-shock curve S2B (U0 ) iff U0 ∈ G3 . More precisely, if U0 ∈ G3 , then λ¯ 2 (U0 ,U) vanishes once at some point U˜ 0 = (ρ˜ 0 , u˜0 ) ∈ S2B (U0 ) ∩ G− 2 such that

λ¯ 2 (U0 , U˜ 0 ) = 0, λ¯ 2 (U0 ,U) < 0, ρ ∈ (ρ0 , ρ˜ 0 ), λ¯ 2 (U0 ,U) > 0, ρ ∈ (ρ˜ 0 , +∞).

(8.3.27)

If U0 ∈ G1 ∪ G2 then λ¯ 2 (U0 ,U) is always positive:

λ¯ 2 (U0 ,U) > 0,

U ∈ S2B (U0 ).

Proof. We only prove i), since similar arguments can be used for ii). It follows from (8.3.2) that

ρ ρ γ − ρ 1/2 ρω1F (U0 , ρ ) − ρ0 u0 0 λ¯ 1 (U0 ,U) = = u0 − κ . ρ − ρ0 ρ0 ρ − ρ0 γ

(8.3.28)

Thus, if U0 ∈ G3 ∪ C − , then u0 < 0 and λ¯ 1 (U0 ,U) < 0. If U0 ∈ G2 , then, since the function ρ → ρ γ is convex and ρ > ρ0 , we deduce from (8.3.20) that

212

8 Compressible fluids in a nozzle with discontinuous . . .

√ λ¯ 1 (U0 ,U) < u0 − κγρ0 2 < 0, γ −1

where the last inequality holds by U0 ∈ G2 . Letting U0 → C + in the last inequality, we obtain λ¯ 1 (U0 ,U) < 0, U = U0 ,U0 ∈ C + . Thus, (8.3.26) is established. Assume now that U0 ∈ G1 . Set l(ρ ) := κ

γ

ρ ρ γ − ρ0 − u20 = 0, ρ ≥ ρ0 . ρ0 ρ − ρ0

Observe that the function l(ρ ) and λ¯ 1 (U0 ,U) have the same roots and take values of opposite signs. Then, a straightforward calculation shows that γ −1

l(ρ0 ) = κγρ0 l (ρ ) = κ

− u20 < 0,

(γ − 1)ρ γ

γ + ρ0

ρ0 (ρ − ρ0 )

l(+∞) = +∞,

> 0.

This shows that there exists a unique value ρ = ρ˜ 0 satisfying l(ρ˜ 0 ) = 0, or

λ¯ 1 (U0 , U˜ 0 ) = 0. Moreover, the other two assertions in (8.3.25) are also satisfied, by the monotonicity of the function l(ρ ). To show that U˜ 0 ∈ G+ 2 , first we observe that from the jump relation [ρ u] = 0, for a Lax shock with zero speed, the value ω1F (U0 , ρ˜ 0 ) defined by (8.2.18) and (8.3.5) must be positive. Since the function ρ → ρ γ is convex, and the function ω1F is decreasing in ρ we have

ρ ρ γ − ρ γ 1/2 γ −1 √ 0 0 = λ¯ 1 (U0 , U˜ 0 ) = u0 − κ > u0 − κγ ρ˜ 0 2 ρ0 ρ − ρ0 γ −1 √ > ω1F (U0 , ρ˜ 0 ) − κγ ρ˜ 0 2 = λ¯ 1 (ψ1 (U0 ), ω1F (U0 ; ρ˜ 0 )). Since ω1F (U0 ; ρ˜ 0 ) > 0, the last inequality indicates that U˜ 0 ∈ G+ 2 . This completes the proof of Proposition 8.3.1.

Notation We will use the following notation. (i) Wi (u1 , u2 ): An i-wave connecting a state u1 and a state u2 , where W = S: Lax shock; W = R: rarefaction wave; W2 = Z, i = 1, 2, 3;

8.3 Shocks and contact waves, and Monotonicity Criterion for contact waves

213

(ii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the left-hand state u1 to the right-hand state u2 is followed by a j-wave from the left-hand state u2 to the right-hand state u3 ; (iii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the right-hand state u1 to the left-hand state u2 is preceded by a j-wave from the right-hand state u2 to the left-hand state u3 ; (iv) U 0 denotes the state resulting from a stationary contact wave from U; (v) Denote by U # ∈ S1F (U) so that λ¯ 1 (U,U # ) = 0, that is, the state with zerospeed 1-shock from U.

8.3.3 Two-parameter sets of composite waves Solutions of the Riemann problem can contain an i-wave followed by a j-wave for i = j and i, j ∈ {1, 2, 3}. The fact that an i-wave is followed by a j-wave for i = j and i, j ∈ {1, 2, 3} forms a composite wave. As seen above, the third eigenvalue λ3 may interchange the order with λ1 across C + and λ3 may interchange the order with λ2 across C − . Let U0 = (ρ0 , u0 , a0 ) be a given state, and a be the cross-section level of the state on the other side of the stationary contact in a Riemann solution. A two-parameter set of composite waves between waves associated with one genuinely nonlinear characteristic field with stationary waves. Thus, a one-parameter family of Riemann solutions can be constructed by letting the wave curve associated with the other genuinely nonlinear characteristic field intersect this set. Indeed, we will illustrate the above observation as follows. Composite waves can be obtained as follows. A Riemann solution can start from a left-hand state UL in the subsonic region G2 by a 1-wave to some state U1 = U1 (ρ ) ∈ G2 , followed I , a > 0 in the supersonic by a stationary jump from U1 to U2 = U2 (a), a ∈ I ⊂ R region G1 by using ϕ1 (U1 , a). Then, the solution uses a second stationary jump into a state U3 = U3 (a) ∈ G2 using ϕ2 (U2 , aL ) to attain the cross-sectional area aR . These composite waves thus have the form W1 (UL ,U1 (ρ )) Z(U1 ,U2 (a)) Z(U2 ,U3 (a)),

ρ > 0, a > 0,

which depend on both parameters ρ and a. So, the set of these composite waves is a two-parameter set. So, the intersection of the backward curve W2B (UR ) with this two-parameter set may yield a one-parameter set of Riemann solution. See Figure 8.2. To avoid this situation, we should therefore prevent any stationary shock that jumps across the sonic curve from the subsonic region to the supersonic region, or vice versa.

214

8 Compressible fluids in a nozzle with discontinuous . . .

Fig. 8.2 A two-dimensional set of composite waves and one-parameter family of solutions

Next, consider another situation, where there may be infinitely many solutions. A solution can start from UL ∈ G2 by a rarefaction wave until it touches C + at U1 . Then, it jumps to U2 = U2 (a) ∈ G1 by a stationary jump to the outside of the interval [aL , aR ], say to a < min{aL , aR } using ϕ1 , then it jumps back to G2 to a state U3 using a zero-speed 1-shock, then it jumps to UR using a second stationary shock. Letting a → 0, then u2 → ∞, and we may get an unbounded sequence of solutions; see Figure 8.3. This observation suggests us that we need to require that the total variation of the cross-section has to be bounded.

Fig. 8.3 Unbounded sequence of solutions

8.4 Solutions of the Riemann problem

215

8.3.4 Admissibility Criterion for stationary contacts As seen in the previous subsection, we need an additional admissibility criterion for selecting admissible stationary waves. Let us impose the following admissibility criterion for stationary contact waves associated with the third characteristic field.

a-monotone Criterion for admissible contact waves (a) Along any stationary curve W3 (U0 ), the cross-sectional area a is monotone as a function of ρ . (b) The total variation of the cross-section component of any Riemann solution must not exceed (and, therefore, is equal to) |aL − aR |, where aL , aR are lefthand and right-hand cross-section levels. Lemma 8.3.4. The a-monotone criterion means that admissible stationary shocks remain in the same subsonic or supersonic region. (i) If U0 is in the supersonic regions G1 and G3 , then the admissible shock is the one using ϕ1 (U0 , a); (ii) If U0 is in the subsonic region G2 , then the admissible shock is the one using ϕ2 (U0 , a). Proof. Taking the derivative (with respect to ρ ) in the identity a2 (ω2 ρ )2 =(a0 u0 ρ0 )2 , we get (8.3.29) a(ρ )a (ρ )(ω2 ρ )2 + 2a2 (ω2 ρ )(ω2 ρ + ω2 ) = 0. To demonstrate the lemma, it is sufficient to show that the last factor of the second term in (8.3.29) remains of a constant sign whenever the curve W3 (U0 ) does not cross the boundary of strict hyperbolicity C± . Indeed, assume for simplicity that u0 > 0, then

ω2 (ρ )ρ + ω2 =

ω 2 − κγρ γ −1 −κγρ γ −1 + ω2 = 2 , ω2 ω2

which remains of a constant sign if and only if W3 (U0 ) does not cross C± . This completes the proof of Lemma 8.3.4.

8.4 Solutions of the Riemann problem Under the transformation x → −x and u → −u, a right-hand state U = (ρ , u, a), u < 0 becomes a left-hand state of the form U = (ρ , −u, a). Therefore, the constructions relying on the initial data around C − can be obtained in a similar way as the ones for

216

8 Compressible fluids in a nozzle with discontinuous . . .

the initial data around C + . Thus, in the following we will construct only Riemann solutions when the initial data UL ,UR are located around C + . Given a0 and a = a0 , let us define the curves of composite waves as follows. The forward composite wave curve W1→3 (U0 ) consists of all the states U ∈ G2 ∪ C = C + ∪ C − , which can be arrived at by an ordered combination of 2 waves: the first 1-wave from the left-hand state U0 to some right-hand state U1 , followed by a second 3-wave from the lefthand state U1 to the right-hand U; the backward composite wave curve W2←3 (U0 ) consists of all states U ∈ G2 ∪ C which can be arrived at by an ordered combination of 2 waves: the first 2-wave from the right-hand state U0 to some left-hand state U1 , preceded by a second 3-wave from the right-hand state U1 to the left-hand state U. That is W1→3 (U0 ) := {U = (h, u, a) ∈ G2 ∪ C : ∃W3 (U1 ,U) shifting a0 to a, U1 = (h1 , u1 , a0 ) ∈ W1F (U0 ), λ1 (U) ≤ 0}, W2←3 (U0 ) := {U = (h, u, a) ∈ G2 ∪ C : ∃W3 (U1 ,U) shifting a0 to a, U1 = (h1 , u1 , a0 ) ∈ W2B (U0 ), λ2 (U) ≥ 0}. (8.4.1) Given left-hand and right-hand states UL = (ρL , uL , aL ),UR = (ρR , uR , aR ), we now discuss the construction algorithm for the corresponding Riemann solution.

8.4.1 Construction 1: supersonic/supersonic This construction is based on the left-hand state UL when it is located in the supersonic region G1 . The solution begins by a 3-stationary contact wave from UL to a state U1 ∈ G1 . This stationary contact will derive the cross-section from the level a = aL to the level a = aR . Let U2 be the intersection point of W1F (U1 ) with W2B (UR ), i.e., {U2 } = W1F (U1 ) ∩ W2B (UR ). Then, the solution continues with a 1-wave from U1 to U2 , followed by a 3-wave from U2 to UR ; see Figure 8.4. Thus, the formula for the solution is given by W3 (UL ,U1 ) W1 (U1 ,U2 ) W2 (U2 ,UR ).

(8.4.2)

The condition for the existence of such a stationary wave W3 (UL ,U1 ) is aR ≥ amin (UL ).

(8.4.3)

Then, the solution (8.4.2) exists provided

λ¯ 1 (U2 ,U1 ) ≥ 0, which means that U2 is above U1# , or coincides with it, and so that the state UR is somewhere in a higher position.

8.4 Solutions of the Riemann problem

217

Fig. 8.4 Riemann solution (8.4.2) by Construction 1

The states U1 and U2 can be calculated as follows. The state U1 is obtained from the equations (8.3.14), where we take the smaller root ϕ1 (UL ; aR ) for the admissible stationary shock from UL to U1 . Precisely, U1 satisfies u1 = ω3 (U0 ; ρ1 ),

(8.4.4) aL uL ρL = 0, aR where ω2 is defined by (8.3.13), and ρ1 satisfies the condition ρ1 < ρmax (ρL , uL ) under the a-Monotonicity criterion. Observe that ρmax (ρL , uL ), as defined from (8.3.18), is given by 1

γ −1 2 γ −1 γ −1 u2L + ρmax (ρL , uL ) := ρL . (8.4.5) κγ (γ + 1) γ +1

Φ (UL ; ρ1 , aR ) = ω3 (UL ; ρ1 )ρ1 −

After finding U1 , we obtain U2 by solving

ω1F (U1 ; ρ ) = ω2B (UR ; ρ ),

(8.4.6)

= ρ2 , where the function ω1F

is defined by (8.2.18) and (8.3.5), and the function for ρ ω2B is defined by (8.2.25) and (8.3.8). Then, u2 = ω1F (U1 ; ρ2 ) = ω2B (UR ; ρ2 ). The configuration of the Riemann solution (8.4.2) in the (x,t)-plane is given by Figure 8.5. Note that this construction is still valid for some part of a subsonic right-hand state. Example 8.4.1. We consider the Riemann problem for the system (8.1.1)–(8.1.2) for an ideal and isentropic fluid so that κ = 1, γ = 1.4. The initial states UL = (ρL , uL , aL ) and UR = (ρR , uR , aR ) are given by Table 8.1. The Riemann solution of the form (8.4.2) at the time t = 0.1 is illustrated by Figure 8.6 in the interval [−1, 1]. The

218

8 Compressible fluids in a nozzle with discontinuous . . .

states U1 and U2 , which determine the elementary waves of the solution (8.4.2), are computed as in Table 8.1.

8.4.2 Construction 2: supersonic/subsonic When UL is in the supersonic region G1 , but the state UR goes downward from Construction 1, say UR is in the subsonic region G2 , there is possibly another type of the solution as follows.

Fig. 8.5 Riemann solution with structure (8.4.2) in the (x,t)-plane

Fig. 8.6 A Riemann solution of the form (8.4.2) in [−1, 1] at the time t = 0.1

8.4 Solutions of the Riemann problem

219

Table 8.1 States determining elementary waves of the Riemann solution (8.4.2) by Construction 1 UL U1 U2 UR

h 0.2 0.160576947931 0.267413085975 1

u 2 2.07584798209 1.62825745687 3

a 1 1 1 1.2

From UL , the solution can jump by a large shock with non-negative speed from UL to some state U1 in the subsonic region G2 . This jump is possible by Proposition 8.3.1. From U1 , the solution can be continued with a stationary shock using ϕ2 to a state U2 ∈ G2 . Then, the solution arrives at UR by a 2-wave; see Figure 8.7.

Fig. 8.7 Riemann solution (8.4.7) by Construction 2

Thus, the formula for the solution is given by W1 (UL ,U1 ) W3 (U1 ,U2 ) W2 (U2 ,UR ).

(8.4.7)

Geometrically, we can determine U1 and U2 as follows. Stationary shocks from a left-hand state U ∈ W1F (UL ) ∩ G2 with a = aL to some right-hand state U ∈ G2 with a = aR . These right-hand states U form a curve denoted by W1→3 (UL ). Let {U2 } = W2B (UR ) ∩ W1→3 (UL ). Then {U1 } = W1F (UL ) ∩ W3 (U2 ). So, the intermediate states U1 ,U2 can be determined.

220

8 Compressible fluids in a nozzle with discontinuous . . .

The states U1 ,U2 can be calculated as follows. We find u1 = ω1F (UL ; ρ1 ), u2 = ω2B (UR ; ρ2 ), u2 = ω3 (U1 ; ρ2 ),

Φ (U1 ; ρ2 , aR ) = ω3 (U1 ; ρ2 )ρ2 −

(8.4.8) aL u1 ρ1 = 0, aR

where the function ω1F is defined by (8.2.18) and (8.3.5), and the function ω2B is defined by (8.2.25) and (8.3.8), the function ω2 is defined by (8.3.13), and Φ is defined by (8.3.14). The value ρ2 is also required to satisfy the condition ρ2 > ρmax (ρ1 , u1 ), since we use ϕ2 for the stationary shock in the subsonic region. The first two equations in (8.4.8) are obtained from the conditions U1 ∈ W1F (UL ),U2 ∈ W2B (UR ), respectively. The last two equations in (8.4.8) are obtained from the fact that U1 and U2 are connected by a stationary shock. The four equations in (8.4.8) can determine the four quantities ρi , ui , i = 1, 2. The configuration of the Riemann solution (8.4.7) in the (x,t)-plane is given by Figure 8.8. Note that this construction still holds for some part of a supersonic right-hand state.

Fig. 8.8 Riemann solution with structure (8.4.7) in the (x,t)-plane

An alternative construction relying on the backward composition can be made, where we can make use of the composite wave curve W2←3 (UR ). Precisely, let W1F (UL ) ∩ W2←3 (UR ) = {U1 }, and let U2 ∈ W2B (UR ) be the corresponding right-hand state of the admissible contact wave from U1 . Then, the solution still has the form (8.4.5). This backward construction is useful when aL > aR .

8.4 Solutions of the Riemann problem

221

8.4.3 Construction 3: resonant waves for supersonic regime When the right-hand state UR is somewhere around the sonic curve C + , there is a very interesting form of the solution which contains three waves of the same (zero) shock speed as follows. The solution can first jump from UL to a state U1 in the supersonic region G1 using a stationary shock with an intermediate cross-section level a = aM . From U1 , the solution can jump to some state U2 in the subsonic region G2 with a zero shock speed, by Proposition 8.3.1, followed by another stationary shock from U2 to a state U3 ∈ G2 to complete the shift of the cross-section to a = aR . Finally, the solution arrives at UR by a 2-wave; see Figure 8.9.

Fig. 8.9 Riemann solution (8.4.9) by Construction 3

The formula for the solution by Construction 3 is thus given by W3 (UL ,U1 ) W1 (U1 ,U2 ) W3 (U2 ,U3 ) W2 (U3 ,UR ).

(8.4.9)

The states U1 ,U2 , and U3 can be calculated as follows. Since U1 ∈ W3 (UL ),U2 ∈ W1F (U1 ),U3 ∈ W3 (U2 ) ∩ W2B (UR ), we find u1 = ω3 (UL ; ρ1 ), Φ (UL ; ρ1 , aM ) = 0,

ρ1 < ρmax (ρL , uL ),

u2 = ω1F (U1 ; ρ2 ),

λ¯ 1 (U1 ,U2 ) = 0,

(8.4.10)

u3 = ω2B (UR ; ρ3 ), u3 = ω3 (U2 ; ρ3 ),

Φ (U2 ; ρ3 , aR ) = 0,

ρ3 > ρmax (ρ2 , u2 ),

222

8 Compressible fluids in a nozzle with discontinuous . . .

where the function ω1F is defined by (8.2.18) and (8.3.5), and the function ω2B is defined by (8.2.25) and (8.3.8), the function ω2 is defined by (8.3.13), and Φ is defined by (8.3.14). λ¯ 1 is given by (8.3.28):

ρω1F (U0 , ρ ) − ρ0 u0 λ¯ 1 (U0 ,U) = . ρ − ρ0 The seven equations in (8.4.10) can determine the seven quantities aM and ρi , ui , i = 1, 2, 3. The above three constructions may be extended if UL is on the sonic curve C + . When UL moves downward to the subsonic region G2 , there are the following constructions for the Riemann solutions. The configuration of the Riemann solution (8.4.9) in the (x,t)-plane is given by Figure 8.10.

Fig. 8.10 Riemann solution with structure (8.4.9) in the (x,t)-plane

8.4.4 Existence and uniqueness properties: supersonic regime Existence for large domain of UR As indicated in Construction 1, a stationary wave from UL to U1 = UL0 and shift a from aL to aR can be followed by a zero-speed 1-shock from U1# = UL0# := (UL0 )# . The point UL0# is a common endpoint of the two curves W1F (U1 ) and L , since we can regard the second stationary wave in Construction 3 to be trivial. As indicated in Construction 2, a zero-speed 1-shock from UL to UL# can be followed by a stationary contact from UL# to (UL# )0 := UL#0 shifting a from aL to aR . The point UL#0 is a common endpoint of the two curves W1→3 (UL ) and L , since we can regard the first stationary wave in Construction 3 to be trivial. Thus, the union W1F (U1 ) ∪ L ∪ W1→3 (UL )

8.4 Solutions of the Riemann problem

223

is a large continuous curve. Then, the backward curve W2B (UR ) can intersect this union of curves for a large domain of UR . This gives us the existence of solutions of the Riemann problem for a large domain of UR .

No local existence Usually, the non-existence of solutions may be observed by letting the left-hand and right-hand states be far from each other. This also can be done for the case when the left-hand state is a supersonic state. However, it is more interesting that the local existence of Riemann solutions is lost if the condition (8.4.3) is violated. In that case, even UR can be made arbitrarily close to UL , or eventually UR = UL and aL > aR ; there are no solutions of the Riemann problem. Indeed, assume that the condition (8.4.3) is violated, i.e., aR < amin (UL ).

(8.4.11)

If (8.4.11) holds, then Construction 1 is not available. Construction 2 and Construction 3 are available for some parts. More precisely, Construction 2 is available when we use a non-positive 1-shock from UL to a subsonic state U1 , followed by a stationary wave from U1 to U2 iff aR ≥ amin (U1 ). In fact, it can be shown that amin (U) → 0 as U tends to the ρ -axis. So, the construction can always be partly made. Construction 3 can be partly made, by letting the first stationary contact from UL to U1 shift a from aL to a ∈ [amin (UL ), aL ]. The form of the solution is the same as before. That is, the solution is a stationary wave from UL to a supersonic U1 with the cross-section a ∈ [amin (UL ), aL ], followed by a zero-speed 1-shock from U1 to a subsonic U2 , followed by another stationary wave from U2 to U3 iff aR ≥ amin (U2 ), and finally followed by a 2-wave from U3 to UR . Since both curves L and W1→3 (UL ) belong to the subsonic region, if the backward curve W2B (UR ) lies entirely in the supersonic region, it cannot meet these two curves. So, there are no solutions of the Riemann problem. That the backward wave curve W2B (UR ) lies entirely in the supersonic region means its u-intercept is nonnegative, i.e., 0 p (ρ ) d ρ ≥ 0, uR + ρ ρR or ρR p (ρ ) uR ≥ dρ . (8.4.12) ρ 0

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8 Compressible fluids in a nozzle with discontinuous . . .

Thus, under the conditions (8.4.11) and (8.4.12), there is no Riemann solution. Since UL is a supersonic state, the local existence of Riemann solutions is lost if (8.4.11) holds.

Multiple solutions If the point UL#0 is located above and the point UL0# is located below the curve W2B (UR ), see Figure 8.11, then the curve W2B (UR ) intersects each of the three curves W1F (U1 ), L and W1→3 (UL ). The Riemann problem accordingly admits three distinct solutions. The fact that the point UL#0 = (ρL#0 , u#0 L ) is located above and the B point UL0# = (ρL0# , u0# L ) is located below the curve W2 (UR ) means B #0 u#0 L > ω2 (UR ; ρL ) B 0# u0#0 L < ω2 (UR ; ρL ),

(8.4.13)

where ω2B is defined by (8.2.25) and (8.3.8). Thus, under the conditions (8.4.13), the curve W2B (UR ) intersects W1F (U1 ), L and W1→3 (UL ) at three points U1 ,U2 , and U3 , respectively. Each of these intersection points is corresponding to the Riemann solution in Constructions 1, 2, and 3 above.

Fig. 8.11 Multiple solutions for supersonic UL under (8.4.13): There may be up to three Riemann solutions given by Constructions 1, 2, and 3 when W2B (UR ) meets all the three curves W1F (U1 ), L and W1→3 (UL ).

Assume now that aR < aL . If a < amin (UL ), for some values a ∈ [aR , aL ], then some part of or the whole curve W1F (U1 ), L , or W1→3 (UL ) may not exist. Thus, the backward wave curve W2B (UR ) may intersect the union W1F (U1 ) ∪ L ∪ W1→3 (UL ) at no point, one point, two points, or three

8.4 Solutions of the Riemann problem

225

points. This shows that the Riemann problem may admit no solution, one solution, two solutions, or three distinct solutions.

Unique solution Under the conditions (8.4.3) and (8.4.13), the backward curve W2B (UR ) intersects W1F (U1 ) at one supersonic point, and it does not intersect the curves L and W1→3 (UL ), since it lies entirely in the supersonic region. Thus, the Riemann problem admits a unique solution, if (8.4.3) and (8.4.13) hold.

8.4.5 Construction 4: subsonic/supersonic There are two types of configurations of a Riemann solution, depending on whether aL < aR or aL ≥ aR . First, consider the case aL < aR . From UL , the solution begins with a 1-rarefaction wave from UL to a state U1 on the sonic curve C + . That is {U1 } = W1F (UL ) ∩ C + . The solution is followed by a stationary shock from U1 to a state U2 ∈ G1 to shift the cross-section to the level a = aR . Let {U3 } = W2B (UR ) ∩ W1F (U2 ). The solution is then followed by a 1-wave from U2 to U3 . Finally, the solution arrives at UR by a 2-wave; see Figure 8.12.

Fig. 8.12 Riemann solution (8.4.14) by Construction 4, case aL < aR

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8 Compressible fluids in a nozzle with discontinuous . . .

Thus, the formula for the solution is given by W1 (UL ,U1 ) W3 (U1 ,U2 ) W1 (U2 ,U3 ) W2 (U3 ,UR ).

(8.4.14)

The states U1 ,U2 , and U3 can be calculated as follows. Since U1 ∈ W1F (UL ) ∩ ∈ W3 (U1 ),U3 ∈ W1F (U2 ) ∩ W2B (UR ), we find

C + ,U2

u1 = ω1F (UL ; ρ1 ), λ1 (U1 ) = 0, u2 = ω3 (U1 ; ρ2 ), Φ (U1 ; ρ2 , aL ) = 0,

ρ2 < ρmax (ρ1 , u1 ),

(8.4.15)

u3 = ω2B (UR ; ρ3 ), u3 = ω1F (U2 ; ρ3 ), where the function ω1F is defined by (8.2.18) and (8.3.5), and the function ω2B is defined by (8.2.25) and (8.3.8), the function ω2 is defined by (8.3.13), and Φ is defined by (8.3.14). The six equations in (8.4.15) can determine the six quantities ρi , ui , i = 1, 2, 3. The configuration of the Riemann solution (8.4.14) in the (x,t)-plane is given by Figure 8.13.

Fig. 8.13 Riemann solution with structure (8.4.14) in the (x,t)-plane

Second, consider the case aL ≥ aR . As seen in Section 8.3, a non-trivial stationary contact from a state U− to a state U+ must satisfy the condition a+ > amin (U− ).

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227

Moreover, if U− is a sonic state, then amin (U− ) = a− . Thus, it is not possible to jump from any sonic state U1 with the cross-section aL to another state with the cross-section aR when aL ≥ aR , as done in the case aL < aR . However, there is another type of configuration of Riemann solution in this case as follows. First, we observe that from any sonic state U ∈ C + with the crosssection aR , it is possible to jump to a subsonic state U˜ with the cross-section aL . The set of these subsonic states U˜ forms a curve, denoted by Ca , which goes along C + . Let the forward 1-wave curve W1F (UL ) intersect this curve at a subsonic point U1 . We can now describe a Riemann solution: the solution begins with a 1-rarefaction wave from UL to a state U1 ∈ G2 at which it jumps to a state U2 ∈ C + by a stationary wave. Let U3 be the intersection point of the curves W1F (U2 ) and W2B (UR ). Then, the solution continues with a 1-rarefaction wave from U2 to U3 , followed by a 2-wave from U3 to UR . So, the solution has the form W1 (UL ,U1 ) W3 (U1 ,U2 ) W1 (U2 ,U3 ) W2 (U3 ,UR ).

(8.4.16)

See Figure 8.14. The configuration of the Riemann solution (8.4.16) in the (x,t)-plane is given by Figure 8.15. Note that this construction still holds for some part of a subsonic right-hand state. Example 8.4.2. Let us take γ = 1.4 and consider the Riemann problem for the system (8.1.1)–(8.1.2), where the initial states UL = (ρL , uL , aL ) and UR = (ρR , uR , aR ) are given by Table 8.2. The Riemann solution of the form (8.4.14) at the time t = 0.1 is illustrated by Figure 8.16 in the interval [−1, 1]. The states U1 ,U2 , and U3 in the solution (8.4.14) are given by Table 8.2.

Table 8.2 States in the Riemann solution (8.4.14) by Construction 4 UL U1 U2 U3 UR

h 4 1.82426015588 0.23094834177 0.514260439794 3

u 0.2 1.33438567909 2.60548307955 1.80945134288 4

a 1 1 1.2 1.2 1.2

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8 Compressible fluids in a nozzle with discontinuous . . .

Fig. 8.14 Riemann solution (8.4.16) by Construction 4, case aL ≥ aR

Fig. 8.15 Riemann solution with structure (8.4.16) in the (x,t)-plane

8.4.6 Construction 5: subsonic/subsonic The left-hand state UL is in the subsonic region G2 or on the sonic curve C+ . From UL the solution begins with a 1-wave from UL to some state U1 ∈ W1F (UL ) ∩ G2 , followed by a stationary shock to a state U2 ∈ G2 . Then, the solution arrives at UR by a 2-wave; see Figure 8.17. Thus, the formula for the solution is given by W1 (UL ,U1 ) W3 (U1 ,U2 ) W2 (U2 ,UR ).

(8.4.17)

As in Construction 2 we can geometrically describe the way to find the intermediate states U1 and U2 as follows. Let

8.4 Solutions of the Riemann problem

Fig. 8.16 A Riemann solution of the form (8.4.14) in [−1, 1] at the time t = 0.1

Fig. 8.17 Riemann solution (8.4.15) by Construction 5

{U2 } = W2B (UR ) ∩ W1→3 (UL ). Then {U1 } = W1F (UL ) ∩ W3 (U2 ). The condition

229

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8 Compressible fluids in a nozzle with discontinuous . . .

W2B (UR ) ∩ W1→3 (UL ) = 0/ means that the right-hand state UR is somewhere around the sonic region. The statesU1 ,U2 can be calculated as follows. SinceU1 ∈ W1F (UL ),U2 ∈ W3 (U1 ) ∩ B W2 (UR ), the four quantities can be calculated using the following four equations u1 = ω1F (UL ; ρ1 ), u2 = ω2B (UR ; ρ2 ), u2 = ω3 (U1 ; ρ2 ), Φ (U1 ; ρ2 , aR ) = 0,

(8.4.18)

ρ2 > ρmax (ρ1 , u1 ),

where the function ω1F is defined by (8.2.18) and (8.3.5), and the function ω2B is defined by (8.2.25) and (8.3.8), the function ω2 is defined by (8.3.13), and Φ is defined by (8.3.14). The configuration of the Riemann solution (8.4.17) in the (x,t)-plane is given by Figure 8.18.

Fig. 8.18 Riemann solution with structure (8.4.17) in the (x,t)-plane

Note that this construction still holds for some part of a supersonic right-hand state. We can also use a backward construction by employing the backward composite wave curve W2←3 (UR ). Indeed, let W1F (UL ) ∩ W2←3 (UR ) = {U1 }, and let U2 ∈ W2B (UR ) be the corresponding right-hand state of the admissible contact wave from U1 . Then, the solution still has the form (8.4.17). This backward construction may be preferred if aL < aR .

8.4 Solutions of the Riemann problem

231

8.4.7 Construction 6: resonant waves for subsonic regime When the right-hand state UR is somewhere around the sonic curve C + , there is still another interesting case, where the solution may contain three waves of the same (zero) shock speed. First, the solution begins by a 1-rarefaction wave from UL to a state U1 on the sonic curve C + , where {U1 } = W1F (UL ) ∩ C + . The solution is followed by a stationary shock from U1 to a state U2 ∈ G1 with an intermediate cross-section level a = aM , followed by a zero-speed 1-shock from U2 to U3 in the subsonic region G2 . The solution then continues by another stationary shock from U3 to a state U4 to complete shifting the cross-section to the level a = aR . / we obtain a Riemann Such states U4 form a set Λ and whenever W2B (UR ) ∩ Λ = 0; solution. This solution arrives at UR from U4 by a 3-wave; see Figure 8.19. The formula for the solution by Construction 6 is thus given by W1 (UL ,U1 ) W3 (U1 ,U2 ) W1 (U2 ,U3 ) W3 (U3 ,U4 ) W2 (U4 ,UR ). (8.4.19) The states Ui , i ∈ {1, 2, 3, 4} and the intermediate cross-section level aM can be calculated as follows. Since U1 ∈ W1F (UL ) ∩ C + ,U2 ∈ W3 (U1 ),U3 ∈ W1F (U2 ),U4 ∈ W3 (U3 ) ∩ W2B (UR ), we have

Fig. 8.19 Riemann solution (8.4.19) by Construction 6

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8 Compressible fluids in a nozzle with discontinuous . . .

u1 = ω1F (UL ; ρ1 ),

λ1 (U1 ) = 0, Φ (U1 ; ρ2 , aM ) = 0, u2 = ω3 (U1 ; ρ2 ), λ¯ 1 (U2 ,U3 ) = 0,

ρ2 < ρmax (ρ1 , u1 ), (8.4.20)

u3 = ω1F (U2 ; ρ3 ), u4 = ω2B (UR ; ρ4 ), u4 = ω3 (U3 ; ρ4 ), Φ (U3 ; ρ4 , aR ) = 0,

ρ4 > ρmax (ρ3 , u3 ),

where the function ω1F is defined by (8.2.18) and (8.3.5), and the function ω2B is defined by (8.2.25) and (8.3.8), the function ω2 is defined by (8.3.13), and Φ is defined by (8.3.14). The nine equations in (8.4.20) can determine the nine quantities aM and ρi , ui , i ∈ {1, 2, 3, 4}. The configuration of the Riemann solution (8.4.19) in the (x,t)-plane is given by Figure 8.20.

Fig. 8.20 Riemann solution with structure (8.4.19) in the (x,t)-plane

8.4.8 Existence and uniqueness properties for subsonic regime As seen from Constructions 4, 5, and 6, due to the continuity of the constructions, the existence of solutions of the Riemann problem can be obtained for a large domain of the right-hand state for the subsonic left-hand state. Let the sonic state U1 be the intersection point of the curves W1F (UL ) and C+ with the cross-section level aL . Let U2 be the supersonic state with the cross-section

8.4 Solutions of the Riemann problem

233

level aR obtained from a stationary jump from the sonic state U1 . This condition implies that aL < aR . If U10 is located above and U2# is located below the backward curve W2B (UR ), then W2B (UR ) intersects with each curve W1→3 (UL ), Λ , and W1F (U2 ); see Figure 8.21. The three intersection points U4 ,U5 , and U6 give us three distinct Riemann solutions in Constructions 4, 5, and 6. Therefore, letting U10 = (ρ10 , u01 ) and U2# = (ρ2# , u#2 ), we have aL < aR u01 > ω2B (UR ; ρ10 ) u#2

(8.4.21)

< ω2B (UR ; ρ2# ),

where ω2B is defined by (8.2.25) and (8.3.8). Thus, under the conditions (8.4.21), the Riemann problem can admit up to three different solutions.

Fig. 8.21 Multiple solutions for subsonic UL under the conditions (8.4.21): each choice of the intersection point U4 ,U5 , or U6 corresponds to a different solution described in Constructions 4, 5, and 6, respectively.

Let us consider the case aR < aL . If a < amin (UL ), for some values a ∈ [aR , aL ], then some part of or the whole curve W1F (U1 ), L , or W1→3 (UL ) may not exist. Consequently, the backward wave curve W2B (UR ) may intersect the union W1F (U1 ) ∪ L ∪ W1→3 (UL ) at no point, one point, two points, or three points. So, the Riemann problem may admit no solution, one solution, two solutions, or three distinct solutions.

234

8 Compressible fluids in a nozzle with discontinuous . . .

8.5 Bibliographical notes We provide a brief selection of the most relevant papers and refer the reader to the bibliography at the end of this monograph for additional references. The Riemann problem for the model of isentropic flows in a nozzle with variable cross-section, assuming a linear equation of state, was considered by Marchesin-Paes-Leme [241]. The equation ∂t a = 0 and the study of nonconservative products were introduced by LeFloch [207, 208]. The Riemann problem for the model of isentropic flows in a nozzle with variable cross-section was solved in LeFloch-Thanh [227]. The Riemann problem for a general fluid flow in a nozzle with variable cross-section was studied by Thanh [301] and Andrianov-Warnecke [20]. The minimum entropy principle for the model of a fluid flow in a nozzle with variable cross-section was established by Kr¨oner-LeFloch-Thanh [201]. Conservation laws with resonances were first analyzed by Isaacson-Temple [174, 175]. The Riemann problem for systems with resonances was considered by Goatin-LeFloch [141]. The Riemann problem for shallow water equations with discontinuous topography was considered by LeFloch-Thanh [228, 229], AlcrudoBenkhaldoun [11], Bernetti-Titarev-Toro [49], Rosatti-Begnudelli [275], and ThanhCuong [308]. The Riemann problem for the isentropic model of two-phase flows was solved by Thanh [305] and Thanh-Cuong [307]. The Baer-Nunziato model of two-phase flows was studied by Andrianov-Warnecke [21] and SchwendemanWahle-Kapila [283].

Chapter 9

Compressible fluids in a nozzle with discontinuous cross-section—General flows

As discussed earlier, the appearance of stationary contact discontinuities is significant in the previous chapter. When the fluid is assumed to be isentropic, the system of governing equations for gas dynamics consists of two conservation laws of mass and momentum. There is no linearly degenerate characteristic field in the isentropic gas dynamics equations. However, contact discontinuities are among the most basic topics in gas dynamics equations for non-isentropic fluids. Both kinds of contact discontinuities are expected to be present in the study of a general fluid model in a nozzle with variable cross-section. It is interesting to see how a Riemann solution contains two kinds of contact discontinuities in the model of non-isentropic fluid flows in a nozzle with variable cross-section. This question will be answered in the current chapter. The model of fluid flows in a nozzle with variable cross-section can be viewed as a simplified model of two-phase flows. The system is not strictly hyperbolic, where characteristic speeds may coincide. The resonance phenomenon occurs when multiple waves associated with different characteristic fields travel with the same shock speed. Solutions to the Riemann problem are made up of a finite number of elementary waves associated with characteristic fields. An admissibility criterion for stationary contact discontinuities is also imposed. The construction of composite wave curves plays an important role in solving the Riemann problem. Wave curves will be projected onto the (p, u)-plane to find intermediate states that separate elementary waves of Riemann solutions. There are still regions where the Riemann problem has no solution. When it does have solutions, it may have exactly one solution in one region but two or three solutions in another region. The Riemann solutions will be presented computationally, where each solution corresponds to a set of algebraic equations. This method can help program exact Riemann solutions using common computational software such as MATLAB. Qualitative properties of entropy solutions will also be investigated. The entropy inequality for standard pairs of entropy in fluid dynamics equations is investigated, © Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 9

235

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

leading to the usual divergence form. This enables us to establish a minimum entropy principle for entropy solutions, indicating that entropy increases with respect to time.

9.1 Basic properties The model of fluid flows in a nozzle with variable cross-section derived in Chapter 2 is given by ∂t (aρ ) + ∂x (aρ u) = 0,

∂t (aρ u) + ∂x (a(ρ u2 + p)) = p(ρ )∂x a, ∂t (aρ e) + ∂x (au(ρ e + p)) = 0, x ∈ R I , t > 0,

(9.1.1)

where the notation ρ , ε , T, S, and p stand for the thermodynamical variables: density, internal energy, absolute temperature, entropy, and pressure, respectively; u is the I , is velocity and e = ε + u2 /2 is the total energy. The function a = a(x) > 0, x ∈ R the cross-sectional area.

9.1.1 Hyperbolicity and non-strict hyperbolicity To investigate basic properties of the system (9.1.1), we can supplement it with the trivial equation ∂t a = 0, to obtain the following system of balanced laws

∂t (aρ ) + ∂x (aρ u) = 0, ∂t (aρ u) + ∂x (a(ρ u2 + p)) = p∂x a, ∂t (aρ e) + ∂x (au(ρ e + p)) = 0, ∂t a = 0,

(9.1.2)

x∈R I , t > 0.

Let us consider the pair of independent thermodynamics variables (ρ , S). The equations of state will then be of the form p = p(ρ , S),

ε = ε (ρ , S),

T = T (ρ , S).

Therefore, the system (9.1.2) can be expanded for the unknown function U = (ρ , u, S, a) as follows. Consider the first equation of (9.1.2), which can be rewritten as at ρ + aρt + a(ρ u)x + ax ρ u = 0, where (.)x = ∂ (.)/∂ x. Using the third equation in (9.1.2), dividing the last equation by a, we get

9.1 Basic properties

237

ρu ax = 0. a The second equation in (9.1.2) can be expanded by ρt + uρx + ρ ux +

(9.1.3)

(aρ )t u + aρ ut + (aρ u)ux + (aρ u)x u + ax p + apx = pax . Simplifying and then re-arranging terms of the last equation gives us 1 u((aρ )t + (aρ u)x ) + aρ ut + uux + (pρ ρx + pS Sx ) = 0. ρ Using the first equation in (9.1.2), we can see that the first term of the last equation vanishes. Discard the first term of the last equation and then divide it by aρ , we get ut +

pρ pS ρx + uux + Sx = 0. ρ ρ

(9.1.4)

Next, the third equation in (9.1.2) can be written as aρ et + (aρ )t e + aρ uex + (aρ u)x e + (aup)x = 0, or ((aρ )t + (aρ u)x )e + aρ (et + uex ) + (aup)x = 0. The first term in the last equation vanishes, thanks to the first equation in (9.1.2). Since u2 e=ε+ , 2 the last equation becomes aρ (εt + uεx ) + aρ u(ut + uux ) + (aup)x = 0. From (9.1.4), we find ut + uux = −

px . ρ

From the last two equations, we get aρ (εt + uεx ) − aupx + (aup)x = 0, or aρ (εt + uεx ) + p(au)x = 0. Using the thermodynamics identity d ε = T dS − pdv, so that

v=

1 , ρ

p ρt , ρ2 p εx = T Sx − pvx = T Sx + 2 ρx , ρ

εt = T St − pvt = T St +

we obtain from the last equation aρ T (St + uSx ) + aρ ·

p (ρt + uρx ) + p(au)x = 0, ρ2

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

or aρ T (St + uSx ) +

ap (ρt + uρx ) + p(au)x = 0. ρ

On the other hand, it follows from (9.1.3) that

ρt + uρx = −(ρ ux +

ρu ax ), a

which, on substitution to the last equation, yields aρ T (St + uSx ) −

ρu ap ax ) + p(au)x = 0. (ρ ux + ρ a

That is, we have u aρ T (St + uSx ) − ap(ux + ax ) + p(ax u + uax ) = 0. a Simplifying the last equation gives us aρ T (St + uSx ) = 0. Dividing both sides of the last equation by aρ T , we obtain St + uSx = 0. Thus, the system (9.1.2) can be written in matrix form by

∂t U + A(U)∂xU = 0, where

⎛ ⎞ ρ ⎜u⎟ ⎟ U =⎜ ⎝S⎠ , a

⎛ u ⎜ ⎜ pρ (ρ , S) ⎜ A(U) = ⎜ ρ ⎜ ⎝ 0 0

ρ

(9.1.5)

0 pS (ρ , S) u ρ 0 u 0 0

uρ ⎞ a ⎟ ⎟ 0 ⎟ ⎟. ⎟ 0 ⎠ 0

(9.1.6)

The characteristic equation associated with the matrix A(U) in (9.1.6) is given by

λ (λ − u) (u − λ )2 − pρ (ρ , S) = 0. Providing that pρ (ρ , S) > 0,

(9.1.7)

the matrix A(U) admits four real eigenvalues,

λ1 (U) = u − c,

λ2 (U) = u,

λ3 (U) = u + c,

λ4 (U) = 0,

(9.1.8)

where c is the sound speed defined by c=

pρ (ρ , S).

(9.1.9)

9.1 Basic properties

239

The corresponding eigenvectors can be chosen as: ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ −ρ −pS ρ ⎜ c ⎟ ⎜ 0 ⎟ ⎜c⎟ ⎟ ⎜ ⎟ ⎜ ⎟ r1 (U) = ⎜ ⎝ 0 ⎠ , r2 (U) = ⎝ pρ ⎠ , r3 (U) = ⎝ 0 ⎠ , 0 0 0

⎛

⎞ ρ u2 ⎜ −upρ ⎟ ⎟, r4 (U) = ⎜ ⎝ ⎠ 0 2 a(pρ − u ) (9.1.10)

where pρ = pρ (ρ , S), pS = pS (ρ , S), and c is the sound speed defined by (9.1.9). One can see that the system may not be strictly hyperbolic, since the eigenvalues may coincide on certain surfaces. In particular,

λ1 (U) = λ4 (U), λ2 (U) = λ4 (U), λ3 (U) = λ4 (U),

U ∈ Σ+ := {U |u = c}, U ∈ Σ0 := {U |u = 0}, U ∈ Σ− := {U |u = −c},

(9.1.11)

where c is given by (9.1.9). Moreover, even under the assumption (9.1.7), the system (9.1.3) may not be hyperbolic. Indeed, let us consider the states U ∈ Σ+ , where λ1 (U) = λ4 (U) = 0. That is, the eigenvalue λ = 0 is a double eigenvalue. This occurs whenever u =

pρ (ρ , S). Then, the eigenvectors r1 (U) and r4 (U) given by (9.1.10) are linearly dependent ⎛ ⎛ ⎞ ⎞ ⎛ ⎞ ρ u2 ρ pρ ρ ⎜ −upρ ⎟ ⎜−cpρ ⎟ ⎜−c⎟ ⎜ ⎟ ⎟=⎜ ⎟ r4 (U) = ⎜ ⎝ ⎠ ⎝ 0 ⎠ = pρ ⎝ 0 ⎠ = pρ r1 (U). 0 0 0 a(pρ − u2 ) We can verify this directly by considering eigenvectors of A(U) given by (9.1.6) associated with the eigenvalue λ = 0. If u = c, the eigenvalue λ = 0 is repeated twice, and its eigenvectors r(x1 , x2 , x3 , x4 ) satisfy A(U)r = 0, or

⎛

uρ ⎞ ⎛ ⎞ a ⎟ x1 ⎜ ⎜ u2 ⎟⎜ ⎟ pS (ρ , S) x2 ⎟ ⎜ 0 ⎟ = 0. ⎜ρ u ⎟⎜ ⎝ ρ ⎜ ⎟ x3 ⎠ ⎝0 0 ⎠ u 0 x4 0 0 0 0 u ρ

0

Expanding this vector equation yields a 4 × 4 linear system of algebraic equations in x1 , x2 , x3 , and x4 . The fourth equation of this system is trivial and so it is discarded. The third equation of this system gives us ux3 = 0.

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Since u = c = 0, the last equation gives us x3 = 0. The first two equations of that system become ux1 + ρ x2 +

uρ x4 = 0, a

u2 x1 + ux2 = 0. ρ

(9.1.12)

Multiplying the second equation in (9.1.12) by ρ /u and subtracting the result from the first equation, we obtain uρ x4 = 0, a so that, since u = c = 0, x4 = 0. Therefore, the two equations (9.1.12) are reduced to the single condition ux1 + ρ x2 = 0, which yields us x2 = − ρu x1 . Thus, the eigenvectors associated with the double eigenvalue λ = 0 of A(U) when u = c are given in the form ⎛ ⎞ ρ x1 ⎜ x1 −u⎟ ⎜ r(U) = ⎝ ⎟ = r1 (U). ⎠ 0 ρ ρ 0 This means that the eigenspace associated with this double eigenvalue λ = 0 has dimension 1. The matrix A(U) therefore does not have any basis of eigenvectors for U ∈ Σ+ . This conclusion holds also for U ∈ Σ− , by a similar argument. So, the system is not hyperbolic on the surfaces Σ± . When U ∈ Σ0 , the system is not strictly hyperbolic, since

λ2 (U) = λ4 (U), that is, u = 0. However, the corresponding eigenvectors r2 and r4 given by (9.1.10) ⎛ ⎛ ⎞ ⎞ −pS 0 ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎟ r2 (U) = ⎜ ⎝ pρ ⎠ , r4 (U) = ⎝ 0 ⎠ 0 apρ are still linearly independent. The matrix A(U) when u = 0 therefore admits a complete set of eigenvectors. So, the system is hyperbolic, but not strictly hyperbolic on the surface Σ0 .

9.1 Basic properties

241

Since

λ1 (U) < λ2 (U) < λ3 (U), for all U, the system is strictly hyperbolic in the regions where the eigenvalue λ4 does not coincide with any of the other eigenvalues λi , i = 1, 2, 3. Thus, the system is strictly hyperbolic apart from the surfaces u = ±c, 0. Precisely, we can describe the strictly hyperbolic regions in which the system is strictly hyperbolic by setting G1 = {U |λ1 (U) > 0}, G2 = {U |λ1 (U) < 0 < u}, G3 = {U |u < 0 < λ3 (U)},

(9.1.13)

G4 = {U |0 > λ3 (U)}. Furthermore, let us introduce some terminologies. A state U is said to be subsonic if |u| < c. A state U is said to be supersonic if |u| > c. A state U is said to be sonic if |u| = c. Thus, G1 and G4 are the supersonic regions, and G2 and G3 are subsonic regions.

9.1.2 Nonlinearity and linear degeneracy of characteristic fields Let us now consider the nonlinearity and the linear degeneracy of the characteristic fields of the system (9.1.3). The 2- and the 4-characteristic fields are linearly degenerate, since Dλ2 (U) · r2 (U) = 0, and Dλ4 (U) · r4 (U) = 0. Moreover, a simple calculation gives us −Dλ1 (U) · r1 (U) = Dλ3 (U) · r3 (U) = c + ρ cρ =

pvv (v, S) , 2cv

(9.1.14)

where v = 1/ρ . Thus, the 1- and the 3-characteristic fields are genuinely nonlinear whenever (9.1.15) pvv (v, S) = 0.

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

9.2 Elementary waves Elementary waves are simple waves that connect two constant states. Elementary waves will be used to construct solutions to the Riemann problem by combining these waves.

9.2.1 Rarefaction waves The first kind of elementary waves we consider are rarefaction waves. These waves are smooth self-similar solutions of the system (9.1.2) of the form U(x,t) = (ρ , u, a)(x,t) = V (ξ ), ξ = x/t. Again, by substituting U(x,t) = V (x/t) into (9.1.5), we obtain 1 x x x x −( 2 )V ( ) + ( )A V ( ) V ( ) = 0, t t t t t or x (A(U(ξ )) − ξ I)U (ξ ) = 0, ξ = . (9.2.1) t Assume that U (ξ ) = 0. Then, there is an index i ∈ {1, 3} such that V (ξ ) = α (ξ )ri (V (ξ )),

λi (V (ξ )) = ξ ,

(9.2.2)

for some function α (ξ ). Differentiating the second equation in (9.2.2) with respect to ξ yields Dλi (V (ξ )) ·V (ξ ) = 1. Multiplying the first equation in (9.2.2) by Dλi (U(ξ )), it follows from the last equation that α (ξ )Dλi (V (ξ )) · ri (V (ξ )) = 1. This implies that

α (ξ ) =

1 . Dλi (V (ξ )) · ri (V (ξ ))

(9.2.3)

Substituting α (ξ ) from (9.2.3) into (9.2.2), we have V (ξ ) =

1 ri (V (ξ )), Dλi (V (ξ )) · ri (V (ξ ))

λi (V (ξ )) = ξ ,

i = 1, 3.

(9.2.4)

Now, let U± be on the integral curve of the first equation in (9.2.4) such that V (λi (U− )) = U− ,

V (λi (U+ )) = U+ .

Then, the rarefaction wave of (9.1.5) connecting the left-hand state U− and the righthand state U+ is given by

9.2 Elementary waves

243

⎧ x ⎪ U− , ≤ λi (U− ), ⎪ ⎪ t ⎨ x x , λi (u− ) ≤ ≤ λi (U+ ), U(x,t) = V ⎪ t t ⎪ x ⎪ ⎩U+ , ≥ λi (U+ ), t

(9.2.5)

is a weak self-similar solution of (9.1.1). From the above argument, we can see that rarefaction waves can be described by the corresponding integral curves. So, let us first consider the integral curves associated with the first characteristic field passing through a given state U0 dU r1 (U) , = dξ Dλ1 · r1 (U) U(ξ0 ) = U0 . That is, we have

dρ dξ du dξ dS dξ da dξ

ρ , c + ρ cρ −c = , c + ρ cρ

(9.2.6)

=

(9.2.7)

= 0, = 0.

The last two equations of (9.2.7) means that the entropy S and the cross-section a remain constant along the integral curve, and therefore, S and a are constant across any 1-rarefaction wave. Whenever c is well-defined and dd ρξ = 0 along the integral curve, the density ρ can be used as a parameter of the integral curve. This observation is similar for u. Thus, under the conditions (9.1.7) and (9.1.15), we can use, for example, ρ as a parameter, and we divide the second equation by the first equation in (9.2.7) side by side to get c du =− . (9.2.8) dρ ρ Observe that we could integrate (9.2.8) to obtain the integral curve associated with the first characteristic field. However, as seen later, it is convenient to find formulas for integral curves parameterized as functions of the pressure p. From (9.2.7), one can see that the entropy S remains constant across a rarefaction wave. So, if the equation of state is given under the form

ρ = ρ (p, S),

and

v = v(p, S),

where v = 1/ρ is the specific volume, then we have through each 1-rarefaction wave issuing from U0 :

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

du c 1 =− =− =− dp ρ∂ρ p ρ ∂ρ p

∂ p ρ (p, S0 ) . ρ (p, S0 )

(9.2.9)

Integrating (9.2.9) yields the integral curve O1 (U0 ) associated with the first characteristic field as p ∂ p ρ (τ , S0 ) dτ . (9.2.10) O1 (U0 ) : u = u0 − ρ (τ , S0 ) p0 Similarly, the integral curve O3 (U0 ) associated with the third characteristic field as p ∂ p ρ (τ , S0 ) O3 (U0 ) : u = u0 + dτ . (9.2.11) ρ (τ , S0 ) p0 Assume in the following that the condition (9.1.7) holds. We can then determine the domain of the curves of rarefaction waves as in the case of the usual Euler equations for fluid dynamics. Precisely, since the characteristic speed λ1 (U) is increasing through any 1-rarefaction wave, it follows from (9.2.10) that the forward curve of 1-rarefaction waves R1F (U0 ) consisting of all right-hand states that can be connected to a given left-hand state U0 by a 1-rarefaction wave is given by p ∂ p ρ (τ , S0 ) F F dτ , R1 (U0 ) : u = u1 (U0 ; p) := u0 − ρ (τ , S0 ) p0 v = vF1 (U0 ; p) = v(p, S0 ), where

p ≤ p0 , p ≥ p0 ,

if if

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

to U to U

along along

O1 (U0 ), O1 (U0 ),

and O1 (U0 ) is defined by (9.2.10). Arguing similarly, we can see that the backward curve of 1-rarefaction waves R1B (U0 ) consisting of all left-hand states that can be connected to a given right-hand state U0 by a 1-rarefaction wave is given by p ∂ p ρ (τ , S0 ) B B dτ , R1 (U0 ) : u = u1 (U0 ; p) := u0 − ρ (τ , S0 ) p0 v = vB1 (U0 ; p) = v(p, S0 ), where

p ≥ p0 , p ≤ p0 ,

if if

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

to U to U

along along

O1 (U0 ), O1 (U0 ),

and O1 (U0 ) is defined by (9.2.10). In the same manner, the forward curve of 3-rarefaction waves R3F (U0 ) consisting of all right-hand states that can be connected to a given left-hand state U0 by a 3-rarefaction wave is given by

9.2 Elementary waves

245

R3F (U0 ) :

u = uF3 (U0 ; p) := u0 +

p p0

∂ p ρ (τ , S0 ) dτ , ρ (τ , S0 )

v = vF3 (U0 ; p) = v(p, S0 ), where

p ≥ p0 , p ≤ p0 ,

if if

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

to U to U

along along

O3 (U0 ), O3 (U0 ),

and O3 (U0 ) is defined by (9.2.11). The backward curve of 3-rarefaction waves R3B (U0 ) consisting of all left-hand states that can be connected to a given right-hand state U0 by a 3-rarefaction wave is given by p ∂ p ρ (τ , S0 ) dτ , R3B (U0 ) : u = uB3 (U0 ; p) := u0 + ρ (τ , S0 ) p0 v = vB3 (U0 ; p) = v(p, S0 ), where

p ≤ p0 , p ≥ p0 ,

if if

∂vv p(v, S) > 0 from U0 ∂vv p(v, S) < 0 from U0

to U to U

along along

O3 (U0 ), O3 (U0 ),

and O3 (U0 ) is defined by (9.2.11). As seen later, it will be convenient to project the wave curves into the (p, u)plane. This can be done as in the case of the usual Euler equations for fluid dynamics. Example 9.2.1. Consider a polytropic and ideal fluid, whose equations of state are given by (9.2.12) p = (γ − 1)ρε , or, using (ρ , S) as the independent thermodynamic variables: S − S∗ ργ , p = p(ρ , S) = (γ − 1) exp Cv where γ > 1 is the adiabatic exponent, Cv > 0 is the specific heat at constant volume, and S∗ is constant. Observe that for a polytropic and ideal fluid, γ and Cv are constants. It is not difficult to check that for a fluid (9.2.12), the forward 1-rarefaction curve R1F (U0 ) consisting of all right-hand states U that can be connected to the left-hand state U0 by a 1-rarefaction wave can be parameterized by the pressure p as p ρ p (τ , S0 ) F F R1 (U0 ) : u = u1 (U0 ; p) = u0 − dτ , p0 ρ (τ , S0 ) S −S 2γ 1/2 (γ −1)/2γ ∗ 0 = u0 − exp ), p ≤ p0 . (p(γ −1)/2γ − p0 2Cv γ (γ − 1)1−1/2γ

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

The backward 3-rarefaction curve R3B (U0 ) of a fluid (9.2.12) consisting of all lefthand states U that can be connected to the right-hand state U0 by a 3-rarefaction wave can be parameterized by the pressure p as p ρ p (τ , S0 ) B B R3 (U0 ) : u = u3 (U0 ; p) = u0 + dτ , ρ (τ , S0 ) p0 S −S 2γ 1/2 (γ −1)/2γ ∗ 0 = u0 + exp ), p ≤ p0 . (p(γ −1)/2γ − p0 2Cv γ (γ − 1)1−1/2γ

9.2.2 Shock waves Consider shock waves of the form

U0 , x < λ t, U(x,t) = U1 , x > λ t,

where U0 ,U1 are the left-hand and right-hand states, respectively, and λ is the shock speed. Applying the Rankine-Hugoniot relation of the third equation in (9.1.2) for this shock −λ [a] = 0, where [a] := a1 − a0 is the jump of the quantity a. The last equation implies that shock waves of (9.1.2) can be classified into two distinct kinds: (i) Shock waves across which the cross-sectional area a remains constant: a1 = a0 , (ii) Shock waves with zero propagation speed λ = 0. First, we consider the case (i). Eliminating a from (9.1.2), we obtain the usual Euler equations for fluid dynamics

∂t ρ + ∂x (ρ u) = 0, ∂t (ρ u) + ∂x (ρ u2 + p) = 0, ∂t (ρ e) + ∂x (u(ρ e + p)) = 0. As usual, the Rankine-Hugoniot relations corresponding to the last Euler equations are given by − λ¯ [ρ ] + [ρ u] = 0, − λ¯ [ρ u] + [ρ u2 + p] = 0, − λ¯ [ρ e] + [u(ρ e + p)] = 0, where [ρ ] = ρ1 − ρ0 , [ρ u] = ρ1 u1 − ρ0 u0 , etc., and λ¯ is the speed of propagation of the discontinuity connecting the states U0 and U1 .

9.2 Elementary waves

247

Thus, the Hugoniot set H (U0 ) can be obtained as the one in the usual Euler equations for fluid dynamics. Precisely, this set consists of all states U1 so that u1 − u0 m := ρ0 (u0 − λ¯ ) = ρ1 (u1 − λ¯ ) = , v1 − v0 p1 − p0 , m2 = − v1 − v0 1 ε1 − ε0 + (p1 + p0 )(v1 − v0 ) = 0. 2

(9.2.13)

It follows from (9.2.13) that the specific volume and therefore the density is given as a function of the pressure and the velocity across a jump associated with the first three characteristic fields (u − u0 )2 . (9.2.14) v1 = v0 − p1 − p0 Moreover, it is derived from (9.2.13) that if m = 0, then u0 = λ¯ = u1 , p0 = p1 . Thus, the discontinuity is exactly the 2-contact discontinuity corresponding to λ¯ = λ¯ 2 = u. Using the same argument as in the case of the usual Euler equations for fluid dynamics, one can see that the specific volume, and therefore, the density can be calculated from the left-hand and right-hand states of the discontinuity using (9.2.14) when the corresponding wave is a shock. Let us now discuss the admissibility criteria for shock waves. When m = 0, we obtain a 1-shock if m > 0 and a 3-shock if m < 0. And so the Hugoniot curves Hi (U0 ), i = 1, 3 associated with the first and the third characteristic fields are defined. If the fluid satisfies the condition (9.1.15) for all U, then the 1st and the 3rd characteristic fields are genuinely nonlinear, and so we can use Lax’s shock inequalities as the admissibility criterion for the shock

λi (U0 ) > λ¯ (U0 ,U1 ) > λi (U1 ),

i = 1, 3,

(9.2.15)

where λ¯ (U0 ,U1 ) is the shock speed. Otherwise, we can use the Liu entropy condition

λ¯ (U0 ,U) ≥ λ¯ (U0 ,U1 ),

for any U ∈ Hi (U0 ) between U0 and U1 ,

i = 1, 3. (9.2.16)

Other admissibility criteria for shock waves can also be applied. As in the usual Euler equations for fluid dynamics, admissible shock waves in the first and third characteristic fields form curves of admissible shocks. Precisely, we define: – The first forward shock curve issuing from U0 , denoted by S1F (U0 ), is the set of all right-hand states that can be connected to a given left-hand state U0 by an admissible 1-shock;

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

– The third forward shock curve issuing from U0 , denoted by S3F (U0 ), is the set of all right-hand states that can be connected to a given left-hand state U0 by an admissible 3-shock; – The first backward shock curve issuing from U0 , denoted by S1B (U0 ), is the set of all left-hand states that can be connected to a given right-hand state U0 by an admissible 1-shock; – The third backward shock curve issuing from U0 , denoted by S3B (U0 ), is the set of all left-hand states that can be connected to a given right-hand state U0 by an admissible 3-shock. The wave curves associated with the genuinely nonlinear characteristic fields are then defined as Wi F (U0 ) := SiF (U0 ) ∪ RiF (U0 ), (9.2.17) Wi B (U0 ) := SiB (U0 ) ∪ RiB (U0 ), i = 1, 3. Furthermore, we will assume that the wave curves Wi (U0 ) can be parameterized by the pressure p as Wi F (U0 ) :

u = uFi (U0 ; p),

v = vFi (U0 ; p),

p ≥ 0,

Wi (U0 ) :

u = uBi (U0 ; p),

= vBi (U0 ; p),

p ≥ 0,

B

v

i = 1, 3.

(9.2.18)

Example 9.2.2. Consider a polytropic, ideal fluid, whose equations of state are given by (9.2.12). The condition (9.1.15) for the fluid (9.2.12) is satisfied for all (v, S). It is easy to check that the following conclusions hold: (a) For a polytropic and ideal fluid (9.2.12), Lax’s shock inequalities for 1-shocks are equivalent to

ρ1 ≥ ρ0 ,

p1 ≥ p0 ,

S1 ≥ S0 ,

u1 ≤ u0 .

(b) For a polytropic and ideal fluid (9.2.12), Lax’s shock inequalities for 3-shocks are equivalent to

ρ1 ≤ ρ0 ,

p1 ≤ p0 ,

S1 ≤ S0 ,

u1 ≥ u0 .

The shock curves are given as follows. Set

μ=

γ −1 . γ +1

For the first forward shock curve S1F (U0 ), the velocity is given by (1 − μ )v0 F , p ≥ p0 , u = u1 (U0 ; p) = u0 − (p − p0 ) p + μ p0 while the specific volume can be obtained by using (9.2.14), and after a straightforward calculation, it is given by

9.2 Elementary waves

249

v = vF1 (U0 ; p) = v0 − =

(uF1 (U0 ; p) − u0 )2 p − p0

v0 (μ p + p0 ) . p + μ p0

The third forward shock curve S3F (U0 ) is given by S3F (U0 ) :

u = uF3 (U0 ; p) = u0 + (p − p0 ) v = vF3 (U0 ; p) =

u = uB1 (U0 ; p) = u0 − (p − p0 ) v = vB1 (U0 ; p) =

p ≤ p0 ,

v0 (μ p + p0 ) . p + μ p0

The first backward shock curve S1B (U0 ) is given by S1B (U0 ) :

(1 − μ )v0 , p + μ p0

(1 − μ )v0 , p + μ p0

v0 (μ p + p0 ) , p + μ p0

where

p ≤ p0 ,

μ=

γ −1 . γ +1

The third backward shock curve S3B (U0 ) is given by S3B (U0 ) :

pu = uB3 (U0 ; p) = u0 + (p − p0 ) v = vB3 (U0 ; p) =

(1 − μ )v0 , p + μ p0

p ≥ p0 ,

v0 (μ p + p0 ) . p + μ p0

It is not difficult to check that for a polytropic and ideal fluid (9.2.12), the functions uF1 (U0 ; p) and uB1 (U0 ; p) are continuous, monotone decreasing in p, and the functions uF3 (U0 ; p) and uB3 (U0 ; p) are continuous, monotone decreasing in p.

9.2.3 Stationary contact waves Next, let us consider the case (ii) that the component a is discontinuous, and therefore, the shock speed vanishes. The solution is independent of time, and it is natural to search for a solution as the limit of a sequence of time-independent smooth solutions. These stationary smooth solutions satisfy the following ordinary differential equations: (aρ u) = 0, (a(ρ u2 + p)) = pa , (au(ρ e + p)) = 0,

(9.2.19)

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

where (.) = d(.)/dx. Solutions of (9.2.19) are constraint to the initial condition (ρ , u, S, a)(x0 ) = (ρ0 , u0 , S0 , a0 ). Recall that the specific enthalpy is given by h = ε + pv, which satisfies dh = T dS + v d p. The following theorem characterizes stationary weak solutions of the system (9.2.19) that satisfy an ordinary differential equation in divergence form. Lemma 9.2.1. The system (9.2.19) can be written in the following divergence form: (aρ u) = 0, u2 + h(ρ , S) = 0, 2 S = 0.

(9.2.20)

Consequently, any jump satisfying the jump relations [aρ u] = 0, u2 + h(ρ , S0 )] = 0, 2 [S] = 0,

[

(9.2.21)

is a weak solution of the system (9.1.2) in the sense of nonconservative products. Furthermore, the curve of discontinuities defined by the jump relations (9.2.21) coincides with the integral curve associated with the 4-characteristic field (λ4 (U), r4 (U)). Proof. Consider the integral curve of (9.2.20) passing through a given state U0 . If u0 = 0, then h (ρ , S) = 0, S = 0. It then holds from the identity dh = T dS + v d p that p (ρ , S) = 0, so that U is also a solution of (9.2.19). Consider u0 = 0. The first equation in (9.2.20) can be expressed as aρ u = a0 ρ0 u0 = C where C is a nonzero constant. Thus, the second equation can be written as (C · u + a · p) = p · a

9.2 Elementary waves

251

or

C · u + a · p = 0.

This yields uu +

p = uu + p v = 0, ρ

v=

1 . ρ

(9.2.22)

Now, the third equation in (9.2.19) can be written as

ε + uu + (pv) = 0.

(9.2.23)

Recall the thermodynamics identity that T dS = d ε + pdv. Since we are considering stationary waves, i.e., solutions independent of time, the thermodynamics identity applied to this kind of wave gives

ε = T S − pv . Substituting this into (9.2.23), we get T S + p v + uu = 0, or, from (9.2.22), we have S = 0. It follows that p v = h (ρ , S), and from (9.2.22), this yields the second equation in (9.2.20). The second statement follows a stability result in [114]. The proof for the third statement is similar to the one in the previous chapter for the isentropic flows, and the system dU(ξ ) = r4 (U(ξ )) dξ coincides with (9.2.20). Lemma 9.2.1 is completely proved.

As seen by Lemma 9.2.1, the jumps (9.2.21) are weak solutions of (9.1.2) and are associated with the 4-characteristic field, which is linearly degenerate. Thus, these jumps are the 4-contact discontinuities. The integral curve of (9.2.20) passing through each point (ρ0 , u0 , S0 , a0 ) can be parameterized by ρ , say ρ → (ρ , u(ρ ), S0 , a(ρ )), and satisfies

aρ u = a0 ρ0 u0 , u2 u2 + h(ρ , S) = 0 + h(ρ0 , S0 ), 2 2 S = S0 .

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Let ρ (x) → ρ1 as x → 0. Set u1 = u(ρ1 ) and a1 = a(ρ1 ). We can see that the states (ρ0 , u0 , S0 , a0 ), (ρ1 , u1 , S0 , a1 ) satisfy the jump relations [aρ u] = 0, u2 + h(ρ , S0 )] = 0, 2 [S] = 0.

[

So we find u=

a0 ρ0 u0 , aρ

u2 u2 + h(ρ , S0 ) = 0 + h(ρ0 , S0 ), 2 2 S = S0 .

(9.2.24)

In general, the equations (9.2.24) locally can define the 4th-wave curve W4 (U0 ) of the form p → U = (u, v, p) = u4 (U0 ; p), ρ ≥ 0. It is easy to check that for a polytropic and ideal fluid (9.2.12), this curve can be parameterized as p → u(p) in the (p, u)-plane, and it is monotone decreasing in p for u0 > 0 and monotone increasing in p for u0 < 0. Next, we aim to solve the equations (9.2.24). Substituting u from the first equation into the second equation in (9.2.24), and then multiplying the result by ρ 2 and re-arranging terms, we get 2 a u ρ 2 u 0 0 0 Φ (ρ ;U0 , a) := h(ρ , S0 )ρ 2 − 0 + h(ρ0 , S0 ) ρ 2 + = 0. (9.2.25) 2 a The equation (9.2.25) may be solved for ρ in terms of a. So, the state U that can be connected to U0 by a stationary wave depends on a.

Criterion for admissible contact waves As in the previous chapter, the Riemann problem for (9.1.2) may admit up to a one-parameter family of solutions. This phenomenon can be avoided by imposing a similar criterion on stationary waves. Motivated by the case of the isentropic model of fluid flows in a nozzle with discontinuous cross-section in the previous chapter, we impose the following admissibility criterion for stationary contact waves to select a unique physical solution.

9.2 Elementary waves

253

a-monotone Criterion for admissible contact waves (a) Along any stationary curve W4 (U0 ), the cross-sectional area a is monotone as a function of ρ . (b) The total variation of the cross-section component of any Riemann solution must not exceed (and, therefore, is equal to) |aL − aR |, where aL , aR are lefthand and right-hand cross-section levels. As in the isentropic case, it is interesting that an admissible contact wave must remain in the closure of the subsonic or supersonic region. Lemma 9.2.2. The a-monotone criterion implies that stationary waves remain in the closure of only one domain Gi , i = 1, 2, 3, 4. We omit the proof of Lemma 9.2.2 since it is just a slight modification of one of the isentropic cases in the previous chapter. It follows from Lemma 9.2.2 that if U0 is in the supersonic region, then ρs (U0 , a) is used, while ρb (U0 , a) is used when U0 is in the subsonic region. The above arguments allow us to define elementary waves of the system (9.1.1) which form Riemann solutions. Definition 9.2.1. Elementary waves for the system (9.2.26) are the following ones: • Admissible shocks, rarefaction waves, contact discontinuities of the usual Euler equations for fluid dynamics corresponding to the case a is constant in (9.1.1). • Admissible stationary contact waves satisfying the a-monotone Criterion.

9.2.4 Example: selection of admissible stationary contacts In this subsection, we restrict our consideration to a polytropic and ideal fluid (9.2.12). Then, the equation (9.2.25) becomes

Φ (ρ ;U0 , a) =

2κγ γ +1 2 2κγ γ −1 2 a0 u0 ρ0 2 ρ − u0 + ρ ρ + = 0, γ −1 γ −1 0 a

where

κ = A(S0 ),

A(S) = (γ − 1)exp

S−S ∗

Cv

(9.2.26)

.

Let us investigate properties of Φ defined by (9.2.26) as a function of ρ , and then characterize its zeros. If u0 = 0, then the equation Φ (ρ , a;U0 ) = 0 gives three roots, and therefore, there

(γ −1)/2 , S0 ) that can be connected are three states (ρ0 , 0, S0 ), (0, ± (2κγ )/(γ − 1)ρ0 to U0 by a stationary wave.

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Assume u0 = 0, the domain of Φ is given by 2κγ γ +1 2 2κγ γ −1 2 ρ − u0 + ρ ρ ≥ 0, γ −1 γ −1 0 or

ρ ≤ ρ¯ (U0 ) :=

γ −1 2κγ

γ −1

u20 + ρ0

1 γ −1

.

(9.2.27)

Thus, we need to investigate the function Φ on the interval [0, ρ¯ (U0 )], only. Finding critical points of Φ , we let d Φ (U0 , ρ ; a) γ +1 γ 2κγ γ −1 = 2κγ ρ − 2 u20 + ρ ρ = 0, dρ γ −1 γ −1 0 which yields a unique critical point

ρ = ρ∗ (U0 ) :=

1 γ −1 2 γ −1 γ −1 u20 + ρ0 . κγ (γ + 1) γ +1

(9.2.28)

This critical point is a minimum point, since d Φ (ρ ; a,U0 ) < 0, dρ d Φ (U0 , ρ ; a) > 0, dρ

ρ < ρ∗ (U0 ), (9.2.29)

ρ > ρ∗ (U0 ).

Moreover, we have

Φ (ρ = 0; a,U0 ) = Φ (ρ = ρ¯ ; a,U0 ) =

a u ρ 2 0 0 0 > 0, a

u0 = 0.

(9.2.30)

From (9.2.28) and (9.2.30), we can see that Φ admits a zero if and only if

Φ (ρ = ρ∗ ; a,U0 ) ≤ 0. Equivalently a ≥ amin (U0 ) :=

a0 ρ0 |u0 | . γ +1 √ 2 κγρ∗ (ρ0 , u0 )

(9.2.31)

If a > amin (U0 ), then there are exactly two values ρs (U0 , a) < ρ∗ (U0 ) < ρb (U0 , a) such that Φ (ρs (U0 , a); a,U0 ) = Φ (ρb (U0 , a); a,U0 ) = 0. By considering (p, S) as the two independent thermodynamics variables and phase domain in the (p, u, S, a) space, we have the following lemma.

9.2 Elementary waves

255

Lemma 9.2.3 (Stationary waves). There exists a stationary contact from a given state U0 = (p0 , u0 , S0 , a0 ) connecting to some state U = (p, u, S = S0 , a) if and only if a ≥ amin (U0 ). More precisely, we have: (i) If a < amin (U0 ), there are no stationary contacts. (ii) If a ≥ amin (U0 ) along the curve W4 (U0 ), there are exactly two points Us := (p = p(ρs (U0 , a), u = a0 ρ0 u0 /(aρs (U0 , a)), S = S0 , a) and Ub := (p = p(ρb (U0 , a), u = a0 ρ0 u0 /(aρb (U0 , a)), S = S0 , a), where ρs (U0 , a) ≤ ρ∗ (U0 ) ≤ ρb (U0 , a) satisfying

Φ (ρs (U0 , a); a,U0 ) = Φ (ρb (U0 , a); a,U0 ) = 0.

(9.2.32)

These two states Us ,Ub coincide only if a = amin (U0 ). Set p∗ (U0 ) = p(ρ∗ (U0 ), S0 ),

ps (U0 , a) = p(ρs (U0 , a), S0 ),

pb (U0 , a) = p(ρb (U0 , a), S0 )

Since the function ρ → p(ρ , S0 ) is increasing, we obtain the following results. Lemma 9.2.4. Given a state U0 = (p0 , u0 , S0 , a0 ). (a) We have ps (U0 , a) < p∗ (U0 ) < pb (U0 , a), p∗ (U0 ) > p0 , U0 ∈ G1 ∪ G4 , p∗ (U0 ) < p0 ,

a > amin (U0 ),

U0 ∈ G2 ∪ G3 ,

(b) The state Us = (ps (U0 , a), u = a0 ρ0 u0 /(aρs (U0 , a)), S = S0 , a) ∈ G1 if u0 < 0, and Us ∈ G4 if u0 > 0. The state Ub := (pb (U0 , a), u = a0 ρ0 u0 /(aρb (U0 , a)), S = S0 , a) ∈ G2 if u0 < 0, and Ub ∈ G3 if u0 > 0. In addition, we have – If a > a0 , then – If a < a0 , then

(c)

ps (U0 , a) < p0 < pb (U0 , a). p0 < ps (U0 , a) for p0 > pb (U0 , a) for

U0 ∈ G1 ∪ G4 , U0 ∈ G2 ∪ G3 .

amin (U, a) < a, amin (U, a) = a,

U ∈ Gi , i = 1, 2, 3, 4, U ∈ Σ+ ∪ Σ− ,

amin (U, a) = 0,

p=0

or

u = 0.

Wave speeds for waves associated with different characteristic fields may change. For a polytropic and ideal fluid with the equation of state (9.2.12), we can calculate from (9.2.13) the shock speeds as

256

9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

ρ u − ρ0 u0 ρ (u − u0 ) + u0 (ρ − ρ0 ) λ¯ = λ¯ (U0 ,U) = = ρ − ρ0 ρ − ρ0 ⎧ p+μ p0 ⎨ u0 − v0 (1−μ )v , for 1 − (forward) shocks u − u0 0 = u0 − v0 = v − v0 ⎩ u0 + v0 p+μ p0 , for 3 − (forward) shocks. (1−μ )v0

Thus, the 1-shock speed λ¯ = λ¯ 1 (U0 ,U) from a given left-hand state U0 to a righthand state U on the Hugoniot set issuing from U0 vanishes if u0 > 0, p = p˜0 :=

(1 − μ )u20 − μ p0 . v0

Besides, Lax’s shock inequalities require that p˜0 > p0 . This means (1 − μ )u20 − μ p0 > p0 v0 or u20 >

1+μ 1 = c2 . p0 v0 = γ p0 v0 = 1−μ ρ p (p0 , S0 )

Since u0 > 0, this is equivalent to

λ¯ 1 (U0 ,U) = u0 − c > 0 which says that U0 ∈ G1 . Similarly, backward 3-shock speed λ¯ 3 (U0 ,U) from a given right-hand state U0 to a left-hand state U vanishes if U0 ∈ G4 and p = We, therefore, arrive at the following proposition.

(1−μ )u20 v0

− μ p0 .

Proposition 9.2.1. (a) The 1-shock speed λ¯ 1 (U0 ,U), (for p > p0 ) may change sign along the forward 1-shock curve S1F (U0 ). More precisely, (i) if U0 ∈ G2 ∪ G3 ∪ G4 , then λ¯ 1 (U0 ,U) remains negative

λ¯ 1 (U0 ,U) < 0,

U ∈ S1F (U0 );

(ii) If U0 ∈ G1 , then λ¯ 1 (U0 ,U) vanishes once at some point U = U˜ 0 ∈ G2 corre(1−μ )u2

sponding to a value p = p˜0 = v0 0 − μ p0 on the 1-shock curve S1F (U0 ) such that λ¯ 1 (U0 , U˜ 0 ) = 0, λ¯ 1 (U0 ,U) > 0, p ∈ (p0 , p˜0 ), λ¯ 1 (U0 ,U) < 0, p ∈ ( p˜0 , +∞). (b) The 3-shock speed λ¯ 3 (U0 ,U), may change sign along the backward 3-shock curve S3B (U0 ) (p > p0 ). More precisely,

9.3 Riemann problem

257

(i) if U0 ∈ G1 ∪ G2 ∪ G3 , then λ¯ 1 (U0 ,U) remains positive

λ¯ 3 (U0 ,U) > 0,

U ∈ S3B (U0 );

(ii) if U0 ∈ G4 , then λ¯ 3 (U0 ,U) vanishes once at some point U = U˜ 0 ∈ G3 corresponding to a value p = p˜0 = S3B (U0 ) such that

(1−μ )u20 v0

− μ p0 on the backward 3-shock curve

λ¯ 3 (U0 , U˜ 0 ) = 0, λ¯ 3 (U0 ,U) < 0, p ∈ (p0 , p˜0 ), λ¯ 3 (U0 ,U) > 0, p ∈ ( p˜0 , +∞).

9.3 Riemann problem In this section, we will establish the global existence and uniqueness of solutions of the Riemann problem for (9.1.2). For simplicity, we assume that the fluid is polytropic and ideal. Under the transformation x → −x,

u → −u,

a right-hand state (p, u), u < 0 becomes a left-hand state of the form (p, −u) on the (p, u)-plane. Therefore, the constructions relying on the initial data around Σ− can be obtained in a similar way as the ones for the initial data around Σ+ . Thus, in the following, we will construct only Riemann solutions when the initial data UL ,UR are located around Σ+ . Given a0 and a = a0 , let us define the curves of composite waves as follows. The forward composite wave curve W1→4 (U0 ) is defined to be the set of all the sonic or subsonic states U which can be arrived at by an ordered combination of 2-waves: the first 1-wave from the left-hand state U0 to some right-hand state U1 , followed by a stationary 4-wave from the left-hand state U1 to the right-hand U. Similarly, the backward composite wave curve W3←4 (U0 ) is defined to be the set of all states U ∈ G2 ∪ Σ which can be arrived by an ordered combination of 2 waves: the first 3-wave from the right-hand state U0 to some left-hand state U1 , preceded by a stationary 4-wave from the right-hand state U1 to the left-hand state U. That is, we have W1→4 (U0 ) := {U = (ρ , u, S, a) : W3←4 (U0 ) := {U = (ρ , u, S, a) :

¯ |u| ≤ c, ∃W4 (U, U) from a0 ¯ |u| ≤ c, ∃W4 (U, U) from a0

to a, U¯ ∈ W1F (UL )}, to a, U¯ ∈ W3B (UL )}.

(9.3.1)

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Notation We will use the following notation: (i) Wi (u1 , u2 ): An i-wave connecting a state u1 and a state u2 , where W = S: Lax shock, W = R: rarefaction wave, W2 = Z, i = 1, 2, 3; (ii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the left-hand state u1 to the right-hand state u2 is followed by a j-wave from the left-hand state u2 to the right-hand state u3 ; (iii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the right-hand state u1 to the left-hand state u2 is preceded by a j-wave from the right-hand state u2 to the left-hand state u3 ; (iv) U 0 denotes the state resulted from a stationary contact wave from U; (v) Denote by U # ∈ S1F (U) so that λ¯ 1 (U,U # ) = 0.

9.3.1 Construction 1: supersonic/supersonic This construction holds for UL ∈ G1 , and UR belongs to G1 ∪ Σ+ and some part of G2 . Let U1 be the state obtained by jumping from UL by a stationary contact from / there is a solution the level aL to the level aR . Whenever W1F (U1 ) ∩ W3B (UR ) = 0, defined as follows. Let {U2 } = W1F (U1 ) ∩ W3B (UR ). Then the solution is W4 (UL ,U1 ) W1 (U1 ,U2 ) W2 (U2 , U¯ 2 ) W3 (U¯ 2 ,UR ).

(9.3.2)

The condition for the existence of such a stationary wave W3 (UL ,U1 ) is aR ≥ amin (UL ).

(9.3.3)

Then, the solution (9.3.2) exists, provided

λ¯ 1 (U2 ,U1 ) ≥ 0, which means that U2 is above U1# , or coincides with it, and so that the state UR is somewhere in a higher position. An illustration of the construction of the solution (9.3.2) is given in Figure 9.1. The states U1 ,U2 , and U¯ 2 that determine the Riemann solution (9.3.2) can be calculated as follows. First, let U1 = (ρ1 , u1 , S1 , a1 ). Then, we have S1 = SL ,

a1 = aR ,

9.3 Riemann problem

259

Fig. 9.1 Construction 1: Riemann solution (9.3.2)

and (ρ1 , u1 ) ∈ G1 are calculated by the equations from (9.2.24) as u1 =

aL ρL uL , aR ρ1

u21 u2 + h(ρ1 , SL ) = L + h(ρL , SL ). 2 2

(9.3.4)

Second, the state U2 = (ρ2 , u2 , p2 , aR ) is the intersection point of W1F (U1 ) and W3B (UR ), so it is determined by u2 = uF1 (U1 ; p2 ) = uB3 (UR ; p2 ),

1/ρ2 = v2 = vF1 (U1 ; p2 )

(9.3.5)

for p = p2 , where the functions uF1 , uB3 , and vF1 are defined by (9.2.18). Third, the state U¯ 2 = (ρ¯ 2 , u2 , p2 , aR ) is determined by 1/ρ¯ 2 = v¯2 = vB3 (UR ; p2 ), where vB3 is defined by (9.2.18). / then there is a vacuum. In fact, let {M} = W1F (U1 ) ∩ If W1F (U1 ) ∩ W3B (UR ) = 0, B {p = 0}, {N} = W3 (UR ) ∩ {p = 0}. The solution is W4 (UL ,U1 ) W1 (U1 , M) W1 (M, N) W3 (M,UR ).

(9.3.6)

The configuration of the Riemann solution (9.3.2) in the (x,t)-plane is given by Figure 9.2. Note that this construction is still valid for some parts of a subsonic right-hand state. Example 9.3.1. Let us take the parameters as

γ = 1.4,

Cv = 1,

S∗ = 1.

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Fig. 9.2 Riemann solution with structure (9.3.2) in the (x,t)-plane

UL U1 U2 U3 UR

ρ 0.9 1.4337159 2.3297883 3.3297883 2.0491005

u 5.3 4.9905286 3.975797 3.975797 3.1270053

p 2 3.8382991 7.6765981 7.6765981 3.8382991

a 1.5 1 1 1 1

Table 9.1 Riemann data and the states separating elementary waves of the Riemann solution (9.3.2)

We consider the Riemann problem for (9.1.2) where the Riemann data UL ,UR and the states U1 ,U2 , and U¯ 2 which separate elementary waves of the Riemann solution (9.3.2) are given in Table 9.1. The Riemann solution is plotted in Figure 9.3.

9.3.2 Construction 2: supersonic/subsonic This construction holds when UL is a supersonic state, and UR is a subsonic state. Consider the wave curve W1F (UL ). Let U˜ L ∈ W1F (UL ) ∩ G2 be the state at which the shock speed vanishes, i.e., λ¯ 1 (UL , U˜ L ) = 0, in view of Proposition 9.2.1. Let U˜ L = ( p˜L , u˜L ). Then, from any point U = (p, u) ∈ W1F (UL ), p ≥ p˜L , which means U is positioned “lower” than U˜ L , a stationary wave jumps from U with a = aL to ¯ Theses states U¯ form a curve a U varies a = aR to some state U¯ using W4 (U, U). F ˜ along W1 (UL ) “downward” from UL . Precisely, set the “composite” curve ¯ W1→4 (UL ) := {U¯ : ∃W4 (U, U) from aL

to aR ,U = (p, u) ∈ W1F (UL ), p ≥ p˜L }. (9.3.7)

9.3 Riemann problem

261

Fig. 9.3 Construction 1: A Riemann solution of the type (9.3.2) in the interval −1 ≤ x ≤ 1 at the time t = 0.1.

Whenever W3B (UR ) ∩ W1→4 (UL ) = 0, / there will be a Riemann solution. In fact, let W3B (UR ) ∩ W1→4 = {U2 } and U1 be the point on W1F (UL ) that corresponds to the stationary wave W4 (U1 ,U2 ) or W4 (U¯ 1 ,U2 ). If u1 ≥ 0, the Riemann solution has the form S1 (UL ,U1 ) W4 (U1 ,U2 ) W2 (U2 , U¯ 2 ) W3 (U¯ 2 ,UR ),

(9.3.8)

where W2 (U2 , U¯ 2 ) is the 2-contact discontinuity from U2 to U¯ 2 . If if u1 < 0 and λ¯ 3 (U2 ,UR ) ≥ 0, the Riemann solution has the form S1 (UL ,U1 ) W2 (U1 , U¯ 1 ) W4 (U¯ 1 ,U2 ) W3 (U2 ,UR ).

(9.3.9)

See Figure 9.4. The states U1 = (p1 , u1 ),U2 = (p2 , u2 ) that determine the Riemann solution (9.3.8) can be calculated using the following equations: u1 = uF1 (UL ; p1 ), S1 = S(vF1 (UL ; p1 ); p1 ), aL ρ1 u1 u2 = , aR ρ2 u22 u2 + h(ρ2 , S1 ) = 1 + h(ρ1 , S1 ), 2 2 ρi = ρ (pi , Si ), i = 1, 2, S2 = S1 , u2 = uB3 (UR ; p2 ),

(9.3.10)

262

9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Fig. 9.4 Construction 2: Riemann solution with structure (9.3.8)

where S = S(v, p) is given as an equation of state, and v1 = vF1 (UL ; p1 ),

v2 = 1/ρ2 .

The state U¯ 2 is calculated using the standard way: the p and u components of U¯ 2 are the same as the ones of U2 , and 1/ρ¯ 2 = v¯2 = vB3 (UR ; p2 ), where vB3 is defined by (9.2.18). A similar argument can be applied for the solution (9.3.9). The configuration of the Riemann solution (9.3.8) in the (x,t)-plane is given by Figure 9.5. Note that this construction still holds for some parts of a supersonic right-hand state.

Fig. 9.5 Construction 2: Riemann solution with structure (9.3.8) in the (x,t)-plane

9.3 Riemann problem

263

An alternative construction relying on the backward composition can be made, where we can make use of the composite wave curve W3←4 (UR ). Precisely, let W1F (UL ) ∩ W3←4 (UR ) = {U1 }, and let U2 ∈ W2B (UR ) be the corresponding right-hand state of the admissible contact wave from U1 . Then, the solution still has the form (9.3.5). This backward construction is useful when aL > aR .

9.3.3 Construction 3: resonant waves for supersonic regime When UL is in the supersonic region and UR is in a medium position between the two cases above, we can observe an interesting phenomenon where wave speeds associated with different characteristic fields coincide. Specifically, there are solutions containing three waves with the same zero speed. The solution begins with a stationary 4-shock from UL to a state U1 in the supersonic region G1 with an intermediate value of cross-sectional area a1 ∈ [aL , aR ], followed by a zero-speed 1-shock to some state U2 ∈ G2 , followed by another stationary 4-shock to U3 = U3 ∈ G2 with the cross-sectional area a = aR . Since u3 > 0, the solution continues with a 2-wave from U3 to the state U¯ 3 , and it finally arrives at UR using a 3-wave. We, therefore, have a solution for the form W4 (UL ,U1 ) S1 (U1 ,U2 ) W4 (U2 ,U3 ) W2 (U3 , U¯ 3 ) W3 (U¯ 3 ,UR ). (9.3.11) See Figure 9.6. The states U1 ,U2 ,U3 , and U¯ 3 can be calculated using the corresponding formula of solution (9.3.11). It is not difficult to check that the mapping [aL , aR ] a → U = U3 (a),

Fig. 9.6 Construction 3: Riemann solution with structure (9.3.11)

(9.3.12)

264

9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

is locally Lipschitz with a deterministic Lipschitz constant K on any compact neighborhood of UL . Set (9.3.13) L (UL , aR ) = {U3 (a)|a ∈ [aL , aR ]}. Observe that the solution (9.3.11) occurs iff W3B (UR ) ∩ L (UL ; aR ) = 0. / The states U1 ,U2 ,U3 that determine the Riemann solution (9.3.11) can be calculated using the following equations: u1 =

aL ρL uL , a1 ρ1

u21 u2 + h(ρ1 , SL ) = L + h(ρL , SL ), 2 2 p2 = (1 − μ )u21 ρ1 − μ p1 , u2 = uF1 (U1 ; p2 ), a1 ρ2 u2 u3 = , aR ρ3 u23 u2 + h(ρ3 , S3 ) = 2 + h(ρ2 , S2 ), 2 2 ρi = ρ (pi , Si ), i = 1, 2, 3, S3 = S2 ,

(9.3.14)

u3 = uB3 (UR ; p3 ), where the functions uF1 , uB3 are defined by (9.2.18), and

μ=

γ −1 . γ +1

The configuration of the Riemann solution (9.3.11) in the (x,t)-plane is given by Figure 9.7.

Fig. 9.7 Construction 3: Riemann solution with structure (9.3.11) in the (x,t)-plane

9.3 Riemann problem

265

9.3.4 Existence and uniqueness properties: supersonic regime Existence for large domain of UR By Construction 1, a stationary wave from UL with the cross-section level a = aL to U1 = UL0 with a = aR can be followed by a zero-speed 1-shock from U1# = UL0# := (UL0 )# . Letting the second stationary wave in Construction 3 be a trivial jump, we can see that the point UL0# is a common endpoint of both curves W1F (U1 ) and L . On the other hand, by Construction 2, a zero-speed 1-shock from UL to UL# can be followed by a stationary contact from UL# with the cross-section level a = aL to (UL# )0 := UL#0 with a = aR . By letting the first stationary wave in Construction 3 be a trivial contact, one can see that the point UL#0 is a common endpoint of the two curves W1→4 (UL ) and L . Thus, the union W1F (U1 ) ∪ L ∪ W1→4 (UL ) constitutes a large continuous curve. This guarantees that the backward curve W2B (UR ) can intersect this union of curves for a large domain of UR . The existence of solutions to the Riemann problem for a large domain of UR is then followed.

No local existence As seen in the isentropic case in the previous chapter, one can see the loss of local existence of solutions of the Riemann problem, if the condition (9.3.3) is violated. In that case, even UR can be made arbitrarily closed to UL , or eventually UR = UL and aL > aR , there are no solutions to the Riemann problem. Indeed, assume that the condition (9.3.3) is violated, i.e., aR < amin (UL ).

(9.3.15)

If (9.3.15) holds, then Construction 1 cannot be performed. Construction 2 and Construction 3 are available for some parts, as in the isentropic case. However, both curves L and W1→4 (UL ) belong to the subsonic region. Therefore, the backward curve W2B (UR ) cannot meet these curves if it lies entirely in the supersonic region, that is, its u-intercept is non-negative 0 ∂ p ρ (τ , SR ) d τ ≥ 0, uR + ρ (τ , SR ) pR or uR ≥

pR ∂ p ρ (τ , SR ) 0

ρ (τ , SR )

dτ .

(9.3.16)

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Thus, under the conditions (9.3.15) and (9.3.16), the Construction 1 is not available and the backward curve W2B (UR ) does not intersect both curves L and W1→4 (UL ). Consequently, there is no Riemann solution in this case.

Multiple solutions It is possible that the curve W3B (UR ) can meet all the three curves W1F (U1 ), L and W1→4 (UL ). This occurs when the point UL#0 is located above the curve W3B (UR ), and the point UL0# is located below the curve W3B (UR ), see Figure 9.8. The fact that the 0# 0# 0# point UL#0 = (ρL#0 , u#0 L ) is located above and the point UL = (ρL , uL ) is located B below the curve W3 (UR ) means B #0 u#0 L > u3 (UR ; ρL ) B 0# u0#0 L < u3 (UR ; ρL ).

(9.3.17)

Thus, under the conditions (9.3.17), the Riemann problem admits three solutions corresponding to the three intersection points U1 ,U2 , and U3 of W3B (UR ) with W1F (U1 ), L , and W1→4 (UL ), respectively. Each of these intersection point U1 ,U2 , and U3 is corresponding to the Riemann solution in Constructions 1, 2, and 3 above. Consider the case where aR < aL . If a < amin (UL ), for some values a ∈ [aR , aL ], then some part of or the whole curve W1F (U1 ), L , or W1→4 (UL ) may not exist. Thus, the backward wave curve W2B (UR ) may intersect

Fig. 9.8 Multiple solutions for supersonic UL under (9.3.17): There may be up to three Riemann solutions given by Constructions 1, 2, and 3 when W3B (UR ) meets all the three curves W1F (U1 ), L , and W1→4 (UL ).

9.3 Riemann problem

267

the union W1F (U1 ) ∪ L ∪ W1→4 (UL ) at no point, one point, two points, or three points. This shows that the Riemann problem may admit no solution, one solution, two solutions, or three distinct solutions.

Uniqueness of Riemann problem. We can obtain uniqueness in several circumstances. For example, by letting the right-hand side UR belong to a neighborhood of the left-hand side UL such that only the solution by Construction N1 occurs. This happens when the curve W3B (UR ) does not intersect the set of waves W1→4 (UL ) ∪ L (UL , aR ). By continuity, this condition is fulfilled provided |aR − aL | is small enough. Moreover, we can see that W3B (UR ) does not intersect the set of waves W1→4 (UL ) ∪ L (UL , aR ) when it lies entirely in G1 . That is, the uniqueness may be obtained if the u-intercept of this curve remains non-negative, i.e., (9.3.16) holds.

9.3.5 Construction 4: subsonic/supersonic This construction holds when UL is a subsonic state, and UR is either a supersonic state, or nearby the sonic surface Σ+ . First, if aL < aR , the solution begins with a 1-rarefaction wave from UL in the subsonic region until it reaches a state U1 on the sonic surface Σ+ , so {U1 } = W1F (UL ) ∩ Σ+ , followed by a stationary 4-shock from U1 to a state U2 ∈ G1 using the smaller zero ρs (U1 , aR ). Let {U3 } = W1F (U2 ) ∩ W3B (UR ). The solution is then continued by a 1-wave from U2 to U3 , followed by a 2-wave W2 (U3 , U¯ 3 ), and finally followed by a 3-wave from U¯ 3 to UR . Thus, the solution has the form W1 (UL ,U1 ) W4 (U1 ,U2 ) W1 (U2 ,U3 ) W2 (U3 , U¯ 3 ) W3 (U¯ 3 ,UR ). (9.3.18) The construction makes sense if λ¯ 1 (U2 ,U3 ) ≥ 0. See Figure 9.9. The states U1 ,U2 , and U3 that determine the Riemann solution (9.3.18) can be calculated using the following equations: u1 = uF1 (UL ; p1 ) = u2 =

aL ρ1 u1 , aR ρ2

1 , ρ p (p1 , SL )

u22 u2 + h(ρ2 , SL ) = 1 + h(ρ1 , SL ), 2 2 ρi = ρ (pi , Si ), Si = SL , i = 1, 2, u3 = uF1 (U2 ; p3 ) = uB3 (UR ; p3 ), where the functions uF1 , uB3 are defined by (9.2.18).

(9.3.19)

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Fig. 9.9 Construction 4: Riemann solution with structure (9.3.18) for aL < aR

Fig. 9.10 Construction 4: Riemann solution with structure (9.3.18) in the (x,t)-plane

The configuration of the Riemann solution (9.3.18) in the (x,t)-plane is given by Figure 9.10. Second, consider the case aL ≥ aR . As seen before, a nontrivial stationary contact from a state U− to a state U+ must satisfy the condition a+ > amin (U− ). Moreover, if U− is a sonic state, then amin (U− ) = a− . Thus, it is not possible to jump from any sonic state U− with the cross-section aL to another state U+ with the cross-section aR when aL ≥ aR , as done in the case aL < aR . There is another form of Riemann solutions which can be constructed as follows. First, we observe that the set of subsonic states with the cross-section aL obtained

9.3 Riemann problem

269

from a stationary jump from any sonic state U ∈ Σ + with the cross-section aR form a curve, denoted by Σa . Let U1 be the intersection point of the forward 1-wave curve W1F (UL ) and Σa . Then, a Riemann solution can begin with a 1-rarefaction wave from UL to a state U1 ∈ G2 at which it jumps to a state U2 ∈ Σ + by a stationary wave. Let U3 be the intersection point of the curves W1F (U2 ) and W2B (UR ). The solution continues with a 1-wave from U2 to U3 , followed by a 2-wave W2 (U3 , U¯ 3 ) and finally followed by a 3-wave from U¯ 3 to UR . So, the solution has the form W1 (UL ,U1 ) W4 (U1 ,U2 ) W1 (U2 ,U3 ) W2 (U3 , U¯ 3 ) W3 (U¯ 3 ,UR ). (9.3.20) See Figure 9.11. The configuration of the Riemann solution (9.3.20) in the (x,t)-plane is given by Figure 9.12. Note that this construction still holds for some parts of a subsonic right-hand state.

Fig. 9.11 Construction 4: Riemann solution (9.3.20) for aL ≥ aR

Fig. 9.12 Construction 4: Riemann solution with structure (9.3.20) in the (x,t)-plane

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Example 9.3.2. Let us take the parameters as

γ = 1.4,

Cv = 1,

S∗ = 1.

We consider the Riemann problem for (9.1.2) where the Riemann data UL ,UR and the states U1 ,U2 ,U3 , and U¯ 2 which separate elementary waves of the Riemann solution (9.3.18) are given in Table 9.2. The Riemann solution is plotted in Figure 9.13.

UL U1 U2 U3 U¯ 3 UR

ρ 2 0.988551622927 0.594463484726 0.985240428091 2.11734844805 1

u 0.5 2.0553599277 2.84826772996 1.83104417825 1.83104417825 0.8

p 8 2.98295765193 1.46360285922 3.01445816905 3.01445816905 1

a 1 1 1.2 1.2 1.2 1.2

Table 9.2 Riemann data and the states separating elementary waves of the Riemann solution (9.3.18)

Fig. 9.13 Construction 4: A Riemann solution of the type (9.3.18) in the interval −1 ≤ x ≤ 1 at the time t = 0.2.

9.3 Riemann problem

271

9.3.6 Construction 5: subsonic/subsonic This construction holds for UL ∈ G2 and UR in G2 ∪ G3 ∪ Σ+ ∪ Σ0 and some part of G1 . Let {A} = W1F (UL ) ∩ Σ+ as in Construction 5, and let B ∈ G2 be the point resulted by a stationary wave W4 (A, B) using pb . Define the “composite” curve ¯ W1→4 (UL ) := {U¯ : ∃W4 (U, U) from aL

to aR ,U = (p, u) ∈ W1F (UL ), p ≥ p˜A } (9.3.21)

Whenever W3B (UR ) ∩ W1→4 = 0, / there will be a Riemann solution. In fact, let W3B (UR ) ∩ W1→4 = {U2 } and U1 be the point on W1F (UL ) that corresponds to the stationary wave W4 (U1 ,U2 ) or W4 (U¯ 1 ,U2 ). Then, the solution can be W1 (UL ,U1 ) W4 (U1 ,U2 ) W2 (U2 , U¯ 2 ) W3 (U¯ 2 ,UR ),

(9.3.22)

if u1 ≥ 0 and W1 (UL ,U1 ) W2 (U1 , U¯ 1 ) W4 (U¯ 1 ,U2 ) W3 (U2 ,UR ),

(9.3.23)

if u1 < 0 and λ¯ 3 (U2 ,UR ) ≥ 0. This construction makes sense whenever U1 ∈ G2 ∪ G3 ∪ ∪Σ+ ∪ Σ0 ∪ Σ− . See Figure 9.14. The states U1 = (p1 , u1 ),U2 = (p2 , u2 ) that determine the Riemann solution (9.3.22) can be calculated using the following equations: u1 = uF1 (UL ; p1 ), S1 = S(vF1 (UL ; p1 ); p1 ), aL ρ1 u1 u2 = , aR ρ2 u22 u2 + h(ρ2 , S1 ) = 1 + h(ρ1 , S1 ), 2 2 ρi = ρ (pi , Si ), i = 1, 2, S2 = S1 ,

Fig. 9.14 Riemann solution with structure (9.3.22) or (9.3.23)

(9.3.24)

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Fig. 9.15 Construction 5: Riemann solution with structure (9.3.22) in the (x,t)-plane

u2 = uB3 (UR ; p2 ), where the functions uF1 , uB3 are defined by (9.2.18). The configuration of the Riemann solution (9.3.22) in the (x,t)-plane is given by Figure 9.15. Note that this construction still holds for some parts of a supersonic right-hand state. We can also use a backward construction by employing the backward composite wave curve W3←4 (UR ). Indeed, let W1F (UL ) ∩ W3←4 (UR ) = {U1 }, and let U2 ∈ W2B (UR ) be the corresponding right-hand state of the admissible contact wave from U1 . Then, the solution still has the form (9.3.14). This backward construction may be preferred if aL < aR . Example 9.3.3. Let us take the parameters as

γ = 1.4,

Cv = 1,

S∗ = 1.

We consider the Riemann problem for (9.1.2), where the Riemann data UL ,UR and the states U1 ,U2 , and U¯ 2 , which separate elementary waves of the Riemann solution (9.3.22), are given in Table 9.3. The Riemann solution is plotted in Figure 9.16.

9.3.7 Construction 6: resonant waves for subsonic regime The solution begins with a 1-rarefaction wave from UL to U1 ∈ Σ+ , followed by stationary 4-shock from U1 to some state U2 ∈ G1 with an intermediate cross-sectional area aM ∈ [aL , aM ], followed by a zero-speed 1-shock from U2 to a state U3 in the

9.3 Riemann problem

273 UL U1 U2 U3 UR

ρ 1.3939394 2 2.0855398 1.0855398 1.4501911

u 1.9325048 1 0.63932289 0.63932289 1.7419716

p 6 10 10.603857 10.603857 15.905786

a 1 1 1.5 1.5 1.5

Table 9.3 Riemann data and the states separating elementary waves of the Riemann solution (9.3.22)

Fig. 9.16 Construction 5: A Riemann solution of the type (9.3.22) in the interval −1 ≤ x ≤ 1 at the time t = 0.1.

subsonic region G2 , then followed by another stationary 4-shock from U3 to U4 in the same subsonic region with the cross-sectional area a = aR , followed by a 2contact discontinuity from U4 to U¯ 4 , since u4 > 0. Such states U4 form a set Λ and / we obtain a Riemann solution. This solution arrives at whenever W3B (UR ) ∩ Λ = 0, UR from U¯ 4 by a 3-wave. Observe that this construction is continued at the two end-points with the constructions 4 and 5. Precisely, if the first stationary shock W4 (U1 ,U2 ) shifts the crosssectional area from aL to aR already, then the construction 6 coincides with Construction 4; and if the first stationary shock W4 (U1 ,U2 ) is trivial, i.e., U1 = U2 and the cross-sectional area a = aL does not change, then Construction 6 coincides with Construction 5. That is, Constructions 4-6 can change continuously. The formula of the solution is thus given by

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Fig. 9.17 Construction 6: Riemann solution with structure (9.3.25)

R1 (UL ,U1 ) W4 (U1 ,U2 ) W1 (U2 ,U3 ) W4 (U3 ,U4 ) W2 (U4 , U¯ 4 ) W3 (U4 ,UR ).

(9.3.25) See Figure 9.17. The states U1 ,U2 ,U3 ,U4 , and U¯ 4 can be calculated using the corresponding formula of solution (9.3.25). The states U1 ,U2 ,U3 , and U4 that determine the Riemann solution (9.3.25) can be calculated using the following equations: u1 = uF1 (UL ; p1 ) = u2 =

aL ρ1 u1 , aρ2

1 , ρ p (p1 , SL )

u22 u2 + h(ρ2 , SL ) = 1 + h(ρ1 , SL ), 2 2 ρi = ρ (pi , Si ), i = 1, 2, 3, 4, p3 = (1 − μ )u22 ρ2 − μ p2 ,

(9.3.26)

u3 = uF1 (U2 ; p3 ), aρ3 u3 u4 = , aR ρ4 u2 u24 + h(ρ4 , SL ) = 3 + h(ρ3 , SL ), 2 2 B u4 = u3 (UR ; p4 ), where the functions uF1 , uB3 are defined by (9.2.18), and

μ=

γ −1 . γ +1

The configuration of the Riemann solution (9.3.25) in the (x,t)-plane is given by Figure 9.18.

9.3 Riemann problem

275

Fig. 9.18 Construction 6: Riemann solution with structure (9.3.25) in the (x,t)-plane

9.3.8 Existence and uniqueness properties: subsonic regime It is not difficult to see that Constructions 4, 6, and 5 change continuously from one to another. Thus, the existence of solutions for the Riemann problem can be obtained for a large domain of the right-hand state when the left-hand state is subsonic. Let us denote by U1 the sonic state which is the intersection point of the curves W1F (UL ) and Σ+ with the cross-section level aL . Let U2 be the supersonic state with the cross-section level aR obtained from a stationary jump from U1 . If U10 is located above and U2# is located below the backward curve W3B (UR ), then W3B (UR ) intersects with each curve W1→4 (UL ), Λ , and W1F (U2 ), see Figure 9.19. The three intersection points U4 ,U5 and U6 give us three distinct Riemann solutions in Constructions 4, 5, and 6. Therefore, letting U10 = (ρ10 , u01 ) and U2# = (ρ2# , u#2 ), we have aL < aR u01 > uB3 (UR ; ρ10 ) u#2

(9.3.27)

< uB3 (UR ; ρ2# ),

Thus, in this case, the Riemann problem can admit up to three different solutions. Consider the case aR < aL . Then, some part of or the whole curve W1F (U1 ), L , or W1→4 (UL ) may not exist, if a < amin (UL ), for some values a ∈ [aR , aL ]. In this case, the backward wave curve W2B (UR ) may intersect the union W1F (U1 ) ∪ L ∪ W1→4 (UL ) at no point, one point, two points, or three points. Then, the Riemann problem may admit no solution, one solution, two solutions, or three distinct solutions.

276

9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Fig. 9.19 Multiple solutions for subsonic UL under the conditions (9.3.27): each choice of the intersection point U4 ,U5 or U6 corresponds to a different solution described in Construction 4, 5, and 6, respectively.

9.4 Quantitative properties 9.4.1 Entropy inequality in divergence form Let us first consider the entropy inequality for the Euler equations of fluid dynamics

∂t ρ + ∂x (ρ u) = 0, ∂t (ρ u) + ∂x (ρ u2 + p) = 0, ∂t (ρ e) + ∂x (u(ρ e + p)) = 0.

(9.4.1)

It was shown by Harten et al [150] that a twice differentiable function Uc of the form (9.4.2) Uc = ρ g(S), is an entropy of (9.4.1) if and only if g(S) satisfies the following two conditions: (i) (ii)

g(S) is strictly decreasing as function of S; g(S) is strictly convex as function of (1/ρ , ε ).

Moreover, the system (9.4.1) is strictly hyperbolic if and only if it admits an entropy of the form (9.4.2). Let us consider now the entropy inequality for the model of a fluid in a nozzle with variable cross-section (9.1.1) by the vanishing viscosity method. Precisely, let us take the following Navier-Stokes equations describing a viscous fluid flow in a nozzle with smooth area function aν = aν (x)

9.4 Quantitative properties

277

∂t (aν ρν ) + ∂x (aν ρν uν ) = 0, ∂t (aν ρν uν ) + ∂x (aν (ρν u2ν + pν )) = pν ∂x aν + ν ∂x (bν ∂x uν ), ∂t (aν ρν eν ) + ∂x (aν uν (ρν eν + pν )) = ν ∂x (bν uν ∂x uν ),

(9.4.3)

where bν = bν (x) ≥ 0 is given and ν denotes the viscosity coefficient. We will consider the limit of solutions of (9.4.3) as ν tends to zero. For simplicity, we drop for the moment the subscript ν . We aim to derive an equation for the specific entropy by assuming that an equation of state of the form ε = ε (ρ , S) is given. Using the equation of conservation of mass in (9.4.3), we can re-write the equation of momentum in (9.4.3) as aρ (ut + uux ) + px a − ν (b ux )x = 0,

(9.4.4)

and the equation of energy in (9.4.3) as aρ et + aρ uex + (aup)x = ν (b uux ))x .

(9.4.5)

From the thermodynamical identity d ε = T dS − pdv,

v = 1/ρ ,

the equation (9.4.5) can be re-written as aρ T (St + uSx ) +

ap (ρt + uρx ) + (aup)x + aρ u(ut + uux ) = ν (bν uux ))x . ρ

Re-arranging terms of the last equation yields aρ T (St + uSx ) +

p

(aρ )t + (auρ )x + u aρ (ut + uux ) + apx − ν (b ux )x = ν b u2x . ρ (9.4.6)

Using the equation of mass in (9.4.3) and of momentum (9.4.4), we obtain the conservation of energy from (9.4.6) as an equation of the specific entropy S as

∂t S + u∂x S =

bν 2 u . aρ T x

Let g(S) be any smooth and decreasing function of S. Multiplying the last equation by aρ g (S), where g (S) = dg(S)/dS, we obtain b aρ∂t g(S) + aρ u∂x g(S) = ν aρ g (S) u2x . aρ T On the other hand, multiplying the conservation of mass of (9.4.3) by g(S) and then adding up with the above equation side by side, we have

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

b u2 . ∂t (aρ g(S)) + ∂x (aρ ug(S)) = νρ g (S) ρT x Now assume that the system (9.4.3) admits a sequence of smooth solution Uν uniformly bounded in amplitude and converging almost everywhere to a limit U = (ρ , ρ u, ρ e) when ν tends to zero. Then, the limit function U satisfies the following entropy inequality

∂t (aρ g(S)) + ∂x (aρ ug(S)) ≤ 0,

(9.4.7)

where g is any function satisfying the conditions (i)-(ii) of the one in (9.4.2). Moreover, it is customary to assume that the fluid is in local thermodynamic equilibrium, so that The function (v, ε ) → ε (v, S), is strictly convex. It is known that this assumption is equivalent to the requirement that the function (v, ε ) → S(v, ε ) is strictly concave. Using the entropy inequality in divergence form (9.4.7), we are going to establish a minimum entropy principle. First, we will show that the generalized entropy inequality for the nonconservative form of the model (9.1.1) coincides with the entropy inequality in divergence form (9.4.7) for the entropy pair (U , F ) = (aρ g(S), aρ ug(S)),

(9.4.8)

where the functions g satisfy the conditions (i)-(ii) of a function g in (9.4.2). Indeed, consider the hyperbolic system in nonconservative form

∂t U + A(U) ∂xU = 0.

(9.4.9)

The entropy inequality for (9.4.9) has the form ∂t U (U) + DU U (U) A(U(.,t))∂xU(.,t) ≤ 0, φ

where φ is a given Lipschitz family of paths, and U is a convex function satisfying DU2 U (U) A(U) = A(U)T DU2 U (U). Basic properties of the nonconservative product imply that if there exists a function F such that (9.4.10) DU U A(U) = DU F (U), then the nonconservative product DU U (U) A(U(.,t))∂xU(.,t) reduces to the φ

usual one in divergence form, and is independent of the path φ . Consequently, the entropy inequality takes the divergence form

9.4 Quantitative properties

279

∂t U (U) + ∂x F (U) ≤ 0

(9.4.11)

in the sense of distributions. We now check that the entropy inequality for (9.1.2) can be reduced to the divergence form (9.4.11) for all entropy pairs of the form (9.4.8). This will establish (9.4.10). The system (9.1.2) can be written in the nonconservative form (9.4.9), where ⎛ ⎞ ⎛ ⎞ aρ w1 ⎜aρ u⎟ ⎜w2 ⎟ ⎜ ⎜ ⎟ ⎟ U =⎝ = aρ e⎠ ⎝w3 ⎠ a w4 and ⎛

⎞ 0 1 0 0 pε pε u ⎜ p − u2 + pε (u2 − e) 2u − −pρ ρ ⎟ ρ ⎜ ⎟ ρ ρ ρ ⎜ ⎟ A(U) = ⎜ 2 2 ⎟ ⎜u(p − e + pε (u − e) − p ) e + p − pε u pε u + u u(p − p ρ )⎟ ρ ⎝ ρ ⎠ ρ ρ ρ ρ 0 0 0 0 (9.4.12) Proposition 9.4.1. Consider the system (9.1.2) in the form (9.4.9), (9.4.12). Let g be any function satisfying the conditions i) and ii) of the one in (9.4.2). Then the function U (U) = aρ g(S)(U),U = (aρ , aρ u, aρ e, a)T , is convex. Moreover, it satisfies DU U A(U) = DU F (U),

F (U) = aρ ug(S).

That is, (U , F ) is a convex entropy pair of the Euler equations for fluid dynamics (9.4.1). Consequently, the entropy inequality in the sense of nonconservative products can be written in the divergence form (aρ g(S))t + (aρ ug(S))x ≤ 0.

(9.4.13)

Proof. First, as is shown in [150], the function ρ g(S) is convex in the variable (ρ , ρ u, ρ e)T . Therefore, the function aρ g(S) = ρ¯ g(S), ρ¯ := aρ , is convex in the variable (ρ¯ , ρ¯ u, ρ¯ e). The function ρ¯ g(S) in fact depend only on the first three variables ρ¯ , ρ¯ u, and ρ¯ eT . So, this function is independent of a. The function ρ¯ g(S) is, therefore, convex in the variable (ρ¯ , ρ¯ u, ρ¯ e, a)T = (aρ , aρ u, aρ e, a)T . Next, using the equation of state of the form S = S(ρ , ε ), one can easily show that w w 1 w2 2 1 3 U (U) = aρ g(S) = w1 g S , , − w4 w1 2 w1

F (U) = uU (U).

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Further, from the thermodynamic identity d ε = T dS − pdv = T dS + we have Sε =

1 T

and Sρ =

−p . T ρ2

Uw3 =

p dρ , ρ2

Observe also that g (S) , T

Uw4 = p

g (S) . T

Therefore, it follows that

Sρ Sε w2 DU U (U) = g(S) + w1 g (S) + ( − w3 ) , w4 w21 2

w2 1 − g (S)Sε , g (S)Sε , −g (S)Sρ 12 2 w4

(9.4.14)

w2 w1 w2 U (U) , uUw3 , uUw4 ). = (uUw1 − 2 U (U), uUw2 + w1 w1

(9.4.15)

g (S) p g (S) g (S) g (S)p = g(S) + (− − e + u2 ), − , , . T ρ 2T T T and DU F (U) = uDU U (U) + U (U)DU

As seen by (9.4.12), (9.4.14), and (9.4.15), we want to show that B = (b1 , b2 , b3 , b4 )T := DU U A(U) − DU F (U) = 0. Actually, we have w2 Sε w2 a21 + w1 g (S) a31 − uUw1 + 2 U w1 w1 w1 w2 w2 = g (S)Sε (a31 − a21 ) − uUw1 + 2 U w1 w1

b1 = −w1 g (S)Sε

= g (S)Sε (

−pw2 w4 w32 w2 w3 w2 + 3 − 2 ) − uUw1 + 2 U . w21 w1 w1 w1

This implies that b1 = g (S)Sε (u3 −

pu w2 − ue) − uUw1 + 2 U ρ w1

g (S) −p g (S)u 2 p (u − − e) − u(g(S) + ( + u2 − e)) + ug(S) T ρ T ρ = 0. =

(9.4.16)

9.4 Quantitative properties

281

Second, we find u u2 p g (S) −p g (S)u g (S) ( (2u − pε ) + (e + − pε ) + u2 − e) − T ρ T ρ T ρ ρ U − (uUw2 + ) w1 U g (S)u2 − uUw2 − = 0. = g(S) − T w1

b2 = g(S) +

Third, we have g (S)u pε g (S) pε u ( + + u) − uUw3 T ρ T ρ g (S)u g (S) + u(p − pρ ρ ) − uUw4 = 0. b4 = pρ ρ T T

b3 = −

These calculations show that (9.4.16) holds, that is DU U A(U) = DU F (U), which completes the proof of Proposition 9.4.1.

9.4.2 Minimum entropy principle First, we need to check a class of entropy pairs satisfying (9.4.2) as follows. Lemma 9.4.1. For any constant p > 1, the function g(S) := (S0 − S) p , where S0 is a constant such that S0 − S > 0 for all S in the domain under consideration, is (i) strictly decreasing and strictly convex as a function of S, (ii) strictly convex as a function of (v, ε ). Proof. We have g (S) = −p(S0 − S) p−1 < 0 and g (S) = p(p − 1)(S0 − S) p−2 > 0, so (i) follow immediately. Next, since the function S(v, ε ) is strictly concave as a function of (v, ε ) for 0 < λ < 1, for (v1 , ε1 ) = (v2 , ε2 ), we have S(λ (v1 , ε1 ) + (1 − λ )(v2 , ε2 )) > λ S(v1 , ε1 ) + (1 − λ )S(v2 , ε2 ). Thus, by (i) it follows

g S(λ (v1 , ε1 ) + (1 − λ )(v2 , ε2 )) < g λ S(v1 , ε1 ) + (1 − λ )S(v2 , ε2 )

< λ g S(v1 , ε1 ) + (1 − λ )g S(v2 , ε2 ) , which establishes (ii). Lemma 9.4.1 is completely proved. We are now in a position to establish the minimum entropy principle for the system (9.1.2).

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9 Compressible fluids in a nozzle with discontinuous cross-section—General flows

Theorem 9.4.1. If U is a bounded entropy solution to the system (9.1.2), then it satisfies the minimum entropy principle: inf S(x,t) ≥

|x|≤R

inf

|x|≤R+t||u||L∞

S(x, 0).

Proof. Consider an arbitrary g = g(S) as in Lemma 9.4.1. We first show that for any bounded entropy solution of (9.1.2) |x|≤R

a(x)ρ (x,t)g(S(x,t))dx ≤

|x|≤R+t||u||L∞

a(x)ρ (x, 0)g(S(x, 0))dx.

(9.4.17)

Indeed, denote by n = (nx , nt ) the unit outer normal of the trapezoid C = {(x,t)||x| ≤ R + (t − τ )||u||L∞ ,

0 ≤ τ ≤ t}.

Integrating the entropy inequality in the divergence form (9.4.13) over C and using Green’s formula yield 0≥

C

((aρ g(S))t + (aρ ug(S))x )dA = =

∂ C

(aρ g(S), aρ ug(S)) · nds aρ g(S)(nt + unx )ds.

∂C

The integrals over the top and bottom lines of ∂ C give the difference between the left- and the right-hand sides of (9.4.17). It follows from the last inequality that this term is bounded from above by −

sides of C

aρ g(S)(nt + unx )ds.

We will show that the last quantity is non-positive. Indeed, on the mantle, we have (nx , nt ) = (1 + ||u||2L∞ )−1/2 (x/|x|, ||u||L∞ ), nt + unx = (1 + ||u||2L∞ )−1/2 ||u||L∞ + ux/|x| ≥ 0.

thus

By the condition g(S) > 0, we obtain the desired conclusion. Now, consider the family of function g(S) = (S0 − S) p , where p > 1 and S0 is a constant satisfying S + S0 > 0; for instance, S0 = ||S||L∞ + 1. The inequality (9.4.17) yields |x|≤R

a(x)ρ (x,t)(S0 − S(x,t)) p dx ≤

|x|≤R+t||u||L∞

a(x)ρ (x, 0)(S0 − S(x, 0)) p dx,

or |x|≤R

a(x)ρ (x,t)(S0 − S(x,t)) p dx

1/p

≤

|x|≤R+t||u||L∞

a(x)ρ (x, 0)(S0 − S(x, 0)) p dx

1/p

.

9.5 Bibliographical notes

283

This means that ||(aρ )1/p (.,t)(S0 − S(.,t))||L p [−R,R] ≤ ||(aρ )1/p (., 0)(S0 − S(., 0))||L p [−R−t||u||L∞ ,R+t||u||L∞ ] . Letting p → +∞ in the last inequality, we obtain ||S0 − S||L∞ [−R,R] ≤ ||S0 − S||L∞ [−R−t||u||L∞ ,R+t||u||L∞ ] . By definition, we can write this in the form sup (S0 − S(x,t)) ≤

|x|≤R

sup |x|≤R+t||u||L∞

(S0 − S(x, 0)),

or, suppressing the large constant S0 , S0 + sup (−S(x,t)) ≤ S0 + |x|≤R

sup |x|≤R+t||u||L∞

(−S(x, 0)).

Eliminating S0 and noting that sup (−S(x,t)) = − inf S(x,t), we obtain the conclusion. The proof of Theorem 9.4.1 is completed. Remark 9.4.1. If the cross-section a remains constant, then the model of fluid flows in a nozzle with variable cross-section is reduced to the usual gas dynamics equations. Thus, one can recover from the above result the original minimum entropy principle for the gas dynamics equations in [299] in the case where the cross-section is constant.

9.5 Bibliographical notes We have listed only a short selection of the most relevant papers for this chapter. We also refer the reader to the bibliography at the end of this monograph for additional references. The additional equation ∂t a = 0 and the connection with nonconservative products was proposed by LeFloch [207, 208]. The Riemann problem for the model of non-isentropic flows in a nozzle with variable cross-section was studied in [20, 301]. LeFloch-Thanh [227, 228] constructed Riemann solutions for the model of isentropic flows in a nozzle with variable cross-section and shallow water equations. The Riemann problem for (9.1.1) was considered by a different approach by Andrianov-Warnecke [20]. The Riemann problem for conservation laws with resonance was considered by Isaacson-Temple [174, 175].

Chapter 10

Shallow water flows with discontinuous topography

In the previous two chapters, we considered models involving a compressible fluid, which flows due to changes in pressure. The pressure is a function of density, and the density in a compressible fluid flow depends on both spatial variables and time. This chapter deals with incompressible fluids, where the density is constant or considered to be constant. Specifically, we consider the Riemann problem for one-dimensional shallow water equations with discontinuous topography. Applications of this model can be seen in water flowing in a river, dam break analysis, open channel flows, etc. Since the bottom is non-flat, the momentum of the flow in the model is affected by variable topography. The momentum equation is thus nonconservative and involves a nonconservative term. Assuming the topography is independent of time supplements the governing equations with a system of balance laws in nonconservative form. We can therefore analyze characteristic properties of the model. It turns out that characteristic speeds may coincide on certain hypersurfaces and are given in explicit form. However, as in the previous two chapters, the system is not strictly hyperbolic. The nonconservative term representing the effect of topography on flow momentum generates stationary contact discontinuities. These stationary waves are admissible if they remain either in a subcritical or supercritical region or just touch but do not cross the critical curve. Elementary waves are defined to be admissible shock waves, rarefaction waves, and admissible stationary contact discontinuities. As usual, solutions to the Riemann problem consist of elementary waves associated with certain characteristic fields. Composite wave curves are constructed to help determine solutions to the Riemann problem. The fact that characteristic fields of the model are given in explicit form suggests that numerical schemes such as Godunov-type methods or Roe-type schemes may be convenient. It will be shown that the Riemann problem for shallow water equations with discontinuous topography may have a unique solution in certain regions but may not have a unique solution in another region. Furthermore, there may exist up to

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 10

285

286

10 Shallow water flows with discontinuous topography

three distinct solutions. Resonant phenomena still occur where multiple waves of the same speed may exist altogether.

10.1 Basic properties 10.1.1 System in nonconservative form Consider the following shallow water equations with discontinuous topography:

∂t h + ∂x (hu) = 0, ∂t (hu) + ∂x h(u2 + gh 2 ) + gh ∂x a = 0,

(10.1.1)

where h is the height of the water from the bottom to the surface, u is the fluid velocity, g is the gravity constant, and a = a(x) is the height of the topography bottom from a given level so that a(x) > 0, x ∈ R I . The function a(x) is assumed to be a piecewise constant of the form aL , x < 0, a(x) = aR , x > 0, where aL , aR are given constants. The system (10.1.1) contains a nonconservative term gh∂x a on the right-hand side. Thus, to investigate the system (10.1.1), we can supplement it with the (trivial) equation ∂t a = 0. (10.1.2) The Riemann problem for (10.1.1)-(10.1.2) is a Cauchy problem corresponding to piecewise constant initial data, of the form (hL , uL , aL ), x < 0, (h, u, a)(x, 0) = (10.1.3) (hR , uR , aR ), x > 0.

10.1.2 Non-hyperbolicity and genuine nonlinearity of characteristic fields The shallow water equations (10.1.1)-(10.1.2) can then be written as a system of balance laws in nonconservative form as

∂t U + A(U)∂xU = 0, where

(10.1.4)

10.1 Basic properties

287

⎛ ⎞ h U = ⎝u⎠ , a

⎛ ⎞ uh0 A(U) = ⎝g u g⎠ . 000

(10.1.5)

Eigenvalues of A(U) satisfy the characteristic equation u − λ h 0 det(A − λ I) = g u − λ g = 0, 0 0 −λ or λ ((u − λ )2 − gh) = 0. The characteristic equation gives us three real eigenvalues of (U):

λ1 (U) := u − gh, λ2 (U) := u + gh, λ3 (U) := 0. (10.1.6) The corresponding eigenvectors can be chosen as ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ gh h √ √h r1 (U) := ⎝− gh⎠ , r2 (U) := ⎝ gh⎠ , r3 (U) := ⎝ −gu ⎠ . 0 0 u2 − gh Note that (λ1 , r1 )(U) ≡ (λ3 , r3 )(U), (λ2 , r2 )(U) ≡ (λ3 , r3 )(U),

U ∈ C+ := U U ∈ C− := U

: :

gh ,

u = − gh . u=

(10.1.7)

(10.1.8)

The system (10.1.1)-(10.1.2) is therefore not hyperbolic on the surfaces C± . Apart from C := C+ ∪ C− , all the eigenvalues are real and distinct, so the system (10.1.1)-(10.1.2) is not only hyperbolic, but also strictly hyperbolic. Let us define regions in which the eigenvalues (10.1.6) keep a constant order: G1 := U | λ1 (U) > λ3 (U) , G2 := U | λ2 (U) > λ3 (U) > λ1 (U) , (10.1.9) G3 := U | λ3 (U) > λ2 (U) . Moreover, we also separate G2 into two subsets depending on the sign of the velocity: G− G+ 2 := {(h, u, a) ∈ G2 , u ≥ 0}, 2 := {(h, u, a) ∈ G2 , u < 0}.

288

10 Shallow water flows with discontinuous topography

In water resources engineering, one may meet the Froude number defined by |u| Fr(U) = √ . gh

(10.1.10)

If Fr(U) = 1, then U is said to be a critical state. If Fr(U) > 1, then U is said to be a supercritical state. If Fr(U) < 1, then U is said to be a subcritical state. Thus, C is the surface of the critical state, and we may call it critical surface for short. Its projection onto the (h, u)-plane is a curve and will be referred to as the critical curve. The regions G1 and G3 consist of supercritical states, and we may refer to them as the supercritical regions. The region G2 consists of subcritical states, and we may refer to it as the subcritical region. See Figure 10.1.

Fig. 10.1 Critical curve, subcritical, and supercritical regions in the (h, u)-plane.

We have −Dλ1 (U) · r1 (U) = Dλ2 (U) · r2 (U) =

3

gh = 0. 2

(10.1.11)

Thus, the first and the second characteristic fields are genuinely nonlinear. Moreover, Dλ3 (U) · r3 (U) ≡ 0, (10.1.12) so that the third characteristic field is linearly degenerate.

10.2 Elementary waves 10.2.1 Rarefaction waves Let us consider rarefaction waves of the system (10.1.4), i.e., smooth self-similar solutions to the system (10.1.4) of the form U(x,t) = (ρ , u, a)(ξ ), ξ = x/t associated with the genuinely nonlinear characteristic fields. By substituting U(x,t) = V (x/t) into (10.1.4), we obtain

10.2 Elementary waves

289

1 x x x x −( 2 )V ( ) + ( )A V ( ) V ) = 0, t t t t t which can be written as (A(U(ξ )) − ξ I)U (ξ ) = 0,

x ξ= . t

Thus, for U (ξ ) = 0, there must be an index i ∈ {1, 2} such that V (ξ ) = α (ξ )ri (V (ξ )),

λi (V (ξ )) = ξ ,

(10.2.1)

for some function α (ξ ). Differentiating the second equation in (10.2.1) with respect to ξ yields Dλi (V (ξ )) ·V (ξ ) = 1. By multiplying the first equation in (10.2.1) by Dλi (U(ξ )), we have from the last equation α (ξ )Dλi (V (ξ )) · ri (V (ξ )) = 1, which implies

α (ξ ) =

1 . Dλi (V (ξ )) · ri (V (ξ ))

(10.2.2)

It is derived from (10.2.1) and (10.2.2) that V (ξ ) =

1 ri (V (ξ )), Dλi (V (ξ )) · ri (V (ξ ))

λi (V (ξ )) = ξ ,

i = 1, 3.

(10.2.3)

Let us take two states U± on the integral curve of the first equation in (10.2.3) such that V (λi (U− )) = U− , V (λi (U+ )) = U+ . We can now define a rarefaction wave of (10.1.4) connecting the left-hand state U− and the right-hand state U+ by (for i = 1, 2) ⎧ x ⎪ U− , ≤ λi (U− ), ⎪ ⎪ t ⎨ x x , λi (u− ) ≤ ≤ λi (U+ ), (10.2.4) U(x,t) = V ⎪ t t ⎪ x ⎪ ⎩U+ , ≥ λi (U+ ). t More precisely, we consider the first equation in (10.2.3) in the case i = 1, which reads dh(ξ ) 2

=− √ h(ξ ), dξ 3 g du(ξ ) 2 = , dξ 3 da(ξ ) = 0. dξ

(10.2.5)

290

10 Shallow water flows with discontinuous topography

The last equation in (10.2.5) shows that the component a remains constant through any rarefaction wave. Moreover, the first two equations of the above system yields

= − gh . Therefore, the integral curve passing through a given point U0 = (h0 , u0 , a0 ) is given by

√ √ u = u0 − 2 g( h − h0 ). du dh

Since the characteristic speed must be increasing through a rarefaction fan, that is, λ1 (U) ≥ λ1 (U0 ), we have h ≥ h0 . The forward curve of 1-rarefaction waves R1F (U0 ) starting from a given left-hand state U0 and consisting of all the right-hand states U that can be connected to U0 by a 1-rarefaction wave associate with the first characteristic field is thus given by

√ √ R1F (U0 ) : u = ω1F (U0 ; h) := u0 − 2 g( h − h0 ), h ≤ h0 . (10.2.6) Arguing similarly, we can obtain the forward curve of 2-rarefaction waves R2F (U0 ) starting from a given left-hand state U0 and consisting of all the right-hand states U that can be connected to U0 by a 2-rarefaction wave associate with the second characteristic field as

√ √ R2F (U0 ) : u = ω2F (U0 ; h) := u0 + 2 g( h − h0 ), h ≥ h0 . (10.2.7) By a similar argument, we also obtain the backward curve of 1-waves R1B (U0 ) starting from a given right-hand state U0 and consisting of all the left-hand states U that can be connected to U0 by a 1-rarefaction wave associate with the first characteristic field as

√ √ R1B (U0 ) : u = ω1B (U0 ; h) := u0 − 2 g( h − h0 ), h ≥ h0 , (10.2.8) and the forward curve of 2-rarefaction waves R2B (U0 ) starting from a given righthand state U0 and consisting of all the left-hand states U that can be connected to U0 by a 2-rarefaction wave associate with the second characteristic field as

√ √ R2B (U0 ) : u = ω2B (U0 ; h) := u0 + 2 g( h − h0 ), h ≤ h0 . (10.2.9)

10.2.2 Shock waves Let us consider a shock wave of the form U0 , x < λ t, U(x,t) = U1 , x > λ t,

(10.2.10)

where U0 ,U1 are the left-hand and right-hand states, respectively, and λ is the shock speed. The Rankine-Hugoniot relation for (10.1.2) is given by −λ [a] = 0, where [a] := a1 − a0 is the jump of the quantity a, which implies that:

10.2 Elementary waves

(i) (ii)

291

either a = constant across the shock, or the shock speed vanishes: λ = 0.

In Case (i), the system (10.1.1)-(10.1.2) reduces to the standard shallow water equations with flat bottom

∂t h + ∂x (hu) = 0, u2 + g(h + a) = 0. ∂t u + ∂x 2 The Rankine-Hugoniot relations hold for the shock: − λ¯ [h] + [hu] = 0,

h −λ¯ [hu] + [h(u2 + g ] = 0, 2

where [h] = h − h0 , etc, and λ¯ = λ¯ (U0 ,U) is the shock speed. A straightforward calculation gives us 1 1 g , (h − h0 ) + u = u0 ± 2 h h0 and along these two curves, we have du g 1 1 =± + − (h − h0 ) dh 2 h h0

2h2

g 1 →± h0 1 1 + h h0

as

h → h0 .

Since the ith-Hugoniot curve Hi (U0 ) is tangent to the eigenvector ri (U0 ), i = 1, 2, we can see that the first Hugoniot curve associated with the first characteristic field is given by 1 g 1 , h ≥ 0, (10.2.11) (h − h0 ) + H1 (U0 ) : u := u1 (h,U0 ) = u0 − 2 h h0 and the second Hugoniot curve associated with the second characteristic field is defined by 1 g 1 H2 (U0 ) : u := u2 (h,U0 ) = u0 + , h ≥ 0. (10.2.12) (h − h0 ) + 2 h h0 Moreover, along the Hugoniot curves H1 (U0 ), H2 (U0 ), the corresponding shock speeds are given by hu − h u h2 g 1,2 0 0 , h ≥ 0, h+ λ¯ 1,2 (U0 ,U) = = u0 ∓ h − h0 2 h0

292

10 Shallow water flows with discontinuous topography

As usual, the admissibility criterion for shock waves is the Lax shock inequalities. So, we require that the shock speed λ¯ i (U0 ,U) satisfy the Lax shock inequalities

λi (U) < λ¯ i (U0 ,U) < λi (U0 ),

i = 1, 2.

(10.2.13)

It follows from (10.2.11), (10.2.12), and (10.2.13) that the i-shock curve Si (U0 ), starting from a left-hand state U0 and consisting of all right-hand states U that can be connected to U0 by a Lax shock associated with the ith characteristic field, i = 1, 2 are given by 1 1 g , h > h0 , (h − h0 ) + S1F (U0 ) : u = ω1F (U0 ; h) := u0 − 2 h h0 1 1 g F F , h < h0 . S2 (U0 ) : u = ω2 (U0 ; h) := u0 + (h − h0 ) + 2 h h0 (10.2.14) Similarly, the backward i-shock curve SiB (U0 ), i = 1, 2, starting from a righthand state U0 and consisting of all left-hand states U that can be connected to U0 by a Lax shock associated with the ith characteristic field i = 1, 2, are given by 1 g 1 B B (h − h0 ) + S1 (U0 ) : u = ω1 (U0 ; h) = u0 − , h < h0 , 2 h h0 (10.2.15) 1 1 g B B S2 (U0 ) : u = ω2 (U0 ; h) = u0 + , h > h0 . (h − h0 ) + 2 h h0 It is not difficult to check that the function h → u = ω1 (h), h ≥ 0, is strictly convex and strictly decreasing; the function h → u = ω2 (h), h ≥ 0, is strictly concave and strictly increasing. Along the shock curves, shock speed may coincide with the speed of stationary contact waves, as seen in the following lemma. The proof is straightforward and is omitted. Lemma 10.2.1. Consider the projection on the (h, u)-plan. To every U = (h, u) ∈ G1 there exists exactly one point U # ∈ S1F (U) ∩ G+ 2 such that the 1-shock speed λ¯ 1 (U,U # ) = 0. The state U # = (h# , u# ) is defined by

−h + h2 + 8hu2 /g uh # h = , u# = # . 2 h Moreover, for any V ∈ S1F (U), the shock speed λ¯ 1 (U,V ) > 0 if and only if V is located above U # on S1F (U). The above argument leads us to define the wave curves as follows. The forward curve of i-waves Wi F (U0 ) consists of all right-hand states U that can be connected to the left-hand state U0 by either an i-shock satisfying the Lax shock inequalities or an i-rarefaction wave, i = 1, 2. The backward curve of j-waves W jB (U0 ) consists of all left-hand states U that can be connected to the right-hand state U0 by either a j-shock satisfying the Lax shock inequalities or a j-rarefaction wave, j = 1, 2. So,

10.2 Elementary waves

293

these wave curves are given by Wi F (U0 ) := SiF (U0 ) ∪ RiF (U0 ) = {U | Ψi (U;U0 ) := u − ωiF (U0 ; h) = 0}, Wi B (U0 ) := SiB (U0 ) ∪ RiB (U0 ) = {U | Φi (U;U0 ) := u − ωiB (U0 ; h) = 0}, (10.2.16) where ⎧ √ √ √ ⎨u0 − 2 g( h − h0 ), h ≤ h0 , ω1F (U0 ; h) := , 1 1 ⎩u0 − g2 (h − h0 ) h > h0 + h h0 , ⎧ √ √ √ ⎨u0 + 2 g( h − h0 ), h ≥ h0 , ω2F (U0 ; h) := , 1 1 ⎩u0 + g2 (h − h0 ) + h < h0 h h0 , (10.2.17) ⎧ √ √ √ ⎨u0 − 2 g( h − h0 ), h ≥ h0 , ω1B (U0 ; h) := , 1 1 ⎩u0 − g2 (h − h0 ) h < h0 h + h0 , ⎧ √ √ √ ⎨u0 + 2 g( h − h0 ), h ≤ h0 , ω2B (U0 ; h) := . 1 1 ⎩u0 + g2 (h − h0 ) h > h0 + h h0 ,

10.2.3 Stationary contact waves In Case (ii), the discontinuity satisfies jump relations and is associated with the 3characteristic field. This is given in the following lemma. Lemma 10.2.2. Any solution of the following system of ordinary differential equations is a stationary smooth solution of (10.1.1)-(10.1.2): (hu) = 0, 2 u + g(h + a) = 0, 2

θ = 0. Any jump satisfying the relations [hu] = 0,

u2 + g(h + a) = 0, 2

(10.2.18)

is a weak solution of (10.1.1)-(10.1.2) in the sense of nonconservative products. Moreover, the curve of jumps defined by (10.2.18) coincides with the integral curve associated with the fourth characteristic field (λ3 (U), r3 (U)).

294

10 Shallow water flows with discontinuous topography

The proof of Lemma 10.2.2 is omitted, since it is a special case of the one in the next chapter, when we consider the shallow water equations with temperature gradients. Thus, a shock wave satisfying the jump relations (10.2.18) is a weak solution in the sense of nonconservative products and is associated with the 3rd characteristic field, which is linearly degenerate. Thus, a wave of this kind is a 3-contact discontinuity. For any state U0 , the stationary 3-contact wave curve passing though U0 defined by (10.2.18) can be parameterized with h: W3 (U0 ) :

h0 u0 , h 2 a = a(h) = 2ga0 + u0 − (ω3 (U0 ; h))2 + 2g(h0 − h).

u = ω3 (U0 ; h) :=

(10.2.19)

The function h → ω3 (U0 ; h) is strictly convex and strictly decreasing function for u0 > 0, and strictly concave and strictly increasing for u0 < 0. The above arguments also show that the a-component of Riemann solutions may change only across a stationary wave. Let us be given a state U0 = (h0 , u0 , a0 ) and a bottom level a for the state on the other side of a stationary shock U = (h, u, a). The values h, u can be found in terms of U0 , a, as follows. Substituting u = h0 u0 /h from the first equation in (10.2.19) to the second equation in (10.2.19), we obtain h0 u0 2 1 2 u0 − + h0 − h = a. a0 + 2g h Multiplying both sides of the last equation by 2gh2 , and then re-arranging terms, we can see that h > 0 is a root of the nonlinear equation

ϕ (h) = ϕ (U0 , a; h) := 2gh3 + (2g(a − a0 − h0 ) − u20 )h2 + h20 u20 = 0. It holds that

(10.2.20)

ϕ (0) = h20 u20 ≥ 0, ϕ (h) = (6gh + 2(2g(a − a0 − h0 ) − u20 ))h, ϕ (h) = 12gh + 2(2g(a − a0 − h0 ) − u20 ).

This yields for h > 0:

ϕ (h) = 0 iff h = h∗ (U0 , a) := u2

u20 + 2g(a0 + h0 − a) > 0. 3g

Thus, if h∗ (U0 , a) < 0, or a > a0 + h0 + 2g0 , then ϕ (h) > 0 for h > 0. Since ϕ (0) = h20 u20 ≥ 0, there is no root for (10.2.20) if

10.2 Elementary waves

295

a > a0 + h0 +

u20 . 2g

a ≤ a0 + h0 +

u20 , 2g

If

then

ϕ > 0 h > h∗ (U0 , a) = ϕ (h) < 0,

u20 + 2g(a0 + h0 − a) ≥ 0, 3g

(10.2.21)

0 < h < h∗ .

Thus, ϕ admits a zero h > 0, and in this case it has two zeros, iff

ϕmin := ϕ (h∗ ) = −gh3∗ + h20 u20 ≤ 0, or h∗ (U0 , a) ≥ hmin (U0 ) :=

h2 u2 1/3 0 0

g

,

(10.2.22)

where h∗ is defined by (10.2.21). We can see that (10.2.22) holds if and only if u2

a ≤ amax (U0 ) := a0 + h0 + 2g0 − 2g31/3 (h0 u0 )2/3 2/3 2 2/3 1 (gh0 )1/3 − u0 = a0 + 2g (2(gh0 )1/3 + u0 ).

(10.2.23)

The formula (10.2.23) yields amax (U0 ) ≥ a0 and amax (U0 ) = a0 only if (h0 , u0 ) ∈ C . Under the condition a ≤ amax (U0 )

(10.2.24)

the function ϕ in (10.2.20) admits two roots denoted by θ1 (U0 ; a) ≤ θ2 (U0 ; a) satisfying θ1 (U0 ; a) ≤ h∗ ≤ θ2 (U0 ; a). Moreover, if the inequality in (10.2.23) is strict, i.e., a < amax (U0 ), then these two roots are distinct: θ1 (U0 ; a) < h∗ < θ2 (U0 ; a). See Figure 10.2. Moreover, we have ϕ (h0 ) = 2g(a − a0 ) < 0 iff a < a0 . So, we have the following comparison between h0 and the roots of ϕ as

θ1 (U0 ; a) < h0 < θ2 (U0 ; a),

if a < a0 .

Consider the case a > a0 . It holds that h∗ − h0 =

u20 − gh0 + 2g(a0 − a) . 3g

296

10 Shallow water flows with discontinuous topography

Thus, if a > a0 and u20 − gh0 + 2g(a0 − a) > 0, then h∗ > h0 , and ϕ (h0 ) > 0. This implies that h0 < θ1 (U0 ; a) < θ2 (U0 ; a),

if a > a0 ,

and u20 − gh0 + 2g(a0 − a) > 0.

if a > a0 ,

and u20 − gh0 + 2g(a0 − a) < 0.

Moreover, it holds that h0 > θ2 (U0 ; a) > θ1 (U0 ; a),

Thus, we arrive at the following lemma.

Fig. 10.2 Graph of the function ϕ = ϕ (h), h ≥ 0 defined by (10.2.20) with g = 9.8, a0 = 1, h0 = 1, u0 = 1, and a = 1.2. The function ϕ admits two zeros in the interval (0, 1).

Lemma 10.2.3. Given U0 = (h0 , u0 , a0 ) and a bottom level a = a0 . The following conclusions hold: (i) amax (U0 ) ≥ a0 , amax (U0 ) = a0 if and only if (h0 , u0 ) ∈ C . (ii) The nonlinear equation (10.2.20) admits a root if and only if the condition (10.2.23) holds, and in this case it has two roots θ1 (U0 ; a) ≤ h∗ ≤ θ2 (U0 ; a). Moreover, whenever the inequality in (10.2.23) is strict, i.e., a < amax (U0 ), these two roots are distinct. (iii) According to the part (ii), whenever (10.2.23) is fulfilled, there are two states Ui (U0 ; a) = (θi (U0 ; a), ui (a), a), where ui (U0 ; a) = h0 u0 /θi (U0 ; a), i = 1, 2 to which a stationary contact from U0 is possible. Moreover, the locations of these states can be determined as follows: U1 (a) ∈ G1 if u0 > 0, U1 (a) ∈ G3 if u0 < 0, U2 (a) ∈ G2 .

10.2 Elementary waves

297

Proof. The parts (i) and (ii) can be easily deduced from the above argument. To prove (iii), it is sufficient to show that along the projection of W3 (U0 ) on the (h, u)plane, the point Umin (U0 ) = (hmin (U0 ), umin (U0 ) := h0 u0 /hmin (U0 )), where hmin (U0 ) is defined by (10.2.22), belongs to C+ if u0 > 0 and belongs to C− if u0 < 0, and that Ui (a) ∈ W3 (U0 ), i = 1, 2, such that U2 (a) ∈ G2 and U1 (a) is located on the other side of U2 (a) with respect to C . Indeed, let us define a function taking values along the stationary curve W3 (U0 ):

σ (h) := u(h)2 − gh =

h20 u20 − gh. h2

Clearly, a point U = (h, u, a) belongs to G1 ∪ G3 if and only if σ (h) > 0 and U belongs to G2 if and only if σ (h) < 0. Since σ (hmin (U0 )) = 0, the point Umin (U0 ) belongs to C . Obviously, Umin (U0 ) ∈ C+ if u0 > 0, and Umin (U0 ) ∈ C− if u0 < 0. Thus, it remains to check that

σ (θ1 (U0 ; a)) > 0,

σ (θ2 (U0 ; a)) < 0.

Since

σ (hmin (U − 0)) = 0,

σ (h) =

−2h20 u20 − g < 0, h3

we can see that the above inequalities hold if

θ1 (U0 ; a) < hmin (U0 ) < θ2 (U0 ; a).

(10.2.25)

On the other hand, we have

ϕ (h) > 0, ϕ (h) < 0,

if h < θ1 (U0 ; a) or h > θ2 (U0 ; a), if θ1 (U0 ; a) < h < θ2 (U0 ; a).

(10.2.26)

And we have

ϕ (hmin (U0 )) = 3(h0 u0 )2 + (2g(a − a0 − h0 ) − u20 )

(h0 u0 )4/3 . g2/3

It is a straightforward calculation to show that the condition a < amax (U0 ) is equivalent to ϕ (hmin (U0 )) < 0. This together with (10.2.26) establish (10.2.25). Lemma 10.2.3 is completely proved.

From Lemma 10.2.3, two-parameter wave sets can be constructed. This situation is similar to the one in the previous two chapters for the model of fluid flows in a nozzle with a discontinuous cross-section. The Riemann problem for (10.1.1) may therefore admit up to a one-parameter family of solutions. To select a unique Riemann solution, we impose an admissibility condition for stationary contacts.

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10 Shallow water flows with discontinuous topography

a-monotone Criterion for admissible contact waves (a) Along any stationary curve W3 (U0 ), the bottom level a is monotone as a function of ρ . (b) The total variation of the bottom level component of any Riemann solution must not exceed (and therefore, is equal to) |aL − aR |, where aL , aR are left-hand and right-hand bottom level levels. Again, it is interesting that an admissible contact wave must remain in the closure of the subcritical or supercritical region. Lemma 10.2.4. The a-monotone Criterion implies that any admissible contact wave must remain in the closure of the subcritical or supercritical region. That is, (i) For any supercritical state U0 , the admissible stationary contact wave under the a-monotone Criterion is the one using = θ1 (U0 ; a); (ii) For any subcritical state U0 , the admissible stationary contact wave under the a-monotone Criterion is the one using = θ2 (U0 ; a). Proof. From the Rankine-Hugoniot relations associated with the linearly degenerate field (10.2.18), we get a = a(h) = a0 + where u = u(h) =

h0 u0 h .

−u2 + u20 − h + h0 , 2g

Then, we find uh0 u0 da −u du = −1 = −1 dh g dh gh2 (u2 − gh) u2 −1 = , = gh gh

which has the same sign as u2 − gh. Thus, a = a(h) is increasing with respect to h in the domains G1 , G3 and is decreasing in the domain G2 . Thus, that a = a(h) is monotone as a function of h implies that the point (h, u, a) has to stay in the closure of the domain containing (h0 , u0 , a0 ). This gives us (i) and (ii).

10.2.4 Monotonicity of the composite wave curves In constructing Riemann solutions, we will project all the wave curves onto the (h, u)-plane. Then, we combine the waves associated with a nonlinear characteristic field with the 3-waves to form a kind of composite wave. Finally, we let the wave curve associated with the remaining characteristic field to intersect this composite wave and obtain the Riemann solution. To determine the uniqueness of Riemann

10.2 Elementary waves

299

solutions, we will show that this composite wave curve has a monotone property in the sense that it can be parameterized under the form where the velocity is given as a monotone function of the height u = u(h). Precisely, given a0 and a = a0 , we first define the curves of composite waves W1→3 (U0 ) := {U = (h, u, a) ∈ G2 ∪ C : ∃W3 (U1 ,U) shifting a0 to a, U1 = (h1 , u1 , a0 ) ∈ W1F (U0 ), λ1 (U) ≤ 0}, W2←3 (U0 ) := {U = (h, u, a) ∈ G2 ∪ C : ∃W3 (U1 ,U) shifting a0 to a, U1 = (h1 , u1 , a0 ) ∈ W2B (U0 ), λ2 (U) ≥ 0},

(10.2.27) In other words, the composite wave curve W1→3 (U0 ) consists of the point U which can be arrived at by an ordered combination of 2 waves: the first 1-wave from the left-hand state U0 to some right-hand state U1 , followed by a second 3-wave from the left-hand state U1 to the right-hand U; the composite wave curve W2←3 (U0 ) consists of all states U which can be arrived at by an ordered combination of 2 waves: the first 2-wave from the right-hand state U0 to some left-hand state U1 , preceded by a second 3-wave from the right-hand state U1 to the left-hand state U. As observed earlier, the right-hand state U = (h, u, a) that can be reached from an admissible stationary contact wave from a given left-hand state U0 = (h0 , u0 , a0 ) satisfies the nonlinear algebraic equations

ϕ (h;U0 ) = 2gh3 + (2g(a − a0 − h0 ) − u20 )h2 + h20 u20 = 0,

u=

u0 h0 . (10.2.28) h

Assume throughout the following that a < a0 . Then, the first equation in (10.2.28) admits two roots θ1 (U0 , a) and θ2 (U0 , a) with

θ1 (U0 , a) < h0 < θ2 (U0 , a). We aim to build W1→3 (U0 ), where the condition λ1 (U) ≤ 0 in (10.2.27) means that the stationary contact must occur in the subcritical region or on the critical curve. Thus, we need to investigate the admissible contact wave using the root θ2 of (10.2.28). Admissible jump along the wave curve W1 (U0 ) ∩ G2 occurs at the left-hand states of the form U1 (h) = (h, u = w1 (U0 ; h), a0 ) ∈ W1 (U0 ) ∩ G2 . For simplicity, we denote in the following U± = (h± , u± , a0 ) = W1F (U0 ) ∩ C± , w1 (h) := ω1F (U0 ; h), U1 (h) = (h, w1 (h), a0 ), θ2 (h) = θ2 (U1 (h), a), U10 (h) = (ϕ2 (U1 (h), a), u = w1 (h)h/ϕ2 (h), a) ∈ G2 .

(10.2.29)

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10 Shallow water flows with discontinuous topography

In particular, U10 (h) is the right-hand state of the admissible stationary wave from U1 (h). Since h = θ2 (h) is a root of (10.2.28), we have G(h) := ϕ (h;U1 (h)) = 2gθ23 (h) + (2g(a − a0 − h) − w21 (h))θ2 (h)2 + h2 w21 (h) = 0. (10.2.30) The h-component of the composite wave curve W1→3 (U0 ) is given by θ2 (h) defined by (10.2.29). Its monotone property is given by the following lemma. Lemma 10.2.5. The h-component of the composite wave curve W1→3 (U0 ) is strictly increasing with respect to h in the domain of this composite wave curve. In other words, (a) If U0 is a subcritical, then the function θ2 (h) is strictly increasing for h ∈ [h+ , h− ]; (b) If U0 is supercritical, then the function θ2 (h) is strictly increasing for h ∈ [h#0 , h− ]. Proof. We need only to prove for part (a) since the one for part (b) is similar. We will establish the monotony property of the function θ2 (h) with respect to h by showing that its derivative is negative. First, differentiating (10.2.30) with respect to h both sides gives us 0 = G (h) = 6gθ22 (h)θ2 (h) + 2(2g(a − a0 − h) − w21 (h))θ2 (h)θ2 (h) + θ22 (h)(−2g − 2w1 (h)w1 (h)) + 2hw21 + 2h2 w1 (h)w1 (h) = θ2 (h)θ2 (h)[6gθ2 (h) + 2(2g(a − a0 − h) − w21 (h))] + 2(hw21 (h) − gθ22 (h)) + 2w1 (h)w1 (h)(h2 − θ22 (h)), where (.) = d(.)/dh. Then, where

θ2 (h)M = N,

(10.2.31)

M = θ2 (h)[3gθ2 (h) + 2g(a − a0 − h) − w21 (h)], N = gθ22 (h) − hw21 (h) + w1 (h)w1 (h)(θ22 (h) − h2 ).

Thus, it is sufficient to show that M > 0,

N > 0.

First, since U1 (h) = (h, w1 (h)) ∈ G2 , using (10.2.30), we have M θ2 (h) = 3gθ23 (h) − (2gθ23 (h) + h2 w21 (h)) = gθ23 (h) − h2 w21 (h) > gh3 − h2 w21 (h) = h2 (gh − w21 (h)) ≥ 0. Provided w1 (h) ≤ 0,

(10.2.32)

10.2 Elementary waves

301

we find gθ22 (h) − hw21 (h) > gh2 − hw21 (h) = h(gh − w21 ) > 0, and

θ22 (h) − h2 > 0. Therefore, we obtain N = gθ22 (h) − hw21 (h) + w1 (h)w1 (h)(θ22 (h) − h2 ) > 0. Now consider the case U0 ∈ G2 ,

0 < w1 (h) < u0 .

This part corresponds to the shock part of the curve u = ω1F (h) is strictly convex in h, we have

W1F (U0 ).

(10.2.33) Since the function

d2 2 d 2w1 (h)w1 (h) = 2(w2 (w1 (h)) = 1 + w1 (h)w1 (h)) > 0. 2 dh dh for h0 ≤ h ≤ h∗ , where h∗ is the h-intercept of the wave curve W1F (U0 ). This implies that the function w1 (h)w1 (h) is increasing, and so g , h0 ≤ h ≤ h∗ . w1 (h)w1 (h) ≥ w1 (h0 )w1 (h0 ) = −u0 h0 Thus, if u0 ≥ 0, then N = gθ22 (h) − hw21 (h) + w1 (h)w1 (h)(θ22 (h) − h2 ) g 2 g 2 2 2 > gθ2 (h) − hw1 (h) − u0 θ2 (h) + u0 h h0 h0

g g 2 2 gh0 − u0 + h u0 ≥ θ2 h − w1 (h) , h0 h0 and thus

g 2 h − w1 (h) N > h u0 h0

≥ h u0 gh0 − w21 (h) ≥ h u20 − w21 (h) ≥ 0,

where the last inequality follows from (10.2.33). If u0 < 0, then N = gθ22 (h) − hw21 (h) + w1 (h)w1 (h)(θ22 (h) − h2 ) g 2 > gθ22 (h) − hw21 (h) − u0 (θ (h) − h2 ) h0 2 g 2 ≥ gh2 (h) − hw21 (h) − u0 (θ (h) − h2 ), h0 2

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10 Shallow water flows with discontinuous topography

which implies that N > gh2 (h) − hw21 (h) + 0 ≥ h(gh(h) − w21 (h)) ≥ 0, since the point (h, w1 (h)) ∈ G2 ∪ C . Finally, consider the case U0 ∈ G2 ,

h+ ≤ h ≤ h0 ,

w1 (h) ≥ u0 .

(10.2.34)

This part corresponds to the rarefaction waves, where

√ √ w1 (h) = u0 − 2 g h − h0 . This implies that

It holds that

and thus,

g w1 (h) = − , h

h ≤ h0 .

(10.2.35)

N = gθ22 (h) − hw21 (h) + w1 (h)w1 (h)(θ22 (h) − h2 ) g 2 (θ (h) − h2 ) = gθ22 (h) − hw21 (h) − w1 (h) h 2

g = θ22 (h) g − w1 w1 (h)h( gh − w1 (h)) h

g 2 + w1 (h)h N = ( gh − w1 (h)) θ2 (h) h ≥ 0,

since the point (h, w1 (h)) ∈ G2 ∪C and w1 (h) > 0. Thus, M > 0 , N > 0 and therefore

θ2 (h) =

d θ2 (U1 (h), a) > 0. dh

This terminates the proof of Lemma 10.2.5. For the u-component of the composite wave curve W1→3 (U0 ) is given by u(h) = w1 (U0 ; h)h/θ2 (U1 (h), a),

(10.2.36)

where U1 (h) = (h, u = w1 (U0 ; h)) ∈ W1F (U0 ) ∩ G2 , and h ≥ h#0 if U0 ∈ G1 . The monotone property of this function on the domain of the composite wave curve is given in the following lemma. Lemma 10.2.6. Along the composite wave curve W1→3 (U0 ) when a < a0 , the u-component is strictly decreasing as a function of the height h. In other words, (a) If U0 is a subcritical, then the function u(h) defined by (10.2.36) is strictly decreasing for h ∈ [h+ , h− ];

10.2 Elementary waves

303

(b) If U0 is supercritical, then then the function u(h) defined by (10.2.36) is strictly decreasing for h ∈ [h#0 , h− ]. Proof. First, consider the case w1 (h) ≥ 0. We will show that the function

ζ (h) = w1 (h)h

(10.2.37)

is strictly decreasing for the corresponding domain of h, that is, h ∈ [h+ , h− ] if U0 is a subcritical state, and h ∈ [h#0 , h− ] if U0 is a supercritical state. Let h∗ be the h-intercept of the shock curve W1F (U0 ). Indeed, assume that U0 is a subcritical state. We find g w1 (h+ ) = − , h+

so

ζ

(h+ ) = w1 (h+ )h+ + w1 (h+ ) = −

g h+ + gh+ = 0. h+

Since ζ (h) < 0, it implies that

ζ (h) < 0,

h+ < h < h∗ .

Thus, ζ is strictly decreasing function for h+ < h < h∗ . Next, assume that U0 is a supercritical or critical state. We have √ 2g h0 h + 4h2 − h20 ζ (h) = u0 − . 4hh0 1h + h10 The function ζ is strictly concave since its second derivative is negative

g/2 8h3 + 12h2 h0 + 3hh0 + h30 ζ (h) = − < 0, 3 1 1 4h3 h30 + h h0

(10.2.38)

(10.2.39)

for h > 0. Thus, the first derivative ζ in (10.2.38) is strictly decreasing on the interval h ∈ (h#0 , h∗ ). This yields in particular ζ (h) < ζ h#0 ,

h > h#0 .

h0 + 16u20 /g − 3 h20 + 8h0 u20 /g g ζ h#0 = u0 − . 8 u0 We will show that ζ (h#0 ) ≤ 0. Indeed, the condition ζ h#0 ≤ 0 is equivalent to

We have

u20 ≤ gh0 + 16u20 − 3g h20 + 8h0 u20 /g,

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10 Shallow water flows with discontinuous topography

g 2 h0 + 8h0 u20 /g . 3g h20 + 8h0 u20 /g ≤ h0

or

Simplifying the last inequality, we obtain u0 2 ≥ gh0 , which is correct, since U0 is a supercritical state. This shows that ζ (h) < ζ h#0 ≤ 0, h#0 < h < h∗ . This means that the function ζ is strictly decreasing for h#0 < h < h∗ . Next, let us define the function u(h) = w1 (h)h

1 θ2 (U1 (h), h)

(10.2.40)

in the domain given in the two cases (a) and (b). It can be seen that the function u(h) in (10.2.40) is the product of the two positive and strictly decreasing functions

ζ (h) = w1 (h)h,

1 θ2 (U1 (h), h)

on the interval h+ < h < h∗ or h#0 < h < h∗ for Case (a) or (b), respectively. Thus, u(h) is also strictly decreasing in the case w1 (h) ≥ 0. Second, we consider the case w1 (h) < 0. Differentiating the equation a0 − a +

1 2 (w (h) − u2 (h)) + h − θ2 (h) = 0 2g 1

gives us 1 (w1 (h)w1 (h) − u(h)u (h)) + 1 − θ2 (h) = 0. g So, we find

u(h)u (h) = w1 (h)w1 (h) + g − gθ2 (h).

Multiplying both sides of the last equation by M, in view of (10.2.31), we have u(h)u (h)M = (w1 (h)w1 (h) + g)M − gM θ2 (h) = (w1 (h)w1 (h) + g)M − gN. Multiply both sides of the last equation by θ2 , substituting M and N from (10.2.31) into the last equation, and using the equation (10.2.30) to simplify the expression, we obtain u(h)u (h)M θ2 = (w1 (h)w1 (h) + g)(gθ23 (h) − h2 w21 (h)) + gθ2 (h)[hw21 (h) − gθ22 (h) + w1 (h)w1 (h)(h2 − θ22 (h))] = w1 (h)w1 (h)h2 (gθ2 (h) − w21 (h)) + ghw21 (h)(θ2 (h) − h), and therefore

10.3 Riemann problem

305

u(h)u (h)M θ2 > w1 (h)w1 (h)h2 (gθ2 (h) − w21 (h)) ≥ w1 (h)w1 (h)h2 (gh − w21 (h)), ≥ 0, where the last inequality follows from the fact that w1 (h) < 0, w1 (h) < 0, θ2 (h) > h, and U1 (h) = (h, w1 (h)) ∈ G2 . Since u(h) = w1 (h)h/θ2 (h) < 0, M > 0 and θ2 > 0, the inequality u(h)u (h)M θ2 > 0 implies that u (h) < 0 by (10.2.40). This completes the proof of Lemma 10.2.6. Similarly, we can establish that along the composite wave curve W2←3 (U0 ), the h-component is increasing, and the u-component is also increasing as a function of the height h. The monotonicity property of the composite wave curves is given as follows. Theorem 10.2.1. Assume that a < a0 . The composite curve W1→3 (U0 ) can be parameterized by a strictly decreasing function u with respect to the height h on its domain. The composite curve W2←3 (U0 ) can be parameterized by a strictly increasing function u with respect to the height h on its domain.

10.3 Riemann problem Under the transformation x → −x, u → −u, a left-hand state U = (h, u, a) in G2 or G3 will be transferred to the right-hand state V = (h, −u, a) in G2 or G1 , respectively. We will combine the 1-wave with the stationary waves, and so the constructions of Riemann solutions will rely on the left-hand state UL , in a forward manner. A similar procedure for backward constructions relying on the right-hand state UR can be made. To construct Riemann solutions of (10.1.1)-(10.1.2), we project all the wave curves on the (h, u)-plane.

Notation We will use the following notation: (i) Wi (u1 , u2 ): An i-wave connecting a state u1 and a state u2 , where W = S: Lax shock, W = R: rarefaction wave, W2 = Z, i = 1, 2, 3; (ii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the left-hand state u1 to the right-hand state u2 is followed by a j-wave from the left-hand state u2 to the right-hand state u3 ;

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10 Shallow water flows with discontinuous topography

(iii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the right-hand state u1 to the left-hand state u2 is preceded by a j-wave from the right-hand state u2 to the left-hand state u3 ; (iv) U 0 denotes the state resulting from a stationary contact wave from U; (v) U # is the state defined in Lemma 10.2.1 so that λ¯ 1 (U,U # ) = 0. Let G¯ 1 denote the closure of G1 . We assume that UL ∈ G¯ 1 , or equivalently λi (UL ) ≥ 0, i = 1, 2, 3.

10.3.1 Construction 1: supercritical/supercritical Consider UL ∈ G1 and UR is located in a “higher” region in the (h, u)-plane. If aL ≥ aR (or aL < aR ≤ amax (UL )), there is a stationary contact upward (downward, respectively) along W3 (UL ) from UL to a state denoted by U1 so that {U1 } = W3 (UL ) ∩ G1 , shifting the level aL directly to the level aR . Under the condition that W1F (U1 ) ∩ W2B (UR ) = 0, /

(10.3.1)

and let {U2 } = W1F (U1 ) ∩ W2B (UR ). a Riemann solution exists, as long as λ¯ 1 (U1 ,U2 ) ≥ 0. Indeed, the solution begins with a stationary wave from UL to U1 , followed by a 1-wave from U1 to U2 . Then, the solution arrives at UR by a 2-wave from U2 . Thus, the solution has the form W3 (UL ,U1 ) W1 (U1 ,U2 ) W2 (U2 ,UR ).

(10.3.2)

The condition for the existence of such a stationary wave W3 (UL ,U1 ) is aR ≤ amax (UL ).

(10.3.3)

Then, the solution (10.3.2) occurs as long as λ¯ 1 (U1 ,U2 ) ≥ 0, which means that U2 is above U1# , or coincides with it, and so that the state UR is somewhere in a higher position. See Figure 10.3. The above construction can be extended if W2B (UR ) lies entirely above W1F (U1 ). In this case, let I and J be the intersection points of W1F (U1 ) and W2B (UR ) with the axis {h = 0}, respectively: {I} = W1F (U1 ) ∩ {h = 0},

{J} = W2B (UR ) ∩ {h = 0},

then the solution can be seen as a dry part Wo (I, J) between I and J. Thus, the solution in this case is W3 (UL ,U1 ) R1 (U1 , I) Wo (I, J) R2 (J,UR ).

(10.3.4)

10.3 Riemann problem

307

Fig. 10.3 Construction 1: A Riemann solution of the form (10.3.2).

The states U1 ,U2 which separate the elementary waves of the Riemann solution (10.3.2) can be calculated as follows. First, the two components of the state U1 = (h1 , u1 , a1 = aR ) is determined by the two equations from (10.2.18) h1 u1 = hL uL ,

u21 + 2g(h1 + aR ) = u2L + 2g(hL + aL ),

(10.3.5)

and the state of the supercritical region is chosen. Since U2 = (h2 , u2 , aR ) is the intersection point of the two wave curves W1F (U1 ) and W2B (UR ), it follows from (10.2.17) that the components of U2 can be calculated by u2 = ω1F (U1 ; h2 ) = ω2B (UR ; h2 ).

(10.3.6)

The configuration of the Riemann solution (10.3.2) in the (x,t)-plane is given by Figure 10.4.

Fig. 10.4 Riemann solution with structure (10.3.2) in the (x,t)-plane

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10 Shallow water flows with discontinuous topography

Note that this construction can be extended for some part of a subcritical righthand state. Example 10.3.1. Consider the Riemann problem for the system (10.1.1)-(10.1.2), where the initial states UL = (hL , uL , aL ) and UR = (hR , uR , aR ) are given in Table 10.1. The gravitational constant is g = 9.8. The Riemann solution of the form (10.3.2) at the time t = 0.1 is illustrated by Figure 10.5 in the interval [−1, 1]. The states U1 and U2 in the solution (10.3.2) are given by Table 10.1.

Fig. 10.5 A Riemann solution of the form (10.3.2) in [−1, 1] at the time t = 0.1

Table 10.1 States in the Riemann solution (10.3.2) by Construction 1 UL U1 U2 UR

h 0.3 0.218158969265 0.323070488094 0.5

u 2 2.75028802172 2.11593705377 3

a 1.1 1 1 1

10.3 Riemann problem

309

10.3.2 Construction 2: supercritical/subcritical The solution begins with a 1-shock wave from the supercritical state UL to a subcritical state U1 , such that λ¯ 1 (UL ,U) ≤ 0, followed by a stationary contact to a subcritical state U2 shifting a from aL to aR . The construction is described as follows: Let 0/ = W2B (UR ) ∩ W1→3 (UL ) = {U2 },

(10.3.7)

and let U1 ∈ W1→3 (UL ) be the corresponding state on the left-hand side of the admissible stationary contact to the right-hand state U2 . Provided λ¯ 2 (U2 ,UR ) ≥ 0, there is a Riemann solution of the form W1 (UL ,U1 ) W3 (U1 ,U2 ) W2 (U2 ,UR ).

(10.3.8)

See Figure 10.6. In the limit case of (10.3.10) where U1 ≡ UL , the solution (10.3.10) coincides with the solution (10.3.8).

Fig. 10.6 Construction 2: A Riemann solution of the form (10.3.8).

The states U1 and U2 which define the Riemann solution (10.3.8) can be calculated by u1 = ω1F (UL ; h1 ), h2 u2 = h1 u1 , u22 + 2g(h2 + aR ) = u21 + 2g(h1 + aL ),

(10.3.9)

u2 = ω2B (UR ; h2 ), where ω1F and ω2B are defined by (10.2.17). The four equations (10.3.9) enable us to calculate the four quantities hi , ui , i = 1, 2. The configuration of the Riemann solution (10.3.8) in the (x,t)-plane is given by Figure 10.7.

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10 Shallow water flows with discontinuous topography

Fig. 10.7 Riemann solution with structure (10.3.8) in the (x,t)-plane

Note that this construction is still valid for some parts of a supercritical righthand state. An alternative construction relying on the backward composition can be made, where we can make use of the composite wave curve W2←3 (UR ). Precisely, let W1F (UL ) ∩ W2←3 (UR ) = {U1 }, and let U2 ∈ W2B (UR ) be the corresponding right-hand state of the admissible contact wave from U1 . Then, the solution still has the form (10.3.8). This backward construction seems to be useful when aL < aR . Example 10.3.2. Consider the Riemann problem for the system (10.1.1)-(10.1.2), where the initial states UL = (hL , uL , aL ) and UR = (hR , uR , aR ) are given in Table 10.2, and g = 9.8. The Riemann solution of the form (10.3.8) at the time t = 0.1 is illustrated by Figure 10.8 in the interval [−1, 1]. The states U1 and U2 in the solution (10.3.8) are given in Table 10.2.

Table 10.2 States in the Riemann solution (10.3.8) by Construction 2 UL U1 U2 UR

h 0.5 1.14047433164 1.34556641834 1.2

u 3 0.595323112844 0.504583586493 0.1

a 1 1 0.800000005796 0.8

10.3 Riemann problem

311

Fig. 10.8 A Riemann solution of the form (10.3.8) in [−1, 1] at the time t = 0.1

10.3.3 Construction 3: resonant waves for supercritical regime In this construction, we will see an interesting phenomenon when wave speeds associated with different characteristic fields coincide. The solution can begin with an intermediate stationary contact W3 (UL ,U1 ) from UL = (h, u, aL ) to some supercritical state U1 = (h1 , u1 , a1 ), for some a1 ∈ [aL , aR ]. Then, the solution continues by a zero-speed 1-shock wave from U1 to a subcritical state U2 , this shock wave is followed by another stationary contact from U2 to a subcritical state U3 , and then followed by a 3-wave to UR . When a1 varies between aL and aR , the points U3 form a set denoted by L . Whenever W2B (UR ) ∩ L = 0/ there is a solution containing three discontinuities having the same zero speed of the form W3 (UL ,U1 ) W1 (U1 ,U2 ) W3 (U2 ,U3 ) W2 (U3 ,UR ). See Figure 10.9.

(10.3.10)

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10 Shallow water flows with discontinuous topography

Fig. 10.9 Construction 3: A Riemann solution of the form (10.3.10).

The states U1 ,U2 , and U3 which determine the Riemann solution (10.3.10) can be calculated as follows: h1 u1 = hL uL , u21 + 2g(h1 + a1 ) = u2L + 2g(hL + aL ), −h1 + h21 + 8h1 u21 /g , h2 = 2 u1 h1 u2 = , h2 h3 u3 = h2 u2 ,

(10.3.11)

u23 + 2g(h3 + aR ) = u22 + 2g(h2 + a1 ), u3 = ω2B (UR ; h3 ), where ω2B is defined by (10.2.17). The seven equations (10.3.11) allow us to calculate the seven quantities hi , ui , a1 , i = 1, 2, 3 so that the states U1 ,U2 and U3 are fully calculated. The configuration of the Riemann solution (10.3.10) in the (x,t)-plane is given by Figure 10.10. Example 10.3.3. Consider the Riemann problem for the system (10.1.1)-(10.1.2), where the initial states UL = (hL , uL , aL ) and UR = (hR , uR , aR ) are given in Table 10.3. The Riemann solution of the form (10.3.10) at the time t = 0.1 is illustrated by Figure 10.11 in the interval [−1, 1]. The states U1 ,U2 and U3 , which determine the elementary waves of the solution (10.3.10), are given in Table 10.3.

10.3 Riemann problem

313

Fig. 10.10 Riemann solution with structure (10.3.10) in the (x,t)-plane. Table 10.3 States in the Riemann solution (10.3.10) by Construction 3 UL U1 U2 U3 UR

h 1 1.02118408711 1.5 1.36577029168 1.56577029168

u 3.2304951685 3.16347973815 1.80369161035 1.98096080431 2.49931084374

a 1 1.00067790166 1.1 1.2 1.2

Fig. 10.11 A Riemann solution of the form (10.3.10) in [−1, 1] at the time t = 0.1

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10 Shallow water flows with discontinuous topography

10.3.4 Existence and uniqueness properties for supercritical regime Existence As seen in Construction 1, a stationary wave from UL with the bottom level a = aL to U1 = UL0 with a = aR can be followed by a zero-speed 1-shock from U1# = UL0# := (UL0 )# . Letting the second stationary wave in Construction 3 be trivial, one can see that the point UL0# is a common endpoint of both curves W1F (U1 ) and L . As seen in Construction 2, a zero-speed 1-shock from UL to UL# can be followed by a stationary contact from UL# with the bottom level a = aL to (UL# )0 := UL#0 with a = aR . Letting the first stationary wave in Construction 3 be trivial, one can see that the point UL#0 is a common endpoint of the two curves W1→3 (UL ) and L . Thus, the union W1F (U1 ) ∪ L ∪ W1→3 (UL ) constitutes a large continuous curve in the (h, u)-plane. This shows that the backward curve W2B (UR ) can intersect this union of curves for a large domain of UR , under the conditions (10.3.1) and (10.3.7). Consequently, solutions to the Riemann problem for a large domain of UR can be constructed by Constructions 1, 2, or 3. Precisely, let P = (0, p) be the intersect point of W1F (U2 ) and the u-axis, and let Q = (hq , uq ) be the lower endpoint of the curve W1→3 (UL ). Then, the backward wave curve W2B (UR ) meets the union of the curves W1→3 (UL ), L , and W1F (U2 ) if P is above and Q is below it, i.e. 0 ≥ ω2B (UR ; p),

uq ≤ ω2B (UR ; hq ).

(10.3.12)

No local existence As observed in the previous chapters, we can have a similar situation of losing local existence of solutions to the Riemann problem when the condition (10.3.3) is violated (10.3.13) aR < amin (UL ). Under the condition (10.3.13), Construction 1 cannot be made. Parts of Construction 2 and Construction 3 can be made and then both curves L and W1→3 (UL ) belong to the subcritical region. Therefore, if the backward curve W2B (UR ) lies entirely in the supercritical region, it cannot intersect these curves and so Constructions 2 and 3 are not available as well. The condition for the backward curve W2B (UR ) lies entirely in the supercritical region is that its u-intercept is non-negative √

uR − 2 g hR ≥ 0,

10.3 Riemann problem

or

315

√

uR ≥ 2 g hR

(10.3.14)

Thus, under the conditions (10.3.13) and (10.3.14), all the Constructions 1, 2, and 3 cannot be made. This yields the non-existence of the Riemann solutions in this case.

Multiple solutions o# B In the case where h#o L < hL , there can be three solutions. The curve W2 (UR ) can F then intersects all the three curves W1 (U1 ), L and W1→3 (UL ) at three distinct points U1 ,U2 and U3 , see Figure 10.12, if Φ2 (UL#o ;UR ) > 0 > Φ2 (ULo# ;UR ), where the function Φ2 (U;UR ) is defined by (10.2.16). Each of these intersection point U1 ,U2 and U3 corresponds to a distinct Riemann solution presented in Construc#0 tions 1, 2, and 3 above. The fact that the point UL#0 = (h#0 L , uL ) is located above and 0# 0# 0# B the point UL = (hL , uL ) is located below the curve W2 (UR ) means B #0 u#0 L > ω2 (UR ; hL ) B 0# u0#0 L < ω2 (UR ; hL ),

(10.3.15)

where ω2B is defined by (10.2.17). Moreover, when aR > aL , it is possible that some part of or the whole curve W1F (U1 ), L , or W1→3 (UL ) may not be available when a > amax (UL ), for some values a ∈ [aL , aR ]. Thus, the backward wave curve W2B (UR ) may intersect the union W1F (U1 ) ∪ L ∪ W1→3 (UL ) at no point, one point, two points, or three points. The Riemann problem therefore may admit no solution, one solution, two solutions, or three distinct solutions.

Unique solution Since L and W1→3 (UL ) belong to the subcritical region, the curve may only meet W1F (U1 ) if it lies in the supercritical region. That is

ω2B (UR ; 0) > 0.

(10.3.16)

Theorem 10.3.1. Let the left-hand state UL be a given supercritical state: (a) (Existence). Under the conditions (10.3.12), the Riemann problem for (10.1.1)(10.1.2) has a solution. (b) (Multiple solutions). Under the conditions (10.3.15), the Riemann problem for (10.1.1)-(10.1.2) has three distinct solutions.

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10 Shallow water flows with discontinuous topography

Fig. 10.12 Multiple solutions for supercritical UL under (10.3.15): There are three Riemann solutions given by Constructions 1, 2, and 3 of the form (10.3.2), (10.3.10), and (10.3.8) with the choice of intersection point U1 , U2 , and U3 , respectively.

(c) (Uniqueness). The Riemann problem for (10.1.1)-(10.1.2) has a unique solution of the form (10.3.2) under the condition (10.3.16).

10.3.5 Construction 4: subcritical/supercritical Consider the case where UL ∈ G2 or on the curve C+ , and UR is located in a “higher” position. There can be two distinct configurations of solutions depending on whether aL ≥ aR . If aL > aR a solution can be constructed as follows. The solution begins from the subcritical state UL with a 1-rarefaction wave until it reaches a critical state U1 ∈ C+ . This rarefaction wave W1 (UL ,U1 ) is followed by a stationary contact from U1 to a supercritical state U2 ∈ G1 . Whenever W1F (U2 ) ∩ W2B (UR ) = 0, /

(10.3.17)

and let U3 be the intersection point of these two wave curves, the solution is continued by a 1-wave from U2 to U3 , and finally the solution arrives at UR by a 2-wave from U3 . Thus, the solution has the form W1 (UL ,U1 ) W3 (U1 ,U2 ) W1 (U2 ,U3 ) W2 (U3 ,UR ).

(10.3.18)

See Figure 10.13. The construction makes sense if λ¯ 1 (U2 ,U3 ) ≥ 0, which means U3 has to be above U2# on W1F (U2 ). The states U1 ,U2 , and U3 which determine the Riemann solution (10.3.18) can be calculated as follows. First, U1 is determined by

10.3 Riemann problem

U1 =

317

u 2 2 1 2

L hL , uL + ghL , aL . √ + 3 g 3 3 3

Second, U2 = (h2 , u2 , aR ) ∈ G1 is calculated from the two equations h2 u2 = h1 u1 ,

u22 + 2g(h2 + aR ) = u21 + 2g(h1 + aL ).

Third, the state U3 is given by u3 = ω1F (U2 ; h3 ) = ω2B (UR ; h3 ). We note that the above construction can also be extended if W2B (UR ) lies entirely above W1F (U2 ). In this case, let I and J be the intersection points of W1F (U2 ) and W2B (UR ) with the axis {h = 0}, respectively: {I} = W1F (U2 ) ∩ {h = 0},

{J} = W2B (UR ) ∩ {h = 0}.

Then, the solution can be seen as containing a dry part Wo (I, J) between I and J. Thus, the solution, in this case, is W1 (UL ,U1 ) W3 (U1 ,U2 ) W1 (U2 , I) Wo (I, J) R2 (J,UR ).

(10.3.19)

Fig. 10.13 Construction 4: A Riemann solution of the form (10.3.18), case aL > aR .

If aL ≤ aR a solution of another type can be constructed as follows. To each U ∈ C+ , a stationary contact to U o ∈ G2 downing a = aR to a = aL is possible, since aR > aL . The set of all these states U o form a curve denoted by Ca . Let {U1 } = W1F (UL ) ∩ Ca .

318

10 Shallow water flows with discontinuous topography

Then, the solution begins by a 1-wave W1 (UL ,U1 ), followed by a stationary jump W3 (U1 ,U2 = U1o ) to U2 ∈ C+ . Let {U3 } = W1F (U2 ) ∩ W2B (UR ).

(10.3.20)

The solution is then continued by a 1-rarefaction wave from U2 to U3 , followed by a 2-wave from U3 to UR . Thus, the solution is given by the formula W1 (UL ,U1 ) W3 (U1 ,U2 ) R1 (U2 ,U3 ) W2 (U3 ,UR ).

(10.3.21)

See Figure 10.14. The construction makes sense if λ1 (U3 ) ≥ 0, or U3 ∈ G¯ 1 . The states U1 ,U2 , and U3 which form elementary waves for the Riemann solution (10.3.21) can be calculated as follows: u1 = ω1F (UL ; h1 ), h2 u2 = h1 u1 , u22 + 2g(h2 + aR ) = u21 + 2g(h1 + aL ),

u2 = gh2 ,

(10.3.22)

u3 = ω1F (U2 ; h3 ) = ω2B (UR ; h3 ), where ω1F and ω2B are defined by (10.2.17). The six equations in (10.3.22) allow us to calculate the six quantities hi , ui , i = 1, 2, 3 so that the states U1 ,U2 , and U3 are fully determined. The construction for the solution (10.3.21) can also be extended if W2B (UR ) lies entirely above W1F (U2 ). In this case, let I and J be the intersection points of W1F (U2 ) and W2B (UR ) with the axis {h = 0}, respectively: {I} = W1F (U2 ) ∩ {h = 0},

{J} = W2B (UR ) ∩ {h = 0}.

Then, the solution can be seen as containing a dry part Wo (I, J) between I and J. Thus, the solution has the form W1 (UL ,U1 ) W3 (U1 ,U2 ) R1 (U2 , I) Wo (I, J) R2 (J,UR ). The configuration of the Riemann solution (10.3.18) in the (x,t)-plane is given by Figure 10.15. Note that this construction can be extended for some part of a subcritical righthand state. Example 10.3.4. Consider the Riemann problem for the system (10.1.1)-(10.1.2), where the initial states UL = (hL , uL , aL ) and UR = (hR , uR , aR ) are given in Table 10.4. The Riemann solution of the form (10.3.18) at the time t = 0.1 is illustrated by Figure 10.16 in the interval [−1, 1]. The states U1 ,U2 and U3 in the solution (10.3.18) are given in Table 10.4.

10.3 Riemann problem

319

Fig. 10.14 Construction 4: A Riemann solution of the form (10.3.21), case aL ≤ aR .

Fig. 10.15 Riemann solution with structure (10.3.18) in the (x,t)-plane

10.3.6 Construction 5: subcritical/subcritical When UL is a subcritical or critical state, a 1-wave from UL to a subcritical state U1 can be followed by a stationary wave from U1 to a subcritical state U2 , and then followed by a 2-wave from U2 to UR . Geometrically, this construction can be made as follows. When U1 varies along W1F (UL ), the corresponding states U2 form a wave curve denoted by W1→3 (UL ). Provided W2B (UR ) ∩ W1→3 (UL ) = 0, /

(10.3.23)

320

10 Shallow water flows with discontinuous topography

Fig. 10.16 A Riemann solution of the form (10.3.18) in [−1, 1] at the time t = 0.1 Table 10.4 States in the Riemann solution (10.3.18) by Construction 4 UL U1 U2 U3 UR

h 1 0.773741058367 0.52715259471 0.620478501078 0.8

u 2 2.75366344567 4.04175658095 3.3318107468 4

a 1.2 1.2 1 1 1

a Riemann solution exists. Indeed, let U2 belongs to this intersection, and let U1 be the corresponding state on the left-hand side of the stationary contact. Then, a Riemann solution can be determined and has the form W1 (UL ,U1 ) W3 (U1 ,U2 ) W2 (U2 ,UR ).

(10.3.24)

The construction (10.3.24) makes sense if λ¯ 2 (U2 ,UR ) ≥ 0, see Figure 10.17. The states U1 and U2 which define the Riemann solution (10.3.24) can be calculated by u1 = ω1F (UL ; h1 ), h2 u2 = h1 u1 , u22 + 2g(h2 + aR ) = u21 + 2g(h1 + aL ), u2 = ω2B (UR ; h2 ),

(10.3.25)

10.3 Riemann problem

321

Fig. 10.17 Construction 5: Riemann solution of the form (10.3.24).

where ω1F and ω2B are defined by (10.2.17). The four equations (10.3.25) enable us to calculate the four quantities hi , ui , i = 1, 2 so that the states U1 and U2 are fully determined. The configuration of the Riemann solution (10.3.24) in the (x,t)-plane is given by Figure 10.18.

Fig. 10.18 Riemann solution with structure (10.3.24) in the (x,t)-plane

Note that this construction can be extended for some part of a supercritical righthand state. We can also use a backward construction by employing the backward composite wave curve W2←3 (UR ). Indeed, let W1F (UL ) ∩ W2←3 (UR ) = {U1 },

322

10 Shallow water flows with discontinuous topography

and let U2 ∈ W2B (UR ) be the corresponding right-hand state of the admissible contact wave from U1 . Then, the solution still has the form (10.3.24). This backward construction is suitable if aL < aR .

10.3.7 Construction 6: resonant waves in the subcritical regime When aL > aR , there is a resonant situation in which a Riemann solution may contain multiple waves of the same zero speed. Indeed, the solution can start with a 1-rarefaction wave from the subcritical state UL to a critical state U1 ∈ C+ , followed by an intermediate stationary wave from U1 to a supercritical state U2 = (h2 , u2 , a2 ) ∈ G1 , for some value a2 ∈ [aR , aL ], followed by a 1-shock with zero speed from U2 to a subcritical state U3 ∈ G2 , followed by another stationary wave to a subcritical state U4 = (h4 , u4 , aR ) ∈ G2 . The set of these states U4 form a curve pattern L . So, whenever W2B (UR ) ∩ L = 0, / and let U4 be the intersection point, the solution arrives at UR by a 2-wave from U4 . Thus, in this case, a Riemann solution containing three zero-speed waves exists and has the form W1 (UL ,U1 ) W3 (U1 ,U2 ) W1 (U2 ,U3 ) W3 (U3 ,U4 ) W2 (U4 ,UR ). (10.3.26) See Figure 10.19.

Fig. 10.19 Construction 6-collision of waves: When aL > aR , there is a solution containing three waves of the same zero speed of the form (10.3.26).

10.3 Riemann problem

323

The states U1 ,U2 ,U3 , and U4 which determine the Riemann solution (10.3.26) can be calculated as follows. First, the state U1 is completely determined by the two equations

u1 = ω1F (UL ; h1 ) = gh1 , where ω1F is defined by (10.2.17). Second, for the states U2 ,U3 and U4 we solve the following equations: h2 u2 = h1 u1 , u22 + 2g(h2 + a2 ) = u21 + 2g(h1 + aL ), −h2 + h22 + 8h2 u22 /g , h3 = 2 u2 h2 u3 = , h3 h4 u4 = h3 u3 ,

(10.3.27)

u24 + 2g(h4 + aR ) = u23 + 2g(h3 + a2 ), u4 = ω2B (UR ; h4 ), where ω2B is defined by (10.2.17). The seven equations (10.3.27) enable us to calculate the seven quantities hi , ui , a2 , i = 1, 2, 3 so that the three states U2 ,U3 , and U4 can also be determined. The configuration of the Riemann solution (10.3.26) in the (x,t)-plane is given by Figure 10.20.

Fig. 10.20 Riemann solution with structure (10.3.26) in the (x,t)-plane

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10 Shallow water flows with discontinuous topography

10.3.8 Existence and uniqueness properties for the subcritical regime Existence Constructions 4, 6, and 5 change continuous from one to another. This indicates that the existence of solutions of the Riemann problem can be obtained for a large domain of the right-hand state UR for the subcritical left-hand state UL . Since the three curves W1→3 (UL ), L , and W1F (U2 ) connect continuously, the backward wave curve W2B (UR ) will intersect one of these three curves, and therefore, a Riemann solution exists if (10.3.17) (if aL ≥ aR ) or (10.3.20) (if aL < aR ) and (10.3.23) hold. Let P = (0, p) be the intersect point of W1F (U2 ) in (10.3.17) (if aL ≥ aR ) or W1F (U3 ) in (10.3.20) (if aL < aR ) with the u-axis, and let Q = (hq , uq ) be the lower endpoint of the curve W1→3 (UL ). Then, the backward wave curve W2B (UR ) meets one of the three curves W1→3 (UL ), L , and W1F (U2 ) if 0 ≥ ω2B (UR ; p),

uq ≤ ω2B (UR ; hq ).

(10.3.28)

Multiple solutions There is a circumstance where three different solutions of the Riemann problem can exist. Indeed, let the critical state U1 with the bottom level aL be the intersection point of W1F (UL ) and C+ . Let U2 be the supercritical state with the bottom level aR obtained from a stationary jump from U1 . Let U1o with the bottom level aR denote the subcritical state obtained from a stationary jump from U1 . Let U2# denote the subcritical state resulted from a 1-shock with zero speed from U2 . If h01 < h#2 , the wave curve W2B (UR ) may intersect all the three curves W1→3 (UL ), L , and W1F (U2 ). This occurs if U10 lies above the curve W2B (UR ) and U2# lies below the curve W2B (UR ). That is, let U10 = (h01 , u01 ) and U2# = (h#2 , u#2 ), aL > aR u01 > ω2B (UR ; h01 ) u#2

(10.3.29)

< ω2B (UR ; h#2 ),

where ω2B is defined by (10.2.17). This means that all solutions described by Constructions 4, 5, and 6 are available. Precisely, each choice of the intersection point U4 ,U5 or U6 leads to a distinct solution given by Construction 4, 5, and 6, respectively. See Figure 10.21. In the case where aR > aL , the Riemann problem may admit no solution, one solution, two solutions, or three distinct solutions. Indeed, we observe that it is possible that some part of or the whole curve W1F (U1 ), L , or W1→3 (UL ) may not be available, if

10.3 Riemann problem

325

a > amax (UL ), for some values a ∈ [aL , aR ]. Thus, the backward wave curve W2B (UR ) may intersect the union W1F (U1 ) ∪ L ∪ W1→3 (UL ) at no point, one point, two points, or three points.

Unique solution If the composite wave curve W1→3 (UL ) can be parameterized where the water velocity is a decreasing function of the water height, then it intersects the backward curve W2B (UR ) at most one time. Thus, if the backward wave curve W2B (UR ) does not intersect the other two curves W1→3 (UL ), L , the Riemann problem for the shallow water equations has only one solution. This occurs when these two wave curves are located above W2B (UR ) in the (h, u)-plane. This means that both states U10 = (h01 , u01 ) and U2# = (h#2 , u#2 ) lie above the curve W2B (UR ). This condition yields u01 > ω2B (UR ; h01 ),

u#2 > ω2B (UR ; h#2 ),

(10.3.30)

where ω2B is defined by (10.2.17).

Fig. 10.21 Multiple solutions for subcritical UL under the conditions (10.3.29): Each choice of the intersection point U4 ,U5 or U6 corresponds to a different solution described in Construction 4, 5, and 6, respectively.

Theorem 10.3.2. Given a subcritical left-hand state UL . (a) (Existence). Under the conditions (10.3.28), the Riemann problem for (10.1.1)(10.1.2) has a solution.

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10 Shallow water flows with discontinuous topography

(b) (Multiple solutions). Under the conditions (10.3.29), the Riemann problem for (10.1.1)-(10.1.2) has three distinct solutions. (c) (Uniqueness). The Riemann problem for (10.1.1)-(10.1.2) has a unique solution of the form (10.3.24) under (10.3.30).

10.4 Bibliographical notes We have listed only a short selection of the most relevant papers for this chapter. We also refer the reader to the bibliography at the end of this monograph for additional references. The hydraulic of open channel flow was studied by Chanson [90]. A tsunami modeling using the shallow water equations was presented by LeVeque-George-Berger [230]. The Riemann problem for shallow water equations with discontinuous topography was systematically investigated in by LeFlochThanh [228, 229]. Riemann solutions for shallow water equations with discontinuous topography were also constructed by Alcrudo-Benkhaldoun [11], Noussair [254], Bernetti-Titarev-Toro [49], and Rosatti-Begnudelli [275]. Shallow water hydrodynamic models for hyper-concentrated sediment-laden flows over erodible bed are studied by Cao-Pender-Carling [74]. Mobile bed flows are considered by Morris-Williams [247], Garegnani-Rosatti-Bonaventura [137]. Hydrodynamics models are also studied by Klingenberg-Schmidt-Waagan [199].

Chapter 11

Shallow water flows with temperature gradient

This chapter deals with the shallow water equations with variable topography and horizontal temperature gradient known as the Ripa system. This system was formulated to model ocean currents and can be derived from the Saint-Venant system of shallow water equations by taking into account horizontal water temperature fluctuations. The system is still nonconservative and can be transformed into a system of balance laws in nonconservative form. The system is not strictly hyperbolic and possesses four characteristic fields, among which two characteristic fields are linearly degenerate. In gas dynamics equations, both particle velocity and pressure remain constant across each contact discontinuity λ = u. This interesting property helps determine intermediate states separating elementary waves in a Riemann solution by projecting wave curves onto the (p,u)-plane and finding the intersection point of the 1-wave forward curve and 3-wave backward curve. However, the Ripa system possesses a different property: only particle velocity remains constant across each contact discontinuity λ = u. This raises a challenging problem for determining intermediate states separating elementary waves in Riemann solutions. To overcome this obstacle, a new kind of composite wave curves is defined: curves of composite waves between waves associated with a nonlinear characteristic field and the linearly degenerate field λ = u. Another kind of curve of composite waves between a nonlinear field and the linearly degenerate field λ = 0 is still needed, as they were shown to be useful in constructing Riemann solutions for other nonconservative models considered earlier. As usual, solutions of the Riemann problem are formed by combinations of a finite number of elementary waves: shocks, rarefaction waves, and contact discontinuities. Each kind of elementary wave must be associated with a certain characteristic field.

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 11

327

328

11 Shallow water flows with temperature gradient

11.1 Basic properties and elementary waves 11.1.1 Hyperbolicity and genuine nonlinearity The governing equations of the Ripa model are given by

∂t h + ∂x (hu) = 0, g ∂t (hu) + ∂x hu2 + h2 θ = −ghθ ∂x a, 2 ∂t (hθ ) + ∂x (huθ ) = 0, x ∈ R I , t > 0.

(11.1.1)

Here, h, u, and θ denote the water depth, the depth-averaged horizontal velocity, and the potential temperature field, respectively; g is the gravitational constant, and a is the bottom topography. The system (11.1.1) is nonconservative, and we again supplement it with the (trivial) equation

∂t a = 0,

x∈R I , t > 0.

(11.1.2)

Let us derive the system (11.1.1)-(11.1.2) in terms of the primitive variables U = (h, u, θ , a). It is convenient to simplify the notation as (.)t = ∂t (.),

(.)x = ∂x (.).

Consider a smooth solutions of (11.1.1). For such a smooth solution, the first equation in (11.1.1) can be written as ht + uhx + hux = 0. The second equation in (11.1.1) can be expressed as uht + hut + huux + u(hu)x +

gh2 θx + ghθ hx + ghθ ax = 0. 2

The first and the fourth terms of the last equation cancel each other, thanks to the first equation in (11.1.1). Now, we may divide both sides of the last equation by h > 0 to get gh ut + gθ hx + uux + θx + gθ ax = 0. 2 Next, the third equation in (11.1.1) can be re-written as hθt + θ ht + huθx + θ (hu)x = 0. Observe that the second and the fourth terms of the last equation cancel each other by using the first equation in (11.1.1). We divide both sides of the last equation by h > 0 to get θt + uθx = 0.

11.1 Basic properties and elementary waves

329

From the above analysis, we see that the system (11.1.1)-(11.1.2) can be written as a system of balance laws in nonconservative form Ut + A(U)Ux = 0, where ⎛ ⎞ h ⎜u⎟ ⎟ U =⎜ ⎝θ ⎠ , a

⎛

u h 0 gh ⎜ ⎜gθ u A(U) = ⎜ 2 ⎝0 0 u 0 0 0

0

⎞

⎟ gθ ⎟ ⎟. 0⎠

(11.1.3)

0

The characteristic equation associated with the matrix A(U) is given by det(A − λ I) = 0, which yields

λ (u − λ ) (u − λ )2 − ghθ = 0.

The matrix A(U) thus admits four real eigenvalues

λ1 (U) = u − c, λ2 (U) = u, λ3 (U) = u + c, λ4 (U) = 0, (11.1.4) √ where c = ghθ . The eigenvector ri (U) corresponding to the eigenvalue λi (U), i = 1, 2, 3, 4 can be chosen as ⎛ ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ ⎞ 1 1 1 1 u ⎜ − ⎟ ⎟ ⎜ ⎜ gθ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎟ ⎜− gθ ⎟ ⎜ h ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ , r , r , r . (U) = (U) = (U) = r1 (U) = ⎜ 2 θ ⎜ ⎜ ⎟ 3 h ⎟ 2 0 ⎟ ⎜ ⎜ h ⎟ 4 ⎜ ⎟ − ⎝ ⎠ ⎠ ⎝ 0 ⎠ ⎝ 2 0 ⎝ u ⎠ h −1 0 0 0 ghθ (11.1.5) √ Note that c = ghθ > 0. So, the first three eigenvalues are always distinct: λ1 (U) < λ2 (U) < λ3 (U) ∀U. However, each of them may coincide with the eigenvalue λ4 (U). Consequently, the system is hyperbolic but not strictly hyperbolic in the entire domain. Let us define the strictly hyperbolic domains of the system (11.1.1)-(11.1.2) by G1 = {U : λ1 (U) > λ4 (U)}, G2 = {U : λ1 (U) < λ4 (U) < λ2 (U)}, G3 = {U : λ2 (U) < λ4 (U) < λ3 (U)}, G4 = {U : λ3 (U) < λ4 (U)}, and non-strictly hyperbolic surfaces by

330

11 Shallow water flows with temperature gradient

C+ = {U : λ1 (U) = 0}, C0 = {U : λ2 (U) = 0}, C− = {U : λ3 (U) = 0}.

(11.1.6)

In accordance with the water resource engineering, we introduce the following terminologies. The generalized Froude number of the Ripa system (11.1.1), denoted by Fr(U) =, and is defined by |u| Fr(U) = √ . ghθ A state U is said to be a critical state, if Fr(U) = 1. If Fr(U) > 1, then U is said to be a supercritical state. If Fr(U) < 1, then U is said to be a subcritical state. A straightforward calculation yields 3 gθ > 0, −∇λ1 (U) · r1 (U) = ∇λ3 (U) · r3 (U) = 2 h and ∇λ2 (U) · r2 (U) = ∇λ4 (U) · r4 (U) = 0, for all U, which means that the 1- and the 3-characteristic fields are genuinely nonlinear, while the 2- and the 4-characteristic fields are linearly degenerate.

11.1.2 Rarefaction waves Let us now consider rarefaction waves of the system (11.1.1), i.e., continuous piecewise smooth self-similar solutions associated with the genuinely nonlinear characteristic fields (λi , ri ), i = 1, 3. Rarefaction waves satisfy the initial-value problem for the ordinary differential equation ri (U) , dξ ∇λi · ri (U) U(ξ0 ) = U0 ,

dU

=

ξ = x/t,

i = 1, 3,

(11.1.7)

for some constant value ξ0 and constant state U0 = (h0 , u0 , θ0 , a0 )T . As seen by (11.1.5), through a rarefaction fan, d θ /d ξ = da/d ξ = 0. This means that the temperature and the bottom level remain constant through any rarefaction wave

θ = θ0 ,

a = a0 .

Furthermore, we deduce from (11.1.7) for i = 1 that du gθ0 =− . dh h

11.1 Basic properties and elementary waves

331

This means that the integral curve associated with the first characteristic field passing through a given point U0 is given by

√ u = u0 − 2 gθ0 ( h − h0 ). Using the condition that the characteristic speed must be increased through a rarefaction fan, we can see that 1-rarefaction waves correspond to the part h ≥ h0 of the integral curve. Thus, we can define the forward curve of 1-rarefaction waves R1 (U0 ) issuing from a given left-hand state U0 , which consists of all the right-hand states U that can be connected to U0 by a rarefaction wave associated with the first characteristic field by

√ R1 (U0 ) : u = u0 − 2 gθ0 ( h − h0 ), h ≤ h0 . (11.1.8) Arguing similarly, we obtain the backward curve of 3-rarefaction waves R3 (U0 ) issuing from a given right-hand state U0 , which consists of all the left-hand states U that can be connected to U0 by a rarefaction wave associated with the third characteristic field as

√ R3 (U0 ) : u = u0 + 2 gθ0 ( h − h0 ), h ≤ h0 . (11.1.9)

11.1.3 Shock waves and material contact discontinuities Recall that a shock wave of the system (11.1.1)-(11.1.2) is a weak solution of the form U− , x < st, U(x,t) = U+ , x > st, where U± are the left-hand and right-hand states, and s is the shock speed. Applying the Rankine-Hugoniot relation to the equation (11.1.2), we have −s[a] = 0.

(11.1.10)

The identity (11.1.10) means that, across a shock, there are two possibilities: (i) either the bottom height a remains constant, (ii) or the shock speed vanishes. In the first case (i), the system (11.1.1)-(11.1.2) is reduced to the following shallow water equations with flat bottom, where the trivial equation (11.1.2) can be neglected: ∂t h + ∂x (hu) = 0, g ∂t (hu) + ∂x hu2 + h2 θ = 0, (11.1.11) 2 ∂t (hθ ) + ∂x (huθ ) = 0, x ∈ R I , t > 0.

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11 Shallow water flows with temperature gradient

Since the system (11.1.11) is conservative, we can apply the Rankine-Hugoniot relations and get − s[h] + [hu] = 0, g − s[hu] + hu2 + h2 θ = 0, (11.1.12) 2 − s[hθ ] + [huθ ] = 0, where the square brackets denote the jump, e.g., [h] = h+ − h− . It will be convenient to fix a state and look for the state on the other side of the shock wave. Let us first fix a left-hand state U− = U0 , and look for all the right-hand states U = U+ of the shock. It holds that [hθ ] = hθ − h0 θ0 = (hθ − h0 θ ) + (h0 θ − h0 θ0 ) = θ [h] + h0 [θ ]. Similarly, [hu] = u[h] + h0 [u],

[huθ ] = θ [hu] + h0 u0 [θ ].

Using these quantities, we can re-write the third equation in (11.1.12) as follows: 0 = −s[hθ ] + [huθ ] = −sθ [h] − sh0 [θ ] + θ [hu] + h0 u0 [θ ] = −sh0 [θ ] + h0 u0 [θ ] = h0 [θ ](u0 − s), where the first equation in (11.1.12) is invoked to simplify the second equation. This yields (11.1.13) [θ ](u0 − s) = 0, which means that either θ = θ0 =constant, or s = u0 . Consider the case θ = θ0 . Then, the first and the third equations in (11.1.12) are equivalent. A straightforward calculation yields 1 gθ0 1 , (11.1.14) (h − h0 ) + u = u0 ± 2 h h0 so that the Hunogiot curves can be parameterized by h. It holds along the curves (11.1.14) that du gθ0 1 1 gθ0 1 →± =± + − (h − h0 ) dh 2 h h0 h0 1 1 + 2h2 h h0 as h → h0 . Since the ith-Hugoniot curve is tangent to the eigenvector ri (U0 ) at U0 , we deduce that the first Hugoniot curve associate with the first characteristic field (λ1 , r1 ) is given by

11.1 Basic properties and elementary waves

H1 (U0 ) :

u := ω1 (U0 ; h) = u0 −

333

1 gθ0 1 (h − h0 ) + , 2 h h0

h ≥ 0, (11.1.15)

and the third Hugoniot curve associated with the third characteristic field (λ3 , r3 ) is given by 1 1 gθ0 (h − h0 ) + H3 (U0 ) : u := ω3 (U0 ; h) = u0 + , h ≥ 0. (11.1.16) 2 h h0 Let us now discuss about the admissibility criterion. From (11.1.12) and (11.1.15), the shock speed along the Hugoniot curve H1 (U0 ) is given by h2 gθ0 . h+ s = s1 (U0 ,U) = u0 − 2 h0 From (11.1.12) and (11.1.16), the shock speed along the Hugoniot curve H3 (U0 ) is given by gθ0 h2 . h+ s = s3 (U0 ,U) = u0 + 2 h0 Shock waves are required to satisfy the Lax shock inequalities

λi (U) < si (U0 ,U) < λi (U0 ),

i = 1, 3.

(11.1.17)

The Lax shock inequalities (11.1.17) select the part h > h0 on H1 (U0 ), and select the part h < h0 on H3 (U0 ). This means that the forward curve of 1-shock waves S1 (U0 ) starting from a left-hand state U0 consisting of all right-hand states U that can be connected to U0 by a Lax shock associated with the first characteristic field is given by 1 1 gθ0 , h > h0 , (11.1.18) (h − h0 ) + S1 (U0 ) : u = ω1 (U0 ; h) = u0 − 2 h h0 Similarly, by reversing the left-hand and right-hand states, we can fix a right-hand state U+ = U0 and look for all left-hand states U = U− of admissible Lax shocks can be obtained. Precisely, the backward curve of 3-shock waves S3 (U0 ) starting from a given right-hand state U0 consisting of all left-hand states U of admissible Lax shocks associated with the third characteristic field is given by 1 1 gθ0 (h − h0 ) + S3 (U0 ) : u = ω3 (U0 ; h) = u0 + . h > h0 , (11.1.19) 2 h h0 Wave curves associated with the two nonlinear characteristic fields can be parameterized by the water height h and are given by

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11 Shallow water flows with temperature gradient

W1 (U0 ) = S1 (U0 ) ∪ R1 (U0 ) : u = ω1 (U0 ; h), ⎧ √ √ √ h0 ), h ≤ h0 , ⎪ ⎨u0 − 2 gθ0 ( h − ω1 (U0 ; h) := 1 gθ0 1 ⎪ (h − h + − ) u , h > h0 , 0 ⎩ 0 2 h h0 W3 (U0 ) = S3 (U0 ) ∪ R3 (U0 ) : u = ω3 (U0 ; h), ⎧ √ √ √ h0 ), h ≤ h0 , ⎪ ⎨u0 + 2 gθ0 ( h − ω3 (U0 ; h) := 1 1 gθ0 ⎪ (h − h0 ) + , h > h0 . ⎩u0 + 2 h h0

(11.1.20)

Next, let us consider the second case s = u0 of (11.1.13). By substituting s = u0 into the first equation in (11.1.12), we get 0 = −u0 [h] + [hu] = −u0 (h − h0 ) + hu − h0 u0 = h(u − u0 ). The last system yields u = u0 , and so u0 = u = λ2 (U0 ) = λ2 (U). Consequently, the shock is a contact discontinuity associated with the second characteristic field (λ2 , r2 ). Across this contact, the water velocity remains constant, and the water height and the temperature are related to each other by the second equation in (11.1.12): g 0 = −s[hu] + hu2 + h2 θ 2 g = −s(u[h] + h0 [u]) + u[hu] + h0 u0 [u] + h2 θ 2 g g 2 2 = −sh0 [u] + h0 u0 [u] + h θ = h θ , 2 2 where u0 = s is invoked. This yields [h2 θ ] = 0, which determines h or θ as a function of the other. Thus, we see that the curve of 2-contact discontinuities issuing from a given state U0 can be parameterized by the water height h and is given by W2 (U0 ) :

u = u0 ,

θ = ω2 (U0 ; h) :=

h20 θ0 . h2

(11.1.21)

Since the 2-contact discontinuities propagate with the water velocity, we refer to what follows this kind of wave as material contacts.

11.1 Basic properties and elementary waves

335

11.1.4 Stationary contact discontinuities In the second case (ii) of (11.1.10), the argument is that the shock propagating with zero speed can be approximated by stationary smooth solutions of the system. Therefore, we first consider stationary smooth solutions of (11.1.1), which are independent of time. For such a stationary smooth solution, the system (11.1.1) becomes (hu) = 0, g hu2 + h2 θ + ghθ a = 0, 2 (huθ ) = 0,

(11.1.22)

where (.) = d(.)/dx. Lemma 11.1.1. Any solution of the following system of ordinary differential equations is a stationary smooth solution of (11.1.1): (hu) = 0, 2 u + gθ (h + a) = 0, 2

(11.1.23)

θ = 0. Furthermore, any jump satisfying the relations [hu] = 0, 2 u + gθ (h + a) = 0, 2

(11.1.24)

[θ ] = 0. is a weak solution of (11.1.1) in the sense of nonconservative products. The curve of jumps defined by (11.1.24) coincides with the integral curve associated with the fourth characteristic field (λ4 (U), r4 (U)). Proof. Let us prove the first statement of Lemma 11.1.1. Consider any integral curve of (11.1.23) passing though a given state U0 = (h0 , u0 , θ0 , a0 )T . If u0 = 0, then the first equation in (11.1.23) yields u = 0. So, the first and the third equations of (11.1.22) are satisfied. Further, the second equation in (11.1.23) can be derived using the second and the third equations of (11.1.23) that (h + a) = 0, which yields the second equation in (11.1.22). Consider u0 = 0. Thanks to the first equation in (11.1.22), the third equation can be re-written as 0 = θ (hu) + huθ = huθ .

336

11 Shallow water flows with temperature gradient

Thus, for u = 0, the third equation in (11.1.22) becomes θ = 0, which means that the temperature remains constant through the solution: θ = θ0 = constant. Therefore, the second equation in (11.1.22) can be simplified as

u2 + gθ (h + a) = 0. 2

Next, let us consider a sequence of stationary smooth solutions Uε of (11.1.1) given by (11.1.23) so that hε → h as ε → 0+, where h is a discontinuity jump. Then it is easy to see that uε → u, aε → a, as ε → 0+, where u and a are the corresponding values of the given jump. By using a stability result, see [114], the jump is also a weak solution in the sense of nonconservative products. This proves the second statement of the lemma. We now consider the integral curve associated with the fourth characteristic field ⎛ ⎞ 1 ⎜ −u ⎟ ⎜ ⎟ dU(ξ ) h ⎟ ⎜ = r4 (U(ξ )) = ⎜ 0 ⎟ . ⎜ 2 ⎟ dξ ⎝ u ⎠ −1 ghθ We have

dθ dξ

= 0, which yields θ = constant. We find du u =− , dh h

so that hdu + udh = 0, or

d(hu) = 0.

This yields us the first equation in (11.1.23). Moreover, it holds for this integral curve that da u2 u du = − 1. −1 = − dh ghθ gθ dh This implies that d u2 ( + a + h) = 0, dh 2gθ or

d u2 ( + gθ (a + h)) = 0, dh 2 which establishes the second equation of (11.1.23). Lemma 11.1.1 is completely proved.

11.2 Admissible stationary contact discontinuities

337

As seen by Lemma 11.1.1, the discontinuities satisfying the jump relations (11.1.24) are weak solutions of the system (11.1.1) associated with the 4th linearly degenerate characteristic field. They are therefore the 4-contact discontinuities. From the jump relations (11.1.24), substitute u=

u0 h0 h

from the first equation and

θ = θ0 from the third equation to the second one, we obtain the curve of 4-contact discontinuities issuing from a given state U0 to be parameterized by the water height h as W4 (U0 ) :

u = ω4 (U0 ; h) :=

u0 h0 , h

θ = θ0 , a = ωa (U0 ; h) := a0 + (h0 − h) +

u20

2gθ0

1−

h20 . h2

(11.1.25)

11.2 Admissible stationary contact discontinuities Since the bottom topography is given, the left-hand and right-hand bottom levels a± are known. The jump relations (11.1.24) for a± may not uniquely determine the stationary jump. As we will see later, sometimes there are two solutions, but sometimes there are none. In this section, we will find the conditions for the existence of such stationary waves and set up an additional admissibility criterion to select a unique solution when there are multiple ones when the state on one side of the contact is given. Precisely, let U0 be a given state, says, the left-hand state, and let a− = a0 and a = a+ be also given. We want to determine the corresponding right-hand state U with the topography level a = a+ . The third equation in the system (11.1.25) can be used to find the water height h in terms of other known quantities. This can be done as follows: multiplying both sides of this equation by 2gθ0 h2 , and then re-arranging terms, we get

ϕ (h) = 2gθ0 h3 + (2gθ0 (a − a0 − h0 ) − u20 )h2 + h20 u20 = 0.

(11.2.1)

This means that the water height is a zero of the function ϕ . Finding zeros of ϕ defined by (11.2.1) and characterizing their properties lead us to investigate the function ϕ . Now, it holds that

ϕ (0) = h20 u20 ≥ 0, ϕ (h) = 6gθ0 h2 + 2(2gθ0 (a − a0 − h0 ) − u20 )h, ϕ (h) = 12gθ0 h + 2(2gθ0 (a − a0 − h0 ) − u20 ).

338

11 Shallow water flows with temperature gradient

Critical points of ϕ satisfy

ϕ (h) = 0,

which gives us a unique critical point h = h∗ , where h∗ =

u20 + 2gθ0 (a0 + h0 − a) . 3gθ0

(11.2.2)

It is clear that in the case h∗ < 0, or equivalently, a > a0 + h0 +

u20 , 2gθ0

we get ϕ (h) > 0 for h > 0, so that the function ϕ is strictly increasing. Since ϕ (0) = h20 u20 ≥ 0, the equation (11.2.1) thus has no root. Next, consider that case h∗ ≥ 0, or, a ≤ a0 + h0 +

u20 , 2g

then ϕ > 0 for h > h∗ and ϕ (h) < 0 for 0 < h < h∗ . Thus, the equation (11.2.1) has two roots h1 ≤ h2 , if ϕ (h∗ ) = −gh3∗ + h20 u20 ≤ 0, or h∗ ≥ hm :=

h2 u2 1/3 0 0

gθ0

.

(11.2.3)

Moreover, the inequality (11.2.3) is equivalent to u2

a ≤ amax (U0 ) := a0 + h0 + 2gθ0 0 − 2(gθ3 )1/3 (h0 u0 )2/3 0

2/3 2 2/3 = a0 + 2g1θ0 (gθ0 h0 )1/3 − u0 (2(gθ0 h0 )1/3 + u0 ).

(11.2.4)

It holds from the formula (11.2.4) that amax (U0 ) ≥ a0 . Furthermore, the inequality is strict if the state U0 belongs to a strictly hyperbolic domain. In this case, i.e., a < amax (U0 ), the two roots are distinct, that is h1 (a) < h∗ < h2 (a). Consequently, whenever amax (U0 ) ≥ a0 holds, there are two states U+ ∈ {U1 ,U2 }, so that a stationary contact discontinuity from U− = U0 to U+ is possible. The positions of these states can be determined in the following lemma.

11.2 Admissible stationary contact discontinuities

339

Lemma 11.2.1. Assume that a < amax (U0 ), where amax (U0 ) is defined by (11.2.4). The function ϕ defined by (11.2.1) admits two distinct roots h1 < h2 . Moreover, the state U1 using the smaller root h1 belongs to G1 if u0 > 0 and belongs to G4 if u0 < 0; the state U2 using the larger root h2 lies in G2 if u0 > 0 and lies in G3 if u0 < 0. Proof. The first statement of Lemma 11.2.1 has been seen earlier. We now prove the second statement of the lemma. It is sufficient to consider the case u0 > 0, since the case u0 < 0 can be proved similarly. Let us define a function along the stationary curve W4 (U0 ) by the formula

σ (h) := u(h)2 − gθ0 h =

h20 u20 − gθ0 h. h2

A straightforward calculation shows that

σ (hm ) =

h20 u20 − gθ0 hm = 0, h2m

where hm is defined by (11.2.3). This implies that the point Um = (hm , um = h0 u0 /hm ) is the intersection point of W4 (U0 ) and C+ in the (h, u)-plane. Furthermore, since the condition a < amax (U0 ) is equivalent to

ϕ (hm ) < 0. Thus, we obtain h1 < hm < h2 . Besides, it is not difficult to check that

σ (h) =

−2h20 u20 − gθ0 < 0, h3

Since σ (hm ) = 0, we see that

σ (h1 ) > 0,

σ (h2 ) < 0,

which establishes the second statement of Lemma 11.2.1. To select a unique Riemann solution, the following admissibility criterion for stationary contact discontinuities is imposed: (MC) Along any stationary curve W4 (U0 ), the bottom level a is monotone as a function of h. The total variation of the bottom level component of any Riemann solution must not exceed |aL − aR |, where aL , aR are left-hand and right-hand bottom levels. Observe that a similar criterion was used [174, 175], [227], and [141]. Lemma 11.2.2. Assume that a < amax (U0 ), where amax (U0 ) is defined by (11.2.4). The Monotonicity Criterion selects the following admissible stationary contact wave:

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11 Shallow water flows with temperature gradient

(i) If U0 ∈ G1 ∪ G4 , then only the stationary contact using the smaller root h1 of ϕ defined by (11.2.1) is selected. (ii) If U0 ∈ G2 ∪ G3 , then only the stationary contact using the larger root h2 of ϕ defined by (11.2.1) is selected. Proof. We need only to prove for the case u0 > 0, as the case u0 < 0 can be treated similarly. Recall from (11.1.25) that u2 h2 a = ωa (U0 ; h) = a0 + (h0 − h) + 0 1 − 02 2gθ0 h = a0 +

u20 − u2 + h0 − h, 2gθ0

where

h0 u0 . h Taking the derivative of a with respect to h, we have u = u(h) =

ωa (U0 ; h) = =

−uu (h) uh0 u0 −1 = −1 gθ0 gθ0 h2 u2 − gθ0 h u2 −1 = , gθ0 h gθ0 h

which has the same sign as u2 − gθ0 h. Thus, a = ωa (U0 ; h) is increasing with respect to h in the domains G1 and is decreasing in the domain G2 . The fact that a = ωa (U0 ; h) is monotone as a function of h implies that the point U on the other side of the admissible contact from U0 must stay in the closure of the domain containing U0 . This establishes the conclusions (i) and (ii). We therefore define elementary waves of the system (11.1.1) as follows. Definition 11.2.1. The admissible elementary waves for the system (11.1.1)-(11.1.2) are the following ones: Shock waves, rarefaction waves, material contact discontinuities, and admissible stationary contact discontinuities.

11.3 Riemann problem In this section, solutions of the Riemann problem for (11.1.1)-(11.1.2) will be constructed. These solutions are weak solutions and consist of a finite number of admissible elementary waves.

11.3 Riemann problem

341

Notation To simplify the presentation, the following notation will be used: (i) Wi (U1 ,U2 ): An i-wave connecting a state U1 and a state U2 , where W = S: Lax shock, W = R: rarefaction wave, W2 = Z, i = 1, 2, 3; (ii) Wi (U1 ,U2 ) W j (U2 ,U3 ): an i wave from the left-hand state U1 to the righthand state U2 is followed by a j-wave from the left-hand state U2 to the righthand state u3 ; (iii) Wi (U1 ,U2 ) W j (U2 ,U3 ): an i wave from the right-hand state u1 to the lefthand state U2 is preceded by a j-wave from the right-hand state U2 to the lefthand state U3 ; (iv) The (forward) curve Wi j (U0 ) consists of all right-hand states U which can be reached from U0 using an i-wave from the left-hand state U0 to some intermediate right-hand state U1 ∈ Wi (U0 ), followed by a j-wave from the left-hand state U1 to the right-hand state U; (v) The (backward) curve Wi j (U0 ) consists of all left-hand states U which can be reached from U0 using an i-wave from the right-hand state U0 to some intermediate left-hand state U1 ∈ Wi (U0 ), preceded by a j-wave from the right-hand state U1 to the left-hand state U.

11.3.1 Riemann problem for flat bottom First, let us consider the Riemann problem in the case of a flat bottom. Then, the source term on the right-hand side of (11.1.1) vanishes. The system has the form of a system of conservation laws

∂t h + ∂x (hu) = 0, g ∂t (hu) + ∂x hu2 + h2 θ = 0, 2 ∂t (hθ ) + ∂x (huθ ) = 0, x ∈ R I , t > 0.

(11.3.1)

The argument in Section 11.2 is still valid, except that the a-component is neglected. Precisely, the system (11.3.1) can be written in the form Ut + A(U)Ux = 0, where

⎛ ⎞ h U = ⎝u⎠ , θ

⎞ u h 0 gh ⎟ ⎜ A(U) = ⎝gθ u ⎠. 2 0 0 u ⎛

(11.3.2)

The matrix A(U) admits three distinct and real eigenvalues λ1 , λ2 , λ3 as (11.1.4). The system is thus strictly hyperbolic.

342

11 Shallow water flows with temperature gradient

The first and the third characteristic fields of (11.3.1) are genuinely nonlinear, while the second one is linearly degenerate. The formulas for these waves are the same as described in Section 2. This is because even in the non-flat bottom, the cross-section level remains constant across these waves. Let us consider the Riemann problem for (11.3.1) with the observation that u = constant across any 2-contact wave. Furthermore, the temperature changes only across 2-contact waves. Therefore, the temperature values on both sides of the 2contact wave are merely the given values from the initial data θL and θR . This suggests us to construct a composite wave curve W12 (UL ), which consists of 1-wave from UL to some state U1 ∈ W1 (UL ), followed by a 2-wave from U1 to U with the same value of velocity, and the temperatures of U1 and of U are θL and θR , respectively. The curve W12 (UL ) can be parameterized by the water height as follows. Let U1 be the state as in the description above, i.e., U1 ∈ W1 (UL ). Then, it holds from (11.1.20) that u1 = ω1 (UL ; h1 ). Let U be the state resulted from a 2-contact wave from U1 , that is, U ∈ W2 (U1 ). Then, it holds from (11.1.21) that u = u1 ,

θR h2 = θL h21 .

This implies that

u = ω12 (UL ; h) := ω1 (UL ;

(11.3.3)

θR h). θL

(11.3.4)

One can check easily that the function u = ω12 (UL ; h) is decreasing as a function h → u, and u → −∞ as h → ∞, and that the curve W3 (UR ) is increasing as a function h → u, and u → ∞ as h → ∞. Therefore, these two curves always intersect at a unique point, provided that we allow the “dried solution”. In other words, we need some condition for the solution not to contain a “dried part”. Indeed, let us denote by uupper the u-intercept of the curve W12 (UL ), which is given by

uupper = uL + 2 ghL θL , and by ulower the u-intercept of the curve W3 (UR ), which is given by

ulower = uR − 2 ghR θR . Whenever ulower < uupper , the two wave curves W12 (UL ) and W3 (UR ) intersect at a unique point in the domain h > 0, denoted by U2 . That is {U2 } = W12 (UL ) ∩ W3 (UR ).

11.3 Riemann problem

343

Since these two wave curves can be parameterized by h, we can solve the nonlinear algebraic equation θR ω1 (UL ; h) = ω3 (UR ; h) (11.3.5) θL for the water height h = h1 , and so U2 can be found. After finding U2 , we can use (11.3.3) to determine U1 , where h = h2 and u = u2 . Thus, a unique solution of the Riemann problem can be determined by W1 (UL ,U1 ) W2 (U1 ,U2 ) W3 (U2 ,UR ).

(11.3.6)

Theorem 11.3.1. Under the condition that

ulower = uR − 2 ghR θR < uupper = uL + 2 ghL θL , the Riemann problem for the system (11.1.1) admits a unique solution, which has the form (11.3.6).

11.3.2 Riemann problem for discontinuous bottom Let us now consider the more general case, where the flat bottom is a jump discontinuity. Solutions of the Riemann problem for (11.1.1)-(11.1.2) can also be constructed. For this purpose, let us consider the projection of wave curves on the (h, u)-plane. Note that under the transformation x → −x, u → −u, a left-hand state U = (h, u) in G2 or G3 will be transferred to the right-hand state V = (h, −u) in G2 or G1 , respectively. Therefore, constructions based on a left-hand state in G2 or G3 will be the same as the ones based on the right-hand state in G2 or G1 , respectively. So, we will only consider the ones based on a left-hand state in G2 or G3 . Waves associated with different characteristic fields may combine and the order of combinations depends on values of the propagation speed. Along a wave curve, the shock speed may change sign, and therefore, change the order in the combinations with the stationary waves. This is descried in the following lemma. Lemma 11.3.1. For every point U = (h, u) ∈ G1 there exists exactly one point U # ∈ S1 (U) ∩ G2 such that the 1-shock speed s1 (U,U # ) = 0. The state U # = (h# , u# ) is defined by

−h + h2 + 8hu2 /gθ uh # , u# = # . h = 2 h Moreover, for any V ∈ S1 (U), the shock speed s1 (U,V ) > 0 if and only if V is located above U # on S1 (U). Since the proof of Lemma 11.3.1 is straightforward, we omit the proof. Note that similar conclusions can be established for S3 (U),U ∈ G4 .

344

11 Shallow water flows with temperature gradient

11.3.3 Construction 1: supercritical/supercritical Let us consider a supercritical left-hand state UL ∈ G1 and UR is also a supercritical right-hand state. Under the condition aL ≥ aR , or aL < aR ≤ amax (UL ), there is a stationary contact upward (downward, respectively) along W4 (UL ) from UL to the state U1 ∈ W4 (UL ) ∩ G1 , shifting the level aL directly to the level aR . As defined earlier, the composite wave curve W32 (UR ) consists of all left-hand state U which can be reached from UR using a combination of a 3-wave from right¯ preceded by a 2-wave from the right-hand hand state UR to some left-hand state U, ¯ state U to the left-hand state U. Suppose that the curve W32 (UR ) and the curve W1 (U1 ) intersect at U2 : {U2 } = W1 (U1 ) ∩ W32 (UR ).

(11.3.7)

By definition of W32 (UR ), the state U2 determines a state U3 = U¯ 2 which can be arrived at by a 2-contact wave, and this 2-contact wave is followed by a 3-wave from U3 to UR . Thus, a Riemann solution is available and of the form W4 (UL ,U1 ) W1 (U1 ,U2 ) W2 (U2 ,U3 ) W3 (U3 ,UR ),

(11.3.8)

provided s1 (U1 ,U2 ) ≥ 0. See Figure 11.1. Let us now discuss about the computation of the states which determine the Riemann solution (11.3.8). As observed in the previous section, the state U1 can always be computed. The states U2 and U3 can be computed from the following system of nonlinear algebraic equations: u2 = ω1 (U1 ; h2 ), u3 = u2 ,

θR h23 = θL h22 , u3 = ω3 (UR ; h3 ). Eliminating u2 , u3 and h3 from the last equations to get a single algebraic nonlinear equation in h2 : θL ω1 (U1 ; h2 ) = ω3 (UR ; h2 ). (11.3.9) θR After computing h2 from the equation (11.3.9), we can use the above nonlinear system to get the states U2 and U3 . A configuration of the Riemann solution (11.3.8) in the (x,t)-plane is given in Figure 11.2. Example 11.3.1. Consider the Riemann problem for the system (11.1.1)-(11.1.2), where the initial states UL = (hL , uL , θL , aL ) and UR = (hR , uR , θR , aR ) are given in Table 11.1, and the gravitational constant is given by g = 9.8. The Riemann solution of the form (11.3.8) at the time t = 0.1 is illustrated in Figure 11.3 in the interval

11.3 Riemann problem

345

[−1, 1]. The states U1 ,U2 , and U3 , which determine the elementary waves of the solution (11.3.8), are also computed as in Table 11.1.

11.3.4 Construction 2: supercritical/subcritical Let us consider the Riemann problem in the case of a supercritical left-hand state UL and a subcritical right-hand state UR . The solution can begin with a 1-shock wave from UL to a some subcritical state U1 , provided s1 (UL ,U1 ) ≤ 0. The shock can be followed by a 4-stationary contact wave to a subcritical state U, which shifts the cross-section from the level aL to the level aR . Such states U form the curve of composite waves W14 (UL ). Suppose that the curves W14 (UL ) and W32 (UR ) intersect at some state U2 : {U2 } = W14 (UL ) ∩ W32 (UR ).

(11.3.10)

Then, a Riemann solution is available. Indeed, let U1 ∈ W1 (UL ) be the left-hand state of the 4-contact wave to the right-hand state U2 ; and let U3 be the right-hand state of the 2-contact wave from the left-hand state U2 . It holds that u = u2 ,

θR h2 = θL h22 .

Under the condition s3 (U3 ,UR ) ≥ 0, there exists a Riemann solution of the form W1 (UL ,U1 ) W4 (U1 ,U2 ) W2 (U2 ,U3 ) W3 (U3 ,UR ).

(11.3.11)

See Figure 11.4. A configuration of the Riemann solution (11.3.11) in the (x,t)plane is given in Figure 11.5. Example 11.3.2. Let us take g = 9.8. The states UL ,U1 ,U2 ,U3 and UR of a Riemann solution of the form (11.3.11) are computed and given by Table 11.2. The Riemann solution at the time t = 0.1 is illustrated in Figure 11.6 in the interval [−1, 1].

11.3.5 Construction 3: resonant waves in supercritical regime If the left-hand state UL is supercritical, and the right-hand state is located around the critical curve, there may exists a resonant phenomenon, where multiple waves propagate with the same speed.

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11 Shallow water flows with temperature gradient

Let us describe this interesting situation as follows. First, the solution begins with a 4-contact wave W4 (UL ,U1 ) from UL = (h, u, aL ) to some supercritical state U1 = (h1 , u1 , a1 ), using a mediate change of cross-section to some value a1 ∈ [aL , aR ]. Then, the solution continues by a 1-shock wave with zero shock speed from U1 to a subcritical state U2 . The shock wave is then followed by another 4-contact wave from U2 to a subcritical state U3 and completes the change in cross-section to aR . The late 4-wave is followed by a 2-wave from U3 to U4 and finally a 3-wave from U4 to UR . In the above construction, the value a1 plays the key role to determine the solution. The state U3 depends on the value a1 , and if we let a1 vary between aL and aR , such points U3 trace out a curve, denoted by L . Suppose that the two curves W32 (UR ) and L intersect at some point U3 : U3 ∈ W32 (UR ) ∩ L .

(11.3.12)

Then, a Riemann solution containing three discontinuities having the same zero speed is available and of the form W4 (UL ,U1 ) W1 (U1 ,U2 ) W4 (U2 ,U3 ) W2 (U3 ,U4 ) W3 (U4 ,UR ). (11.3.13) See Figure 11.7. A configuration of the Riemann solution (11.3.13) in the (x,t)plane is given in Figure 11.8. Example 11.3.3. The states UL ,U1 ,U2 ,U3 ,U4 , and UR of a Riemann solution of the form (11.3.13) are computed and given in Table 11.3. The Riemann solution at the time t = 0.1 in the interval [−1, 1] is illustrated as in Figure 11.9. Remark 11.3.1. Note that from the above argument, the Riemann problem for (11.1.1)-(11.1.2) may have up to three distinct solutions, which have the form in Constructions 1-3. Precisely, the number of Riemann solutions coincides with the one of intersection points between the curve U3 ∈ W32 (UR ) and the union of the curves W1 (U+ ) as in (11.3.7), W14 (UL ) as in (11.3.10), and L as in (11.3.12). A unique solution can also be determined. For example, if we require the curve U3 ∈ W32 (UR ) remains in the supercritical region G1 , then it may intersect only W1 (U1 ), which leads to a unique solution.

11.3.6 Construction 4: subcritical/supercritical Let us now consider the Riemann problem when the left-hand state UL is either critical or subcritical, and the right-hand state UR is supercritical. There will be two types of solutions corresponding to the case aL > aR and aL ≤ aR . First, let us assume that aL > aR . The solution can begin from the subcritical state UL with a 1-rarefaction wave until it reaches a critical state U1 ∈ C+ . This

11.3 Riemann problem

347

rarefaction wave W1 (UL ,U1 ) is followed by a stationary 4-contact wave from U1 to a supercritical state U2 ∈ G1 . Suppose that the curves W1 (U2 ) and W32 (UR ) intersect at some state U3 : {U3 } = W1 (U2 ) ∩ W32 (UR ).

(11.3.14)

Then, the solution can continue by a 1-wave from U2 to U3 , followed by a 2-wave from U3 to U4 , and finally the solution arrives at UR by a 3-wave from U4 . This construction is valid if s1 (U2 ,U3 ) ≥ 0, which means that the state U3 has to be above U2# on W1 (U2 ) in the (h, u)-plane. Thus, a Riemann solution is available if W1 (U2 ) and W32 (UR ) intersect and s1 (U2 ,U3 ) ≥ 0, and it has the form W1 (UL ,U1 ) W4 (U1 ,U2 ) W1 (U2 ,U3 ) W2 (U3 ,U4 ) W3 (U4 ,UR ). (11.3.15) See Figure 11.10. A configuration of the Riemann solution (11.3.15) in the (x,t)plane is given in Figure 11.12. Example 11.3.4. The states UL ,U1 ,U2 ,U3 ,U4 , and UR of a Riemann solution of the form (11.3.15) are computed as in Table 11.4. The Riemann solution at the time t = 0.1 is illustrated in Figure 11.11 in the interval [−1, 1]. Next, consider the case aL ≤ aR . A solution of another type can also be constructed. Let us describe a “boundary” curve as follows. Consider 4-waves from a state U ∈ C+ to some subcritical state U o , which decreases the cross-section level from aR to aL . The set of all these subcritical states U o form a curve, denoted by Ca . Assume that the curves W1 (UL ) and Ca intersect at some state U1 : {U1 } = W1 (UL ) ∩ Ca . Then, the solution can begin by a 1-wave W1 (UL ,U1 ), followed by a 4-stationary contact wave W4 (U1 ,U2 = U1o ) to U2 ∈ C+ . Let {U3 } = W1 (U2 ) ∩ W32 (UR ).

(11.3.16)

The solution is then continued by a 1-rarefaction wave from U2 to U3 , followed by a 2-wave from U3 to U4 and finally a 3-wave from U4 to UR . Thus, provided

λ1 (U3 ) ≥ 0, a Riemann solution is available, and of the form W1 (UL ,U1 ) W4 (U1 ,U2 ) R1 (U2 ,U3 ) W2 (U3 ,U4 ) W3 (U4 ,UR ). (11.3.17) See Figure 11.13.

348

11 Shallow water flows with temperature gradient

One can see that the above construction can be extended for some part of a subcritical right-hand state.

11.3.7 Construction 5: subcritical/subcritical Let us deal with the Riemann problem for a subcritical left-hand state UL and subcritical right-hand state UR . Assume that the curves W14 (UL ) and W32 (UR ) intersect at some state U2 : {U2 } = W14 (UL ) ∩ W32 (UR ).

(11.3.18)

Let U1 ∈ W1 (UL ) be the left-hand state of the 4-contact wave to the right-hand state U2 . Let U3 be the right-hand state of the 2-contact from the left-hand state U2 . Then, a Riemann solution is available, and of the form W1 (UL ,U1 ) W4 (U1 ,U2 ) W2 (U2 ,U3 ) W3 (U3 ,UR ).

(11.3.19)

The construction (11.3.19) makes sense if u2 ≥ 0, see Figure 11.14. The configuration of the Riemann solution (11.3.19) in the (x,t)-plane is given in Figure 11.15. Example 11.3.5. The states UL ,U1 ,U2 ,U3 , and UR of a Riemann solution of the form (11.3.19) are computed as in Table 11.5. The Riemann solution at the time t = 0.1 is illustrated in Figure 11.16 in the interval [−1, 1]. Note that this construction can be extended for some parts of a supercritical righthand state.

11.3.8 Construction 6: resonant waves in subcritical regime In the case where aL > aR , the Riemann problem with a subcritical left-hand state UL and a right-hand state UR near critical curve may possess a resonant phenomenon. That is, the Riemann solution may contain multiple waves which propagate with the same (zero) speed. Precisely, the solution may start with a 1-rarefaction wave from the subcritical state UL to a critical state U1 ∈ C+ , followed by an intermediate stationary 2-contact wave from U1 to a supercritical state U2 = (h2 , u2 , a2 ) ∈ G1 , which shift the crosssection level from aL to some value a2 ∈ [aR , aL ]. This wave can be followed by a 1-shock with zero speed from U2 to a subcritical state U3 ∈ G2 , followed by another stationary 4-contact wave to a subcritical state U4 ∈ G2 and completes the change of the cross-section from aL to aR . The set of these states U4 trace out a curve L . Assume that the two curves W32 (UR ) and L intersect at U4 :

11.4 Bibliographical notes

349

U4 ∈ W32 (UR ) ∩ L .

(11.3.20)

Then, the solution continues by a 2-wave from U4 to U5 , and finally it arrives at UR by a 3-wave from U5 . Thus, a Riemann solution containing three zero-speed waves is available, and of the form W1 (UL ,U1 ) W4 (U1 ,U2 ) W1 (U2 ,U3 ) W4 (U3 ,U4 ) W2 (U4 ,U5 ) W3 (U5 ,UR ). (11.3.21) See Figure 11.17. A configuration of the Riemann solution (11.3.21) in the (x,t)plane is given in Figure 11.18. Example 11.3.6. The states UL ,U1 ,U2 ,U3 ,U4 ,U5 , and UR of a Riemann solution of the form (11.3.21) are computed as in Table 11.6. The Riemann solution at the time t = 0.1 is illustrated in Figure 11.19 in the interval [−1, 1]. Remark 11.3.2. We also note that from the above argument, the Riemann problem for (11.1.1)-(11.1.2) may have up to three distinct solutions, which have the form in Constructions 4-6. Indeed, each intersection point of the curve W32 (UR ) with each curve W1 (U2 ) as in (11.3.14), W14 (UL ) as in (11.3.18), and L as in (11.3.20) can determine a particular Riemann solution.

11.4 Bibliographical notes We list only a short selection of the most relevant papers for this chapter, and also refer the reader to the bibliography at the end of this monograph. The Ripa model was introduced in [268, 269]. The Riemann problem for the shallow water equations with horizontal temperature gradients was investigated in [306]. Numerical schemes for the Ripa system were constructed in [93, 148, 277, 317]. Godunov-type and van Leer-type schemes for various models involving nonconservative terms were presented in [229, 278, 283].

Fig. 11.1 A Riemann solution of the form (11.3.8) in Construction 1.

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11 Shallow water flows with temperature gradient

Fig. 11.2 Riemann solution with structure (11.3.8) in the (x,t)-plane

Fig. 11.3 A Riemann solution of the form (11.3.8) in [−1, 1] at the time t = 0.1 Table 11.1 States that separate the elementary waves of a Riemann solution of the form (11.3.8) in Construction 1 h u θ a UL 0.1 4 3 1.2 U1 0.0739038916804 5.41243486513 3 1 U2 0.263663341277 2.38424088748 3 1 U3 0.228339151592 2.38424088748 4 1 UR 0.2 2 4 1

11.4 Bibliographical notes

351

Table 11.2 States that separate the elementary waves of a Riemann solution of the form (11.3.11) in Construction 2 h u θ a UL 0.2 3 3 1 U1 0.422296649273 0.68651600364 3 1 U2 0.191219321444 1.51613030431 3 1.2 U3 0.141224828283 1.51613030431 5.5 1.2 UR 0.112979862627 0.930860434074 5.5 1.2

Fig. 11.4 A Riemann solution of the form (11.3.11) in Construction 2.

Fig. 11.5 Riemann solution with structure (11.3.11) in the (x,t)-plane

352

11 Shallow water flows with temperature gradient

Table 11.3 States that separate the elementary waves of a Riemann solution of the form (11.3.13) in Construction 3 h u θ a UL 0.18 2.8 3 1 U1 0.138666886761 3.63460961568 3 0.95 U2 0.290419684743 1.7354195548 3 0.95 U3 0.357916139807 1.4081510833 3 0.9 U4 0.292237304411 1.4081510833 4.5 0.9 UR 0.233789843529 0.646609928404 4.5 0.9

Table 11.4 States that separate the elementary waves of a Riemann solution of the form (11.3.15) in Construction 4 h u θ a UL 0.6 2 4 1.2 U1 0.387976014993 3.89982817413 4 1.2 U2 0.230006086181 6.57825981598 4 1 U3 0.308991050587 5.615266934 4 1 U4 0.436979334392 5.615266934 2 1 UR 0.349583467514 4.99444830273 2 1

Fig. 11.6 A Riemann solution of the form (11.3.11) in [−1, 1] at the time t = 0.1

11.4 Bibliographical notes

Fig. 11.7 A Riemann solution of the form (11.3.13) in Construction 3.

Fig. 11.8 Riemann solution with structure (11.3.13) in the (x,t)-plane

353

354

11 Shallow water flows with temperature gradient

Fig. 11.9 A Riemann solution of the form (11.3.13) in [−1, 1] at the time t = 0.1

Fig. 11.10 A Riemann solution of the form (11.3.15), case aL > aR , in Construction 4.

11.4 Bibliographical notes

Fig. 11.11 A Riemann solution of the form (11.3.15) in [−1, 1] at the time t = 0.1

Fig. 11.12 Riemann solution with structure (11.3.15) in the (x,t)-plane

355

356

11 Shallow water flows with temperature gradient

Fig. 11.13 A Riemann solution of the form (11.3.17), case aL ≤ aR , in Construction 4.

Fig. 11.14 Riemann solution of the form (11.3.19) in Construction 5.

11.4 Bibliographical notes

357

Fig. 11.15 Riemann solution with structure (11.3.19) in the (x,t)-plane

Fig. 11.16 A Riemann solution of the form (11.3.19) in [−1, 1] at the time t = 0.1 Table 11.5 States that separate the elementary waves of a Riemann solution of the form (11.3.19) in Construction 5 h u θ a UL 0.9 3 2 1 U1 1.35 1.08297104873 2 1 U2 1.13779949093 1.28494600977 2 1.2 U3 1.01767880259 1.28494600977 2.5 1.2 UR 0.814143042073 0.225705300652 2.5 1.2

358

11 Shallow water flows with temperature gradient

Fig. 11.17 Collision of waves in Construction 6, when aL > aR , there is a solution containing three waves of the same zero speed of the form (11.3.21).

Fig. 11.18 Riemann solution with structure (11.3.21) in the (x,t)-plane

11.4 Bibliographical notes

359

Fig. 11.19 A Riemann solution of the form (11.3.21) in [−1, 1] at the time t = 0.1

Table 11.6 States that separate the elementary waves of a Riemann solution of the form (11.3.21) in Construction 6 h u θ a UL 1 3.1 2 1.4 U1 0.810131180162 3.98479248282 2 1.4 U2 0.556372903811 5.80223194677 2 1.2 U3 1.13202382168 2.85171086948 2 1.2 U4 1.40475925014 2.29804832144 2 1 U5 1.14698112477 2.29804832144 3 1 UR 0.917584899813 1.06619766975 3 1

Chapter 12

Baer-Nunziato model of two-phase flows

12.1 Introduction In this chapter, we will investigate the Riemann problem for a model of two-phase flows. The model under study is obtained from the well-known Baer-Nunziato model, which is used for studying deflagration-to-detonation transition in porous energetic materials [33, 73]. In this model, it is observed that intergranular pores form an interconnected region occupied by gas. The model is postulated using several principles of continuum-mixture theory as follows. First, each phase is assigned a density ρk , a specific volume vk = 1/ρk , a particle velocity uk , a specific internal energy ek , a temperature Tk , and a volume fraction αk , where the subscript k can be s or g, which denote the quantities in the solid phase and the gas phase, respectively. The volume fractions satisfy the saturation condition αs + αg = 1. The density and the volume fraction define the mass fractions bk =

αk ρk , αs ρs + αg ρg

k = s, g.

Second, the principle of phase separation holds. Each phase is assumed to be in local thermodynamic equilibrium and is characterized by equations of state. Third, the motion of each constituent is described by balance laws of mass, momentum, and energy as in the case of a single phase. These laws represent the evolution of local, phase-averaged density, momentum, and energy. The interaction among constituents is represented by source terms that contain nonconservative terms. In general, multi-phase flow models are governed by a large system of balance laws and often possess very complicated properties. Apart from nonconservative terms, many multi-phase flow models do not have characteristic fields in explicit form. This presents a significant challenge for investigating the Riemann problem. Fortunately, the two-phase flow model by Baer-Nunziato [33, 73] has all characteristic fields in explicit form. This important property motivates us to investigate elementary waves and the Riemann problem. Rarefaction waves can then be given © Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7 12

361

362

12 Baer-Nunziato model of two-phase flows

by explicit formulas. The system is closed by the governing equations and the compaction dynamics equation. For the isentropic case, a phase decomposition approach may be used to separate shock waves in each phase. Solid contacts will then relate these two phases. Solutions of the Riemann problem in either subsonic or supersonic regions will be constructed computationally, where each Riemann solution corresponds to a set of algebraic equations. Mathematical modeling for two-phase flows has been carried out by many authors using different approaches. For example, one may consider two-phase flow models by employing gas dynamics equations and an evolution equation of massdensity fraction, where pressure depends on both specific volume and mass-density fraction, such as in the bubble-liquid model. Furthermore, even for two-phase flow models using an average-phase process, there is one additional equation needed to obtain the closure form of the system. There are many choices for this additional equation. In particular, one may choose to require that the two pressures coincide, leading to one-pressure models of two-phase flows. Although this seems to reduce the size of unknowns, the system may lack hyperbolicity—a basic condition for solving the Riemann problem in the standard way. There are still many open questions on the Riemann problem for two-phase flow models: resonant cases, nonisentropic cases, and models other than the famous BaerNunziato model.

12.2 Preliminaries 12.2.1 Two-phase flow models Consider the following model of two-phase flows which consists of six governing equations representing the balance of mass, momentum, and energy in each phase, namely,

∂t (α1 ρ1 ) + ∂x (α1 ρ1 u1 ) = 0, ∂t (α1 ρ1 u1 ) + ∂x (α1 (ρ1 u21 + p1 )) = P(U)∂x α1 , ∂t (α1 ρ1 e1 ) + ∂x (α1 u1 (ρ1 e1 + p1 )) = −P(U)∂t α1 , ∂t (α2 ρ2 ) + ∂x (α2 ρ2 u2 ) = 0, ∂t (α2 ρ2 u2 ) + ∂x (α2 (ρ2 u22 + p2 )) = −P(U)∂x α1 , ∂t (α2 ρ2 e2 ) + ∂x (α2 u2 (ρ2 e2 + p2 )) = P(U)∂t α1 , x ∈ R I ,t > 0,

(12.2.1)

see [119]. The notation αk , ρk , uk , pk , εk , Sk , Tk , ek = εk + u2k /2, k = 1, 2, respectively, stand for the volume fraction, density, velocity, pressure, internal energy, specific entropy, temperature, and the total energy in the k-phase, k = 1, 2, respectively. The volume fractions satisfy α1 + α2 = 1.

12.2 Preliminaries

363

The system (12.2.1) is underdetermined, since the number of unknowns is larger than the number of equations by one. There are basically two approaches to make the system determined. The first approach is to seek for an additional governing equation. In this approach, many authors propose an additional equation under the form ∂t α1 + Q(U)∂x α1 = 0, x ∈ R I ,t > 0. (12.2.2) In particular, taking in (12.2.1)-(12.2.2) P(U) = p2 ,

Q(U) = u1 ,

(12.2.3)

one obtains the well-known Baer-Nunziato model, see [33]. Another choice in the first approach is given by Saurel-Abgrall [278], where they propose in (12.2.1)(12.2.2) P(U) = α1 p1 + α2 p2 ,

Q(U) =

α1 ρ1 u1 + α2 ρ2 u2 . α1 ρ1 + α2 u2

(12.2.4)

In this approach, the system is often hyperbolic. The second approach to make the system (12.2.1) determined is to reduce the size of the unknowns by algebraic equations. For example, it has been found in the bubble-droplet model, which has applications in the cooling of nuclear reactors that the difference in the pressure of the two phases is negligible. Therefore, the size of the unknowns can be reduced by one by assuming that P(U) = p1 = p2 .

(12.2.5)

However, the system could then fail to be hyperbolic, which leads to challenging problems, see [194, 303]. The first approach therefore demonstrates certain advantages for the study, since the system is hyperbolic in certain regions. Indeed, let us investigate the hyperbolicity of the model (12.2.1)-(12.2.2). Suppose that the two fluids have the following equations of the state of stiffened gases:

εk =

pk + γk p∞,k , ρk (γk − 1)

p = (γ − 1)ρ (ε − ε∗ ) − γ p∞ ,

(12.2.6)

where γ , p∞ , and ε∗ are constants, k = 1, 2. The pressure pk and the enthalpy can be written as functions of the density and the specific entropy by γ

pk (ρk , Sk ) = κk (Sk )ρk k − pk,∞ , hk (ρk , Sk ) =

κk (Sk )γk γk −1 ρ + εk,∗ , γk − 1 k

(12.2.7)

364

12 Baer-Nunziato model of two-phase flows

where

κk (Sk ) :=

ck,v (γk − 1)Tk,∗

Sk − Sk,∗ ck,v

,

k = 1, 2.

(12.2.8)

The local sound speeds in each phase are given by ck = ∂S pk (ρk , Sk ) = γk (pk + p∞,k )/ρk ,

k = 1, 2.

(12.2.9)

γ −1

k ρk,∗

exp

The model (12.2.1)-(12.2.2) can then be written as a system of balance laws in nonconservative form as ∂t U + A(U)∂xU = 0, (12.2.10) where U is the primitive variable ⎛ u ρ ⎜ 1 1 ⎜ ⎜0 u ⎜ 1 ⎜ ⎜ ⎜ ⎜ 0 ρ1 c21 ⎜ ⎜ A(U) = ⎜ ⎜0 0 ⎜ ⎜ ⎜ ⎜0 0 ⎜ ⎜ ⎜ ⎜0 0 ⎝ 0 0

⎛

⎞ ρ1 ⎜ u1 ⎟ ⎜ ⎟ ⎜ p1 ⎟ ⎜ ⎟ ⎟ U =⎜ ⎜ ρ2 ⎟ , ⎜ u2 ⎟ ⎜ ⎟ ⎝ p2 ⎠ α1

and vk =

1 , ρk

c2i,k =

0 0

0

0

v1 0

0

0

u1 0

0

0

0 u2 ρ2 0 0 0 u2 v2 0 0 ρ2 c22 u2 0 0

0

0

⎞ ρ1 (u1 − Q(U)) ⎟ α1 ⎟ p1 − P(U) ⎟ ⎟ α1 ρ1 ⎟ ⎟ 2 ρ1 ci,1 ⎟ (u1 − Q(U)) ⎟ ⎟ α1 ⎟ ⎟, ρ2 − (u2 − Q(U)) ⎟ ⎟ α2 ⎟ ⎟ p2 − P(U) ⎟ − ⎟ α2 ρ2 ⎟ 2 ⎟ ρ2 ci,2 − (u2 − Q(U))⎟ ⎠ α2 Q(U)

P(U)(γk − 1) + pk + γk πk , ρk

k = 1, 2.

The matrix A(U) admits seven eigenvalues:

λ1 (U) = u1 − c1 , λ4 (U) = u2 − c2 , λ7 (U) = Q(U).

λ2 (U) = u1 , λ5 (U) = u2 ,

λ3 (U) = u1 + c1 , λ6 (U) = u1 + c2 ,

(12.2.11)

12.2.2 Properties of the isentropic Baer-Nunziato model From now on, we consider the isentropic Baer-Nunziato model by assuming that the fluids are isentropic in both phases. Precisely, the isentropic Baer-Nunziato under consideration is given by

12.2 Preliminaries

365

∂t (αg ρg ) + ∂x (αg ρg ug ) = 0, ∂t (αg ρg ug ) + ∂x (αg (ρg u2g + pg )) = pg ∂x αg , ∂t (αs ρs ) + ∂x (αs ρs us ) = 0, ∂t (αs ρs us ) + ∂x (αs (ρs u2s + ps )) = −pg ∂x αg , ∂t ρs + ∂x (ρs us ) = 0, x ∈ R I ,t > 0,

(12.2.12)

where the subscripts g, s stand for the quantities in the gas phase and solid phase, respectively. The system (12.2.12) can be written as

ρg (ug − us ) ∂x αg = 0, αg ∂t ug + hg (ρg )∂x ρg + ug ∂x ug = 0, ∂t ρs + us ∂x ρs + ρs ∂x us = 0, pg − ps ∂t us + hs (ρs )∂x ρs + us ∂x us + ∂x αg = 0, αs ρs ∂t αg + us ∂x αg = 0, x ∈ R I ,t > 0, ∂t ρg + ug ∂x ρg + ρg ∂x ug +

where hk is given by hk (ρ ) =

pk (ρ ) , ρ

k = s, g,

and the dash denotes the derivative of the function under consideration. Thus, the system (12.2.12) has the form of a nonconservative system of balance laws

∂t U + A(U)∂xU = 0,

(12.2.13)

where ⎛

⎞

ρg ⎜ ug ⎟ ⎜ ⎟ ⎟ U =⎜ ⎜ ρs ⎟ , ⎝ us ⎠ αg

⎛ ug

ρg

0

0

⎜ ⎜ ⎜hg (ρg ) ug 0 0 ⎜ A(U) = ⎜ 0 0 us ρs ⎜ ⎜ 0 hs (ρs ) us ⎝ 0 0

0

0

0

ρg (ug − us ) ⎞ ⎟ αg ⎟ ⎟ 0 ⎟ ⎟. 0 pg − ps ⎟ ⎟ ⎠ αs ρs us

The matrix A(U) in (12.2.13) admits five real eigenvalues λ1 (U) = ug − pg , λ2 (U) = ug + pg , λ3 (U) = us − ps , λ4 (U) = us + ps , λ5 (U) = us . The corresponding right eigenvectors can be chosen as

(12.2.14)

(12.2.15)

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12 Baer-Nunziato model of two-phase flows

⎛ ⎛ ⎞ ⎞ −ρg ρg ⎜ ⎜ ⎟ ⎟ ⎜ pg (ρg )⎟ ⎜ pg (ρg )⎟ ⎜ ⎜ ⎟ ⎟ r2 (U) = μ ⎜ r1 (U) = μ ⎜ ⎟, ⎟, 0 0 ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠ ⎠ 0 0 0 ⎞ 0 ⎞ ⎛ ⎛ 0 0 ⎜ ⎜ ⎟ ⎟ 0 0 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ r4 (U) = ν ⎜ ρs ⎟ r3 (U) = ν ⎜−ρs ⎟ , ⎟, ⎝ ps (ρs )⎠ ⎝ ps (ρs )⎠ 0 0 ⎛ ⎞ −(ug − us )2 ρg αs ps (ρs ) ⎜ ⎟ (ug − us )pg (ρg )ps (ρs )αs ⎜ ⎟ 2 − p (ρ ))α ⎟ (p ( ρ ) − p ( ρ ))((u − u ) r5 (U) = ⎜ g g g s g⎟ , g g ⎜ s s ⎝ ⎠ 0 ((ug − us )2 − pg (ρg ))αg αs ps (ρs ) where

μ=

2 pg (ρg ) pg (ρg )ρg + 2pg (ρg )

,

(12.2.16)

2 ps (ρs ) . ν = ps (ρs )ρs + 2ps (ρs )

It is not difficult to check that the eigenvectors ri , i = 1, 2, 3, 4, 5 are linearly independent. Thus, the system is hyperbolic. Furthermore, it holds that λ3 < λ5 < λ4 .

Fig. 12.1 Illustration of the subsonic, supersonic regions, and the sonic surface

However, the eigenvalues λ5 may coincide with either λ1 or λ2 on a certain hypersurface of the phase domain, called the resonant surface. Due to the change of order of these eigenvalues, we set

12.2 Preliminaries

367

Ω1 := {U | λ1 (U) > λ5 (U)}, Ω2 := {U | λ1 (U) < λ5 (U) < λ2 (U)}, Ω3 := {U | λ2 (U) < λ5 (U)}, Σ+ := {U | λ1 (U) = λ5 (U)}, Σ− := {U | λ2 (U) = λ5 (U)}.

(12.2.17)

The system is thus strictly hyperbolic in each domain Ωi , i = 1, 2, 3, but fails to be strictly hyperbolic on the resonant surface

Σ := Σ+ ∪ Σ− . A state is subsonic if |ug − us | < cg :=

(12.2.18)

pg ,

a state is supersonic if |ug − us | > cg , and a state is sonic if |ug − us | = cg . Thus, the subsonic set is the region Ω2 , the supersonic set is the regions Ω1 and Ω3 , and the sonic set is the surface Σ , see Figure 12.1. On the other hand, it is not difficult to verify that Dλi (U) · ri (U) = 1,

i = 1, 2, 3, 4,

Dλ5 (U) · r5 (U) = 0,

(12.2.19)

so that the first, second, third, fourth characteristic fields (λi (U), ri (U)), i = 1, 2, 3, 4, are genuinely nonlinear, while the fifth characteristic field (λ5 (U), r5 (U)) is linearly degenerate.

12.2.3 Rarefaction waves Next, let us look for rarefaction waves of the system (12.2.13). These waves are the continuous piecewise-smooth self-similar solutions of (12.2.12) associated with nonlinear characteristic fields, which have the form U(x,t) = V (ξ ),

x ξ= , t

t > 0, x ∈ R I.

Substituting this into (12.2.13), we can see that rarefaction waves are solutions of the following initial-value problem for ordinary differential equations dV (ξ ) = ri (V (ξ )), dξ V (λi (U0 )) = U0 .

ξ ≥ λi (U0 ),

i = 1, 2, 3, 4,

(12.2.20)

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12 Baer-Nunziato model of two-phase flows

Thus, the integral curve of the first characteristic field is given by −2 pg (ρg ) d ρg (ξ ) = ρg (ξ ) < 0, dξ pg (ρg )ρg + 2pg (ρg ) 2 pg (ρg ) dug (ξ ) = pg (ξ ) > 0, dξ pg (ρg )ρg + 2pg (ρg ) d ρs (ξ ) dus (ξ ) d αg (ξ ) = = = 0. dξ dξ dξ

(12.2.21)

This implies that ρs , us , αg are constant through 1-rarefaction waves, ρg is strictly decreasing with respect to ξ , and ug is strictly increasing with respect to ξ . Moreover, since ρg is strictly monotone though 1-rarefaction waves, we can use ρg as a parameter of the integral curve pg (ρg ) − dug = . (12.2.22) d ρg ρg The integral curve (12.2.22) determines in the gas phase the forward curve of 1-rarefaction wave R1F (U0 ) consisting of all right-hand states that can be connected to the left-hand state U0 using 1-rarefaction waves ρg pg (y) R1F (U0 ) : ug = ω1 ((ρg,0 , ug,0 ); ρg ) := ug,0 − dy, ρg ≤ ρg,0 , y ρg,0 (12.2.23) where ρg ≤ ρg,0 follows from the condition that the characteristic speed must be increasing through a rarefaction fan. Similarly, ρs , us , αg are constant through 2rarefaction waves. The backward curve of 2-rarefaction wave R2B (U0 ) consisting of all left-hand states thatcanbeconnectedtotheright-handstateU0 using2-rarefaction waves is given by ρg pg (y) B R2 (U0 ) : ug = ω2 ((ρg,0 , ug,0 ); ρg ) := ug,0 + dy, ρg ≤ ρg,0 . y ρg,0 (12.2.24) In the same way, ρg , ug , αg are constant through 3- and 4-rarefaction waves. In the solid phase, the forward curve of 3-rarefaction wave R3F (U0 ) consisting of all right-hand states that can be connected to the left-hand state U0 using 3-rarefaction waves is given by ρs ps (y) F dy, ρs ≤ ρs,0 . R3 (U0 ) : us = ω3 ((ρs,0 , us,0 ); ρs ) := us,0 − y ρs,0 (12.2.25) The backward curve of 4-rarefaction wave R4B (U0 ) consisting of all left-hand states that can be connected to the right-hand state U0 using 4-rarefaction waves is given by

12.3 Shock waves and solid contacts

R4B (U0 ) :

369

us = ω4 ((ρs,0 , us,0 ); ρs ) := us,0 +

ρs ρs,0

ps (y) dy, y

ρs ≤ ρs,0 . (12.2.26)

12.3 Shock waves and solid contacts 12.3.1 Shock waves Let us consider a shock wave of the form

U0 , x < λ t, U(x,t) = U1 , x > λ t,

(12.3.1)

where U0 ,U1 are the left-hand and right-hand states, respectively, and λ is the shock speed. This shock is a weak solution so that it satisfies the generalized RankineHugoniot relations for a given family of Lipschitz paths. We will show that these generalized Rankine-Hugoniot relations will be reduced to the usual ones. Let us first consider the conservative equations in the solid phase of (12.2.12), which are the equation of conservation of mass and the compaction dynamics equation. Since these equations are of conservative form, the generalized RankineHugoniot relations corresponding to any family of Lipschitz path must coincide with the usual ones. This means that the following jump relations hold −σ [αs ρs ] + [αs ρs us ] = 0,

−σ [ρs ] + [ρs us ] = 0,

where σ is the shock speed, [A] = A+ − A− , and A± denote the values on the right and left of the jump on the quantity A. The last equations imply that [αs ρs (us − σ )] = 0,

[ρs (us − σ )] = 0,

or

ρs (us − σ ) = M = constant,

M[αs ] = 0.

(12.3.2)

The second equation of (12.3.2) implies that either M = 0 or [αs ] = 0. Since ρs > 0, one obtains the following conclusion: across any discontinuity (12.3.1) of (12.2.12) - either [αs ] = 0,

or us = σ = constant.

(12.3.3)

It is derived from (12.3.3) that if [αs ] = 0, then the volume fractions remain constant across the discontinuity. The volume fractions can therefore be eliminated from the governing equations. So, the system (12.2.12) is reduced to the two independent sets of isentropic gas dynamics equations in both phases

370

12 Baer-Nunziato model of two-phase flows

∂t ρg + ∂x (ρg ug ) = 0, ∂t (ρg ug ) + ∂x (ρg u2g + pg ) = 0, ∂t ρs + ∂x (ρs us ) = 0, ∂t (ρs us ) + ∂x (ρs u2s + ps ) = 0, x ∈ R I ,t > 0.

(12.3.4)

The Rankine-Hugoniot relations for the shock read − λ [ρg ] + [ρg ug ] = 0, − λ [ρg ug ] + [ρg u2g + pg ] = 0, − λ [ρs ] + [ρs us ] = 0, − λ [ρs us ] + [ρs u2s + ps ] = 0. The first two equations of the above Rankine-Hugoniot relations determine the Hugoniot curves in the gas phase, while the last two equations determine the Hugoniot curves in the solid phase. These curves behave like the ones in the isentropic fluids. Across a 1-shock wave or a 2-shock wave, the solid density, solid velocity and the volume fractions are constant; across a 3-shock wave or a 4-shock wave, the gas density, gas velocity, and volume fractions are constant. As usual, assume that the fluids in the two phases are isentropic and ideal. Then, we can use Lax’s shock inequalities to select admissible shock waves. Given a left-hand state U0 , let us denote by SiF (U0 ), i = 1, 3 the forward shock curves consisting of all right-hand states U that can be connected to the left-hand state U0 by an i-Lax shock, i = 1, 3, and by S jB (U0 ), j = 2, 4 the backward shock curves consisting of all left-hand states U that can be connected to the right-hand state U0 by a j-Lax shock, j = 2, 4. These curves are given by (pg − pg,0 )(ρg − ρg,0 ) 1/2 , ρg > ρg,0 , ρg,0 ρg 1/2 (pg − pg,0 )(ρg − ρg,0 ) ug = ω2 ((ρg,0 , ug,0 ); ρg ) := ug,0 + , ρg > ρg,0 , ρg,0 ρg 1/2 (ps − ps,0 )(ρs − ρs,0 ) us = ω3 ((ρs,0 , us,0 ); ρs ) := us,0 − , ρs > ρs,0 , ρs,0 ρs 1/2 (ps − ps,0 )(ρs − ρs,0 ) us = ω4 ((ρs,0 , us,0 ); ρs ) := us,0 + , ρs > ρs,0 . ρs,0 ρs

S1F (U0 ) : S2B (U0 ) : S3F (U0 ) : S4B (U0 ) :

ug = ω1 ((ρg,0 , ug,0 ); ρg ) := ug,0 −

(12.3.5) Next, we can define the forward wave curves issuing from U0 by W1F (U0 ) = R1F (U0 ) ∪ S1F (U0 ),

W3F (U0 ) = R3F (U0 ) ∪ S3F (U0 ). (12.3.6) The backward wave curves issuing from U0 are defined by W2B (U0 ) = R2B (U0 ) ∪ S2B (U0 ),

W4B (U0 ) = R4B (U0 ) ∪ S4B (U0 ). (12.3.7) These curves are parameterized in such a way that the velocity is given as a function of the density in each phase, under the form u = ωi (U0 ; ρ ), ρ > 0, i = 1, 2, 3, 4. It is

12.3 Shock waves and solid contacts

371

not difficult to check that ω1 , ω3 are strictly decreasing; and that ω2 , ω4 are strictly increasing. From the above argument, we get the following result. Lemma 12.3.1. Across a 1- or a 2-wave, the solid density, solid velocity and the volume fractions are constant; across a 3-wave or a 4-wave, the gas density, gas velocity, and volume fractions are constant.

12.3.2 Solid contact waves Let us now consider the second of (12.3.3) where [αs ] = 0. We will show that this is the case of a contact discontinuity associated with the fifth characteristic field, which will be called a solid contact. Theorem 12.3.1. Let U be a contact discontinuity of the form (12.3.1) associated with the linearly degenerate characteristic field (λ5 , r5 ), that is, [αs ] = 0 and U± belong to the same trajectory of the integral field of the 5fth characteristic field. Then, U is a weak solution of (12.2.12) in the sense of nonconservative products and independent of paths and the solid velocity remain constant across this solid contact wave (12.3.8) us,0 = us± = σ . Moreover, the following jump relations in the usual form hold across the solid contact: [αg ρg (ug − us )] = 0, (ug − us )2 + 2hg = 0, (12.3.9) [mug + αg pg + αs ps ] = 0, where m is a constant given by m = αg ρg (ug − us ).

(12.3.10)

Proof. Consider the integral curve corresponding to the 5fth family passing through U− dV (ξ ) = r5 (V (ξ )), ξ ≥ ξ0 := λ5 (U− ), (12.3.11) dξ V (ξ0 ) = U− . A (unique) solution V = V (ξ ) of (12.3.11) always exists on a certain interval [ξ0 , ξ1 ], where ξ1 > ξ0 . Let U+ = V (ξ1 ). Since [αs ] = 0, it follows from (12.3.2) that the equation of (12.3.8) holds. Moreover, it is derived from (12.2.19) that d λ5 (V (ξ )) dV (ξ ) = Dλ5 (V (ξ )) · = Dλ5 (V (ξ )) · r5 (V (ξ )) ≡ 0. dξ dξ

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12 Baer-Nunziato model of two-phase flows

This means that the 5fth characteristic speed remains constant through trajectory of I be any smooth function. We define a function (12.3.11). Next, let η : [ξ0 , ξ1 ] → R U(x,t) = V (η (x − σ t)) = W (x − σ t).

(12.3.12)

Then U is the classical solution of (12.2.13) with the smooth initial data U(x, 0) = V ◦ η (x) = W (x),

x ∈ [ξ0 , ξ1 ].

Actually, it holds that, for ξ = x − σ t Ut + A(U)Ux = −Vξ (ξ )η σ + A(V (ξ ))Vξ (ξ )η (ξ ) = (−I σ + A(V (ξ )))Vξ (ξ )η (ξ ) = (−I λ5 (V (ξ )) + A(V (ξ )))r5 (V (ξ ))η (ξ ) = 0, where the last equation is deduced from the fact that λ5 remains constant across the trajectory (12.3.11). For such a smooth solution U as in (12.3.12), the system (12.2.12) becomes −σ (αg ρg ) + (αg ρg ug ) = 0, −σ (αg ρg ug ) + (αg (ρg u2g + pg )) = pg αg , −σ (αs ρs ) + (αs ρs us ) = 0, −σ (αs ρs us ) + (αs (ρs u2s + ps )) = −pg αg , −σ ρs + (ρs us ) = 0.

(12.3.13)

The first equation of (12.3.13) can be re-written in the divergence form as (αg ρg (ug − σ )) = 0,

(12.3.14)

(αg ρg (ug − σ )) ≡ m = constant.

(12.3.15)

which implies The second equation of (12.3.13) is expressed as (αg ρg ug (ug − σ )) + αg pg + pg αg = pg αg . Using (12.3.14) and (12.3.15), and canceling the terms, we get from the last equation (mug ) + αg pg = 0 or, since m is constant,

mug + αg pg = 0.

Using (12.3.15) in a reverse way, we obtain from the last equation (αg ρg (ug − σ ))ug + αg pg = 0. Canceling αg > 0 from the last equation to get

12.3 Shock waves and solid contacts

373

ρg (ug − σ )ug + pg = 0, or

(ug − σ )ug + pg /ρg = 0.

The last equation yields the following equation in divergence form:

(ug − σ )2 + 2hg = 0,

hg = pg /ρg .

(12.3.16)

The third and the fifth equations of (12.3.13) are trivial, since σ = us ≡ constant, we discard them. Consider the fourth equation of (12.3.13). Adding up the second equation to the fourth equation of (12.3.13) we obtain the conservation of momentum of the mixture −σ (αg ρg ug ) + (αg (ρg u2g + pg )) − σ (αs ρs us ) + (αs (ρs u2s + ps )) = 0. The last equation can be simplified using (12.3.14) and that us = σ as follows: (αg ρg ug (ug − σ )) + (αg pg ) + (αs ρs (us − σ )) + (αs ps ) = 0, or

(mug + αg pg + αs ps ) = 0.

(12.3.17)

We have established that the system (12.2.12) for the traveling wave solution is reduced to the system (12.3.14)-(12.3.17): (αg ρg (ug − σ )) = 0, (ug − σ )2 + 2hg = 0, (mug + αg pg + αs ps ) = 0,

(12.3.18)

where m is given by (12.3.15). Since (12.3.18) has a divergence form, it is independent of paths, if U is considered as a weak solution in the sense of nonconservative products. Now, let ηε be a sequence of smooth functions such that ξ0 , for x < 0, ηε (x) → (12.3.19) as ε → 0. ξ1 , for x > 0, Then, it holds that Wε (x) = V ◦ ηε (x) → W0 (x) =

V (ξ0 ) = U− , V (ξ1 ) = U+ ,

for x < 0, for x > 0.

(12.3.20)

As seen in the above argument, we obtain the sequence of corresponding smooth solutions Uε (x,t) = Wε (x − σ t) = V ◦ ηε (x − σ t) of (12.2.12) that satisfy the equations (12.3.18). Let U still stand for Uε (x,t) in (12.3.18). Then, passing to the limit of (12.3.18) for U = Uε (x,t) as ε → 0, we obtain the jump relations (12.3.10). Due to the stability result of weak solutions in the sense of nonconservative products,

374

12 Baer-Nunziato model of two-phase flows

the limit function, where the jump relations (12.3.10) hold, is a weak solution of the system (12.2.12) and it is independent of paths. In the sequel, we fix one state U0 and look for any state U that can be connected with U0 by a contact discontinuity. As seen above, the state U satisfies the equations

αg ρg (ug − us ) = αg,0 ρg,0 (ug,0 − us ) := m, (ug − us )2 + 2hg = (ug,0 − us,0 )2 + 2hg,0 , and ps =

αs,0 ps,0 − [mug + αg pg ] . αs

(12.3.21)

(12.3.22)

The quantities in the gas phase can be calculated using (12.3.21). Then, the solid pressure is given by (12.3.22), and then the calculations of other quantities in the solid phase follow: Definition 12.3.1 (Elementary waves). The elementary waves of the system (12.2.12) are • i-Lax shock waves i = 1, 2, 3, 4; • i-rarefaction waves, i = 1, 2, 3, 4; • Solid contact waves.

12.4 Riemann problem based on a phase decomposition Notation We use the following notation: (i) Wi (u1 , u2 ): An i-wave connecting a state u1 and a state u2 , where W = S: Lax shock, W = R: rarefaction wave, W2 = Z, i = 1, 2, 3; (ii) Wi (u1 , u2 ) W j (u2 , u3 ): an i wave from the left-hand state u1 to the right-hand state u2 is followed by a j-wave from the left-hand state u2 to the right-hand state u3 .

12.4.1 Phase decomposition Recall from Lemma 12.3.1 that (i) Through i-Lax shock waves and i-rarefaction waves, i = 1, 2, the quantities in the solid phase ρs , us and αs remain constant; (ii) Through j-Lax shock waves and j-rarefaction waves, j = 3, 4, the quantities in the gas phase ρg , ug and αg remain constant;

12.4 Riemann problem based on a phase decomposition

375

(iii) Through the solid contact (the 5-wave), the solid velocity us remains constant. Precisely, quantities in the solid phase involve only in the 3rd, 4th, and 5fth characteristic fields, where λ3 (U) < λ5 (U) < λ4 (U), for all U in the phase domain. In the solid phase, the Riemann solution is always a 3-wave from VL to V− , followed by a solid contact from V− to V+ , and then the solution arrives at VR by a 4-wave; see Figure 12.2

Fig. 12.2 The construction of Riemann solutions in the solid phase

One has in the solid phase V− = (ρs− , us− ) ∈ W3F (UL ),

V+ = (ρs+ , us+ ) ∈ W4B (UR ).

(12.4.1)

Therefore, since us remains constant across the 5-contact, it holds that us− = ω3 ((ρsL , usL ); ρs− ) = us , us+ = ω4 ((ρsR , usR ); ρs+ ) = us ,

(12.4.2)

where ω3 , ω4 are given by (12.2.25)-(12.2.26) and (12.3.5). Since the solid velocity us remains constant across the solid contact associated with λ5 = us , the construction of Riemann solutions in the gas component can be performed on the hyperplane us = us,0 , where us,0 is the value of the solid velocity on either side of the solid contact of the Riemann solution. This can be done when we can determine the value us,0 . In general, it is determined from the whole set of algebraic equations for all states of the Riemann solutions. The construction of Riemann solutions will then be relied on the location of the solid contact when it belongs to the supersonic region or subsonic region.

376

12 Baer-Nunziato model of two-phase flows

12.4.2 Solutions containing a supersonic solid contact We want to find a Riemann solution containing the solid contact in the supersonic region, Ω1 for example. This suggests that the solution in the gas phase should start with a solid contact, and is followed by the two waves in the gas phase, see Figure 12.3. For simplicity, we can denote P = (ρg , ug ) the state in the gas phase and V = (ρs , us ) the state in the solid phase, and U− ,U+ the states on the left-hand and right-hand state of the solid contact in the Riemann solution.

Fig. 12.3 Case A: A structure of Riemann solutions when the solid contact is in the supersonic region

Let us find the equations for the states of this Riemann solution as follows. First, we will find the two states on both sides of the solid contact. Since the solution starts with the solid contact in the gas phase, (ρg , ug )− = (ρg , ug )L . Moreover, since the solid velocity remains constant across the solid contact, us,± = us,0 . Thus, we are reduced to find five quantities ρs,− , us,0 , ρg,+ , ug,+ and ρs,+ . This requires five algebraic equations. Now, the jump relations across the solid contact yields three equations

αg,R ρg,+ (ug,+ − us,0 ) = αg,L ρg,L (ug,L − us,0 ) := m (ug,+ − us,0 )2 + 2hg (ρg,+ ) = (ug,L − us,0 )2 + 2hg,L

(12.4.3)

αs,R ps,+ − αs,L ps,− + m(ug,+ − ug,L ) + (αg,R pg,+ − αg,L pg,L ) = 0. In addition, the two equations in (12.4.3) become us,− = ω3 ((ρs,L , us,L ); ρs,− ) = us,0 , us,+ = ω4 ((ρs,R , us,R ); ρs,+ ) = us,0 , where ω3 , ω4 are given by (12.2.25)-(12.2.26) and (12.3.5).

(12.4.4)

12.4 Riemann problem based on a phase decomposition

377

It has been shown that the equations (12.4.3)-(12.4.4) always admits one solution, provided that |αR − αL | is small enough, see [307]. Thus, the two states U− and U+ on both sides of the solid contact is determined. In particular, the value us,0 of the solid velocity of the solid contact is known. Next, we will describe the Riemann solution in each phase as follows. In the gas phase, the solution starts with a solid contact from PL = (ρg , ug )L to P+ = (ρg , ug )+ , followed by a 1-wave from P+ = (ρg , ug )+ to P1 = (ρg , ug )1 , where P1 is the intersection point of W1F (U+ ) and the back wave curve of 2-waves W2B (UR ) on the hyperplane us = us,0 , and finally the solution arrives at PR = (ρg , ug )R by a 2-wave. That is, the solution in the gas phase has the form W5 (PL , P+ ) W1 (P+ , P1 ) W2 (P1 , PR ),

(12.4.5)

see Figure 12.4.

Fig. 12.4 Case A: The construction of Riemann solutions in the gas phase

In the solid phase, the solution begins with a 3-wave from VL to V− , followed by a solid contact from V− to V+ , and then the solution arrives at VR by a 4-wave. That is, the solution is the solid phase has the form W3 (VL ,V− ) W5 (V− ,V+ ) W4 (V+ ,VR ).

(12.4.6)

Example 12.4.1. Consider the Riemann problem for the two-phase flow model (12.2.12), where the Riemann data are given by Table 12.1. The parameters of the equations of state are chosen to be

γg = 1.4,

κg = 0.4,

γs = 1.6,

κs = 1.0.

(12.4.7)

The exact Riemann solution begins by a 1-shock wave from the left-hand state UL to some state U1 , followed by a 2-rarefaction wave from U1 to a state U2 , then followed by a 3-shock wave from U2 to U3 . The solution is continued with a solid contact from U3 to U4 . And finally it arrives at UR by a 4-shock wave from U4 . These states are

378

12 Baer-Nunziato model of two-phase flows

given in Table 12.1. Observe that the solid contact between U3 and U4 are located in the supersonic region and has a negative velocity (us = −0.2). UL U1 U2 U3 U4 UR

ag 0.5 0.5 0.5 0.5 0.45 0.45

pg 0.0233694796177 0.0319391283772 0.0741361020681 0.0741361020681 0.0865232682942 0.0865232682942

ug -3.21917689263 -3.33333333333 -3 -3 -2.98602293711 -2.98602293711

ps 0.230831984945 0.230831984945 0.230831984945 0.329876977693 0.307166246473 0.214940111545

us 0.0225365057109 0.0225365057109 0.0225365057109 -0.2 -0.2 -0.419580006017

Table 12.1 Riemann data and the states separating elementary waves of the Riemann solution (12.4.5)-(12.4.6)

The exact solution is plotted in Figure 12.5.

Fig. 12.5 Case A: An example of a Riemann solution corresponding to the initial data 12.1 at the time t = 0.2

12.4 Riemann problem based on a phase decomposition

379

12.4.3 Solutions containing a subsonic solid contact Next, we will construct Riemann solutions which contain a solid contact in the subsonic region. As before, we denote by U = (ρg , ug ) the state in the gas phase, by V = (ρs , us ) the state in the solid phase, and by (U− ,V− ), (U+ ,V+ ) the states on the left-hand and right-hand state of the solid contact in the Riemann solution, see Figure 12.6. Since the solid in the subsonic region, we may find a solution such that in the gas phase it begins with a 1-wave from UL to U− , followed by a solid contact from U− to U+ , and it then arrives at UR by a 2-wave. Note that in the solid phase, the solution also has the form (12.4.6). Let us now find states of this Riemann solutions.

Fig. 12.6 Case B: A structure of Riemann solutions when the solid contact is in the subsonic region

First, the relations between the gas velocities ug,± and the gas densities ρg,± give us two equations ug,− = ω1 (UL ; ρg,− ),

ug,+ = ω2 (UR ; ρg,+ ),

(12.4.8)

since U− ∈ W1F (UL ), and U+ ∈ W2B (UR ). Second, the relations between the solid velocities us,± and the solid densities ρs,± give us two equations us,0 = ω3 (VL ; ρs,− ),

us,0 = ω4 (VR ; ρs,+ ),

(12.4.9)

since V− ∈ W3F (VL ), and V+ ∈ W4B (VR ), and the solid velocity remains constant across the solid contact. Third, the jump relations across the solid contact read

380

12 Baer-Nunziato model of two-phase flows

αg,R ρg,+ (ug,+ − us,0 ) = αg,L ρg,− (ug,− − us,0 ) := n (ug,+ − us,0 )2 + 2hg (ρg,+ ) = (ug,− − us,0 )2 + 2hg,− αs,R ps,+ − αs,L ps,− + n(ug,+ − ug,− ) + (αg,R pg,+ − αg,L pg,− ) = 0.

(12.4.10)

It can be shown that the seven equations (12.4.8)-(12.4.10) determine the states U± ,V± . Thus, the solution in the gas phase has the form W1 (UL ,U− ) W5 (U− ,U+ ) W2 (U+ ,UR ),

(12.4.11)

see Figure 12.7. The solution in the solid phase has the form

Fig. 12.7 Case B: The construction of Riemann solutions in the gas phase

W3 (VL ,V −) W5 (V− ,V+ ) W4 (V+ ,VR ).

(12.4.12)

Example 12.4.2. Consider the Riemann problem for the two-phase flow model (12.2.12). The parameters of the equations of state are also chosen to be

γg = 1.4, γs = 1.6,

κg = 0.4 κs = 1.

(12.4.13)

In this case, the solution is a 1-shock from UL to U1 , followed by a 3-shock from U1 to U2 = (U− ,V− ), and then by the following waves in the order from left-toright: a 5-contact wave from U2 to U3 = (P+ ,V+ ), a 2-rarefaction wave from U3 to U4 = (PR ,V+ ), a 4-shock wave from U4 to UR . These states are given in Table 12.2. The exact solution is plotted in Figure 12.8. Remark 12.4.1. The above two constructions of Riemann solutions are typical. Other types of Riemann solutions also exist, but the argument is expected to be more involved. A complete set of Riemann solutions for two-phase flow models is still an open question.

12.5 Bibliographical notes UL U1 U2 U3 U4 UR

ag 0.5 0.5 0.5 0.52 0.52 0.52

pg 0.772374571078 1.05560632862 1.05560632862 1.05878605628 1.57258218911 1.57258218911

381 ug 1.18815942627 1 1 1.00810500996 1.25810500996 1.25810500996

ps 0.844866354024 0.844866354024 5.7995461348 5.9971426759 5.9971426759 4.19651094647

us 3.16306898236 3.16306898236 1.2 1.2 1.2 0.816668212236

Table 12.2 Riemann data and the states separating elementary waves of the Riemann solution (12.4.11)-(12.4.12)

Fig. 12.8 Case B: An example of a Riemann solution corresponding to the initial data 12.2 at the time t = 0.2

12.5 Bibliographical notes We list here only a short selection of the most relevant papers for this chapter. We also refer the reader to the bibliography at the end of this monograph for additional references. Multi-phase flow models are derived in the book by Drew-Passman [119]. The Baer-Nunziato model for the study of deflagration-todetonation transition in granular materials was first proposed by Baer-Nunziato in [33]. The compaction dynamics equation of the Baer-Nunziato model was modified by Bzil-Menikoff-Son-Kapila-Steward [73]. For the numerical modeling of twophase flows, we refer to Ransom-Hicks [265], Stewart-Wendroff [296], AbgrallSaurel [278], Gallouet-H´erard-Seguin [136], and Taherzadeh-Saidi [300]. A two-

382

12 Baer-Nunziato model of two-phase flows

phase model with logistic growth was studied by Ekrut-Cogan [120]. A two-fluid model for avalanche and debris flows was presented in Pitman-Le [262]. A twophase shallow debris flow model with energy balance is considered by BouchutFernandez Nieto-Mangeney-Narbona Reina [61]. A method for the unsteady coupling of two-phase flow models is considered by Ambroso-Herard-Hurisse [17]. Phase field models are formulated for interfacial dynamics in [56]. The Riemann problem for two-phase flow models was considered by AbadpourPanfilov [1], Chalons [84], Thanh [305], and Thanh-Cuong [307], Toro-Castro [315]. Riemann solutions of the Baer-Nunziato model of two-phase flows were also studied by Andrianov-Warnecke [21] and by Schwendeman-Wahle-Kapila [283]. Numerical solutions of the Riemann problem for a model of two-phase flows were given by Rosatti-Zugliani [276]. The Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition was investigated by Hantke-Dreyer-Warnecke [149]. The Riemann problem and numerical modeling for two-phase flow models were studied by Zeidan [331]. A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension was studied by Rohde-Zeiler [274]. A global existence result of weak solutions for a viscous two-phase model was established by Evje-Karlsen [122, 123].

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Index

E Entropy, 23, 72, 74, 78 Euler equations, 22, 27, 32, 46, 87, 116

N Nonconservative system, 195, 206, 278, 286, 328 S Shallow water equations, 34, 47, 285, 327

K Kinetic function, 155, 184, 191

W Weak solutions, 60, 63, 71

© Springer Nature Switzerland AG 2023 P. G. LeFloch and M. D. Thanh, The Riemann Problem in Continuum Physics, Applied Mathematical Sciences 219, https://doi.org/10.1007/978-3-031-42525-7

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