Discrete mechanics : concepts and applications 9781119482826, 1119482828, 9781119575146, 1119575141

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Discrete Mechanics

Series Editor Roger Prud’homme

Discrete Mechanics Concepts and Applications

Jean-Paul Caltagirone

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Jean-Paul Caltagirone to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2018962477 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-283-0

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

List of Symbols

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xxi

Chapter 1. Fundamental Principles of Discrete Mechanics . . . . . .

1

1.1. Definitions of discrete mechanics . . . . . . 1.1.1. Notion of discrete space–time . . . . . . 1.1.2. Notion of a discrete medium . . . . . . . 1.2. Properties of discrete operators . . . . . . . . 1.3. Invariance under translation and rotation . . 1.4. Weak equivalence principle . . . . . . . . . . 1.5. Principle of accumulation of stresses . . . . 1.6. Duality-of-action principle . . . . . . . . . . 1.7. Physical characteristics of a medium . . . . 1.8. Composition of velocities and accelerations 1.9. Discrete curvature . . . . . . . . . . . . . . . 1.10. Axioms of discrete mechanics . . . . . . .

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1 1 4 6 9 11 13 14 16 20 23 28

Chapter 2. Conservation of Acceleration . . . . . . . . . . . . . . . . . .

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2.1. General principles . . . . . . . . . . . . . . 2.2. Continuous memory . . . . . . . . . . . . . 2.3. Modeling the compression stress . . . . . . 2.3.1. Compression experiment . . . . . . . . 2.3.2. Modeling the stress in a solid . . . . . . 2.3.3. Modeling the stress in a fluid . . . . . . 2.3.4. Compression with small time constants

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2.3.5. Modeling the accumulation of the normal stress . . . . . . . 2.3.6. The energy formula, e = m c2 . . . . . . . . . . . . . . . . 2.4. Modeling the rotation stress . . . . . . . . . . . . . . . . . . . . 2.4.1. Couette’s experiment . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Behavior over time . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Rotation stress in solids . . . . . . . . . . . . . . . . . . . . . 2.4.4. Rotation stress in fluids . . . . . . . . . . . . . . . . . . . . . 2.4.5. Stresses in a porous medium, Darcy’s law . . . . . . . . . . 2.4.6. Modeling the accumulation of the rotation stress . . . . . . . 2.4.7. Rotation in Couette and Poiseuille flows . . . . . . . . . . . 2.5. Modeling other effects . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Gravitational effects . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Inertial effects . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Discrete equations of motion . . . . . . . . . . . . . . . . . . . . 2.6.1. Geometric description . . . . . . . . . . . . . . . . . . . . . . 2.6.2. Derivation of the equations of motion . . . . . . . . . . . . . 2.6.3. Dissipation of energy . . . . . . . . . . . . . . . . . . . . . . 2.7. Coupling conditions . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Formulation of the equations of motion at a discontinuity . . . . 2.9. Other forms of the equations of motion . . . . . . . . . . . . . . 2.9.1. Curl and vector potential formulation . . . . . . . . . . . . . 2.9.2. Conservative form of the equations of motion . . . . . . . . 2.10. Incompressible models derived from the discrete formulation . 2.10.1. Kinematic projection methods . . . . . . . . . . . . . . . . 2.10.2. Incompressibility in discrete mechanics . . . . . . . . . . . 2.11. Consequences on the dynamics of the vorticity . . . . . . . . .

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42 43 44 44 45 46 47 47 48 49 49 50 52 56 56 58 60 63 65 66 67 69 70 70 74 74

Chapter 3. Conservation of Mass, Flux and Energy . . . . . . . . . . .

77

3.1. Conservation of mass in a homogeneous medium . . . . 3.1.1. In continuum mechanics . . . . . . . . . . . . . . . . 3.1.2. In discrete mechanics . . . . . . . . . . . . . . . . . . 3.2. Transport within multicomponent mixtures . . . . . . . . 3.2.1. Classical approach . . . . . . . . . . . . . . . . . . . . 3.2.2. Discrete model for the transport of chemical species 3.2.3. Equilibrium in a binary mixture . . . . . . . . . . . . 3.3. Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Conservation of flux . . . . . . . . . . . . . . . . . . . . . 3.4.1. General remarks . . . . . . . . . . . . . . . . . . . . . 3.4.2. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Conservation of energy . . . . . . . . . . . . . . . . . . . 3.5.1. Conservation of total energy . . . . . . . . . . . . . . 3.5.2. Conservation of kinetic energy . . . . . . . . . . . . . 3.5.3. Conservation of internal energy . . . . . . . . . . . .

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77 78 80 81 81 84 86 88 89 89 90 93 93 94 95

Contents

3.5.4. Monotonically decreasing kinetic energy 3.6. A complete system of equations . . . . . . . 3.7. A simple heat conduction problem . . . . . . 3.7.1. Case of anisotropic materials . . . . . . . 3.8. Phase change . . . . . . . . . . . . . . . . . . 3.8.1. The Stefan problem . . . . . . . . . . . . 3.8.2. Condensation . . . . . . . . . . . . . . . .

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Chapter 4. Properties of the Discrete Formulation

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Chapter 5. Two-Phase Flows, Capillarity and Wetting . . . . .

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97 98 99 101 102 103 108

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4.1. Fundamental properties . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Limitations on the velocity . . . . . . . . . . . . . . . . . . 4.1.2. Inverting the formulas Vφ = ∇φ and Vψ = ∇ × ψ . . . 4.1.3. Material frame-indifference . . . . . . . . . . . . . . . . . 4.1.4. Fundamental invariants . . . . . . . . . . . . . . . . . . . . 4.2. System of equations . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Differences from continuum mechanics . . . . . . . . . . . . . 4.3.1. Differences from the Navier-Lamé equations . . . . . . . . 4.3.2. Differences from the Navier-Stokes equations . . . . . . . 4.3.3. Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Compatibility conditions for the Navier-Stokes equations . 4.4. Examples of analytic solutions of the equations of motion . . 4.4.1. Rigid rotational motion . . . . . . . . . . . . . . . . . . . . 4.4.2. Planar Couette flow . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4. Radial flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Incompressible motion . . . . . . . . . . . . . . . . . . . . . . 4.5.1. The Green-Taylor vortex . . . . . . . . . . . . . . . . . . . 4.5.2. Lid-driven cavity . . . . . . . . . . . . . . . . . . . . . . . 4.6. Compressible fluids and perfect fluids . . . . . . . . . . . . . . 4.6.1. Generalized Bernoulli equation . . . . . . . . . . . . . . . 4.6.2. Propagation of linear waves . . . . . . . . . . . . . . . . . 4.6.3. Sod shock tube . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Statics of fluids and solids . . . . . . . . . . . . . . . . . . . . 4.8. Conditions for modeling a rigid solid . . . . . . . . . . . . . . 4.9. Flows in a porous medium . . . . . . . . . . . . . . . . . . . . 4.10. Stretching of space-time and Hugoniot’s theorem . . . . . .

5.1. Formulation of the equations at the interfaces 5.1.1. Modeling the curvature . . . . . . . . . . . 5.1.2. Formulation of the equations of motion . . 5.2. Two-phase flows . . . . . . . . . . . . . . . . . 5.2.1. Two-phase Poiseuille flow . . . . . . . . .

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vii

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5.2.2. Sloshing of two immiscible fluids 5.3. Capillarity-dominated flows . . . . . 5.3.1. The Laplace problem . . . . . . . 5.3.2. Oscillating ellipse . . . . . . . . . 5.3.3. Marangoni-type flow in a droplet 5.3.4. Interacting bubbles . . . . . . . . 5.3.5. Simulating foam in equilibrium . 5.4. Partial wetting . . . . . . . . . . . . . 5.4.1. Droplet in equilibrium on a plane 5.4.2. Spreading of a droplet . . . . . . . 5.4.3. Droplet acted upon by gravity . . 5.4.4. Flows within a lens . . . . . . . . 5.4.5. Capillary ascension in a tube . . .

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181 187 187 188 190 192 194 195 198 200 204 205 206

Chapter 6. Stresses and Strains in Solids . . . . . . . . . . . . . . . . . 209 6.1. Discrete solid medium . . . . . . . . . . . . . . . . . . . . . . 6.2. Stresses in solids . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Discrete equations . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Material frame-indifference . . . . . . . . . . . . . . . . . 6.2.3. Solid statics equations . . . . . . . . . . . . . . . . . . . . . 6.2.4. Calculating the displacement . . . . . . . . . . . . . . . . . 6.3. Properties of solid media . . . . . . . . . . . . . . . . . . . . . 6.3.1. In continuum mechanics . . . . . . . . . . . . . . . . . . . 6.3.2. In discrete mechanics . . . . . . . . . . . . . . . . . . . . . 6.4. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 6.5. Rigid motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Validation of the model on examples . . . . . . . . . . . . . . 6.6.1. Simple example of a monolithic fluid–structure interaction 6.6.2. Mechanical equilibrium of sloshing . . . . . . . . . . . . . 6.6.3. Beam under extension . . . . . . . . . . . . . . . . . . . . . 6.6.4. Multimaterial compression . . . . . . . . . . . . . . . . . . 6.6.5. Planar shearing . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.6. Flexing beam . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.7. Settling of a block under gravity . . . . . . . . . . . . . . . 6.6.8. Mechanical equilibrium of a solid object . . . . . . . . . . 6.6.9. Extension to other constitutive laws . . . . . . . . . . . . . 6.7. Toward a unification of solid and fluid mechanics . . . . . . . Chapter 7. Multiphysical Extensions

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209 211 212 214 215 217 219 220 223 225 228 230 230 233 235 237 238 239 240 242 243 246

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7.1. Deflection of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.1.1. Description of the physical phenomenon . . . . . . . . . . . . . . . 250 7.1.2. Deflection of light by the Sun in Newtonian mechanics . . . . . . . 252

Contents

7.1.3. Deflection of light by the Sun according to the duality-of-action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4. Deflection of light by the Sun in a one-dimensional approach . . . 7.2. On a discrete approach to turbulence . . . . . . . . . . . . . . . . . . . 7.2.1. General remarks about the approach . . . . . . . . . . . . . . . . . . 7.2.2. Dynamics of the vorticity in two spatial dimensions . . . . . . . . . 7.2.3. Analysis of a turbulent flow in a planar channel . . . . . . . . . . . 7.2.4. Model of the turbulence in discrete mechanics . . . . . . . . . . . . 7.2.5. Application to a flow in a channel with Reτ = 590 . . . . . . . . . 7.3. The lid-driven cavity problem with Re = 5,000 . . . . . . . . . . . . . 7.4. Natural convection into the non-Boussinesq approximation . . . . . . 7.5. Fluid–structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . References Index

ix

256 257 262 262 264 266 271 272 280 283 286

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Preface

Nature is complex; we must accept it. A concept or a theory might initially develop as a series of reproducible experiments, previously established principles, carefully conducted thought experiments and so on. Any theory must then be subjected to repeated verification, usually of an experimental nature, to validate or disprove its hypotheses, axioms and principles. Most key milestones in the history of science were intuitively understood by their authors before they were formalized in terms of equations; the equations describing a theory can be adjusted over time as the state of mathematics progresses. This was, for example, the case with Newton’s laws of dynamics, which were only rephrased into the modern formulation later, by Euler. Some phenomena continue to defy our efforts to fully understand them, such as turbulence; we are not currently capable of giving an accurate a priori description of the mechanisms underlying the interactions of vortices, or the processes by which they are created and dissipated. In some of the more accessible cases, simulations are our only tool for understanding turbulence at every scale and analyzing the physical behavior of turbulent flows in more depth. Simulation-based approaches rely on the equations of fluid mechanics, which are derived from Newton’s second law. This perspective is expanded into a method: first, an equation is derived, which is validated in simple cases; this equation is then solved to explore the behavior of more complex phenomena. Over time, with enough simulation and exploration, the model is refined and our understanding of the phenomenon improves alongside it. The equation becomes more “intelligent” than its author. But, in mechanics, the laws of dynamics and their modern counterparts – the equations of motion – still remain shrouded in some mystery, with hidden surprises for the attentive observer. The equations of motion are capable of describing the behavior of many problems involving fluids, solids and waves. However, unification has not yet been achieved; for example, the formulations used for solids and fluids do not coincide, even though performing mechanics on an underlying continuum was

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supposed to guarantee precisely this type of consistency. The equations of continuum mechanics hold hidden structural flaws that are unraveled in this book; the origins of each issue vary, but our most important objection will be raised against the concept of a continuum itself. The results themselves offered by continuum mechanics are usually not invalid, given that they are perfectly consistent with experimental observations; however, the theory is afflicted by deeper problems that are effectively covered up by redundancy in its principles and laws. Discrete mechanics abandons the notion of a continuum, the principle of local thermodynamic equilibrium, the state equations and constitutive equations, second-order tensors, differentiation at a point, analysis and so on. Building from the ideas proposed by Galileo, the equivalence principle and the notion of relativity, the discrete approach to mechanics begins by defining a geometric basis of space that cannot be reduced to a single point. The discrete equations of motion are then formally derived from a set of hypotheses, axioms and principles. Jean-Paul C ALTAGIRONE November 2018

Introduction

The laws of physics were established from experimental observations by finding simple relations to describe these observations; for example, the hypothesis of linearity between the flux and the force was established as a general principle of thermodynamics. Once a relation has been chosen, a coefficient of proportionality is selected. This coefficient is typically viewed as an intrinsic physical property, even though it depends directly on the choice of equations. If the law is overly simplistic, inconsistencies will be encountered when we attempt to generalize to parameter intervals larger than those originally considered. Fortunately, the complexity of the observed physical behavior typically allows us to reconstruct enough consistency to cover a broad spectrum of the parameter space. If only the thermodynamic coefficients themselves have any true physical significance, why should we need to introduce an ideal gas law between the variables, other than for convenience? The laws of mechanics are some of the oldest laws in physics; throughout history, Galileo, Issac Newton and Albert Einstein each left a prodigious mark on our understanding of the universe around us. The concepts of force, mass and acceleration – linked together by Newton’s second law – have not changed in modern mechanics. Over the past three centuries, a great succession of physicists, mathematicians and engineers gradually perfected the laws and equations of mechanics, which today seem inexorably set in stone. Leonhard Euler, Maurice Couette, Daniel Bernoulli, Ernst Mach, George Gabriel Stokes, Henri Navier, Clifford Truesdell and many others contributed to formalizing the laws of mechanics within a modern mathematical framework. Much has been written on the epistemology of the connections between the theories of these famous individuals, and occasionally the personal relationships between them. The path taken by scientific thought over time seems natural and logical if we consider the path taken by its equations, from classical mechanics to relativistic mechanics. Sometimes, it is not entirely clear where exactly new ideas came from, especially during periods where publication could not always be taken for granted and religion sometimes

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interfered with the propagation of new theories. Together, this scientific work led to equations of mechanics that have been frozen in time for decades or perhaps even centuries; today, these equations allow us to simulate all kinds of motion of solids, fluids, propagating waves, etc., extremely realistically. Yet, the equations of each type of motion are very different, even though continuum mechanics was supposed to establish a unified representation of all physical phenomena within the same set of equations. How might we have established the laws of mechanics from scratch if we had access to every observation in history? This book attempts to give an answer. We start with the following question: why did Isaac Newton formulate his fundamental laws of dynamics as an equality of forces, given that he was aware of the equivalence between inertial mass and gravitational mass? This equivalence principle is one of the cornerstones of Albert Einstein’s theory of general relativity. Discrete mechanics also relies heavily on this principle, using it to eliminate the concept of mass, which is not required to describe an equality between accelerations in mechanics. The notions of rest mass and relativistic mass at the heart of relativistic mechanics clearly characterize mass as equivalent to energy. Similarly, the fundamental law of dynamics can be formulated as an equality between the acceleration experienced by a particle and the sum of the accelerations applied to this particle. The intrinsic nature of the acceleration is not the same as that of the velocity; we can apply an absolute acceleration even if the absolute velocity of a body is not known. The classical approach to finding the absolute velocity is to consider an inertial frame of reference. We must accept that it is pointless to attempt to understand this concept of absolute velocity, as well as the trains, station masters and elevators considered by the various thought experiments of the last century. It must be abandoned; we cannot detect uniform motion. By contrast, we can now measure acceleration to extremely high accuracy, corroborating ever further that equality between masses truly represents a fundamental principle. Discrete mechanics also abandons the idea of a continuum; the differential calculus and differentials introduced by Gottfried Wilhelm Leibniz and Isaac Newton played an essential role in formulating the laws of mechanics into their modern differential form. To reduce every variable to a point, we are forced to construct an inertial frame of reference so that we can compute derivatives in specific directions of space. If we abandon the continuum, we must therefore also abandon classical differentiation and integration. We can still scale the discrete topology down geometrically to orders of magnitude that preserve compatibility with our macroscopic view of matter. The operations of differentiation and integration are replaced by operators based on discrete differential geometry. Inertial frames of reference are replaced by local frames, in which each point of the domain only perceives an immediate neighborhood of points, edges and faces with known distances and orientations. Every point is connected by causality links defined within

Introduction

xv

space-time. Nature provides plenty of examples of collective behavior: schools of sardines, flocks of starlings, etc. The presence of limits or obstacles is perceived by each object through the behavior of its immediate neighborhood over a period of time that reflects the causality between them. The distinction between the material velocity of the medium and the wave velocity (celerity) can be preserved by viewing the latter as a parameter determined by local conditions, whereas the material velocity itself is simply a secondary variable, defined as a velocity field known only up to uniform motion. The wave velocity is bounded by the speed of light in a vacuum by the axiom proposed by Albert Einstein, but is a function of the local conditions in general. The material velocity itself is not bounded a priori, since none of our axioms directly imply that any such bound exists. Even though the laws of discrete mechanics can be applied to problems derived from cosmology, they will chiefly be applied to areas of classical mechanics where cause-and-effect relations are associated with finite wave velocities. The local equilibrium hypothesis is also set aside; none of our axioms imply that an arbitrary medium is in local equilibrium, and various counterexamples can be found. Hence, state equations become useless; applying these equations generates artifacts and violates conservation equations such as the law of conservation of mass. In continuum mechanics, the state equations are typically used to close the system of equations so that it has one equation for each variable. Today, this approach seems simplistic and reveals a lack of understanding of the degree of autonomy of the equations of motion. Only some of the physical properties are worth knowing at any cost; these properties should influence the solution of course, but only via the relation that exists between the variables in the conservation equations. Including other constitutive equations among the laws of mechanics, such as rheology describing a material’s behavior, is no longer justified. Like any other properties, these equations should simply be known locally and instantaneously, even if they depend on the problem variables. Thus, a drastic distinction is drawn and preserved between properties and conservation equations. Is the tensor formulation of the equations of mechanics strictly necessary? Tensors, introduced by Woldemar Voigt in their modern form and adopted by many other renowned thinkers over the last century, were in particular employed by Albert Einstein to formulate his theory of relativity, with help from Marcel Grossmann. In the past, physicists and engineers have legitimately needed a representation of the properties of certain materials that varies as a function of the direction. But can we justify viewing tensors as an integral part of the laws of mechanics? After all, Maxwell’s equations can be expressed equivalently in either vector or tensor form. Tensors are required to define the stress as the product of the gradient of the velocity or the displacement and a coefficient. Cauchy’s symmetrization of the velocity tensor made matters worse. The generalization of vectors to tensors can of course be justified mathematically. But in the equations of motion, the use of tensors generates

xvi

Discrete Mechanics

artifacts that require us to impose additional constraints to resolve the resulting indeterminacy. For example, in solid mechanics, compatibility conditions must be imposed before the displacement can be computed from the stress. Over the past two centuries, other artifacts of similar type have been introduced into the equations of continuum mechanics. Discrete mechanics adopts a different perspective of the notion of stress. Shearing in a plane can, for example, be induced by a rotational stress in the direction normal to this plane; continuum mechanics would describe the same shearing as a stress in the plane itself. Eliminating tensors from the formulation of the equations of motion is a fundamental aspect of the discrete mechanical approach. Setting aside the concepts of frame of reference and the dimensionality of space allows us to introduce the operators of differential geometry ad hoc. The inappropriate usage of certain laws of physics can obscure inconsistencies in the system of equations; in fluid mechanics, the equations of motion are necessarily accompanied by the equations of conservation of mass. The Navier–Stokes equations do not conserve mass when used autonomously. There is a simple reason for this: the velocity-dependent contribution to the accumulation of the pressure is missing. The Navier–Stokes equations are incomplete, whereas the Navier–Lamé equations for solids are self-sufficient. Applying conservation of mass compensates for this deficiency, but it should not need to be invoked here in principle. It is common practice in mathematics and mechanics to couple together the equations of the boundary conditions and the initial conditions. According to Jacques Hadamard (1902), a problem is said to be well posed if it satisfies certain force or displacement conditions formulated in terms of partial derivatives at the boundary of the domain. In particular, Hadamard remarks that the displacement field can only be expressed up to rigid motion. This perspective is incompatible with discrete mechanics; it is difficult to apply boundary conditions to the displacement or the velocity when these quantities are only defined up to a constant. It will, therefore, be essential to introduce any conditions on the boundary or within the domain directly into the discrete formulation itself, phrased in terms of the only persistent quantities, the stresses; these stresses are applied using discrete operators that filter out uniform translational and rotational motion. The boundary conditions are an integral element of the mathematical formulation and are no longer viewed as external conditions, as is the case in continuum mechanics. Similarly, and for the same reasons, the initial conditions cannot be imposed directly upon the velocity variable. The initial state of a system is entirely defined by the state of its stresses, corresponding to a state of mechanical equilibrium. This equilibrium state is defined as the state that satisfies the equations of motion exactly; the evolution of a physical system takes the form of a succession of equilibrium states. The primary objective when establishing a physical model or an equation is to predict the future from a fixed current state. The predictive strength of a model is limited by various factors, for example relating to the quality of our knowledge of

Introduction

xvii

this current state, the inherently chaotic and turbulent nature of the evolution of nonlinear dynamic systems and so on. Because of progress in mathematics over the past century, we now have a solid understanding of these questions, but our current perspective of the physical model is still problematic. How can we predict the state in a fixed neighborhood of space-time given a perfectly determined state? In continuum mechanics, this cannot be done without introducing constitutive equations to connect the variables; for example, the state equations are used to close the system of equations and connect the states of the system at different times, even though this diverges from the intended role of these equations. This approach implicitly adopts the hypothesis of local equilibrium, which violates conservation of mass over time. To establish a deterministic prediction that is consistent with conservation principles, we must guarantee continuity in time as well as in space, together with causality in time to establish a continuous history of the evolution of the system. Discrete mechanics introduces the principle of accumulation of stresses: the variations in the velocity or the displacement modify the pressure and shear stresses, which preserves the continuous history of the system. The variable in the equations of the motion, the velocity, is an instantaneous quantity that is only used to accumulate the stresses. Thus, the discrete equations of motion are autonomous and do not depend on any state equations or conservation equations, e.g. a separate law describing the conservation of mass. The current notions of a continuum, constitutive equations, boundary conditions and thermodynamics led to a physical model that attempts to mimic reality as accurately as possible with a system of partial differential equations. But once this system has been established, what do we do with it? Since the system is formulated at a point in accordance with the hypothesis of a continuum, before we can apply it to a space, such as a three-dimensional space, we must first discretize the partial derivatives by finite differences or some other suitable variational formulation. Regardless of the formulation, this numerical discretization step is necessary to transform the problem into a linear problem that can be solved using a topology with a fixed or variable mesh; this step introduces new sources of error. The behavior of the numerical model cannot always be improved by applying higher order numerical schemes, and the same is true of the final simulation. By contrast, discrete mechanics does not require an additional discretization of the space, and the discrete equations of motion are ready to be used directly. The initial topology of the formulation of the model is the same as that of the discretization, and the discrete operators are the same as the operators of both the equations of motion and the boundary conditions. The properties of continuous operators also coincide with those of the discrete topology; this is not true, in general, for classical numerical methods on unstructured topologies. The principle of objectivity or material frame-indifference introduced by Clifford Truesdell states that the reaction of a material under a stress is independent of the direction of observation; the response of the material is invariant under change of

xviii

Discrete Mechanics

reference. The Cauchy stress tensor is objective, but the rate-of-rotation tensor is not; this is one of the obstacles presented by formulations based on the curl of the velocity. The curl is directly related to rigid-body translational and rotational motion. For rigid rotational motion, the rotation tensor can be expressed in terms of its dual, the curl vector. This point, as well as the objectivity of the discrete equations of motion, will be discussed later, but our formulation of the problem will ultimately diverge from continuum mechanics, where the answers given in the literature vary and the discussion remains open. The fact that the equations of motion do not require any constitutive equations or global frame of reference will greatly simplify our answer to this question. Some authors consider thermodynamics a science in its own right; in particular, its extension to the thermodynamics of irreversible processes by Ilya Prigogine (1977) led to an agitated yet fruitful confrontation with the ideas of rational mechanics formulated by Clifford Ambrose Truesdell and Walter Noll. Truesdell’s thoughts on the history of science and, in particular, the field of mechanics offer insight that still remains relevant in modern times. Some aspects of this mechanical–thermodynamic interaction raise obstacles that are incompatible with deterministic views of the process of deriving equations. In particular, the Clausius–Duhem inequality for the entropy leads to a condition on the viscosity, compression and shear coefficients. The symmetry conditions required for a description of an isotropic medium lead to the same result: indeterminacy in the volume viscosity. Stokes’ choice to hypothesize that the volume viscosity is zero has proven inadequate to say the least. However, the ramifications of this decision have been fully mitigated by applying the law of conservation of mass in connection with the equations of motion. Discrete mechanics is able to eliminate the indeterminacy in the volume viscosity by showing that the compression and shear stresses are two distinct concepts, each with its own coefficient that is unrelated to the other. The derivation of the conservation laws in a discrete medium does not involve any equations describing the behavior of the material, any constitutive equations or any thermodynamics. The thermophysical properties must simply be known. Surface or shock discontinuities can pose major difficulties when modeling physical phenomena; their nature can vary greatly: fissures in solid materials, faults in porous materials, phase changes, interfaces between immiscible fluids, shock waves in compressible flows, etc. To formulate these problems mathematically, equations are established on each subdomain of the medium, with jump conditions imposed on certain scalar or vector variables at the interfaces. Implementing these jumps is always tricky and often depends on the numerical methodology. In discrete mechanics, the jumps are fully integrated into the formulation, within the conservation equations. The discontinuity is formulated as the gradient of a phase function in the equations of motion or in the equations describing the conservation of flux. For example, this allows us to find an exact solution for the Laplace problem of

Introduction

xix

a capillary pressure jump in a droplet. The discrete equations of motions are capable of describing compressible flows and, in particular, the propagation of shock waves. Helmholtz-Hodge decomposition is not an axiom as such but rather a mathematical result, the fundamental theorem of vector calculus, established by Hermann Ludwing Ferdinand von Helmholtz, and extended by the work of William Vallance Douglas Hodge on differential geometry. The theorem states that any vector may be decomposed into a component with zero divergence (the solenoidal part) and another component with zero curl (the irrotational part). The theorem introduces notions of scalar and vector potentials that have clear physical interpretations in some fields of physics, including electromagnetism. Strangely, even though it has been used in mechanics to project onto a divergence-free field, the theorem itself has never been viewed as an intangible principle. Although some connectedness assumptions are required, Helmholtz-Hodge decomposition can be used as an axiom to derive the conservation equations; this is the approach taken by the theory of discrete mechanics developed in these pages. The Navier-Lamé equations are to some extent analogous – they give a decomposition into the sum of a gradient and a curl – but much like the Navier-Stokes equations, they only hold within the framework of continuum mechanics. In discrete mechanics, the acceleration decomposes naturally into the two components cited above, and the equations of motion can be presented as a Helmholtz-Hodge extractor for the scalar and vector potentials. The field of discrete mechanics offers a new paradigm founded on the concept of a discrete medium in which space consists of oriented edges at every scale of observation. The concepts of continuum, derivatives, and global frames of reference are abandoned. The equations of discrete mechanics may be derived axiomatically from the clearly identified principles described below, leading to a consistent formulation that gives a unified description of multiple distinct domains of physics.

List of Symbols

· × ⊗ : ∇ ∇· ∇× ∇p × ∇d × ∇2 (∗), tr d dt ∂ ∂t α αl αt β δ ε χT χS δij εij φ

scalar product (inner product) vector product (cross product) tensor product contracted tensor product nabla operator, gradient divergence curl primal curl dual curl ∇ · ∇(∗), Laplacian trace of a tensor material derivative, total derivative partial derivative with respect to time isothermal expansion coefficient attenuation factor of longitudinal waves attenuation factor of transverse waves thermal expansion coefficient length of an edge in the primal topology local porosity of a porous medium isothermal compressibility coefficient isentropic compressibility coefficient Kronecker delta components of the strain tensor scalar potential of the acceleration

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Discrete Mechanics

γ κ κl κt λ ϕ μ μdm μsm μt ν ρ ρm ρv σ τ ψ γ ε ω ωo σ τ ξ ψ ψo

heat capacity ratio, surface tension mean curvature of an interface longitudinal curvature of the primal topology at a point transverse curvature on a face compression viscosity heat flux density shear viscosity on a face discrete shear eddy viscosity eddy viscosity turbulent viscosity kinematic viscosity density density on a face density on an edge surface tension per unit mass, Poisson coefficient time constant stream function acceleration strain rate tensor shear-rotation stress potential equilibrium shear-rotation stress potential stress tensor viscous stress tensor phase indicator vector potential of the acceleration equilibrium vector potential of the acceleration

Σ Φ Φ Γ Ω Ω Ψ

local discontinuity on the edge Γ dissipation function heat flux edge in the primal topology, curvilinear contour volume of a domain rate-of-rotation tensor, spin tensor vector potential

(x, y, z) (r, θ) (r, θ, z) (r, θ, ϕ) (e1 , e2 , e3 ) A D L

Cartesian coordinates polar coordinates cylindrical coordinates spherical coordinates unit vectors area of a surface domain, control volume linear operator

List of Symbols

L M N P V S

Lamb vector molar mass non-linear operator power volume surface of a face in the primal topology

a cp cv cl ct d e f k h m p po p∗ pB q qm qv r s t v

thermal diffusivity specific heat at constant pressure specific heat at constant volume longitudinal celerity, longitudinal wave velocity transverse celerity, transverse wave velocity distance between the points or vertices [a, b] specific internal energy scalar function thermal conductivity, turbulent kinetic energy specific enthalpy mass pressure, scalar potential equilibrium scalar potential dynamic pressure Bernoulli pressure heat production per unit volume mass flow rate volume flow rate ideal gas constant specific entropy, curvilinear coordinates time specific volume

D Dh E J R L S T To T0 V0

flow rate hydraulic diameter Young’s modulus, total energy Jacobian of the transformation molar gas constant reference distance, latent heat entropy temperature equilibrium temperature reference temperature reference velocity

xxiii

xxiv

Discrete Mechanics

f g q t v v D F I K M N T V W W W

body force acceleration due to gravity, mass force momentum unit tangent vector fluctuation in the velocity perturbation in the velocity strain rate tensor force identity matrix or tensor permeability tensor mobility tensor outward normal to a free surface stress component of the velocity along the edge Γ modulus of the velocity velocity averaged velocity

Bi Da M Ma Ra Re We

Biot number Darcy number Mach number Marangoni number Rayleigh number Reynolds number Weber number

1 Fundamental Principles of Discrete Mechanics

This chapter is dedicated to the foundations of discrete mechanics. The notion of space is defined directly as a set of topological elements: edges and surfaces. These geometric elements exist at every scale and cannot be reduced to a point like in a continuum; as a result, we must abandon the concept of local differentiation, as well as inertial and non-inertial frames of reference. Some of the classical principles of physics can be kept, such as the weak equivalence principle and the principle of relativity, and some new physical principles are encountered for the first time, such as Hodge–Helmholtz decomposition. We also require new axioms and hypotheses: the accumulation of stresses and the duality of mechanical actions of all kinds. 1.1. Definitions of discrete mechanics 1.1.1. Notion of discrete space–time A method of positioning ourselves within space and time is essential if we wish to represent the universe around us, whether the universe of our daily lives, or the wider universe governed by the laws of general relativity. Positioning systems (GPS and Galileo) have become indispensable tools for many human activities such as transportation, well-drilling and so on. But in fact, to move toward a nearby target, we do not need to know our position exactly with respect to some absolute reference; we simply need to know the path to our target. The various theories of mechanics (Newtonian, quantum, continuum, relativistic, etc.) do not contradict each other – quite the opposite – but the connections between them have not yet been definitively established. Each theory of mechanics only describes a part of reality. The concepts, analysis tools and hypotheses of each theory vary. Ultimately, a unified theory of mechanics might not be strictly necessary.

Discrete Mechanics: Concepts and Applications, First Edition. Jean-Paul Caltagirone. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

2

Discrete Mechanics

The theory of discrete mechanics presented here assumes that there exists a time, the present, that describes the state of a physical system instantaneously. Although this image of the present exists as such, an observer located within space can only perceive its environment at later moments in time, since waves (light, sound, tidal waves) travel at finite velocity. The present can, therefore, only be perceived by an exterior observer in the form of a mathematical model that provide an instantaneous description of every phenomenon in the physical system.

Figure 1.1. Light cone in space–time. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Figure 1.1 shows the configuration of space–time in discrete mechanics; this model is borrowed from cosmology, where the present only makes sense for events unfolding at the origin. Any event that can influence or be influenced by an event unfolding at the origin is contained in the two cones whose summits are joined at the origin: the lower cone, which represents the past, and the upper cone, which represents the future. This light cone defines a so-called causal structure. For example, since the distance between the Earth and the Sun is large, we only receive light from the Sun 8 min after it was emitted. Any light signal emitted from the Earth would take just as long to reach the Sun. Events that occur during this period of time cannot be perceived by an observer; these events are said to be located elsewhere. In cosmology, the displacement A B of a point in space–time is represented by the four-vector (c dt, dx), and the present is restricted to the origin. In both discrete and continuum mechanics, the present unites all elements within a single causal structure; in Figure 1.1, the boundary of this structure is a circle, which we shall call the discrete horizon. Every event in this space is linked by cause and effect, the radius of the circle rh is independent of time and every event within the circle is known instantaneously. Even if a specific observer located at some point of this space cannot directly perceive every event unfolding in the present instantaneously, the

Fundamental Principles of Discrete Mechanics

3

instantaneous field of all problem variables exists and can be represented by a mathematical model. Time is assumed to unfold linearly. Figure 1.2 shows two spaces. The first has a finite horizon as its boundary, and the second is a sphere without a boundary but which is nonetheless finite; in both cases, all events unfolding on these surfaces are connected by the propagation of various types of waves through space. On the space with a boundary, events will necessarily be influenced by boundary conditions, which will be defined later. On the sphere, interactions will cumulate as the system evolves over time.

Figure 1.2. Space with a discrete horizon as a boundary (left) and a sphere without a boundary but which is nonetheless finite (right)

Thus, the discrete horizon defines a space on which separate phenomena can be described by a mathematical model at the same moment in time. For example, atmospheric models give an image of the present time and can be used to predict the weather over the next few days. Neglecting absorption, sound waves require over 300 h to travel the 40, 000 km of the circumference of the Earth, and light waves require slightly over 0.1 s. Even if an event at one point is not perceived instantaneously from another point, we can still construct an instantaneous image of the atmospheric currents. To make accurate predictions, we need a good representation of the present, which can be achieved by collecting a large amount of precise data. However, the chaotic and to some extent random nature of the turbulent evolution of atmospheric flows limits the prediction range of the model to just a few days. The approach adopted by discrete mechanics draws heavily from the classical view of mechanics, where every interaction is defined directly. The interactions conventionally described as “actions at a distance”, such as variations in gravity due to the Moon and Sun, are predictable and can be taken into account in the mathematical model. However, we cannot represent the cause-and-effect relations of more rapid events, such as the collapse of two black holes producing the gravitational waves predicted by general relativity.

4

Discrete Mechanics

Nevertheless, Newtonian mechanics is an alternative that is compatible with reality. Newtonian mechanics is widely thought to only be valid at velocities far below the speed of light, but in fact Newton’s theory has remarkable properties when extended to the propagation of waves. The velocity is not the only relevant distinction between the relativistic and Newtonian theories of mechanics; relativistic kinematics and dynamics are other examples. Ultimately, the objective of the discrete perspective presented here is to investigate whether Newtonian mechanics is capable of describing all types of fluid and solid behavior, as well as the propagation of all types of waves. 1.1.2. Notion of a discrete medium The perspective presented and explored here abandons the hypothesis of a continuum, which defined all of the problem variables, physical properties, etc., at every point. In continuum mechanics, Newton’s law of dynamics, also known as Newton’s second law, is formulated at a point. To express the spatial variation of the vector quantities on which the theory is based, we are forced to define the concepts of frame of reference and differentiation. Newton himself contributed to the development of infinitesimal calculus, even though he originally represented vectors as bipoints [NEW 90]. Later work expanded this continuous approach, which has various advantages, but also disadvantages that can generate artifacts.

Figure 1.3. Construction of a discrete medium from points, edges, surfaces and volumes

The field of discrete mechanics is built upon connected objects such as those shown in Figure 1.3. First, we consider points that are not absolutely positioned within space; we shall work within a local frame of reference that positions objects relatively to one other. Two points and a straight line define an edge, or bipoint. This introduces two important ideas: the distance d between the points and a direction, also defined in relative terms. From multiple edges, we can construct a surface, which is necessarily planar; to represent non-planar surfaces, we can reduce them to triangles, which are planar by definition. Finally, by assembling multiple planar surfaces, we can construct volumes. The concept of dimension (one, two or three) is abandoned. For example, a (2D) plane constructed from three points simply defines

Fundamental Principles of Discrete Mechanics

5

two unit vectors: one associated with any given edge and the other normal to the plane. In three-dimensional space, this pair of vectors would necessarily be orthogonal. These topologies are all described as primal topologies; we shall then need to define a dual topology based on localizations such as the barycenter of a face or a volume. Consider the elementary primal topology shown in Figure 1.4. This example will allow us to introduce notation for later. The three edges Γ with unit vector t form a planar surface S with normal vector n; the vectors t and n are necessarily orthogonal, t · n = 0. This condition is strictly necessary to establish the desired properties of the operators of the primal and dual topologies. The orientations of the unit vectors t and n are arbitrary but mutually dependent.

Figure 1.4. The elementary topology of mechanics in a discrete medium: three straight edges Γ between vertices, defining a planar face S. The normal vector n of this face and any of the vectors along Γ are orthogonal; in other words, t · n = 0. The edge Γ can optionally be intercepted by a discontinuity Σ at the point c located between the vertices a and b of Γ. The quantities Φ and Ψ denote the scalar and vector potentials, respectively. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The edge Γ of length d lies between its two vertices a and b. This is the edge on which the equations of motion will later be defined. Scalar quantities such as the scalar potential φ are defined at points, whereas the vector potential is defined orthogonally to the plane S. The plane Σ that intersects Γ at c divides the edge [a, b] into two parts, for example representing two media of different types or with different physical properties. The interface Σ could represent a discontinuity, a shock wave, or a contact discontinuity. The velocity vector is defined on the edge Γ as a scalar constant directed from a to b. The notion of a continuum is set aside by this approach. The topologies – the edge, surface and volume shown in Figure 1.3 – can never be reduced to a single point. The concept of differentiation is also abandoned. The measure d of the length of the edge Γ can be made arbitrarily small, but everything else scales proportionally, conserving angles. Thus, even though the medium is no longer continuous, we can nonetheless

6

Discrete Mechanics

consider scales as small as permitted by the continuity properties of the macroscopic approach. The mean free path of individual molecules is always a lower bound for the scale. 1.2. Properties of discrete operators The discrete operators are defined straightforwardly from the elementary topology in Figure 1.4. First, the discrete gradient is simply defined as a difference. For example, the gradient of the scalar potential φ is ∇φ = (φb − φa )/d. The gradient vector is not equivalent to the analogous concept in a continuum. Here, the gradient vector is an oriented scalar with direction t. The primal curl ∇p × V of the vector V is defined on the unit vector n as the circulation around the edges of the oriented surface S. The divergence, for example of a vector, can be defined at a given point in terms of the (net) flux of every edge leading to or away from this point. The fourth operator that we shall define is the dual curl ∇d × ψ; the components of ψ are orthogonal to each primal surface S. The gradient and dual curl operators can be used to project the action of various phenomena onto each oriented edge Γ, which also serves as the basis for conservation of momentum and various other vector quantities, including the components of the velocity V. With this topological structure, some of the operators defined above are exact in the sense that the numerical error is zero. This is the case for the gradient, which is defined as a difference, as well as the primal curl, which is computed from Stokes’ theorem as the line integral of the velocity vector around the contour Γ. The two other operators, the divergence and the dual curl, incur numerical errors whose magnitude depends on the quality of the mesh and the construction of the dual space. Although classical mechanics relies heavily on the divergence theorem to redefine the flux of a surface onto a volume and then at a point, discrete mechanics derives the equations of motion from the fundamental theorem of analysis and corollaries such as Stokes’ theorem. These key theorems are briefly recalled below. If F(x) is uniformly and continuously differentiable on [a, b], then the fundamental theorem of analysis, also known as the fundamental theorem of integral and differential calculus, states that: ⎧  F (x) = f (x) ⎪ ⎪ ⎨  b ⎪ ⎪ ⎩ f (t) dt = F (b) − F (a) a

[1.1]

Fundamental Principles of Discrete Mechanics

7

Stokes’ theorem is a corollary of this result. It allows us to compute the curl of a surface as the line integral around a contour: 

 Γ

V · t dl =

Σ

∇ × V · n ds.

[1.2]

The scope of this theorem is remarkable. We can compute the curl of a surface without explicitly knowing the velocity itself; we simply need to know its components on some closed contour. Armed with this result, the concept of a frame of reference, needed to define the velocity vector at specific points in absolute terms, becomes less essential. Furthermore, since the curl is not defined at points or on curves, this operator can only be defined by taking limits in continuum mechanics. The divergence theorem, also known as the Green–Ostrogradski theorem, is another corollary of Stokes’ theorem that allows us to quantify the source or sink behavior of any quantity defined at a point: 

 Σ

V · n ds =

Ω

∇ · V dv.

[1.3]

Since the discrete equations of motion are derived on an edge Γ, we need to know how to interpret the product of two functions on this edge. The generalized mean value theorem for definite integrals (which follows from Rolle’s theorem) states that ∃ c ∈ [a, b] such that: 



b

b

f (x) g(x) dx = f (c) a

g(x) dx. a

This theorem, which only holds in one dimension, will be extremely useful to us. When working with two-phase flows, this theorem can be used to bound the density in terms of its values at the endpoints a and b. Two key properties from continuum mechanics, ∇ × ∇p = 0 and ∇ · ∇ × V = 0, will prove especially important with discrete media. Labeling discrete quantities with the subscript h, we can easily show that the discrete curl of a discrete gradient is zero on the primal topology:

8

Discrete Mechanics

⎧  b ⎪ ⎪ ∇p · t dl = pb − pa ⎪ ⎪ ⎪ a ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ ∇p · t dl = 0 Γ ⎪  ⎪ ⎪   ⎪ ⎪ ∇ × ∇p · n ds = 0 ⎪ ⎪ ⎪ S ⎪ ⎪   ⎩ ∇h × ∇h p = 0

[1.4]

Similarly, the discrete divergence of the discrete primal curl computed on the dual volume is zero: ⎧ n n  ⎪ ⎪ ⎪ Γ = ∇ × V · n ds = 0 i ⎪ ⎪ ⎪ s i=1 i=1 ⎪ ⎪ ⎪  ⎪ ⎪   ⎨ ∇ × V · n ds = 0 S ⎪  ⎪ ⎪   ⎪ ⎪ ⎪ ∇ · ∇ × V ds = 0 ⎪ ⎪ ⎪ V ⎪ ⎪   ⎩ ∇h · ∇h × V = 0

[1.5]

Figure 1.5 illustrates why the property ∇ × ∇φ = 0 holds on the primal topology, and why ∇ · ∇ × ψ = 0 holds on the dual topology.

Figure 1.5. Properties of operators: ∇ × ∇φ = 0 (left); ∇ · ∇ × ψ = 0 (right)

Fundamental Principles of Discrete Mechanics

9

These properties allow the discrete equations of motion to express the acceleration as the sum of a gradient and a curl, or in other words as a Hodge–Helmholtz decomposition. Hodge–Helmholtz decomposition is usually encountered when solving the Navier–Stokes equations, where it is used to separate the irrotational and solenoidal parts of various vectors. In this way, the velocity correction associated with the irrotational term can be used to construct a divergence-free field [ANG 12, CAL 15c]. More generally, Hodge–Helmholtz decomposition gives two orthogonal terms satisfying certain boundary conditions that are useful in various other fields, such as imaging, fingerprint recognition and so on. Since any vector can be decomposed into these two components, it often makes sense to find expressions for each component of a physical vector a priori. We can do this for both the velocity and the acceleration. Some operators and combinations of the above operators, including the Laplacian ∇2 φ = ∇ · ∇φ or ∇2 ψ = ∇ · ∇ψ, will not be used; they can lead to artifacts or, for vectors, increase the tensor order of the operators. There is no suitable representation for tensors of orders two or higher in the discrete setting. For example, the gradient of a vector ∇V, which has a clearly defined meaning in a continuum, cannot be represented in discrete mechanics. One of our key objectives is therefore to determine whether the operators cited above suffice to describe all possible physical behavior. It might not be immediately obvious whether tensors are strictly necessary to describe all behavior and physical phenomena observed at macroscopic levels. Second-order tensors provide a valid description of the behavior of anisotropic materials, but these tensors do not necessarily reflect the underlying laws of mechanics. Solid mechanics, general relativity, fluid mechanics and other fields frequently include higher order tensors in their laws. For example, in solid mechanics, shear stresses are defined as the gradient of a vector; however, they could alternatively be expressed just in terms of the curl. Over time, the distinction between constitutive laws and the fundamental laws describing the underlying physics has faded. The laws of physics can in fact be phrased equivalently in terms of vectors or tensors. Maxwell’s equations are a good example of this; each formulation has specific advantages and disadvantages. 1.3. Invariance under translation and rotation The equations of motion must satisfy Galileo’s principle of relativity; we should not be able to distinguish between the physical phenomena of a system in uniform motion and those of the same system at rest. If an object is held by an observer aboard a train traveling at velocity Vt , it will fall at his or her feet if dropped without acceleration. The velocity of the uniform motion must therefore somehow “cancel out” in the equations of motion. The problem of uniform rotation is similar but trickier than linear motion. In particular, any shear stresses in a uniformly rotating

10

Discrete Mechanics

object must be independent of the frame of reference. This is known as the principle of material frame-indifference and was introduced by Truesdell [TRU 65]. Consider a translational motion Vt and a rotational motion Vr : ⎧ ⎨ Vt = u0 ex + v0 ey + w0 ez ⎩

[1.6] Vr = Ω × r

where (u0 , v0 , w0 ) are the constant components of the translation vector and Ω represents the uniform rotation. Let us now apply the discrete operators with the same properties as their differential counterparts to these vectors. In particular, the properties ∇ · ∇ × V = 0 and ∇ × ∇f = 0 are satisfied, where V is a vector and f is a scalar. Applying these operators to these uniform velocities yields: ⎧ ∇ · Vt = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∇ · Vr = 0

∇ × Vt = 0

⎪ ⎪ ∇ × Vr = 2 Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∇×Ω=0

[1.7]

The case of translational motion is straightforward. The divergence and primal curl of Vt are identically zero; if a harmonic field is present, it can be eliminated in the same way. The divergence of the rotation field is indeed zero, but the primal curl is merely constant throughout the domain, provided that the latter is connected. The circulation around an arbitrary contour (that does not enclose the origin) reduces to the curl of the rotational velocity, which is constant. By equation [1.7], the dual curl of the primal curl is zero. At least for some of the terms of the equations of motion, we can therefore hope to establish complete invariance between frames of reference. The principle of material frame-indifference cited by Truesdell is automatically satisfied if the shear stresses are expressed in terms of these discrete operators. The non-linearity of the inertial term in the equations of motion creates extra terms. If we split the velocity field into V = V + Vr = V + Ω × r, the acceleration can be written as: ⎧ dV ∂V 1 2 ⎪ ⎪ ⎪ ⎨ γ = dt = ∂t − V × ∇ × V + 2 ∇ |V| ⎪   ⎪ ⎪ ⎩ γ = dV = ∂V − V × ∇ × V + 1 ∇ |V |2 + 2 Ω × V + Ω × Ω × r dt ∂t 2

[1.8]

Fundamental Principles of Discrete Mechanics

11

where 2 Ω × V is the Coriolis acceleration and Ω × Ω × r is the centrifugal acceleration. The

centrifugalterm Ω × Ω × r derives from a scalar potential and is 2 equal to −∇ |Ω × r| /2 . For uniform rotational motion, the centrifugal acceleration is locally and instantaneously opposite to the centripetal acceleration. The derivatives dV/dt = dV /dt are identical in an absolute frame of reference and a moving frame. The Coriolis and centrifugal accelerations only need to be considered when changing between frames of reference. In Newtonian mechanics, interactions between two frames are assumed to be instantaneous. General relativity is required for a more realistic description of actions at a distance. Here, in the discrete setting, we shall work in a local frame, and every interaction is defined by the wave celerity, which is necessarily finite. Any uniform translational celerity Vt or rotational motion Vr is imperceptible and cannot be detected by an observer attached to the local frame. This is both a disadvantage and an opportunity; we cannot describe absolute motion, but on the other hand our velocity and position with respect to the universe ultimately do not matter anyway. In our approach, the universe will, therefore, be limited to a horizon that is described as discrete (Figure 1.1). Inertial analysis plays a special role in discrete analysis and is discussed in Chapter 2. 1.4. Weak equivalence principle Since Galileo’s time, various experiments have been performed to investigate the action of gravity on two different types of mass that appear to be accelerated identically, regardless of the internal structure or composition of the object being accelerated – this is the equality between gravitational mass and inertial mass (also known as the Weak Equivalence Principle, or WEP). Einstein would later extend the WEP into a stronger equivalence principle that relates to the fact that the velocity is always bounded by the celerity of light in a vacuum in special relativity. Today, the weak equivalence principle has been repeatedly verified by an impressive array of experiments. The equivalency is quantified by the Eötvös ratio η = 2 |γ1 − γ2 | / |γ1 + γ2 |, where γ1 and γ2 are the accelerations of the two masses. The acceleration can be measured independently from any frame of reference to extremely high accuracy. The modern view is that the WEP holds exactly, with an Eötvös ratio of η < 10−14 ; the article by Will [WIL 09] presents the various experiments that have been conducted over the last century. Other experiments led by France and the United States are also currently underway. It is hoped that they will

12

Discrete Mechanics

achieve even greater levels of precision (10−15 or 10−18 ) to confirm (or refute) the exactness of the Weak Equivalence Principle (WEP). The equivalence principle allows us to rewrite the fundamental law of dynamics established by Newton in his Principia [NEW 90] in the following form: γ = g.

[1.9]

This equality might seem self-evident, since an isolated observer cannot distinguish the effect of gravity from his own specific acceleration. The equality had been known since Galileo, but Newton formulated his second law as m γ = F even though gravity was the most important force at the time. This law would prove to be somewhat problematic in the field of electromagnetism, but would be adopted by the field of dynamics nonetheless. The next question is the underlying meaning of the force per unit mass g. Is it exclusive to gravitational forces, or can we view g as the sum of the forces acting upon any particle of matter? The distinction is irrelevant anyway if gravity is the only force exerted by a body of mass M , viewed as a point, on a particle with (possibly zero) mass m. In this case, the acceleration due to gravity is equal to −G M/r2 , derived from the potential G M/r, where G is the universal gravitational constant and r is the distance between the particle and the body of mass M . If the gravitational force exerted by a body reduced to a point mass M is small, we can consider the Taylor expansion at G M/r c2 c, then ρ = ρb . We find that: 

1 ρv = d



c

ρa dx + a



b

ρb dx .

[1.17]

c

After defining a partition function ψ ∈ [0, 1], we have that: ρv = ψ ρa + (1 − ψ) ρb ,

[1.18]

where ψ = [ac]/[ab]. Any thermodynamic coefficients can simply be defined directly in terms of the mass, volume, heat, etc. We shall define the following quantities classically: – the isochoric expansion coefficient α; – the isobaric expansion coefficient β; – the isothermal compressibility coefficient χT ; – the specific heat at constant pressure and constant volume, cp and cv ; – the latent heat from a phase change at constant temperature L. The definitions of these quantities are recalled in the following:

⎧ 1 ∂p ⎪ ⎪ α = ⎪ ⎪ p ∂T ρ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ cp =



∂h ∂T

β=−

1 ρ

cv = p



∂e ∂T

∂ρ ∂T

∂ρ ∂T

p

∂T ∂p

ρ

∂p ∂ρ

= −1, T

χT = p

We know that:

L= ρ

1 ρ



T dp Δρ dT

∂ρ ∂p

T

Fundamental Principles of Discrete Mechanics

19

and so these coefficients satisfy the relation: β = 1. α p χT The notion of tensor is needed here if we wanted to express the variation of a quantity at a point in terms of the direction of observation. However, if the direction of observation Δ is fixed, the various quantities, mechanical properties, stresses, etc., can be simply described as scalars or oriented vectors on Γ. Some materials have tensor properties, such as the thermal conductivity, the permeability, and so on, as well as certain mechanical properties. Like the components of the heat flux, displacement, or velocity, any tensor quantities are defined on the edge Γ by projection. For example, consider the diffusion of the heat flux Φ in an anisotropic material. The matrix [Λ] representing the thermal conductivity tensor is diagonalizable. The eigenvectors define the principal directions of the tensor Λ; the matrix is diagonal, λ = (λ1 , λ2 , λ3 ), with respect to its basis of eigenvectors. The flux on the edge t can be formally defined as Φ · t = −k (TR − TP )/L, where k is the scalar representing the conductivity on the edge and (TR − TP )/L is the discrete gradient. We can now identify the heat flux along t and compute the value of k: k = λ · t,

[1.19]

where λ is written with respect to the basis of eigenvectors. This property is constant over the whole edge. Note that any quantity defined on an oriented edge with unit vector t can equivalently be written either as a scalar that is constant over the edge or as an oriented vector. These constitutive laws, state laws, etc., are only required to describe the behavior of a medium, whether fluid or solid, in terms of specific scalar or vector variables, such as the temperature, the pressure, the mechanical stress and so on. If the properties of the medium vary as a function of the direction (the anisotropic case), it can be useful to summarize the behavior of the medium with a symmetric tensor, which allows us to easily compute the stress in any direction. However, this tensor is always defined by a basis, typically an orthonormal basis, and therein lies the problem. The extremely general nature of this approach introduces difficulties that will need to be addressed later, for example using the principle of material frame-indifference, which allows the tensor properties to be introduced into the conservation equations. The Cauchy stress tensor defined at a point has six independent coefficients, which are expressed in terms of the velocity or the displacement. This results in 81 coefficients for the elasticity tensor in the stress–strain relationship defined by

20

Discrete Mechanics

Hooke’s law. In an isotropic medium, the number of coefficients is reduced to just two, the Lamé coefficients. Whether these Lamé coefficients are representative depends on whether the material is fluid or solid. This robust connection between the constitutive equations and the conservation equations can, however, be broken without undermining the representativeness of the model constructed from the fundamental equations of motion. Any anisotropy or inhomogeneity in the medium does not directly affect these fundamental equations. The physical properties – the viscosity of the primal surface and the compressibility coefficient defined at the endpoints of each edge – are functions of the scalar variables of pressure, temperature and density, or vector variables such as the vector potential. Anisotropy is handled analogously to inhomogeneity, viscosity is defined on planes, and compressibility is defined at points; these properties can vary over space and of course over time. 1.8. Composition of velocities and accelerations Consider a particle following a rectilinear trajectory along some axis Δ. At time to , the velocity of this particle is given by Vo and can be viewed as a scalar V o defined on the oriented axis Δ; the acceleration is assumed to be constant and equal to γ. After a period dt has elapsed, the velocity of the particle is V = Vo + γ dt, provided that dt is sufficiently small – it can be deduced by integrating with respect to time if not. In discrete mechanics, this period dt is chosen to be as small as necessary. Suppose now that an additional acceleration δ is applied to the particle, say δ = γ: the acceleration of the particle is now equal to 2γ and its velocity at time t + dt is V = Vo + 2γdt. Thus, when the acceleration is doubled, the velocity is modified by an amount that depends on dt. Although the acceleration and the velocity are both vectors, they perform completely different roles in mechanics. This distinction is essential if we hope to understand the laws of physics, especially when summing vectors. In mathematics, the sum of two vectors v1 and v2 is simply defined as the vector v = v1 + v2 . This still holds in Newtonian mechanics, but in relativistic mechanics a Lorentz transformation is required, meaning that the equation cited above no longer holds. Acceleration and velocity are independent concepts. The sum of two accelerations is always a priori equal to γ1 + γ2 in a single frame of reference but the same is not true for two velocities. For example, consider a single particle: it can be acted upon by two accelerations simultaneously, but it can only have one single instantaneous velocity. We need to distinguish between the quantities applied to the particle, the external accelerations and the internal quantities of the particle, its own acceleration and velocity. Some of the thought experiments performed in the early 20th Century explored the idea of two trains passing each other by. Newtonian mechanics needed to be

Fundamental Principles of Discrete Mechanics

21

questioned in light of the newly proposed postulate that the velocity is necessarily bounded by the speed of light in a vacuum. This postulate is now widely accepted but is not an axiom. The trains considered by these thought experiments moved at constant speed, so with zero acceleration, and the fact that they are passing by one another is not especially significant if they do not interact, for example in a two-way tunnel. To conduct thought experiments with two particles, the causality principle must hold; in other words, the particles must interact and share the same discrete space–time. Therefore, we shall assume axiomatically that the accelerations sum additively. By contrast, the velocities are additive, free from any constraints and shall be viewed as secondary variables. This illustrates the benefit of formulating the equations of motion in terms of the acceleration rather than the velocity or the displacement. Any constraints on the velocity necessarily arise from the fact that the acceleration tends to zero; if so, we need to know the velocity at previous times to know the velocity at the present time. Consider the case of steady motion where the particle has zero acceleration, γ = 0. By Newton’s first law, there is no force acting on the particle, or more precisely the sum of all forces acting upon the particle is zero; we can rephrase this in terms of the acceleration: “Every body remains at rest or in uniform rectilinear motion unless acted upon by some acceleration which forces it to change its state”. More precisely, it is the sum of the accelerations that must be zero: for example, in the case of a rigidly rotating flow with constant velocity, the non-zero centripetal and centrifugal accelerations compensate one another exactly as two equal gradients of scalar potentials. The velocity V = Ω × r is constant in the absence of external action. This solution can be superimposed with any other solution without modifying the latter, provided that there are no strong interactions between the two. As well as constant translational motion (Galilean frame of reference), this rotational motion needs to be eliminated from the field of solutions of the equations of motion; in fact, this can only be done up to the gradient of a scalar potential. Two theoretical difficulties need to be addressed: the mathematical behavior at infinity, as well as the physical notion of time, which is necessary for this type of flow to arise. Indeed, to generate this flow, we must take the transverse speed to be infinite or wait for an infinite period of time. Let v1 and v2 be two vectors viewed as mathematical objects, and let V1 and V2 be two material velocities. The mathematical and mechanical composition rules of these vectors are not the same: – Galilean transformation: v = v 1 + v2 ;

[1.20]

22

Discrete Mechanics

– relativistic transformation: V=

V1 + V2 . V1 V 2 1+ c2

[1.21]

The second relation is more general. However, there is an issue – we are comparing a material velocity against a celerity (wave velocity). It does not seem justifiable to conflate the velocity and the celerity in a term of the form V/c, since at a fundamental level these quantities relate to completely different concepts, despite having the same units. The celerity is an intrinsic property of the medium under fixed conditions (solid, gas, vacuum); it determines the velocity of the wavefront. The material velocity, on the other hand, governs the advection of a physical quantity, whether massive or massless. The two concepts are quite simply distinct, yet the Lorentz transformation unites them into a dimensionless ratio that is assumed to be universally valid. By contrast, the acceleration of a particle is always equal to the sum of the accelerations applied to this particle: γ = γ1 + γ2 . The composition of accelerations simply reduces to the conservation of acceleration, i.e. Newton’s second law. For example, the local acceleration of an object subject to gravitational attraction from both the Earth and the Moon is equal to the vector sum of the accelerations induced by each body. The relativistic composition rule for the acceleration γ involves the Lorentz factor γ, whenever it is specified in terms of the velocity: ⎞3 ⎛ 2 1 − Vc2 ⎠ . γ =γ⎝ 1 − vcV2

[1.22]

This expression gives a description of the acceleration and may differ from one inertial frame to another. With a local frame, the law of dynamics F = mγ is identical in both Newtonian and relativistic mechanics, provided that m is the mass that is undergoing the motion. The velocity V is in fact a Lagrangian that is updated by the acceleration V = Vo + γ dt. Like the pressure, the density, the temperature or the shear-rotation stress, the velocity is computed from its previous value at time to . Adopting the perspective that the velocity is the accumulation of the acceleration offers one important advantage; it eliminates unphysical solutions such as those which arise from superimposing velocities exceeding the speed of the light; the velocity Vo is a

Fundamental Principles of Discrete Mechanics

23

mechanical equilibrium and V is another equilibrium state, since the acceleration vanishes as the velocity approaches its limit value, the celerity c. Moreover, the velocity does not appear directly in the expression of the acceleration but only indirectly via operators such as the divergence and the curl that eliminate any uniform motion. 1.9. Discrete curvature The concept of curvature is essential in mechanics; it is encountered with various phenomena such as inertia, gravity, capillarity and so on. The notions of principal curvature, mean curvature, Gaussian curvature, etc., were originally introduced in planar geometry in an intuitive form before being formalized into a rigorous mathematical framework for more general spaces. Curvature is defined by the Riemann curvature tensor; this tensor specifies the acceleration at which two neighboring geodesics split from one another on a curved space. In general relativity, gravity is viewed as a manifestation of the curvature of space–time. The work performed by Gauss allowed a formula to be established for the curvature K of a surface. In Riemann coordinates, this is equivalent to finding an expression for the completely covariant Riemann tensor Rxyxy . In two dimensions, this tensor can be stated as follows: Rxyxy

1 =− 2



∂ 2 gxx ∂ 2 gyy + ∂y 2 ∂x2

= K,

[1.23]

where gxx and gyy denote the coefficients of the metric in Riemann coordinates. Most work on the curvature of space has relied on metrics defined for specific frames of reference, both inertial and non-inertial; this allows the curvature to be expressed in terms of tensors of orders greater than two in a manner consistent with the concepts of a continuum and differentiation at a point. The fundamental principles of analysis, differential geometry and vector analysis that are conventionally used to introduce the notion of curvature are not compatible with the discrete approach, nor is performing an a posteriori discretization of the results of the continuous approach a good solution. Alternative approaches in the field of discrete differential geometry have considered meshed topologies, but they are mostly intended for computations and numerical simulations, without attempting to fully detach themselves from the equations of a continuum. Discrete mechanics [CAL 15a] revisits the equations of mechanics from a different perspective: we assume the existence of objects, edges and oriented surfaces that cannot be reduced to a single point by scaling. The usual notions of frame of reference, differentiation, etc., are no longer suitable and must therefore be abandoned. Like the

24

Discrete Mechanics

other quantities used to establish the equations of motion, the curvature requires a discrete definition valid for any primal and dual topologies.

(a)

(b)

Figure 1.7. Longitudinal and transverse curvatures, calculated as the divergence of the vectors t and m in curvilinear coordinates. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Consider the elementary stencils shown in Figure 1.7; Figure 1.7a shows a discrete curvilinear contour of edges. Each edge Γ has unit vector t and is rectilinear between its endpoints a and b. Figure 1.7b shows the curvilinear edge orthogonal to Γ with unit vector m; at the point c, the vector n satisfies n = t × m. The discrete curvature of the edge Γ has two components: the longitudinal curvature κl and the transverse curvature κt . The first, defined at the point a or b, represents the variation in the direction of the vector t in discrete curvilinear coordinates, whereas the second, defined at the point c, defines the variation of the curvature along the axis with unit vector m. As in continuous differential geometry, the mean curvature is defined as the sum of the principal curvatures κ = κl + κt . Like the other quantities of the discrete approach, κt can be viewed either as a vector or a scalar defined on the unit vector n. The curvature κl is a scalar defined at the points of the primal topology. If the primal topology is known, we can compute each unit vector t from the local coordinates of the points. The principal curvatures are expressed in terms of discrete operators on the primal topology as follows: ⎧ ⎨ κl = ∇ · t ⎩

[1.24] κt n = ∇ × t

Fundamental Principles of Discrete Mechanics

25

These are formal definitions that introduce metrics that must be computed from the points of the primal topology where the unit vectors t and n are defined. Whenever the curvature is encountered in a physical phenomenon, our objective is to find the acceleration γc of the phenomenon; for example, in the context of an interfacial effect, the curvature is used to find the capillary acceleration γc ; this quantity may be obtained by projecting the action of variations in the surface curvature onto Γ using the gradient and dual curl operators. The capillary acceleration is therefore given by: γc = ∇ (σl ∇ · t) + ∇ × (σt ∇ × t) ,

[1.25]

where σl and σt are the surface tension per unit mass of the two media separated by the interface Σ. Even with constant curvature, the interface may experience motion due to the Marangoni effect. Note that these effects are fully defined on the interface Σ and the overpressure is simply the consequence of their action on a closed surface such as a drop or a bubble. In fact, the formulation of equation [1.25] is perfectly generic and can be extended to gravity, inertial effects and so on. Figure 1.8 shows how the acceleration γc is defined on the edge Γ; the two actions represented by the divergence of t and the curl of t are both projected onto the principal edge Γ.

Figure 1.8. Longitudinal and transverse curvatures, computed as the divergence of the vector t and the primal curl of t, respectively, on the contour of Σ. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The velocity induced by each effect on the edge Γ does not need to be known directly; the velocity V also depends on all other effects and the value Vo of the mechanical equilibrium at time to and can be updated accordingly, V = Vo + γdt. If we compute the curvature of a circle of radius R represented by equidistant points connected by edges using the expression from equation [1.24], we recover the exact

26

Discrete Mechanics

solution κ = 1/R regardless of the number of markers in the primal topology. In three dimensions, we recover the notion of mean curvature κ = κl + κt from differential geometry. As we saw elsewhere, these two effects can double up; for example, the mean curvature of a sphere is twice that of a cylinder. They can also cancel, for example in surfaces with zero curvature such as catenoids. The minimal surface of a soap film pressing against two circles of radius R takes the form of a catenary rotated around an axis – also known as a catenoid. Assuming atmospheric pressure on either side of the film, this surface has zero mean curvature. Equation [1.25] is stated in a generic form; any vector may be written as the sum of a gradient and a dual curl, defining any internal actions in terms of the divergence and primal curl operators. This decomposition is the cornerstone of discrete mechanics and in particular the equations of motion. In practice, the nature of the problem of computing the curvatures κl and κt changes according to whether the interface is defined analytically by markers, by a phase function, by an indicator function, etc. We will revisit the purely numerical aspects of this problem in Chapter 5. As a demonstration, let us compute the curvature of a zero-curvature surface – the catenoid from classical differential geometry. In a parametric form, the catenoid may be expressed as x = cosh(k u) cos v/k, y = cosh(k u) sin v/k, z = u, where k is a real constant. Writing r(u, v) for a point on the surface, the normal vector n at this point is given by:

n=

∂r ∂u ∂r | ∂u

⎧ cos v ⎪ − ⎪ ⎪ cosh(k u) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂r sin v × ∂v − = ∂r cosh(k u) × ∂v | ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sin(k u) ⎪ ⎪ ⎩ cosh(k u)

[1.26]

After computing the metrics [gij ] on the surface, the principal curvatures are as follows: ⎧ k ⎪ ⎪ κ1 = − ⎪ 2 ⎪ ⎨ cosh (k u) ⎪ ⎪ ⎪ ⎪ ⎩ κ2 =

[1.27] k 2 cosh (k u)

and we find that the mean curvature κ = κ1 + κ2 is locally zero, as expected.

Fundamental Principles of Discrete Mechanics

27

On a discrete surface for which a parameterization is not available, the computation is performed directly from equation [1.24]. The computation of the longitudinal curvature from the expression of κl is relatively straightforward; similarly, the transverse curvature κt can be computed as the variation of the vector m along the curvilinear edge. In the general case, for arbitrary primal topologies, the mean curvature can be computed using existing techniques from discrete differential geometry or based on the ideas presented here. Figure 1.9 shows two surface geometries, a sphere and a catenoid, constructed from a relatively regular triangulation.

Figure 1.9. Surface topologies of a sphere and a catenoid based on a regular triangulation

With this example topology, we recover the theoretical (continuous) value of the mean curvature of the sphere to a high degree of accuracy, even though the edges Γ are oriented arbitrarily. In the case of the catenoid, we recover the value of zero predicted by differential geometry from its parametric function. It is then easy to compute the unit vectors t from the coordinates of the vertices of each triangle and deduce the curvatures κl and κt . The duality hypothesis allows us to write the acceleration on the edge Γ as the sum of the gradient of a scalar potential and the dual curl of a vector potential. Techniques based on elaborate concepts from differential geometry and numerical methods of evaluating the curvature are not all so accurate nor easy to implement. Here, we only require the values of the curvatures κl and κt at the points a and c; depending on the methodological context, the best way to evaluate them is as follows: γc = ∇ (σl κl ) + ∇ × (σt κt n) .

[1.28]

Defining a single non-localized mean curvature is not equivalent to defining the two longitudinal and transverse curvatures from equation [1.28] at the points and barycenters of the cells of the primal topology. The formulation presented here aligns with the approach adopted by [VIN 04], who evaluate the mixed partial derivatives to compute the Hessian of the principal curvatures, even though these derivatives are

28

Discrete Mechanics

not considered by classical methodologies. The mixed derivatives are evaluated at the center of each face of an MAC (marker-and-cell)-type mesh. The use of a single curvature, defined on the points of the primal mesh, leads to a loss of information and accuracy. Returning to the standard continuum approach, consider a point M on a curved surface S defined implicitly by the equation f (x, y, z) = 0. The principal radii of curvature of S are computed on two orthogonal planes that are perpendicular to the tangent plane at the point M . If t1 = (a1 , b1 , c1 ) and t2 = (a2 , b2 , c2 ) are two unit vectors of the tangent plane, and n = ∇f is the normal vector of the interface at the point M , the radius of curvature of the surface in the plane (n, t1 ) is given by the ratio R1 = n/Hes(n, t1 ), where Hes(n, t1 ) = tt1 Ht1 is the Hessian of the surface at the point M and H is the Hessian matrix of second-order partial derivatives of f : 

R1 =

2

2

∂f ∂x

2

+

∂f ∂y

2

2 2

+

∂f ∂z

2 2

2

∂ f ∂ f ∂ f a21 ∂∂xf2 + b21 ∂∂yf2 + c21 ∂∂zf2 + 2a1 b1 ∂x∂y + 2a1 c1 ∂x∂z + 2b1 c1 ∂y∂z

.

[1.29]

The second tangent vector t2 can then easily be determined from its vector products with n and t1 , since it satisfies t1 · t2 = 0 and n · t2 = 0. The second radius of curvature R2 is obtained similarly by repeating this approach in the second plane (n, t2 ). The mean curvature of the surface at the point M is given by κ = 1/R1 + 1/R2 . Let f be the vector of derivatives ∂f /∂xj − ∂f /∂xk for each of the three components i. After observing that ∇f · ∇ × f = 0, we can define one of the tangent vectors t as the unit vector of f by setting t = f /f . The notion of discrete curvature is consistent with these results from differential geometry, as is the fact that we need to define different quantities at both the points and the barycenters of each cell of the primal topology. The mixed second derivative terms in equation [1.29] are taken into account by the discrete approach in the expression of the transverse curvature. An important result from differential analysis shows that the curvature of the surface is independent of the choice of basis (t1 , t2 ); this is another cornerstone of discrete mechanics, where the unit vector t of the edge Γ is defined with respect to a local frame of reference. 1.10. Axioms of discrete mechanics In summary, the above-described principles – WEP and the causality principle – are assumed without reservation; these principles are consistent with every physical experiment that has ever been performed to date. Furthermore, Hodge–Helmholtz

Fundamental Principles of Discrete Mechanics

29

decomposition allows us to decompose any physical vector into an irrotational component and a solenoidal component. We define scalar and vector potentials for the acceleration in order to accumulate the compressive and shear stresses. This is known as the principle of accumulation of stresses. The other axioms of discrete mechanics may be stated as follows: – the fundamental principle of dynamics takes the form of a composition rule for accelerations; the acceleration γ of a particle is equal to the sum of the accelerations g acting upon this particle, γ = g; – the acceleration is an absolute quantity in a single frame of reference; – the velocity is defined as the accumulation of the acceleration V = Vo + γ dt; – like the velocity, the quantities of pressure, stress, density and temperatures are defined as accumulators; – the propagation of relaxed longitudinal waves is directed and has velocity equal to the celerity; – the effects of longitudinal and transverse propagation are disjoint; – absolute time is assumed to exist and unfolds in the positive direction; – there is a discrete horizon within which all points are connected by causality; – any vector may be written as the sum of a solenoidal component and an irrotational component (Hodge–Helmholtz); – for physical phenomena derived from a potential, there always exists both a direct effect and a dual effect; these effects can either cumulate or cancel. Discrete mechanics moves away from various ideas that served as milestones in the history of the evolution of classical mechanics. The following concepts are abandoned: – the notion of a continuum; – the hypothesis of a local equilibrium; – the necessity of using tensors; – the composition rule of velocities viewed as vectors; – the restriction of the velocity to the speed of light in a vacuum. We shall set aside quantities such as force, momentum and energy for now while deriving the equations of motion; like other quantities discussed above, these secondary quantities will be derived later by integrating the primary variables. The hypothesis that the material velocity is bounded by the speed of light is not rejected as such by the field of discrete mechanics; it is simply not required as part of the theoretical framework.

2 Conservation of Acceleration

The principle of equivalence between inertial and gravitational mass is extended to every other acceleration applied to a particle or a medium. Newton’s second law is rephrased as an equality of accelerations: the acceleration experienced by the medium is equal to the sum of every force per unit mass applied to the medium. A series of fundamental experiments are performed to establish models for each acceleration, as well as the roles of physical properties of media such as solids and fluids. The derivation of the discrete laws of mechanics expresses the acceleration formally as a Hodge–Helmholtz decomposition. 2.1. General principles Isaac Newton’s second law, also known as the fundamental law of dynamics, was originally formulated as an axiom and then confirmed experimentally. Rephrased in a more modern form by Leonhard Euler, it states that the inertial force m γ is equal to the sum of all other forces F applied to a body. The scientific community had known about the equivalence of gravitational mass and inertial mass since Galileo, if not earlier, and Isaac Newton himself was no doubt aware of it. At the beginning of the last century, Albert Einstein postulated that the velocity cannot exceed the speed of light. This was viewed as a stronger form of the equivalence principle. Is this equality specific to gravity, or can it be extended to every force? To answer this question, we must define physical quantities that are physically absolute in nature, or in other words quantities that are inherently measurable at every place and every time, independent of any frame of reference, without knowledge of the history of the physical system over time. In mechanics, the acceleration γ has this property. The acceleration of an element of matter, a particle or a physical system can be measured regardless of its current position or velocity. We are currently capable of measuring the acceleration to an accuracy of one part in 1017 ; the weak equivalence principle (WEP) could plausibly be extended to be an axiom in its own right if the accuracy is improved any further.

Discrete Mechanics: Concepts and Applications, First Edition. Jean-Paul Caltagirone. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

32

Discrete Mechanics

The model presented here views the acceleration as an absolute quantity that satisfies mathematical vector addition: γ = γ1 + γ2 ,

[2.1]

where γ denotes the acceleration of the element, and the terms γ1 and γ2 represent two accelerations applied to it. This property has been widely debated in the literature; in discrete mechanics, it is adopted as a postulate and used to derive the equations of motion. It should be noted that none of the accelerations are associated with any specific frame of reference; both accelerations γ1 and γ2 are simply applied to the same edge Γ in local coordinates. The velocity does not satisfy this property. Two velocities V1 and V2 do not sum additively unless we remain strictly within the scope of classical mechanics. The arithmetic sum of two velocities close to the speed of light can exceed it. Therefore, it would contradict the theory of special relativity if velocities were simply additive. The velocity is not an absolute variable; we have always known this, but the equations of mechanics rely heavily on the velocity nonetheless. It seems a priori inappropriate to compare a velocity to a celerity, even if both quantities share the same units; relying on the velocity creates the need for frames of reference and transformations such as the Lorentz transformations of special relativity. In discrete mechanics, the velocity is simply viewed as an accumulator of the acceleration. It satisfies the expression V = Vo + γ dt, where dt is the time elapsed between two observations of the physical system. To know the instantaneous value of the velocity, we need to know its mechanical equilibrium value Vo at time to , then integrate the instantaneous acceleration with respect to time. Hence, if the acceleration is known at every point in time, we can simply integrate it to track the motion of the particle and deduce the instantaneous velocity, provided that Vo is known. It is unnecessary to assume that the velocity is bounded by the speed of light at an axiomatic level in discrete mechanics. The velocity continues to change so long as the acceleration remains non-zero. This effectively rephrases the problem: if the velocity is bounded by the celerity of the medium, the acceleration must necessarily tend to zero. The problem no longer focuses on the velocity or its boundedness; instead, the equations of motion are reformulated in terms of the acceleration. We do not need to impose any a priori restrictions on the acceleration, but its value naturally tends to zero as the velocity approaches the celerity of the medium. We will see later that the bound V < c only holds for rectilinear motion when the curvature of space is non-zero. Many other variables such as the position are also viewed as non-absolute quantities; the displacement itself is the simply the accumulator of the velocity, U = Uo + V dt. The role played by the force as a variable in the equations of

Conservation of Acceleration

33

motion must be reconsidered. Forces are omnipresent in mechanics, but this is merely the remnant of a distant past where the idea of muscular force was deeply ingrained and inescapable. Forces are encountered in every phenomenon – gravity, mechanical tests, capillary phenomena, rotation, viscous effects, etc. But speaking in terms of a gravitational force rather than a gravitational acceleration does not fundamentally alter the nature of the phenomenon of gravity and does not improve our understanding of it. The modern equations of fluid mechanics (Navier–Stokes) and solid mechanics (Navier–Lamé) are phrased in terms of forces or stresses, neglecting to model the history of the system in favor of an instantaneous equilibrium state. As a result, there remains an unknown quantity, such as the pressure, which remains indeterminate, forcing us to invoke some other principle to obtain a closed system. The modern equations of mechanics cannot describe the evolution of a medium with continuous memory. Relativistic mechanics faces the same obstacles. By comparing the velocity to the speed of light, we conclude that the force required to achieve this velocity tends to infinity. This reasoning is precisely what we are questioning: any imposed acceleration, regardless of its value, should cause the velocity to approach the celerity of the medium – in this case the vacuum. The force F in classical mechanics and relativistic mechanics neglects the history of the motion. As a simple analogy, consider a bar that initially has length L subject to a constant stress; the bar begins to stretch at some velocity. As the cumulative sum of all stretching stresses increases, the velocity tends to zero, and the displacement is given by dU. The acceleration is then also zero, whereas the stretching force remains constant. It is easy to see that the relation F = m γ cannot describe this phenomenon directly. In general, zero acceleration does not imply that the sum of the forces is zero; the sum of the applied forces must therefore be extended by a term describing the history of the evolution of the system. In the 18th Century, the concept of celerity was not yet understood; the ideas of velocity, ether and relative motion were still controversial; and the simultaneity of events was taken for granted. An explanation for the experiments performed by Albert Michelson and Edward Morley was given by Ernst Mach; this explanation was then formalized into the theory of special relativity by Albert Einstein based on the assumption that the “speed of light” is identical in every direction. In discrete mechanics, the velocity is not limited a priori. The secondary definition of the velocity as the accumulation of the acceleration prevents us from imposing any bounds on it. Under certain conditions, discussed later, the velocity can exceed the celerity of the medium – both in the case of light and in classically known problems involving fluids. A simplified analysis of modern “classical” mechanics might lead us to expect that the velocity of a medium should become infinite whenever a constant acceleration is applied. However, this is not accurate: for a unidirectional compressible flow, the material velocity of the medium is bounded by the local celerity of sound. The velocity only exceeds this threshold if the curvature of the tube containing the fluid is decreasing. In the case of rectilinear motion, when the curvature is equal to zero, the material velocity cannot exceed the celerity of a wave;

34

Discrete Mechanics

the latter is always a function of the medium through which the wave is propagating, for example c0 for light propagating through a vacuum. The parallels between fluid dynamics and the propagation of light have not been recognized formally, but it is worth noting that the work by Ernst Mach in fluid dynamics acted as a source of inspiration for Albert Einstein’s special relativity. In summary, we can simply eliminate the mass from Newton’s second law because of WEP. Furthermore, the sum of the accelerations g imposed on the system to obtain the acceleration γ should necessarily include a term describing the history of the motion. The fundamental law of discrete dynamics can accordingly be stated as follows: γ = g,

[2.2]

where g is the sum of the applied body forces. In the discrete formalism, γ and g are simply scalar values attached to the oriented edge Γ; these quantities represent the components of the induced and applied acceleration vectors. Equation [2.2] might seem self-evident or redundant; it simply states that the induced acceleration is equal to the imposed acceleration. However, it is no less universal than the original version of the fundamental law of dynamics formulated by Isaac Newton himself. We shall assume that every phenomenon can be modeled by an acceleration in the equations of motion. Besides gravitational effects, which we already know can be represented by an acceleration because of the equivalence principle, any other action induced by viscosity, rotation, a capillary effect, etc. will also be described by an acceleration, or the component of an acceleration on the edge Γ where applicable. Suitable forms are already known for some effects, such as the centrifugal acceleration, but we will need to revisit the underlying physics of other phenomena such as capillary effects to rephrase their action in terms of an acceleration instead of a force. The equations of motion are derived at an abstract level, before specializing into the various subfields mentioned above, for example fluid flows, solid behavior and wave propagation. Instead of Newton’s second law, we shall start with equation [2.2]. The concept of an absolute frame of reference is abandoned. 2.2. Continuous memory Every law of physics represents a guarantee that two experiments performed in two different places by two different observers will produce the same results under identical conditions. In other words, the laws of physics predict the result of future

Conservation of Acceleration

35

repetitions of an experiment. The initial state of a physical system plays a crucial role when predicting the behavior of this system. The initial state must be known; the further into the future we wish to predict, the more accurately we must know the initial state. Classically, the laws of mechanics are applied instantaneously, disregarding the history of the medium. If the initial conditions are not known, we must appeal to some other physical principle, for example constitutive equations or state equations. Alternative workarounds include invoking conservation of mass even though it may not be strictly necessary. In some cases, ad hoc techniques such as these simply generate new problems like material frame-indifference later. They are, nonetheless, capable of producing accurate and consistent results even when their applicability is not fully justified – just for the wrong reasons. Discrete mechanics adopts the axiom of “accumulation of stresses”. Nearby states in time and space are calculated from an earlier reference state; every state is a mechanical equilibrium by definition. More precisely, a state is said to be a mechanical equilibrium if it satisfies the equations of motion perfectly at some fixed point in time. The state of each stress is defined by a scalar equilibrium potential φo together with a vector potential ψ o ; knowledge of these scalar and vector fields at every moment in time yields a formulation with continuous memory. State equations or other secondary equations are no longer required. Only the physical properties of the system, including the celerity, must be known. In the general case, we define both the longitudinal celerity cl and the transverse celerity ct , representing, in fluids, compression effects and rotation-shear effects, respectively. Compression effects are always one-directional. Transverse rotation-shear effects can be polarized. Consider a fluid and a solid from the perspective of classical mechanics. In both cases, the scalar potential φ is defined to be equal to the pressure, or more precisely the ratio of the pressure and the density φ = p/ρ; the units of φ are m2 s−2 . The vector potential ψ is defined to be equal to the shear stress ω divided by the density ψ = ω/ρ. In an elastic solid, the shear stress is fully accumulated over time, whereas in a Newtonian fluid the shear effects are relaxed, with very low time constants. Can the state of a physical system be known without causally linking it to another state at some other time or position? In general, the answer is no. Consider, for example, the Navier–Stokes equations. The primary variable is the velocity V, but there is also a secondary quantity p/ρ. Regardless of whether the motion is compressible or incompressible, the pressure can only be deduced by invoking some other equation. The Navier–Stokes equations are not autonomous; they must always be paired with conservation of mass to correct the shortcomings of their original derivation. The principle of conservation of mass should only need to be applied when the mass varies for reasons other than the motion, for example variations in the temperature or the concentration. Specifying the scalar potential φ, or more precisely

36

Discrete Mechanics

the deviator that is used to update the potential at each moment in time, tells us the history of the motion and makes the equations of motion autonomous. The density does not need to be known a priori. In the equations of motion, the density is simply a physical parameter that is evaluated on the edge Γ. The principle of accumulation evaluates the mechanical disequilibrium at each moment in time by calculating the instantaneous acceleration and updating the potentials φo and ψ o ; the latter are known as the “mechanical equilibrium potentials”. If the equations of motion are perfectly satisfied and the acceleration is zero, a steady state is established and the potentials are independent of time. To describe an evolution in time, another step is added to compute the potentials at time t + dt. The potentials φo and ψ o represent the memory of the evolution of the physical system. The equilibrium state is fully determined by these two quantities and the instantaneous velocity; the new values can then be used to calculate the next evolution in time and space. The stresses p and ω (or the potentials φ and ψ) at time to + dt are updated from the equilibrium stresses at time to , which are denoted as po and ω o (or φo and ψ o ), using the deviators of the compression and shear effects. Below, we shall determine an appropriate model for these deviators by performing a few simple and fundamental experiments. The modeling approach adopted by discrete mechanics is straightforward: the model must satisfy the fundamental principle of discrete mechanics; second, it must reproduce any known results from existing theories that have been observed and confirmed across the various domains of mechanics. The equations derived by the approach must be identical for every medium, whether a compressible or an incompressible fluid, a fluid with a complex rheology, a solid, a porous medium, a multiphase mixture, a vacuum, etc. Likewise, the properties used by the theory must be general in nature; most importantly, they must be perfectly measurable. The longitudinal and transverse celerities cl and ct have these qualities regardless of the medium. These quantities will therefore be preferred over the various coefficients used by engineers, such as the Young’s modulus and the viscosity. The celerities are not necessarily constant in general; they might, for example, depend on the pressure or the temperature. We shall validate this unified theory of mechanics on examples from both fluid mechanics and solid mechanics, as well as the propagation of waves. An extension to relativistic mechanics is briefly mentioned but not discussed in any further detail. It is trivial to show that relativistic mechanics reduces to classical mechanics when the material velocity of the medium is small relative to the speed of light, but we would need to do more than simply recover Newton’s second law when |V| cl ). By replacing dt with dt = dx/V , we can reformulate this equation in terms of the curvature κ and the divergence of the velocity, which corresponds to the local rate of variation of the volume:

c2 1− l2 |V|

dt ∇ · V +

dκ = 0. κ

[4.84]

This can be rewritten as: 1−

c2l |V|2

κ∇·V+

dκ = 0. dt

[4.85]

In one spatial dimension, the equation becomes:

1 dκ dv c2l =− 1− 2 . κ dt V dx

[4.86]

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Discrete Mechanics

This equation describing the flow within a channel with variable cross-section enables us to draw the same conclusions as from Hugoniot’s equation: a flow can only increase to supersonic levels if the curvature is decreasing. Consider a one-dimensional compressible motion along the curvilinear axis s represented in Figure 4.18 during an acceleration phase where the cross-section is increasing.

  







 




Figure 4.18. Stretching of space-time

At each moment in time, the velocity at b is greater than the velocity at a, and this is also true of the divergence of the local velocity, ∇ · Vb > ∇ · Va . The gradient of the divergence of the velocity on the segment [a, b], which we can view as an acceleration, is non-zero. Now, let us return to the Lagrangian formulation of the equations of motion:   dV = −∇ φo − dt c2l ∇ · V + g. dt

[4.87]

In the case of a one-dimensional and compressible flow with zero curvature, equation [4.87] permits the velocity to attain the celerity cl , assuming that the upstream conditions allow it. However, the velocity cannot exceed the celerity. In a shock tube, the limiting velocity attained during the relaxation phase is the local celerity of sound. Models with two or three spatial dimensions work the same way. The curvature is defined by the topology of the channel, e.g. de Laval nozzle [4.19]. In a one-dimensional description of the flow, the velocity cannot exceed the celerity unless the model explicitly includes a variation of the cross-section or the curvature in the equations of motion. In vector notation, the one-dimensional equations of motion can therefore be stated as:



dV κ = −∇ φo − dt c2 1 + o ∇ · V + g, dt κ

[4.88]

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167

where κo is the curvature of space at time to . This formulation reduces to Bernoulli’s equation for a compressible flow, as well as Euler’s equation for a compressible flow with variable cross-section.

 
















Figure 4.19. Stretching of space-time in a divergent nozzle in one spatial dimension. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

5 Two-Phase Flows, Capillarity and Wetting

There are key differences between discrete mechanics and continuum mechanics for two-phase motion, where the physical properties can change abruptly at the interfaces separating different media. Immiscible fluid flows create surface discontinuities in the form of jumps, which are described extremely well by the discrete formalism. This chapter will review some concepts such as curvature from differential geometry to reinterpret their meaning in a discrete context. The static and dynamic effects of capillary phenomena, such as partial wetting, can be described fully consistently by the Hodge–Helmholtz decomposition of the capillary acceleration. 5.1. Formulation of the equations at the interfaces The two-phase flows of immiscible fluids considered here are challenging due to the presence of capillary effects, but also because of the abrupt variations in the properties of the medium: the density, the viscosity or in some cases even the compressibility. Discrete mechanics offers a clear solution for such problems by distinguishing between the two fluids with a phase indicator ξ. Some of the specific difficulties encountered with two-phase flows relate to the local curvature at the interfaces and the concept of wettability at the triple line where the solids, gases or liquids meet. Since these capillary effects depend on other inertial and viscous effects, they need to be integrated into a unified unsteady formulation.

Discrete Mechanics: Concepts and Applications, First Edition. Jean-Paul Caltagirone. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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5.1.1. Modeling the curvature

Figure 5.1. Boundary conditions between two media; the plane Σ is defined by the tangent vector t and the normal to the interface n. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The local curvature is a key quantity that must be evaluated and incorporated into the capillary acceleration term in the equations of motion. In differential geometry, the curvature is calculated directly on the relevant surface. In general, we consider the Frenet frame of reference, which consists of three unit vectors that form a direct orthonormal basis (t, n, m). The vector t is tangent to the surface Σ at the point M , whereas the vector n is orthogonal to the surface at this point. The vector m, known as the binormal vector, is defined as the vector product m = t×n. The plane spanned by (n, t) is the osculating plane, the plane (n, t) is the normal plane and the plane (m, t) is the rectifying plane. The curvature is defined as the divergence per unit surface area of the normal vector of the interface. The radius of curvature of the plane (n, t) is denoted as r, as shown in Figure 5.1, and the radius of torsion is denoted as t; these radii are defined in terms of the derivatives of the basis vectors: ⎧ dt 1 ⎪ ⎪ = n ⎨ ds r(s) dm 1 ⎪ ⎪ = n ⎩ ds t(s)

[5.1]

where s is the arclength along Γ. The curvature κ = 1/r and the torsion τ = 1/t are defined as the inverses of the radius of curvature and the radius of torsion, respectively. The so-called Darboux vector ω of the basis (t, n, m) is defined as: ω = τ t + κ m.

[5.2]

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This vector allows us to express the Frenet formulas more succinctly: ⎧ dt ⎪ ⎪ =ω×t ⎪ ⎪ ⎪ ⎨ ds dm =ω×m ⎪ ds ⎪ ⎪ ⎪ ⎪ ⎩ dn = ω × n ds

[5.3]

where ω is a function of the arclength s. By applying these basic ideas from differential geometry to contours or surfaces and combining them with other concepts, we can construct a series of specific tools for Lagrangian geometries. The normal vector, the mean curvature, the intersection of a line and a surface or two surfaces, etc., can be computed extremely accurately and independently of the topology on which the equations of motion are solved. From a practical point of view, most purely geometric operations in both Euclidean and non-Euclidean settings can be reduced to scalar and vector products.

Figure 5.2. Diagram of the interface detection process on the primal topology. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The Eulerian approach to calculating quantities such as the curvature introduces ideas such as level sets and volume-of-fluid functions that do not align with the interfaces of a two-phase flow. The density, the viscosity and the compressibility are discrete quantities that are assigned to the fluids on either side of the interface. When associated with the Navier–Stokes equations, this perspective leads to approximations, interpolations and smoothing – various types of error that undermine the representativeness of the model.

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For an interface defined in terms of markers in two spatial dimensions, the curvature can simply be found as the inverse of the radius of the osculating circle by considering three consecutive points on the interface. For a simple presentation of this methodology, consider a planar domain and an unstructured triangular mesh, as shown in Figure 5.2. Generalizing to more complex tessellations is perfectly straightforward. Initially, the chain of initial markers is superimposed on the primary mesh. Each marker belongs to exactly one primal cell. The markers can be assigned to their cells using a very fast ray-tracing algorithm. Each chain of markers has certain connectivity properties to track which pairs of markers are neighbors. At the same time, each node of the primal mesh is assigned to either the inside or the outside of a closed chain; when working with an open chain instead, each node is assigned to the left or the right of the (oriented) chain. This defines a phase function ξ on the nodes that takes values zero and one according to whether the node is outside or inside the n-sided polygon formed by the chain of n markers. There are highly efficient algorithms that can be used for this step. We can now very easily and efficiently determine which edge between two markers of the chain intercepts the edge Γ of the primal mesh. This also works in three dimensions simply by finding which plane intercepts the edge. The position of the interface is defined by the markers of the oriented edge Γ, which allows us to specify the value of a parameter α ∈ [0, 1] that will be used to calculate the value of the density ρv assigned to this edge, among other things. The local curvature is calculated according to the Lagrangian approach directly from the normal vector defined by two markers. This procedure yields an exact value in the case of a circle or a sphere; in general, it is of second order in a characteristic parameter describing the distance between two markers. In any case, it achieves a level of accuracy that far exceeds any alternative Eulerian method. The value of the curvature calculated in this way is then assigned to the edge Γ of the primal topology that intercepts the chain of markers. In the general case of an interface that is not a cylinder or a sphere, any variations in the curvature necessarily generate motion due to the non-zero solenoidal component of the capillary term. The next section examines the spatial rate of convergence of the process that determines the curvature from the positions of the markers. To evaluate this rate of convergence, consider the example of an ellipse with radii a = 1 and b = 0.75. The analytic solution of this problem is known. The theoretical curvature of the ellipse is given by: κ=

ab (a2

sin2 θ

+ b2 cos2 θ)

3/2

,

for θ = [0, 2 π], x = cos θ and y = sin θ.

[5.4]

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Figure 5.3 shows the convergence of the curvature calculated from three consecutive markers in the chain with respect to the L2 -norm as a function of the number of markers n.

Figure 5.3. Second-order spatial convergence of the curvature of an ellipse with a = 1, b = 0.75. The quantity ε represents the error with respect to the L2 -norm

The calculation is based on the vector product, like the osculating circle method. It gives an exact solution for a circle with constant curvature, and a solution that is accurate to second order for the ellipse considered here. The order of convergence depends on the characteristic length describing the distance between two markers. Since increasing the number of markers in the chain does not reduce the representativeness of the motion of the fluid and does not significantly increase the computational cost, the accuracy of the curvature can always be chosen to be greater than the accuracy of the description of the motion of the fluid. The curvature of a three-dimensional surface may be calculated analogously by considering every face adjacent to a given point of the surface, as shown in Figure 5.4. There are robust algorithms that yield extremely accurate values of the curvature on surfaces based on triangular tessellations. The connectivity properties of the surface allow us to track the evolution of the surface as it deforms, in addition to calculating the curvature. Meyer et al. [MEY 03] define the normal curvature κN (θ) in terms of the principal curvatures κ1 and κ2 by κN (θ) = κ1 cos2 (θ) + κ2 sin2 (θ); the mean curvature κH is the integral of the normal curvature divided by 2 π and hence satisfies the definition κH = (κ1 + κ2 )/2. In practice, the mean curvature is calculated on a stencil, such as the one shown in Figure 5.4, formed from triangles in space. For deeper reasons

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from differential geometry, the authors define an operator K(xi ) that returns the mean normal curvature at the point xi : K(xi ) =

1 2A



(cotg αij + cotg βij ) (xi − xj ) ,

[5.5]

j∈N1 (i)

where A is the area of the Voronoi surface attached to the central point, the (xi , xj ) are the endpoints of each edge, and the αij and βij are the angles opposite the vector (xi − xj ). The mean curvature κH is computed as the absolute value of the quantity [5.5] divided by two. This expression produces an excellent value of the local curvature for any triangle-based three-dimensional surface.

Figure 5.4. Representation of the stencil used to calculate the curvature of a three-dimensional surface. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

In discrete mechanics, the concept of curvature takes on a slightly different meaning, as discussed in Chapter 1. Although there are various methods to evaluate the mean local curvature of an interface, the discrete curvature must be defined as the variation of the unit vectors t and m along their respective curvilinear axes. The two principal curvatures corresponding to these vectors are called the longitudinal curvature κl and the transverse curvature κt ; as in differential geometry, the mean curvature is defined as the sum of these two curvatures, κ = κl + κt . The longitudinal curvature is assigned to each point, and the transverse curvature is localized on each face of the primal topology. 5.1.2. Formulation of the equations of motion Our objective in this section is to establish a formulation of the capillary effects that includes the pressure generated specifically by these effects, any variations in the

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surface tension per unit mass σ as a function of the temperature (Marangoni effect), any variations in the local curvature κ along the curvilinear axis of the interface and the wettability of the substrate, which is defined by the partial equilibrium wetting angle θ. In dynamic problems, every single one of these effects is justified and necessary, and they must all be combined into a single source term which will be added to the equations of motion. This model must reproduce any indisputable physical experiments, such as overpressure within a droplet or a bubble, the shape of a small droplet on a plane, gravitational ascension in a small capillary or the shape of a lens between two other fluids, Marangoni or Bénard–Maragoni instability and so on. Each of these phenomena is well understood and can be explained in terms of general principles, such as Laplace’s law, Jurin’s law, Plateau’s equation and the Young–Dupré equation. Alongside the contact angle, we can introduce the spread parameter S = γSG − (γSL + γLG ) (where each medium is, respectively, denoted by L for a liquid, G for a gas and S for a solid) to characterize the wetting: if S ≥ 0, then perfect wetting occurs and the fluid spreads fully over the substrate; if S < 0, the wetting is partial. We have two problems to solve: first, we must determine whether these equations are exact or approximate; second, we need to know whether they can be established more generally as an exact solution of the equations of motion. As a general rule, each equation carries certain hypotheses and approximations that many authors overlook or neglect to mention. For example, in the case of capillary ascension, the solution of a static equilibrium (the rest state satisfying Jurin’s law) leads to an interface that is not strictly speaking a spherical cap. Jurin’s law is only satisfied in the limit as the capillary diameter approaches zero, or in other words, as the theoretical height tends to infinity [CAL 15b, CAL 16b]. Introducing other types of forces, such as gravitational or rotational forces, into the problem changes the mechanical equilibrium and calls some of the hypotheses into question. Simply observing these physical phenomena and finding a rudimentary description is not enough to establish mechanical equations that accurately reproduce the above results both at static equilibrium and in the presence of dynamics. A good model (a representative equation) must be capable of reproducing the correct behavior directly in a simulation. Our objective here is to establish a capillary model that correctly describes every relevant effect, including partial wetting. The capillary forces arising from capillary pressure differences, variations in the curvature at the interface, or even the Marangoni effect, can be defined extremely precisely in a frame associated with the interface; these forces are related to the quantities: γ κ δ;

γ

dκ ; ds

dγ dγ dT = , ds dT ds

where ds is the arclength and γ is the usual notion of surface tension.

[5.6]

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Discrete Mechanics

Representing these curvilinear surface effects within a volume is more difficult. This is the objective of so-called continuum surface tension (CSF) modeling, which leads to parasitic current artifacts. In discrete mechanics, the forces are defined on the edges Γ of the primal topology; thus, it is perfectly logical to project each capillary effect onto the edge on which the acceleration is calculated. Regardless of the choice of methodology, whether it is designed to align with the interface or not, this vector description can easily be applied to reproduce the exact classical solutions without any parasitic currents. To begin, we shall introduce the physical parameter σ = γ/ρ, where γ is the classical surface tension, with units of N m−1 . The quantity σ is, therefore, the surface tension per unit mass. In light of the equivalence principle, it seems sensible to introduce a density-independent parameter into the system [3.38]. From a physical point of view, we are not doing anything particularly special. Each medium will simply have its own value for this parameter, and the ratio will be constant at the interface between the two fluids. Given the same observations, different authors might propose different interpretations and definitions for the relevant properties. The property σ, as well as the parameter γ, could be allowed to depend on the temperature. A second indispensable parameter is the curvature κ, which describes the local shape of the interface. The curvature is somewhat more difficult to formulate and its effects are relatively subtle. For example, two surfaces that are homothetic do not necessarily have the same curvature. These two ideas then lead us to consider a third – a jump in some quantity to describe the discontinuity of the variables on either side of the interface. In the equations of classical mechanics, surface or shock discontinuities are presented as conditions that must be considered in parallel to the equations of motion. In the one-fluid model, the jump is described by a Heaviside function. In discrete mechanics, we only need to introduce a phase indicator function ξ, e.g. taking the value 0 in one fluid and 1 in the other. If the interface cuts through the edge Γ, as shown in Figure 1.4, the indicator function would take the values 1 at a and 0 at b. Given that every vector can be represented as the sum of the gradient of a scalar potential and the curl of a vector potential, we can construct a consistent formulation to describe the capillary effects. The various parameters that we need can in fact be combined into a single scalar potential, the capillary potential φc = σ κ. By analogy with the gravitational potential φg and the inertial potential φi , there must exist a direct effect described by a gradient ∇φc together with a dual effect described by a curl ∇ × (φc n). The capillary acceleration can, therefore, be written as: γc = ∇ (σ κl ) + ∇ × (σ κt ) ,

[5.7]

where κt is the vector defined at the center of the cells κt = (κt n) and κl is the curvature at the points of the primal topology. This description is perfectly consistent

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177

with the rest of the discrete formulation, where every effect is presented as a discrete Hodge–Helmholtz decomposition. This formalism is also consistent with the postulate that every effect defined by a potential (here, the potential φc ) can be split into a direct effect and a dual effect. As we have seen in other situations, the two effects sometimes align to produce a doubling effect, like for the mean curvature of a sphere, which is twice that of a cylinder. In other cases, they cancel each other out, like for surfaces with zero mean curvature, such a catenoid, the minimal surface between two circles. Figure 5.5 illustrates the definition of the acceleration γc on the edge Γ; the two actions represented by the divergence of t and the curl of t are both projected onto the principal edge Γ.

Figure 5.5. Capillary acceleration, calculated as the sum of an irrotational term ∇(σl κl ) and a solenoidal term ∇ × (σt κt n), where κl is the longitudinal curvature assigned to the point a (or b) and κt is the transverse curvature localized at c (or c ). The quantities σl and σt denote the surface tension per unit mass at each point. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Unlike the more widespread concept of mean curvature used in continuum mechanics, the discrete mechanical approach only defines the mean curvature on the points of the primal topology, where it is denoted κ and calculated as the sum of the principal curvatures determined by the gradient and its dual. As a result, we only need a single operator to describe the acceleration [5.7] – we shall choose the gradient. The capillary term is therefore given by γc = ∇ (σ κ ξ). In the equations of motion, the capillary acceleration is defined as the gradient of this potential. Every mechanism associated with a capillary effect is incorporated into this term: the pressure jump on either side of an interface, any motion arising from variations in the curvature and any motion induced by the Marangoni effect. Even though the concept of mean curvature itself is more oriented toward the classical equations of continuum mechanics, the form [5.7] is extremely compatible with the discrete approach, where the acceleration γ is simply defined on the edge Γ. This physical model satisfies the most important conventional laws of physics when the relevant hypotheses are given. For example, it satisfies Laplace’s law for a droplet at rest with constant surface tension and constant curvature κ = 1/R; in this

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case, the equations of motion is reduced to ∇pc = ∇(γ κ ξ), i.e. pc = γ/R up to a constant. Similarly, Jurin’s law [CAL 15b] holds as the capillary radius tends to zero. We do not need to introduce any new physical parameters to describe the concept of partial wetting; any parameters that we did attempt to introduce would not achieve the desired level of generality. In fact, the curvature alone suffices to give an account of the experimental observations. For a droplet in static equilibrium, the local curvature is equal everywhere on the surface of the droplet, κ = 1/R. For a hemispherical droplet placed on a horizontal planar surface, the curvature is always equal to 1/R – hence, the curvature along the contact line is also equal to κ = 1/R; the gradient of the curvature is zero in the immediate neighborhood of the triple intersection. Assigning any other value to the curvature κc along the triple line necessarily generates a flow that cannot be eliminated except by setting the curvature to κc everywhere on the interface. The parameter κc is known as the “contact curvature”. The formulation of the capillary term, however, remains unchanged. The equations of motion can, therefore, be stated as follows:     dV = −∇ φo − dt c2l ∇ · V + ∇ × ψ o − dt c2t ∇ × V + ∇ (σ κ ξ) . dt

[5.8]

The equilibrium of forces defined by the Young–Dupré equation for a rest state is correctly represented by equation [5.8], which allows us to find the equilibrium state, but also gives a description of any evolutions of the system, which occur when a disequilibrium is introduced. Even if the form [5.7] is more harmonious with the discrete mechanical approach, equation [5.8] based on the mean curvature still reproduces each of the relevant effects, including wetting. One alternative to solving the vector equation [5.8] is to separate the equations of motion from the incompressibility constraint. Given a field with non-zero divergence predicted by the equations of motion from a pressure field pn calculated at time n, the projection method type approaches project the velocity field onto a divergence-free field [GUE 06]. A pressure correction is calculated first, followed by a velocity projection which ultimately gives the divergence-free velocity field. The discrete approach reverses these steps, calculating the divergence-free field first and then deducing the pressure. This method is called the kinematic scalar projection (KSP) method [CAL 15c]. The KSP method can be extended to work with the capillary term. Consider the oriented edge Γ, defined by the two vertices a and b, intercepted by an interface at

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c. Given a non-divergence-free field V∗ , derived from an explicit or semi-implicit prediction, the KSP method with capillary effects can be stated as follows: ⎧ ⎪ ∇2 φ = −∇ · V∗ ⎪ ⎪ ⎨  ⎪ ⎪ ⎪ ⎩ pb = pa −

c



b

ρ1 ∇φ · t dl −

a

c



b

ρ2 ∇φ · t dl −

[5.9] ∇ (γ κ ξ) · t dl

a

As usual when we encounter the integral of a gradient, we can apply the integral version of the mean value theorem to obtain: pb = pa + ρv [φ]ba + ρv [σ κ ξ]ba ,

[5.10]

where ρv is calculated as a function of the linear fraction occupied by the fluids on the edge Γ. 5.2. Two-phase flows Two-phase flows of immiscible fluids are a good opportunity to investigate the differences between continuum mechanics and discrete mechanics; they require us to handle any variations in the physical properties (viscosity and density) with extreme care, while also paying sufficient attention to other concepts that are specific to capillary effects. The relative size of the inertial and capillary terms, characterized by the Weber number, allows us to distinguish between flows where the capillary effects may be neglected and flows that are primarily driven by capillarity. 5.2.1. Two-phase Poiseuille flow We have already studied single-phase Poiseuille flows in some depth. With a structured Cartesian topology, we obtain the exact solution regardless of the number of points; with an unstructured mesh, the exact solution can be obtained if additional constraints are imposed on the metrics. The numerical and theoretical difficulties posed by the flow of two immiscible fluids in a planar channel, known as “two-phase Poiseuille flow”, are much more significant. Consider a parallel flow of two immiscible fluids with viscosities μ1 and μ2 in a horizontal channel without gravitational effects. The motion is driven by a constant axial pressure gradient Δp. We shall look for a steady-state solution, which therefore does not depend on the density of the fluids. Since the problem is incompressible, steady and one-dimensional, the solution will only depend on the axial component

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of the velocity u(y). Suppose that the channel has length L = 1, height H and the second fluid occupies the lower section of the channel of height h. Then the solution is as follows: ⎧ Δp 2 μ2 Δp h (μ1 − μ2 − (H 2 /h2 ) μ1 ) ⎪ ⎪ u (y) = y + y ⎪ 1 ⎪ 2 μ1 2 μ1 (μ22 − μ1 μ2 + (H/h) μ1 μ ⎪  2) ⎪ ⎨ 2 2 Δp 2 Δp h μ1 − μ2 − (H /h ) μ1 y + u2 (y) = y ⎪ 2 μ2 (μ22 − μ1 μ2 + (H/h) μ1 μ2 ) ⎪ ⎪ ⎪ Δp h H (μ1 − μ2 − (H/h)2 μ1 ) ⎪ Δp h H 2 ⎪ ⎩ − − 2 μ2 2 (μ2 μ22 − μ1 μ2 + (H/h) μ1 μ2 )

[5.11]

We shall consider a test case with parameter values μ1 = 5 · 10−4 ; μ2 = 1.85 · 10−5 ; h = 10−2 ; H = 2 · 10−2 . Table 5.1 lists the absolute value of the residual between the theoretical and numerical solutions for a structured Cartesian mesh. Note that the result does not depend on the spatial approximation. The absolute error is always of the order of the machine error. N

ε

2

1.77733 · 10−14

4

2.60264 · 10−15

8

1.43179 · 10−14

Table 5.1. Two-phase Poiseuille flow, absolute value of the error for a structured Cartesian mesh

The same behavior is found when the simulations are repeated with coarser approximations. For N = 2, with two cells, there is only one test point on the vertical interface; the other two test points are on the walls, which have zero velocity. Even so, the velocity on the interface is consistent with the theoretical prediction. The comparison shown in Figure 5.6 visually confirms that the numerical and analytic solutions coincide; in particular, the change in the slope of the solution at the interface is inversely proportional to the ratio of the viscosities, as expected. If the free surface separating the two media does not lie on an edge of the mesh, we can apply a homogenization technique to the viscosity within the cell by modifying the properties to satisfy the conditions at the interface exactly and implicitly. Consider the same case as before, except with a height h that does not align with an edge of the mesh. The parameter values are μ1 = 5 · 10−4 ; μ2 = 1.85 · 10−5 ; h = 1.2 · 10−2 ; H = 2 · 10−2 . After making suitable choices for the viscosities, the error is of the order of the machine error.

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u 0.2

0.15

0.1

0.05

0

0

0.005

0.01

0.015

z

0.02

Figure 5.6. Two-phase Poiseuille flow. The figure shows the velocity profiles of the theoretical (points) and numerical (dashes); z is the vertical coordinate

If the topology is unstructured, the error depends on the topology. If the topology aligns with the physical interface, we recover the case of two single-phase flows and the solution is exact. If the mesh does not align with the physical interface, the errors accumulate; in general, the order of spatial convergence is two. Figure 5.7 shows the numerical solutions of simulations with aligning and non-aligning meshes for the parameter values: μ1 = 5 · 10−1 ; μ2 = 1 · 10−1 ; h = 0.5; H = 1. The discrete formulation resolves many of the difficulties arising from the physical properties by localizing these properties on specific elements of the topology. Despite this, although some of the discrete operators like the gradient and primal curl are exact, other operators produce results that depend on the metric. 5.2.2. Sloshing of two immiscible fluids The system [3.38] can of course represent acoustic waves with celerity c, but also various other phenomena, depending on the external actions that are introduced into the system: gravity, capillary forces, rotation, etc. In the presence of a constant and uniform gravitational force, different types of gravity waves can form and persist over large time scales. This is for example the case with solitary waves, ocean waves and so on.

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Figure 5.7. Two-phase Poiseuille flow. The pressure field is shown with a few streamlines superimposed on triangular meshes. The mesh on the left aligns with the interface, whereas the mesh on the right does not. The interface is located at the half-way point of the channel. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

We shall consider the case of a liquid sloshing within a cavity partially filled with gas. The principal mechanisms driving this motion are inertia and gravity. The motion of the two fluids, although formally compressible, can be viewed as an incompressible at large time scales. Let L be the length of the cavity, and 2 H its height. The cavity contains a fluid with density ρ2 and viscosity μ2 lying beneath another fluid with density ρ1 and viscosity μ1 . A slight perturbation is applied to the interface separating the two immiscible phases. Under the effect of gravity, the interface then oscillates around its horizontal equilibrium position. When the bottom fluid is at rest, it occupies a region of height H. The concept of viscosity is not particularly relevant to this problem; varying the viscosity of one of the fluids only affects the rate at which the waves attenuate. We shall, therefore, assume that both viscosities are equal to zero, μ1 = μ2 = 0, eliminating the corresponding term from the equations of motion. The evolution of the system over time is therefore determined by a competition between the inertia of the fluid and gravity. In the continuum-based formulation, the gravitational force is ρ g. The latter term does not derive from a scalar potential. The Hodge–Helmholtz decomposition of ρ g = ∇Φ + ∇ × Ψ yields two separate non-zero contributions. The irrotational part modifies the scalar potential po to describe the static gravitational effects; the vector potential modifies the mechanical equilibrium. In the discrete formulation, g does derive from a scalar potential, the potential φ = p/ρ; both formulations lead to the same results up to the interpolation errors of the Navier–Stokes equations. The coupling with the inertia generates a sloshing motion whose frequency can be calculated using the theory of linear stability. If we define the initial perturbation of the interface in terms of Fourier

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modes, i in the longitudinal direction and j for the transverse modes, linear theory gives us the following explicit expression for the frequency [LAM 93]:       i2 i2 j2 j2 1   πg + 2 th π H + 2 , f= 2π L2 l L2 l

[5.12]

where L is the length of the domain in the x-direction and l is the width in the y-direction. The angular frequency ω and the period T are also defined as follows: ω =2πf =

2π . T

[5.13]

The theoretical expression of the frequency [5.12] is established by a linear theory describing the stability of a fluid with density ρ2 without a second fluid lying on top. If the densities ρ1 and ρ2 are close, we need to introduce a correction term [LAN 59], leading to the relation:       2 2 ρ −ρ 2 2 1  i i j j 2 1 π g f= + 2 th π H + 2 . 2π L2 l ρ2 + ρ1 L2 l

[5.14]

The problem is tackled using this system of equations and a specific Arbitrary Lagrangian Eulerian (ALE)-type methodology, developed to separate the two phases by means of a mobile segment of the mesh. This model does not involve any interpolation, neither in the viscosity nor the density, which avoids many of the difficulties encountered by Eulerian algorithms for two-phase flows. Related complications, such as calculating the curvature of the structured mesh, further reduce the representativeness of the simulated phenomena. We shall consider water and air, with densities ρ2 = 1, 000 and ρ1 = 1.1728. The fluids are contained in a square cavity with dimensions L = 2 H = 0.1 m. We will only test the first two-dimensional mode, so i = 1 and j = 0. The time step is 10−3 s, which achieves sufficient accuracy for the oscillation frequency. The initial perturbation is sinusoidal with an amplitude of 1 mm. Figure 5.8 shows the periodic evolution in the height of fluid 2 on a boundary of the domain, at x = 0. The gravitational forces induce downward motion under the highest points of the free surface. If there are no viscous forces, the oscillatory motion is governed by the confrontation of gravity and inertia. The oscillations persist for long periods of time without attenuating noticeably. We also observe that the attenuation of the waves decreases as the time step becomes smaller. This example tests and validates

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all of our methodology collectively: the equations of motion, the spatial discretization, the ALE process, etc.

Figure 5.8. Height of the liquid over time at a point half-way up the left boundary of the domain, for two fluids with densities ρ1 and ρ2 . The viscosity is set to 0, μ1 = μ2 = 0

To quantify the errors introduced in each phase of the modeling and simulation process, we can compare the frequency calculated numerically against the theoretical frequency [5.12]. Table 5.2 shows the period of the oscillations. The numerical and theoretical results are highly consistent, which validates both the formulation [3.38] and the numerical methodology. This example also demonstrates that the kinetic energy is conserved by this formulation when the viscous effects are neglected. Even though there is no viscosity-induced momentum transfer between the primal surfaces, the curl is not zero.

Period

Theoretical

Simulation

0.3742

0.3748

Table 5.2. Period of oscillation of the sloshing in a square cavity, for the first mode of linear stability theory. The numerical value was calculated using the discrete formulation.

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Next, we shall apply the same formulation to simulate a flow generated by a free surface that is not in mechanical equilibrium. Water and air, each occupying one-half of a cavity measuring 4 × 1 m, are subjected to the gravity field g = −9.81 ey . The initial condition of the interface, which is described by the relation y = 0.5 + 0.3 cos(4 π x/L) exp(−x/2), is shown in Figure 5.9. The standard values are used for the physical properties of water and air. The simulation was performed for a total duration of 20 s and a time step of 2 · 10−3 . The deformable mesh is of Lagrangian–Eulerian type, with 64 × 256 cells, including a refinement around the interface to properly capture the rotational layers in its immediate neighborhood. The surface tension is taken into account but has very little significance, given the size of the system.

Figure 5.9. Nonlinear sloshing of two fluids, air and water after with imposed initial conditions (top) and after a duration of t = 20 s (bottom). The simulation uses an ALE approach

The evolution of the system over time is shown in Figure 5.10, which plots the elevation of the free surface at the left boundary of the cavity. The behavior of the solution as a whole remains good over time despite the significant constraints on the simulation: large dimensions, low viscosities, strong gravitational effects, extremely thin rotation layers, etc. The ALE method for simulating nonlinear waves is highly compatible with this formulation and the system [3.38]. This technique avoids any form of interpolation of the density and the viscosity, which likely lies at the root of many of the numerical problems encountered elsewhere. A simulation of sloshing in three spatial dimensions was also performed, initiated from the mode (2, 1) in a parallelepiped-shaped cavity with dimensions 0.2 × 0.1 × 0.05 m and a density ratio ρ2 /ρ1 of around 1,000. Table 5.3 gives an overview of the difference between the theoretical and numerical solutions. Figure 5.11 shows the physical interface, where the vertical velocities take both positive and negative values. The cross-section in the (x, z)-plane plots the isovalue lines of the velocity along the y-axis.

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Figure 5.10. Nonlinear sloshing: height of the liquid over time at half-way point on the left boundary of the domain

Period

Theoretical

Simulation

Mode (2,1)

0.3360

0.3365

Table 5.3. Period of oscillation for sloshing in a hexahedral cavity for the mode (2,1). The numerical value was calculated using the discrete formulation

Figure 5.11. Sloshing in a cavity with dimensions 0.2 × 0.1 × 0.05 m filled with water and air for the mode (2,1). The vertical velocity is shown on the free surface, and the cross-section shows the velocity along y. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

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These examples of flows with free surfaces in cavities and either linear or non-linear perturbations demonstrate that the kinetic energy is preserved; when the viscosity is zero, the integral of the kinetic energy over the cavity as a whole must be stable over time. The 2D and 3D simulations performed with the discrete formalism confirm that this property holds, and that the motion is neither attenuated nor amplified; however, since the rate of convergence of the formalism is of order two, numerical errors will inevitably crop up over long time scales, resulting in a numerical attenuation of the kinetic energy. 5.3. Capillarity-dominated flows 5.3.1. The Laplace problem The most emblematic example of capillary effects is the problem of finding the equilibrium of a cylindrical or spherical droplet. In this section, we shall examine the collective ability of the methodologies described above to maintain a long-term rest state. In this state, by considering the Hodge–Helmholtz decomposition, we see that the surface tension σ and the curvature κ must be constants. If the surface tension was not constant, Marangoni-type currents would arise at the surface of the droplet. A non-constant curvature would lead to currents orthogonal to the interface. Consider the case of a constant surface tension γ = 1. The markers are prearranged into a circle. There are two possible cases – either the curvature is constant up to the machine error, which is, for example, the case if the markers are arranged exactly into a circle of radius R, or there are errors, in which case the markers move under the effect of the curvature differences. In the latter case, our methodology will reposition the markers back onto the circle. If the curvature is calculated exactly using a chain of markers, the source term ∇(σ κ ξ) is injected into the equations of motion. Since the curvature is constant and non-zero, the droplet collapses on itself with velocity equal to the capillary velocity. However, the medium is assumed to be incompressible and there is an equal and opposite force acting on the interface. The equilibrium is therefore obtained almost instantaneously and the pressure is uniform in the interior of the droplet, equal to pc = γ/R in two dimensions, pc = 2 γ/R in the case of a sphere, and pc = γ κ in general. Figure 5.12 shows two examples. The first has a structured mesh that does not align with the circle, and the second has a mesh that aligns with the circular geometry. In both cases, the curvature is exactly equal to κ = 1/R, the pressure is equal to pc = 400 Pa after a single iteration, and the velocity both inside and outside the droplet is strictly equal to 0. The same result is obtained in the case of a sphere represented as a triangulated surface within a cube meshed by regular hexahedra; Figure 5.13 simply shows the isosurface of constant pressure within the droplet.

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Figure 5.12. Capillary pressure in a droplet of radius R = 2.5 · 10−3 m and constant surface tension γ = 1. On the left, the mesh does not align with the disk and the disk is represented by a chain of markers. On the right, the mesh is unstructured but aligns with the disk. In both cases, the pressure is exact and the velocity is strictly equal to zero

Figure 5.13. Pressure in a sphere inside a cube. The pressure is homogeneous throughout the sphere and the pressure difference on either side of the surface is σ κ ∇ξ. The velocity is strictly equal to zero in both media

5.3.2. Oscillating ellipse In a system that is not in mechanical equilibrium, the motion is generated by the imbalance between the gradient of the scalar potential (or pressure) and the capillary acceleration γc = ∇(σ κl ) + ∇ × (σ κt n). In two spatial dimensions, the capillary effects need to be reintroduced into the plane, making the form γc = ∇(σκξ), where σ is the mean curvature, more suitable. As a two-dimensional example, we shall consider the case of an elliptic droplet of water within a square cavity of size d = 10−3 m, filled with air. At the initial moment in time, the large radius of the ellipse is a = 3 · 10−4 m and the small radius is a = 2 · 10−4 m. The density of water is 103 kg m−3 , and the viscosity is set to μ = 10−2 Pa s to reduce the oscillation time; the density of air is 1 kgm−3 , with a viscosity of μ = 2 · 10−5 Pa s. The surface tension is γ = 0.073 Nm−1 . We shall use a method based on an ALE formulation. The triangular mesh is

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189

relatively coarse, but since it is consistent with the geometry of the droplet at each moment in time, it achieves very satisfactory accuracy. Figure 5.14 shows two states in the evolution of the ellipse over time. The first is shortly after the initial state, and the other portrays the static equilibrium after convergence is complete.

Figure 5.14. Evolution of an ellipse over 10 iterations, p ∈ [−172, 352] and 1,000 iterations, p ∈ [−63.3, 234.6], with dt = 10−6 s. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The capillary forces rapidly start to act on the droplet, causing it to deform. Incompressibility is ensured by the compressibility coefficient of approximately 0.444 · 10−9 (corresponding to cl = 1, 500 m s−1 ), the volume (and the mass) is conserved exactly. After the system is allowed to evolve for a relatively long period of time, the droplet becomes circular, and its radius of curvature becomes constant and equal to R = 2.45 · 10−4 m, which is consistent with the constant pressure difference of Δp = 297.9 Pa between the two fluids. Since the initial perturbation was relatively large, the internal and external flows are made somewhat more complex by the inertial effects. Another simulation was also performed for a similar case involving a circle of radius R = 10−3 m and an initial perturbation of 0.1% of the diameter of the circle. The domain, which is approximately circular, oscillates around its average position and seems to converge to the theoretical solution at equilibrium. However, in this case, the viscosity (μ = 10−2 Pa s) means that an extremely large convergence time is required. The evolution of a point on the surface of the circle is plotted in Figure 5.15. The attenuation of the oscillations is slow. In principle, we could compare the frequency of the oscillations against a prediction from linear stability theory [LAN 59]; however, since this example involves viscosity, a full study would be required, which exceeds the scope of this book.

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d

t Figure 5.15. Evolution of the position over time of a point on an ellipse with mean radius 10−3 m according to a linear regime; the initial perturbation is 0.1% of the diameter of the droplet and the time step is dt = 10−6 s. The properties are those of water (μ = 10−3 Pa s) and air, and the surface tension is γ = 0.073 N m−1

5.3.3. Marangoni-type flow in a droplet Temperature variations on the surface of a bubble or a droplet generate rapid, high-amplitude motion as a result of the Marangoni effect. The surface tension can be expressed as follows as a function of the temperature: γ = γ0

T 1− Tc

n ,

[5.15]

where Tc is the critical temperature and n is an exponent that is close to one (more specifically, 11/9 = 1.22). Thus, the surface tension vanishes at the critical point where the meniscus disappears in Natterer’s experiment. Note that the surface tension decreases as the temperature increases. This variation is the underlying cause of Marangoni instability, or Bénard–Marangoni instability, depending on the value of gravity. In the case of water, we have:

11/9 T γ = 0.147 1 − . Tc

[5.16]

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Figure 5.16 shows a simple example without a reference solution that simply intends to demonstrate some of the possibilities and versatility of the formulation. The surface currents induced by the Marangoni effect can be calculated outside, inside and on the sphere.

Figure 5.16. Marangoni currents in a droplet of water with radius R = 1 mm. The trajectories on and within the pseudo-sphere are shown on the left; the free surface is shown on the right, after multiplying the amplitude of the variations in the radius by a factor of 1,000 for visibility. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Even if the initial curvature of a droplet is constant, variations in the capillary potential φc = σ κ ξ along the interface can generate motion directly on the interface itself. This occurs, for example, if the interfacial tension varies due to temperature fluctuations – this is the Marangoni effect. The motion within the droplet is not only at the surface. Indeed, the divergence is non-zero on the surface; sources and sinks can easily be identified wherever there is orthogonal motion at the surface of the droplet, causing it to lose its spherical shape. The surface deformations are very small in amplitude but generate curvature gradients and hence motion in the fluid. The example in question considers a droplet of water with radius R = 1 mm. A periodic temperature variation with amplitude ΔT = 10◦ K is imposed at the surface of the droplet. Figure 5.16 shows some of the trajectories of the fluid within the liquid and at its surface, as well as the deformation of the free surface over the first few moments in time. The variations in the radius of the pseudo-sphere are in phase with the temperature variations and of the order of a fraction of a micron. The tessellation of periodic waves at the interface are also shown in the figure. The initial sphere has 450, 000 tetrahedra to mesh the volume; the surface mesh used to calculate the properties of the surface (normal vectors, curvature, etc.) has 9100 triangles that are aligned with the volume mesh. The capillary source term ∇(σ κ ξ) is introduced into the equations of motion on the edges of the surface. Here, the phase indicator function ξ is equal to one everywhere, since the external medium

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is not meshed; an explicit model of the external medium is necessary here, since the air is assumed to have uniform pressure. Given the solution of the problem at each time step, represented as a velocity–pressure pair (V, p), we can calculate the normal velocity of the interface and hence readjust the position of the interface over time. The metrics are recalculated at every time step, but this does not noticeably increase the computation time. The volumetric divergence is set to zero throughout the simulation to ensure that volume is locally conserved. 5.3.4. Interacting bubbles The methodology described above can be applied to simulate various problems that are of interest in practice. Simulating the recombination of structures, bubbles or droplets by fragmentation or coalescence is more delicate. It is perfectly feasible to manage the chains of markers as evolving structures; this has already been done by several authors (see, for example, [PRO 07]). The case of interacting bubbles, droplets or multimaterials can be formulated straightforwardly by chains of markers and 3Dmeshed surfaces to an accuracy that cannot be rivaled by volume-of-fluid or level-set methods. It has proven relatively easy to manage different materials within the same cell of a structured or unstructured mesh. When using a discrete mechanical methodology, the evolution of the interface is determined by the markers and their positions within the cells. The density ρv is calculated exactly on each edge Γ, and the viscosity is only defined on the faces. Unlike the approaches used by continuum mechanics, interpolation is not needed to evaluate these quantities at other localizations in the stencil. Figure 5.17 shows the result of a static simulation whose initial conditions are given by a rest state with zero pressure and velocity. The equations of motion describe the collapse of the bubbles, contracted by the incompressibility imposed on each medium, in a single iteration. Instantaneously, the pressure is equal to its theoretical value, and the geometry is perfectly conserved, since the calculated value of the velocity is zero. If the simulation is continued, the rest state is conserved. Plateau’s equation predicts a contact angle of 120◦ degrees between the surfaces of this geometry, which is reproduced exactly by the simulation. An extension of the same problem to a 3D surface is shown in Figure 5.18. There are two conceivable approaches to managing the motion of a 2D or 3D structure. The first method embeds the geometry into a structured or unstructured mesh (as shown in Figure 5.17), then calculates the motion inside and outside of the bubbles by solving the equations of motion. Alternatively, we could apply the concept of capillary velocity to the 3D case shown in Figure 5.18 to compute the evolution of the markers from the incompressibility alone, without calculating the motion of the fluid. Unsurprisingly, different approaches work better for different problems. When

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looking for the equilibrium state of a complex structure, the second method will typically be more suitable.

Figure 5.17. Two immiscible droplets in static equilibrium within a third medium. The pressure in each structure is equal to (pi − pj )/κ = const. The left image shows the two chains of markers on a Cartesian mesh; the right image shows a Lagrangian reconstruction. The pressure values obtained by the simulation are exact. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Figure 5.18. Two immiscible droplets in a static equilibrium, defined by 3D-meshed surfaces. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The methodology developed in this book is particularly effective for problems involving bubbles. The principal difference relative to a droplet is that there are two interfaces, which doubles the capillary effects. The gas can either be the same or different on either side of the interface. The film is not described explicitly and has zero thickness. We could, however, assign a thickness to the film and allow it to vary under the effect of forces, such as gravity and inertia, by solving the equations of motion on a 3D surface. The divergence per unit surface area is not necessarily zero, so the thickness of the film can be allowed to vary over time.

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The following examples were calculated with the assumption of zero thickness. Figure 5.19 shows an equilibrium with three bubbles, in two and three dimensions. In each case, the contact angles between the bubbles satisfy Plateau’s equation, with θ = 120◦ . Each surface in contact with the external medium is either a circular arc or a spherical cap. The bubbles have the same volume, so the internal pressure is the same in each of them and any internal surfaces are planar. The contact angle between a bubble and a plane is 90◦ . This requires two interfaces and a wettable surface – a bubble cannot continue to exist on a dry surface.

Figure 5.19. Three 2D bubbles of same radius in equilibrium (left), and three 3D bubbles on a perfectly wetting plane (right). For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

In any structure assembled from bubbles, the pressure in the gas within each bubble will be nearly uniform. Gravity plays a minor role but is not completely negligible. With these assumptions, we can calculate the evolution of the assembly without needing a mesh. 5.3.5. Simulating foam in equilibrium The same methodology can also be extended to simulate the evolution of foam, or at least dry foam. Given the thickness of the films making up this type of foam, it would be hopeless to attempt to simulate these structures by a purely Eulerian method. However, it is conceivable to assign a single pressure point to each bubble, then calculate the evolution of the foam using the velocity on the edges connecting the bubbles between themselves. After accounting for the incompressibility of the medium, each bubble could be allowed to deform while keeping the same volume. The rearrangement of the foam can then be computed by calculating the local curvature of the film. Since each film is a circular arc, it can be defined with just three points. Any perturbations are compensated by a motion of the interface, and consequently the entire foam, to ensure that the incompressibility constraint remains

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satisfied. A diagram of the foam model is shown in Figure 5.20. The equation governing the evolution of a dry foam may be stated as follows: dV =∇ dt



po − dt c2l ∇ · V + ∇ (σ κ ξ) . ρv

[5.17]

Figure 5.20. Diagram of the model of a dry foam; the curvilinear membranes of the foam are shown in blue, the localization of the pressure is shown by red points and the black points identify the markers used to calculate the curvature. The velocity is calculated on the edges between the pressure points. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The curvature κ is computed according to a Lagrangian approach using chains of markers; the velocity V represents the velocity of the interface, since, by definition, the liquid film is impermeable to the gas. The interfaces deform over time to satisfy a new mechanical equilibrium that leads to a rest state; each surface is a spherical cap (or a circular arc in two dimensions). At rest, the pressure in each bubble satisfies the condition (pi − pj )/κ = const., where pj is the pressure in the bubbles adjacent to the ith bubble. The ripening of the foam is more difficult to simulate – we need to describe the curvature between each pair of interfaces; at the relevant point, the curvature is greater than in two adjacent interfaces. The capillary forces at this point are higher, and the films experience drainage toward the triple line, where the accumulation of liquid modifies the curvature. This drainage reduces the thickness of the film until the interface bursts, and the foam reassembles into bubbles with larger volumes. 5.4. Partial wetting To describe the wetting of a realistic surface, we must overcome a range of obstacles associated with defining, characterizing and modeling the capillary effects

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in the presence of three separate media – usually a gas, a liquid and a solid – which meet at the so-called triple line. Flows involving capillary effects depend on two characteristic properties: the surface tension σij between the ith and jth media, and the local curvature of the free surface Σ of the immiscible media. Do we really need to introduce another physical parameter to describe the concept of wetting? For the past few decades, the proposed solution has been to observe the contact angle θ between two of the media, typically a liquid surface and a solid substrate. This contact angle represents a static measurement that is deduced from an array of theoretical principles and experimental methodologies. Much has been written about contact angles in the literature, but since the perspective developed here diverges significantly from past work, we shall focus on a presentation that aligns with the discrete approach. When the interface is in motion, past approaches have proceeded by introducing a time-dependent contact angle known as the dynamic contact angle; various observation-inspired laws have been formulated to describe the evolution of the angle θ over time under certain specific circumstances. We shall tackle the problem at a global level from the very start – since we ultimately want to know the acceleration γ of the fluid on the edge Γ, it must be possible to phrase any actions on this edge as a sum of contributions. First of all, there must be inertial and viscous accelerations, a gravitational acceleration and finally a capillary acceleration γc . The motion of the triple line depends on all of these effects; it does not make sense to define a dynamic contact angle that depends exclusively on time a priori. The model proposed by discrete mechanics is based on the following observation: if the interface Σ with the substrate is allowed to move freely, it will naturally attempt to recover the equilibrium position defined by the static contact angle θ. The dynamics themselves are governed by the acceleration γ. The physical behavior of the interface in the neighborhood of the triple line is determined by the need to establish a mechanical equilibrium from the accelerations of each pair of media. The curvature of the interface in this region is in fact the most effective parameter for describing the tendency of the system to return to equilibrium. If the so-called contact curvature κc is known, then, although the values of the instantaneous curvature κ depend on each of the various effects, they will tend toward κc at mechanical equilibrium. For example, in the absence of gravity, a droplet of liquid on a non-deformable solid substrate will tend toward having the curvature κc that corresponds to the static contact angle θ; at equilibrium, the entire interface will have curvature κc . The relation between the contact curvature κc and the contact angle θ is established by considering the geometry of the interface. The key advantage of the proposed model is that it avoids introducing a new parameter, since the curvature is already part of the model of γc . We can simply require the curvature to be equal to κc on the triple line and allow the dynamics of the system to be governed naturally by the equations of motion themselves. Figure 5.21 shows the various equilibria obtained for partial wetting by solving the system [5.8]. The spherical-cap shape is modeled directly and the curvature on the

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primal topology can be calculated by a formula from differential geometry. In the case of a spherical cap, the curvature is exact (κ = 1/R) up to the machine error. Since the contact curvature is κc = κ, the velocity is zero, but the capillary pressure is equal to pc = γ κ, as expected. We will consider the case of the unsteady evolution of a droplet on a plane later. Any droplet of liquid on a plane that does not satisfy the equilibrium −∇po + ∇ (γ κ ξ) = 0

[5.18]

will necessarily generate motion and oscillations due to the inertial effects that lead it toward the equilibrium state with pressure pc = γ κ.

Figure 5.21. Simulation of the wetting of a droplet on a planar surface, from left to right. The contact curvature is equal to the mean curvature. The final velocity is zero and the pressure is pc = γ κ

In the presence of gravity, but without hysteresis or inertial effects, the velocity of the droplet on a plane inclined at an angle of ϕ is constant and the acceleration is zero; the equation governing the mechanical equilibrium in this case may be stated as follows: −∇po + ∇ (γ κ ξ) + ρ sin ϕ g = 0.

[5.19]

Gravity introduces an asymmetry between the advancing contact angle and the receding contact angle. As a result, the droplet no longer has the shape of a spherical cap or a circular arc. The solution can only be found by solving equation [5.8]. The notions of contact angle and contact curvature are of course linked – bijectively, in the case of a static equilibrium without external forces. From a physical point of view, the contact curvature is a local and intrinsic property of the surface; it is the maximum curvature that the interface can sustain before a flow is generated to find another equilibrium position. For example, in the case of the lotus effect, the mean surface can be viewed as non-wetting, but this property arises from the high curvature of the interface around the hairs of the leaf. Any gravitationally induced effects are compensated by variations in the curvature, which is essentially constant at a

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sufficient distance from the contact points. In the general case, it seems impossible to predict the resting contact angle in the presence of forces such as gravity. The static and dynamic equilibria themselves are physically complex and it is difficult to give an answer that is both simple and general. The strategy adopted here is to construct a set of equations of motion that can account for any potential capillary effects, then apply this model to perform direct simulations. The term ∇(σ κ ξ) seems to incorporate all of the desired effects. The examples given in the following help to justify this perspective. By using this concept of contact curvature, we can guarantee that the local curvature in the neighborhood of the triple line is equal to κc at the static equilibrium. Without gravity, in the case of perfect wetting, κc is equal to zero. For a perfectly non-wetting surface, the contact curvature must be smaller or equal to the inverse of the radius of the sphere that would no longer be in contact with the surface. Again, the desired physical phenomena can only be reproduced if the equations of motion themselves are representative. The contact curvature directly describes the equilibrium or disequilibrium in terms of the resultant of the accelerations on the triple line. Phenomena involving hysteresis are not addressed here, but they can be approached with the same formalism by replacing the advancing and receding contact angles with two distinct contact curvatures. 5.4.1. Droplet in equilibrium on a plane This section studies the equilibrium of a droplet on a plane without gravitational effects using a marker-based method on a regular structured marker-and-cell Cartesian mesh that does not align with the interface. Our objective is to verify that a rest state is attained when the contact curvature κc is equal to the constant curvature 1/R of the droplet in the case of a circular planar geometry. The surface tension is assumed to be constant. The radius of the osculating circle passing through any three markers is exactly constant throughout the entire marker chain, so the calculated value of the curvature is exact up to the machine error. The term ∇(σ κ ξ) reduces to σ κ ∇ξ, so the capillary pressure jump is equal to pc = γκ. The simulation is performed with zero pressure and zero velocity, like the initial conditions. Equation [3.38] is solved in a single time step for two quasi-incompressible media, χT = 10−10 . The solution obtained after one step has 0 velocity up to the machine error and pressure equal to the exact theoretical value. This favorable behavior can be traced back to the fact that the incompressibility constraint is implicitly satisfied. In conclusion, if the local contact curvature is equal to the curvature of the droplet, we obtain a static equilibrium, as expected. Figure 5.22 shows the solution obtained when the internal pressure of the droplet is higher than that of the external medium. Figure 5.23 shows the final result of an

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unsteady simulation where an initially hemispherical droplet on a solid plane takes on its equilibrium shape after setting the contact curvature to θ = 30◦ . The numerical methodology applies the front-tracking method to a structured mesh that does not align with the interface; since the curvature of a circle is exact up to the machine error, the velocity at the mechanical equilibrium is strictly zero.

Figure 5.22. Modeling the wetting of a droplet on a planar surface; the radius of curvature is equal to R = 10−3 m. The position of the center is (0, −1.5 · 10−3 ). The capillary pressure difference is Δpc = 103 Pa, up to the machine error. The velocity is strictly zero

Let us now simulate the same problem with an unstructured mesh based on regular triangles. The droplet and its surroundings are represented as two distinct domains separated by an initially circular interface. As shown in Figure 5.24, the phase indicator is set to 0 or 1 depending on the fluid; the nodes of the interface are 1 The interface intersects the edges of the triangles arbitrarily assigned to medium . 2 at as small a distance as possible, set to the machine error in adjacent to medium  practice. This procedure eliminates any interpolation on the density and the viscosity.

Figure 5.23. Droplet on a plane without gravity. Above, the equilibrium state at the initial moment in time, where the contact angle is 90◦ ; the pressure is exactly equal to its theoretical value pc = 100 and the system is at rest (V ≡ 0). Below, the mechanical equilibrium for a contact angle θ of 30◦ ; the pressure is pc = 24.0143, and the velocity is also 0. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

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The triple line represented by the black point in Figure 5.24 belongs to both the interface and the solid wall. We can assign a special value of the curvature to this point, the contact curvature κc . If this value is the same as the rest of the interface, then, like for the simulation with a structured mesh, the velocity is 0 and the overpressure in the droplet is therefore equal to pc = γ κ. If κc = κ, the forces are no longer balanced on the triple line, and the resultant generates motion that spreads to the rest of the droplet by continuity. In conclusion, whether or not the interface aligns with the mesh, the equilibrium of a droplet is guaranteed whenever κc = κ in the absence of gravity, and the contact angle θ can also be used to describe the static equilibrium. If gravity or some other force is present, the contact angle relation is no longer satisfied, but the concept of contact curvature remains applicable at static equilibrium.

Figure 5.24. Model of the wetting of a droplet on a planar surface. The red points 2 where are on the interface, where ξ = 1 and κ∗ = κ, the blue points are in fluid , ξ = 0, and the black point is on the triple line, where ξ = 1 and κc = κ. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

5.4.2. Spreading of a droplet The evolution of a droplet of water on a partially wetting plane is a classical problem that has the advantage of taking on a rest state at equilibrium. The initial shape of the droplet is a semicircle of radius R = 1 cm, placed on a horizontal plane. Gravity is not taken into account; the surface tension is constant and equal to γ = 1. From the initial moment onward, the pressure in the droplet is uniform and equal to pc = 100 Pa. Motion is induced by imposing a contact curvature of κc = 24.0143 onto the two points of the domain representing the contact line; for the geometry of this problem, this value is equivalent to a contact angle of θ = 30◦ . The gradient of the curvature around the triple line induces a motion of the fluid outside and inside the droplet, causing the droplet to spread out. By the assumption of incompressibility, the volume of the droplet must remain constant, so its height decreases. Figure 5.25 shows an instantaneous snapshot of the shape and motion of the fluid. However, at the moment shown, the coupling condition is not satisfied. There

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cannot be instantaneous coupling and the contact angle cannot be imposed. Thus, the contact angle varies as the droplet spreads out as a function of the inertia forces, the viscosity and the capillary forces of the problem. The intermediate state is shown in Figure 5.25. Convergence to this state is somewhat slow, but once it is attained, the shape of the interface becomes a circular segment whose curvature is indeed equal to κc , with zero velocity, and a final pressure of pc = 24.0143 P a. From a dynamical point of view, the problem studied here is that of a slipping surface where V · n = 0. We could of course impose no-slip conditions for the fluid on the wall, V = 0, but, at the scale of the problem, given that the viscosity of water is small, it would not significantly change the dynamics of the spread. The same would not be true at smaller scales, where the dimensions of the dynamic boundary layer of the droplet are of the same order as the droplet itself. The dynamic constraint cannot be imposed on the progression of the triple line itself, which simply slides freely over the wall according to the forces acting upon it.

Figure 5.25. Spreading of a droplet over a slipping surface. Initially, the droplet is a semicircle. The colors show the magnitude of the horizontal velocity and the streamlines show the circulation of the fluid within the cavity. The values of the pressure in the initial and final states are shown in Figure 5.23. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Note that, in the absence of external forces such as gravity, the contact angle and the contact curvature satisfy simple geometric relations. In this case, the surface area S of the droplet is initially fixed. If the initial surface area of the spherical cap is S = π R2 /2, the same surface area must be maintained throughout the rest of the simulation; the methodology of incompressible problems conserves both volume and mass. The action of the capillary stresses on the triple line modifies the local curvature. However, at equilibrium, the shape of the droplet is a spherical cap whose characteristics are straightforward to calculate. Writing r for the radius of the spherical cap and θ for the contact angle, we have: S=

r 1 (2 θ − sin(2 θ)) = (2 θ − sin(2 θ)) . 2 2κ

[5.20]

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Since S is constant, we can calculate the contact angle with Newton’s method by solving equation [5.20]. In particular, when θ = √ 0, the curvature is zero. When the contact angle is θ = 180◦ , the curvature is κ = 2/R. The source term of this problem derives from just one scalar potential, the capillary pressure. This would no longer be the case if we took gravity into account: even if gravity itself derives from another potential, the equilibrium surface is no longer a circle and the term ∇(σ κ ξ) + g would include a non-zero solenoidal component. The static equilibrium satisfies the condition imposed on the contact curvature κc , but the shape of the interface is determined by the equilibrium of forces, which can only be reproduced from equation [3.38]. Let us now simulate the same dynamic problem using the ALE method in two dimensions with an adaptive mash based on regular triangles. A droplet shaped like a semicircle of radius R = 0.625 · 10−3 m is placed on a plane whose wettability is allowed to vary over time in terms of the contact curvature κc . The density of the fluid is ρ = 1, 000, and its viscosity is μ = 10−2 kg m−1 s−1 . The external medium is taken to be air. The curvature of the initial semicircle is equal to κ = 1, 600; after the initial moment in time, we immediately impose a contact curvature κc = 750 to simulate a surface with high wettability. After time t = 0.2 s, this value is increased to κc = 2, 230, so that the wettability of the surface decreases. Table 5.4 lists the characteristic properties of the droplet in each step of the simulation. Curvature (m−1 ) Radius of the meniscus (m) Contact angle (◦ ) Capillary pressure (Pa) 1600 750 2230

0.625 · 10−3

90

112

1.333 · 10

−3

48.2

52.5

0.415 · 10

−3

150

155.04

Table 5.4. Evolution of a droplet on a plane; characteristics of the meniscus in each step of the simulation

Figure 5.26 shows the shape of the droplet in each quasi-steady state with negligible velocity. In each case, the pressure inside the droplet is nearly constant. As soon as κc is modified, the forces acting on the triple line lead the droplet toward an equilibrium state where the contact curvature κc is satisfied. Over time, the velocity evolves within the domain, first causing the droplet to spread until a contact angle of 48.2◦ is attained, at which point the velocity field is similar to the fields shown in Figure 5.25, and then causing the interface to contract until the contact angle becomes 150◦ .

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Figure 5.26. Simulation of the evolution of a droplet on a planar surface as a function of the curvature applied to the wall. In the top image, the droplet is in equilibrium, and the initial conditions specify a curvature of κc = 1,600 and contact angle of θ = 90◦ . In the middle, a curvature of κc = 750 (θ = 48.2◦ ) is imposed on the triple line. The bottom image shows the equilibrium obtained after imposing κc = 2,230 (θ = 150◦ ) at time t = 0.2 s. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Figure 5.27 shows the evolution of the pressure difference between the interior and the exterior of the droplet. In fact, the term ∇(σ κ ξ) is an acceleration; it takes high values whenever the contact curvature changes and becomes equal to zero when the state approaches equilibrium.

pc

t Figure 5.27. Evolution of the mean pressure difference pc (in Pa) inside and outside of a droplet on a planar surface as a function of time. The curvature at the wall is modified at time t = 0.2 s. The capillary pressure asymptotes are pc = 52.5 and pc = 155

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5.4.3. Droplet acted upon by gravity When gravity is present, the static equilibrium of a rest state is more delicate to describe. Exact solutions are much less common; in the general case, a solution can only be given by solving the equations of motion [3.38] numerically. Consider a semicircular droplet initially at equilibrium on a plane with a contact angle of θ = 90◦ . At the initial moment, gravity is increased from 0 to g = −100 m s−2 . The weight of the droplet causes it to flatten out and significantly changes its shape. The curvature, initially equal to κ = 1600, becomes variable along the interface, but the maximum curvature gradually tends toward the imposed value of κc = 1600 over time. Figure 5.28 shows a rest state very close to the static equilibrium. Gravity exerts a noticeable effect on the pressure field. Unlike the contact angle θ, the concept of contact curvature κc is extremely robust. It simply describes the action of any effects applied to the triple line. In the description of a discrete medium, any forces are replaced by accelerations, which provides a number of advantages. Furthermore, we do not need to introduce any other information than is already present in the discrete equations of motion. The concept of curvature, which is already defined on any interface to describe the capillary effects, can also be used to describe phenomena involving the triple line.

Figure 5.28. Equilibrium of a droplet of radius R = 6.25 · 10−4 under gravity g = -100 m s−2 from an initial state where the droplet is hemispherical and has 0 velocity. The maximum pressure variation in the pressure field is Δp = 188.9 Pa, and the curvature σ takes values in [835, 1574]. Conservation of mass is satisfied up to the machine error. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The system [3.38] does not require any boundary conditions, neither on a vector variable such as the velocity component V nor a scalar variable. Every constraint is incorporated implicitly [CAL 15a] in the same way as partial wetting.

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5.4.4. Flows within a lens The spread of a droplet of liquid between two other immiscible fluids is a slightly different problem from a droplet on a solid substrate; the latter has intrinsic surface properties that can be described as wetting or non-wetting. In the case of three immiscible fluids, the position of the triple line reflects an equilibrium between the fluids, all three of which are viewed as perfectly wetting. Another difference is that the normal displacement of the wall is negligible for a solid substrate, but not for three fluids. Consider the problem of a two-dimensional droplet that is initially circular and has radius R. The points of the trace of the triple line are placed at (R, 0) and (R, π). The density of the fluid in the droplet is ρ = 1000 kg m−3 , and its viscosity is taken to be μ = 10−2 P a s to reduce the time needed to return to equilibrium. The surface 1 is denoted tension between the fluid in the droplet and the upper fluid (interface ) σ12 , and, similarly, the surface tension between the fluid in the droplet and the lower 2 is denoted σ13 . The surface tension between the upper and lower fluid (interface ) fluids, outside the droplet, is denoted σ23 . The lens can only be in equilibrium if the Young–Dupré equation is satisfied: σ31 t31 + σ12 t12 = σ32 t32 ,

[5.21]

where the vector t denotes the units vectors on the triple line. This equality describes the static equilibrium of the three fluids. This problem can be approached either by considering a domain occupied by the three fluids or by only modeling the droplet fluid and its two interfaces directly, introducing the effect of the third fluid via a force acting on the triple line; in this 3 between fluids (2) and (3) is not represented explicitly – only case, the interface  its action is modeled. We shall adopt this second approach. The impact of the pair of fluids (32) on the droplet is regulated by the surface tension σ32 or the curvature κc imposed on the triple line. In our case, the surface tensions are taken to have values σ12 = 0.05 and σ13 = 0.1. Near the static equilibrium, the curvatures are equal to κ1 = 571 and κ2 = 690, compared to a curvature of κ = 1000 for the initial circle. The corresponding tangent vectors are given by t12 = (−0.364, 0.931), t31 = (−0.62, 0.78), from which it follows that t32 = (0.984, −0.151). As expected, the interface (32) is not horizontal. In cases with gravity, this interface still becomes planar far away from the triple line, but the curvature along it is no longer a constant. If there are any external forces such as gravity, the interfaces are no longer circular segments or spherical caps; the contact angles will differ from the angles observed without these forces.

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Figure 5.29 shows the evolution of the lens over time; initially, it has the shape of a disk of radius R = 10−3 m. Since the surface tension σ31 is larger than σ12 , the lens loses its symmetry relative to the horizontal axis, and the fluid tends to rise within the lens, generating recirculation due to the curvature gradients. The motion attenuates very slowly and the system converges toward a rest state.

Figure 5.29. Evolution of a lens from an initial state where the droplet is circular (left). The action of the two other fluids generates motion that tends to cause the points of the triple line to move apart (the vertical velocity is shown in the center). The right-hand figure shows the final state, which is a static equilibrium. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

5.4.5. Capillary ascension in a tube Capillary ascension in a tube or between two planes is an especially interesting problem: the static equilibrium and the height attained by the fluid in the tube depend on the characteristics of the problem (surface tension, radius of the tube, contact angle), but also on gravity. Classically, the ascension height h is calculated by Jurin’s law: h=

γ , ρ g r cos θ

[5.22]

where r is the radius of the tube and θ is the contact angle of a partial wetting. In fact, this equation is merely an approximation. It cannot be satisfied exactly, and the shape of the interface cannot be a perfect spherical cap. The curvature could only be constant if the following equation were satisfied: −∇p + ∇ (γ κ ξ) + ρ g = 0.

[5.23]

Since the interface is not orthogonal to gravity, motion is generated, and the disequilibrium generates a curl. Furthermore, since the surface tension and the curvature are constant, the second term of [5.23] is a pure gradient, and the pressure

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can only be equal to p = γ κ. As a result, the solenoidal component is not compensated, and the equation is not satisfied. The only way that the interface can be a spherical cap is if there exists an equilibrium vector potential ω o that balances out the solenoidal component of ρ g. If so, the equation describing the equilibrium is as follows: −∇p + ∇ (γ κ ξ) + ∇ × ω o + ρ g = 0.

[5.24]

This hypothesis was proposed and developed in [CAL 15b]. This problem represents a limiting case that remains valid even when persistent rotation effects are allowed, for example, in a fluid with a complex rheology that conserves shearing. Without attempting a microscopic description, one possible hypothesis is that the rheology of the liquid is modified at very small scales in the immediate neighborhood of the triple line. In the classical case of a purely Newtonian fluid, the vector potential ψ o does not exist. Accordingly, the free surface of the meniscus cannot be a spherical cap and Jurin’s law is only an approximation that holds when the radius of the tube tends to zero. Geometrically, the shape of the interface is not straightforward to characterize – to find it, we must solve the equations describing its evolution: dV = −∇ dt



po + dt c2l ∇ · V + ∇ (σ κ ξ) + g, ρv

[5.25]

where the viscous term has been neglected. The term in ∇ · V ensures that the incompressibility constraint is met at every moment in time. After converging, the acceleration is zero and equation [5.23] is satisfied. Performing an accurate validation of this problem using Jurin’s law is more complex than some other studies on the topic might seem to suggest. The unsteady problem described in a model by Washburn [WAS 21] is even more challenging. Clearly, the physical solution should lead to a complete rest state; however, according to the results obtained from the Navier–Stoke equations, high-intensity counter-rotating vortices continue to form in the immediate neighborhood of the triple line, regardless of the duration of the simulation.

6 Stresses and Strains in Solids

The primary objective of discrete mechanics is to unify various laws from different areas of physics, such as fluid mechanics and solid mechanics. This same objective was also pursued by continuum mechanics, but the latter has not been entirely successful in accomplishing it. By choosing the acceleration as the primary variable, we can express the velocity and the displacement simply as quantities that accumulate over time. This gives a dual representation in terms of both the strain and the stress that does not require any additional compatibility conditions. The resulting formulation is able to describe the motion of complex media under large deformations and large displacements. 6.1. Discrete solid medium There are no fundamental differences between fluids and solids. The same equations govern the behavior of both types of medium – only their properties change. Solids have properties that cause them to accumulate rotational stress, whereas fluids do not. Isaac Newton (1643–1727) and Robert Hooke (1636–1703) laid the first foundations of our understanding of the behavior of fluids and solids by introducing the well-known laws of physics which bear their names – defining the stress as linearly proportional to the velocity gradient in fluids, and as a function of the deformation in solids. An attempt to unify these two types of mechanics was only made much later; it does not seem to have ever been successfully concluded. The fundamental law of dynamics, also known as Newton’s second law of motion, was originally formulated in the Principia in 1759 [NEW 90]. Since then, the field of mechanics has changed. The 18th Century was especially significant, introducing a rapid succession of new concepts to improve the models of solid and fluid behavior. The notion of a continuum and tensors, the introduction of thermodynamics and the invention of constitutive equations gradually became

Discrete Mechanics: Concepts and Applications, First Edition. Jean-Paul Caltagirone. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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established as immutable principles, resulting in the Navier–Lamé equations for solids and the Navier–Stokes equations for fluids. At a fundamental level, the concept of a continuum [GER 95, TRU 74] introduces degeneracy in every quantity at the points of the material – whether this quantity is a scalar, a vector or a higher order tensor. A continuum requires a local description in some absolute frame of reference or a relative description in a Galilean frame. Vectors are represented by their components in a three-dimensional space. There are various flaws inherent to this local approach of continuum mechanics. For example, since the stresses are defined using a second-order tensor, the Cauchy tensor, we are forced to take the divergence to reduce their order to that of the velocity itself. This also introduces a large number of coefficients that can only be eliminated by applying external arguments, such as the Clausius–Duhem principle, even though it is just an inequality. Over time, the equations of mechanics were gradually differentiated further, for example by introducing constitutive laws into the equations of motion. This, in turn, made it necessary to formalize new constraints, such as the principle of material frame-indifference or objectivity, which states that the information of the system should be preserved when a rotation is applied. Another example is the use of nonlinear tensors to describe large deformations in solid mechanics. There are many other reasons to question the current form of continuum mechanics, as explored in [CAL 15a] from the perspective of a new type of mechanics. Discrete mechanics [CAL 15a] is founded upon the observations described above. The concept of vector is redefined as an oriented bipoint, very similar to the original vision adopted by Newton. This is one aspect of a quantity that we can conceptually still view as a velocity, except that knowledge of a specific absolute frame of reference is not required. If the edge itself has an orientation, the velocity is simply a scalar quantity. The definition of a discrete medium does not mean that we are working at a microscopic scale. Rather, the discreteness of the approach arises from the notion of a bipoint (a line segment defined by two endpoints) that exists at every scale of observation. In this sense, much like a continuum, the discrete perspective is only valid when the length of the bipoint is greater than the mean free path of the molecules of the problem. However, unlike a continuum, there exists a physical length – this simply represents the scale at which we wish to examine the evolution of the system. Another key difference between the mechanics of a continuum and a discrete medium is the time scale at which the physical system is observed. The notion of time is absent from the Navier–Stokes equations; in other words, the behavior of a medium is independent of time. But a medium like water can be viewed as

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incompressible at large time scales, whereas it behaves like a compressible medium whose waves propagate with finite celerity when the system is considered at scales around the order of a microsecond. The objective here is to very briefly review the essential foundations of the theory of discrete mechanics [CAL 15a] in order to establish a system of equations that fully describes the behavior of both fluid and solid media. This system will allow us to naturally switch between solid and fluid states according to a unified approach. In particular, we shall redefine the concept of mechanical equilibrium in terms of a scalar potential and a vector potential, which are defined as the components of the Hodge– Helmholtz decomposition of the acceleration vector. 6.2. Stresses in solids In continuum mechanics, the stress is defined at a point using a either a nonlinear tensor or a linear tensor such as the Cauchy tensor in the case of small strains and deformations. In an isotropic medium, the six independent components of the Cauchy tensor allow us to compute the normal and tangential components in each of the three directions of space. In discrete mechanics, these three privileged directions of space are not defined. There is no Galilean or inertial frame of reference, and the concept of tensor is no longer meaningful. The stress is constructed in the form of two separate components that do not interact: the normal stress and the tangential stress, which is called the shear-rotation stress. As in previous chapters, the elementary topology shown in Figure 1.4 can be used to define the pressure stress and the shear stress on the edge Γ. Figure 6.1 illustrates the pressure stress p and the rotation stress ω more precisely. The normal vector n is the only privileged direction that is necessary and sufficient to fully specify the local stress state of an arbitrary point in a material. The pressure stress is normal to the surface, as in continuum mechanics, but the shear-rotation stress is also defined along the normal vector as the flux of the velocity vector, calculated as the circulation of this vector around a closed contour. The two other directions x and y such that x · n = 0 and y · n = 0 are never needed to represent the stress state at a point in the medium. Therefore, we never need tensors to specify components for each possible direction. Conceptually, shearing has a specific direction in the plane with normal vector n, whereas rotation (or local torsion) is slightly more general in scope. Simple shearing of a sample in the x-direction can, for example, be represented by a vector ω orthogonal to x with magnitude proportional to ∇ × V. It is worth noting that the concept of discrete medium has nothing to do with the structure of the fluid or solid material. For instance, the length of the edge Γ should

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be assumed to be large relative to the molecular mean free path. Instead, the essence of the discrete perspective is that the edge Γ has a direction that remains constant under scaling even if the size of the edge approaches 0. Vectors are never reduced to points like in continuum mechanics. Furthermore, we never attempt to address the deeper physics of solid and fluid materials, and any connections between molecules and the structure of matter are not directly taken into account. We only ever define the coefficients that are strictly necessary to give a macroscopic description of the mechanical behavior of this matter.

Figure 6.1. Diagram of the pressure and rotation stresses, both defined along the normal vector n of the plane S. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

6.2.1. Discrete equations Discrete mechanics [CAL 15a] establishes the equations of conservation of momentum as a Hodge–Helmholtz decomposition where the pressure p is the scalar potential of the acceleration and ω is the vector potential of the rotational effects. These two potentials act as accumulators for the pressure and viscosity stresses, recovering all or part of the energy dispensed over the evolution of the system. Newton’s second law is rewritten as follows in terms of the acceleration: γ = −∇φ + ∇ × ψ + g,

[6.1]

where p = ρ φ is the so-called thermodynamic pressure, which combines any relevant contributions (dynamic, thermal, mass-driven, etc.); the term ω = ρ ψ is the accumulator of the shear-rotation stress; φ is the scalar potential of the acceleration γ, and ψ is its vector accumulator. The source term g includes any other forces per unit mass that are relevant to the problem: gravity, rotation, surface tension, etc. The Hodge–Helmholtz decomposition can be formulated as an inherent part of the fundamental law of dynamics [6.1] or alternatively in a form that can simply be applied to any arbitrary vector. For example, the velocity vector can, of course, be

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decomposed into an irrotational part and a solenoidal part, but so can the source term, g = ∇φg + ∇ × ψg . The nature and the significance of these two decompositions are not the same; φ and ψ represent the two fundamental components of the acceleration, which play an essential role in defining the dynamic equilibrium. In the topology shown in Figure 1.4, both components of the velocity V coexist on the edge Γ. The component Vφ derives from the gradient of the scalar potential φ, whereas the component Vψ arises from the dual curl of ψ. One of the discrete equations of motion describes the mechanical equilibrium on the edge Γ, alongside other equations describing the accumulation of the scalar and vector potentials; the velocity and the displacements are themselves defined as the first- and second-order accumulators of the acceleration: ⎧     γ = −∇ φo − dt c2l ∇ · V + ∇ × ψ o − dt c2t ∇ × V + g ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ φ = φo − dt c2l ∇ · V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ o 2 ⎪ ⎪ ⎨ ψ = ψ − dt ct ∇ × V ⎪ ⎪ ⎪ ρ = ρo − ρ dt ∇ · V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V = Vo + γ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ U = Uo + V dt

[6.2]

where γ is the acceleration, V is the velocity and U is the displacement; the vector g represents the sum of the components of any accelerations applied to the edge Γ. The discrete equations of motion can be solved in a Lagrangian formulation, followed by the transport of the quantities (φ, ψ, V, U), as well as the density ρ where applicable. Equivalently, they can be solved directly in a Eulerian formulation by incorporating the inertial terms into the system. The quantity p represents the thermodynamic pressure, but effects resulting from inertia, source terms, etc., can also be incorporated into the gradient term. Thus, pB = po + 1/2 ρ|V|2 is the Bernoulli pressure; the term ρ|V|2 effectively represents a pressure. If the density is allowed to vary globally, each oriented edge Γ on the streamline is simply assigned values for the velocity and the density. The driving pressure can be defined as p∗ = (pB + ρ φi ), where φi is the inertial potential. The boundary conditions are applied to the potentials or the velocities themselves. In system [6.2], the potentials are simply updated from the divergence and the curl

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of the velocity, which are perfectly well-defined at the boundaries of the domain, without needing any boundary conditions. To discuss the behavior of fluid and solid media, consider again a simplified form of the system [6.2] where the inertial terms are reincorporated into the total derivative. We can now introduce the two factors αl and αt to describe the attenuation of longitudinal and transverse waves, respectively: ⎧     dV ⎪ ⎪ = −∇ φo − dt c2l ∇ · V + ∇ × ψ o − dt c2t ∇ × V ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎨ φ = αl φo − dt c2l ∇ · V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ = α ψ o − dt c2 ∇ × V t t

[6.3]

In a fluid medium, there is practically zero accumulation of the shear-rotation stresses (αt ≈ 0). With the exception of fluids with complex rheologies, fluids relax these stresses over very short time scales by reorganizing the structure of their matter. The accumulation term ψ o can, therefore, be eliminated from the equations describing evolution of the system; the instantaneous stress in the fluid is then given by the relation ψ = −ν ∇ × V. However, ψ o can still be employed to define the initial conditions of a system in mechanical equilibrium. In the presence of a source term such as gravity, a zero pressure field does not satisfy the mechanical equilibrium defined by the statics of fluid law. The original system [6.2] is the best form of the equations of motion in the sense that it includes conservation of mass; solving this equation automatically satisfies the constraint of conservation of mass in a single step. These equations can be used to formulate incompressible or compressible flows (including with shocks), one-phase or two-phase flows, multicomponent flows, etc. It is also capable of describing phase change phenomena if coupled with vector equations on the heat flux. 6.2.2. Material frame-indifference The principle of material frame-indifference [TRU 74] states that the mechanical stresses within an object should not depend on the observer; in particular, if the object is rotated, the stress should remain the same in the object’s own frame of reference. Suppose that a rigid rotation is applied to the discrete system of equations. Consider a rotation with non-zero velocity Ω around some arbitrary axis. The velocity could, for example, be strictly constant or constant over some sufficiently short period. The local velocity is given by V = Ω r. Let Π be the potential of the centrifugal effects within the domain, so that Π = 1/2 (Ω × r)2 . If the velocity is constant and the object does not need to satisfy any other constraints, the mechanical

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215

equilibrium is described by the equality ∇ Π = ∇ × V, since ∇ Π is the only contribution of the inertial terms in the equations of motion: −∇

1 2 (Ω × r) 2

+ ∇ × (Ω r) = 0.

[6.4]

If the system performs a simple rotation over some period τ through an angle of θ = Ω dt and then stops moving, the velocity is indeed equal to zero, and consequently the centrifugal force also becomes instantaneously zero. This operation was described by Truesdell [TRU 74]; Truesdell only considered a rigid rotation through a certain angle, but as we can see here, the same principle can be extended to continuous rotations. Returning to the equations of motion and introducing the rigid rotation gives: ⎧     dV ⎪ ⎪ = −∇ Π + φo − dt c2l ∇ · V + ∇ × Ω r + ψ o − dt c2t ∇ × V ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎨ φ = αl φo − dt c2l ∇ · V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ = α ψ o − dt c2 ∇ × V t

[6.5]

t

By equation [6.4], the stresses of the accumulators (φo , ψ o ) are left unchanged by a discrete or continuous rotation. If we also translate the system at a constant velocity V0 in addition to the rigid rotation, we recover the idea of a Galilean or inertial change of reference. Translational motion is neglected by the equations of motion, since changes in the velocity are only captured by the intermediary of the acceleration γ. At high rotational velocities, the centrifugal effects may become non-negligible, for example causing the solid to stretch. However, this is a strong interaction that is unrelated to the principle of material frame-indifference. In summary, both types of uniform motion – translation at a constant velocity of V0 and rotation at a constant velocity of Ω – do not directly modify the mechanical stress in the medium. 6.2.3. Solid statics equations The fluid statics equations describe the mechanical equilibrium of a fluid with density ρ under a gravity field g. They can be stated as follows: −∇po + ρ g = 0.

[6.6]

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Discrete Mechanics

In incompressible problems, the only unknown is the equilibrium pressure po . In a compressible medium, we typically need to introduce other relations such as state equations to calculate the solution (po , ρ) of the problem. In discrete mechanics, the state equations are excluded from the equations of motion. The density is calculated directly from the equations of conservation of mass, whereas the pressure is updated from the compressibility coefficient. Thermal effects can be modeled using the ratio γ of the specific heat at constant pressure and the specific heat at constant volume. In the general case where the properties of the medium also depend on the pressure, equation [6.6] cannot be solved explicitly; the pressure and the density must be calculated by the incremental process described above. The equilibrium of a solid under a gravity field is not usually given much attention outside of a few traditional problems, such as the behavior of an elastic cable under its own weight. In problems involving objects, the intermolecular bonds enable us to view the body as a rigid medium. In the general case, we will need to calculate the stresses in solid materials, as well as any displacements if an initial position is defined. Without attempting to give a microscopic explanation of the underlying interactions, we can assume that there exists a state of mechanical stress in which the solid is at rest (V = 0) or in uniform motion. We shall characterize this state by the compression and rotation stresses (po , ω o ). Note that these stresses both have the same units. The unified statics equations, valid for both fluids and solids, may be stated as follows: −∇po + ∇ × ω o + ρ g = 0.

[6.7]

The rotation stress is zero in fluids that do not accumulate the rotation of the medium. The stress exists instantaneously, but is not accumulated over time. Equation [6.7] is a generalization of the law of fluid statics. It can be understood as a Hodge–Helmholtz decomposition [ANG 13, LEM 14] of the vector ρ g. If the density and gravity are constant, the latter derives from just a scalar potential, the pressure p. If the density is non-constant and ∇ρ × g = 0, the mechanical equilibrium is no longer possible and ρ g includes both a scalar potential and a vector potential associated with the motion of the fluid that is completely separate from the vector potential of the solid ω o . Note that this condition is defined locally and does not have the usual meaning from the geodynamics of barocline flows. Outside of a few simple cases where an analytic solution is possible, equation [6.7] can be solved directly and highly efficiently with a suitable range of discrete-mechanical methodology. The potentials (po , ω o ) are calculated in a single step and the equations are solved up to the machine error. Equation [6.7] can, of course, also be expressed in the equivalent

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form −∇φo + ∇ × ψ o + g = 0 by writing the pressure stress as po = ρ φo and the shear stress as ω o = ρ ψ o . 6.2.4. Calculating the displacement The displacement U can be calculated directly from the fundamental law of dynamics by integrating twice with respect to time. However, solving the equations of motion only gives the velocity, or more precisely its components V, in a local frame attached to the edge Γ. This velocity is only defined up to two other uniform velocities, a translational velocity Vt and a rigid rotation Ω × r. The velocity of a particle in an absolute frame is, therefore, given by: Vp = V + Vt + Ω × r.

[6.8]

This formalism extends the notion of a Galilean frame to uniform rotational motion. As we shall see, this means that we will need to define a scalar potential for the centrifugal effects. Returning to the discrete equations of motion, if V = Vp , then: ∂V −∇ ∂t

o



p 1 1 2 2 2 − dt cl ∇ · V |V| n + ∇ |V| = −∇ 2 2 ρv

o ω − dt c2t ∇ × V . [6.9] +∇ × ρv

×

Observe that the uniform translation field has the properties ∇ · Vt = 0 and ∇ × Vt = 0; any contributions will, therefore, vanish from the equations of motion. The rotation field Ω × r, on the other hand, is solenoidal, ∇ · (Ω × r) = 0, but has constant curl, ∇ × (Ω × r) = 2 Ω. Note also that ∇ × ω o = 0, since the accumulation of a constant curl is a constant field with zero curl; the viscous term is also zero. Introducing the velocity field Ω × r into equation [6.10] gives: −2 V × Ω = −∇

po 1 2 + V . ρv 2

[6.10]

writing φr = po , we obtain the potential: φr =

1 2 (Ω × r) . 2

[6.11]

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Discrete Mechanics

A rigid rotation requires us to introduce a scalar potential into the equations of motion, but the motion itself cancels out. Thus, the notion of a Galilean frame of reference is generalized to all linear combinations of uniform motions. In a moving frame where ∇φr = Ω × Ω × r (the centrifugal term), we of course recover the classical expression of the equations of motion. In the general case, adding a gradient of any regular function to the equations of motion does not modify the motion governed by this equation; it simply redefines the equilibrium pressure. In the case of a uniform rotation, we have po + φr . If the thermomechanical properties are independent of any pressure variations, the contribution φr can be ignored. We can, therefore, calculate the displacement U of a fluid or solid particle from the velocity Vp directly, by integrating with respect to time over an interval dt that separates the observations of two mechanical equilibrium states. This principle of accumulation of stresses lies at the heart of the entire discrete-mechanical approach, leading from a system in equilibrium at time t0 to another equilibrium state at time t0 + dt. It gives us a simple way to tackle the large deformations and displacements of continuum mechanics. Various important differences between the two approaches are described in [CAL 15a]; for example, the use of second-order tensors to describe the stresses and deformations of a medium is abandoned. This is motivated by a simple reason: if we can describe the motion of a mechanical system using simple differential operators such as the divergence, curl and gradient, then why construct a second-order tensor only to reduce its order later by taking the divergence anyway? This fairly trivial remark regarding the usage of tensors nonetheless allows us to understand why additional conditions are required in continuum mechanics, such as the symmetry condition on the Cauchy stress tensor, the principle of material frame-indifference introduced by [TRU 74], the Clausius–Duhem condition on the Lamé conditions, 3 λ + 2 μ ≥ 0, or even compatibility conditions on the deformations. None of these conditions are necessary in discrete mechanics, since the principles that they describe are automatically satisfied. Consider the case of compatibility conditions on the strain tensor ε. These conditions require the existence of a displacement field U such that ε = 1/2 (∇U + ∇t U). The tensor ε has six components, whereas the displacement only has three components in an orthonormal coordinate system [COI 07]. We encounter similar problems when finding a potential Φ from a vector field V. This time, the condition is that the field must derive from a potential, namely ∇ × V = 0. These compatibility relations can be summarized by ∇l × ∇l × ε = 0 or ∇r × ∇r × ε = 0, where l and r denote the two forms (left and right) of the curl of a tensor.

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In discrete mechanics, the displacement and the stress are assigned to each edge Γ of the topology shown in Figure 1.4 and the frame is always local. Furthermore, the displacement U and the stress (po , ω o ) are calculated in a single step by the process that governs the evolution of the system between equilibrium states. In practice, the displacement and the deformation are calculated by integrating the velocity with respect to time after deducing the velocity from the equations of motion in the context of a continuous-memory medium. 6.3. Properties of solid media The properties of the medium, whether fluid or solid, must be known and incorporated into the conservation equations; these properties might be determined theoretically, experimentally, from a model, etc. They do not directly interfere with the equations and cannot violate the conservation process described by them. Examples of such physical characteristics include elasticity coefficients, the Young’s modulus, the Poisson coefficient or even the Lamé viscosity coefficients, λ and μ; we could also cite thermodynamic coefficients such as the isothermal compressibility coefficient χT or the thermal expansion coefficient β. In general, these quantities can depend on variables such as the temperature or the pressure, but they are always simply assumed to be known. Some variables should not be viewed as just properties; this is, for example, the case for the density ρ. The density has its own specific conservation equation – the conservation of mass. This variable is a function of a single quantity, the divergence of the velocity. Any attempt to fix the value of the density by applying state equations or otherwise would be a mistake that introduces an inconsistency. Similarly, the notion of pressure is complex and cannot be calculated from any state equations. The pressure is the scalar potential of the acceleration and its evolution over time is solely fixed by the equations of motion. Continuum mechanics has blurred the lines between fundamental variables, characteristic quantities, state laws, constitutive laws, etc. Counting the number of variables and conservation equations of the continuous formalism shows that additional equations are necessary to close the system, namely the state equations. But this perspective is ultimately flawed; these equations are not needed. In discrete mechanics, the density is viewed as the potential associated with the conservation of mass, and the pressure is the scalar potential associated with the conservation of acceleration. Historically, the properties of an arbitrary medium have been fraught with uncertainty and controversy; consider, for instance, the second Lamé coefficient λ for a fluid. The value of this coefficient has supposedly been measured by various authors, e.g. for water, but their results vary wildly and are far from consistent with Stokes’ hypothesis. As we shall see, the measurement methods introduce a bias that makes these measurements inconsistent. Since the earliest days of mechanics,

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straightforward techniques have been used to measure the elasticity coefficients; by stretching a steel sample, we can measure its elongation, then simply apply the linear Hooke’s law to deduce the Young’s modulus. While doing so, we observe a natural shrinking of the cross-section, which allows us to deduce the Poisson coefficient. If we chose a sample with greater width than length, we would find different results, so the experiment has a de facto dependence on its environment. The objective of the formulation of a discrete medium is to only introduce the characteristics that are strictly necessary, while ensuring that they are perfectly measurable. For a more homogeneous presentation, we shall write the properties identically for both solids and fluids. The only difference is the variable with which they are associated, the velocity or the displacement. Thus, for example, the dynamic viscosity of a fluid has units of (Pa · s), whereas the shear modulus or Coulomb’s modulus G has units of (Pa); the former is associated with a velocity and the latter is associated with a displacement. If we consider Couette’s experiment on the shearing of a fluid medium and a solid medium for some fixed period dt, the elongation of the solid medium is equal to that of the fluid medium, V0 dt, where V0 is the drag velocity. The only difference is the accumulation process of the stresses within the solid. We will choose to work primarily from the perspective of fluids, where the vector variable is the velocity V. The displacement is directly fixed by the product V dt, where dt again denotes the observation interval of the system between two states of mechanical equilibrium. The discrete equations of motion [3.38] are, therefore, expressed in terms of velocities and stresses (po , ω o ) or potentials of the acceleration (φo , ψ o ). 6.3.1. In continuum mechanics In continuum mechanics, the fourth-order elasticity tensor Cijkl relating the stress to the strain rates, which starts with 81 independent coefficients, can be reduced to just 21 coefficients in the general case by accounting for symmetries. Using Voigt notation to write this tensor as a 6 × 6 matrix, we can perform further reductions in the isotropic case to obtain a matrix with just two independent coefficients such that C11 = C22 = C33 = λ + 2 μ, and every other non-zero term is equal to C44 = μ. This shows that the compression and shearing effects are intertwined in continuum mechanics; λ and μ are the two independent Lamé coefficients, but C11 , which represents the directional compression, also depends on μ. The resulting coefficient is known as the compression wave modulus in the literature: M = λ + 2 μ.

[6.12]

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221

In solids, the shear modulus is denoted G. Effectively, M and G = μ are two independent coefficients that can be found by measuring the Young’s modulus E and the Poisson coefficient σ. If we consider the propagation of waves within a solid medium, we can write wave equations on the displacement in terms of cl and ct , the longitudinal celerity and the transverse celerity:  ⎧ ⎪ λ+2μ ⎪ ⎪ cl = ⎪ ⎪ ⎨ ρ ⎪  ⎪ ⎪ μ ⎪ ⎪ ⎩ ct = ρ

[6.13]

These expressions only hold in the case of an unbounded medium. In a medium of finite width, the longitudinal celerity cl depends on the width of the sample, and the celerities become:  ⎧ ⎪ E ⎪ ⎪ cl = ⎪ ⎪ ⎨ ρ ⎪  ⎪ ⎪ μ ⎪ ⎪ ⎩ ct = ρ

[6.14]

The Young’s modulus depends on the environment of the sample, and so does ct . By performing a few calculations from the definitions of the celerities and the Poisson coefficient, it can be shown that:  ct = cl

(1 − 2 σ) . 2 (1 − σ)

[6.15]

√ This ratio is always less than 1/ 2. Figure 6.2 shows the evolution of (ct /cl )2 as a function of the Poisson coefficient σ. Excluding a few very special cases, the Poisson coefficient usually takes values in the interval [0, 0.5]. Values of zero are typically assigned to “highly compressible” media that do not contract laterally when compressed (like cork), and values of 0.5 are for “incompressible” media – which tend to conserve their volume, i.e. which contract when extended (like rubber). The uniform nature of the law shown in Figure 6.2 allows us to directly relate σ to the ratio of the celerities, allowing us to work with the latter parameter instead, since the celerities can be measured more

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accurately. At this point, the continuum approach encounters its first obstacle: dependency on the environment – some coefficients depend on the way in which they are measured. For example, certain precautions are required when measuring the ratio of the dimensions of a sample under traction to determine the Young’s modulus and the Poisson coefficient.

Figure 6.2. Evolution of the ratio (ct /cl )2 as a function of the Poisson coefficient

Fluids present another challenge: the second Lamé coefficient λ is not well-defined for a gas or a liquid. The hypothesis of an incompressible medium where σ = 0.5 is inconsistent and leads to infinite values of λ, as well as an infinite celerity of sound. The kinetic theory of gases and thermodynamics based on the Clausius–Duhem inequality have not been able to produce an acceptable solution to this problem. The inequality 3 λ + 2 μ ≥ 0 allows λ to take negative values. Examining the invariants of the stress tensor in three dimensions leads to the same conclusion. Stokes’ hypothesis, 3 λ + 2 μ = 0, which can be established for a monatomic gas using the kinetic theory of gases with the hard-sphere model, is far from providing a satisfactory answer. Since μ is small in liquids, e.g. (10−3 ) in water, there are multiple equivalent values of λ that have strictly no effect on the results. If we calculate the product μ χT , we find values of the order of 10−12 , so why not choose λ = −2/3 μ. Full or partial incompressibility is guaranteed by conservation of mass anyway. Finally, the concept of tensor significantly complicates the derivation of the equations of motion. Introducing the hypothesis of a continuum makes it necessary to

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reduce every other quantity – variables, properties, etc. – to a single point by applying the divergence theorem and then the local equilibrium hypothesis. Although these hypotheses can be justified for scalar quantities, reducing a vector to a point requires us to construct a frame of reference to express the components of this vector. Anisotropic materials can of course still be described by tensors but, as we shall see, it is not necessary to do so. 6.3.2. In discrete mechanics In discrete mechanics, the coefficients describing the behavior of a fluid or a solid are perfectly measurable. The dynamic viscosity coefficient μ for a fluid and the shear modulus for a solid are analogous in each medium – only the type of potential changes: displacement for a solid and velocity for a fluid. The second independent coefficient is the compressibility coefficient χT , which is defined as follows: χT =

1 ρ



∂ρ ∂p

,

[6.16]

T

where the variations of ρ and p are assumed to be known at each point of the primal topology (Figure 1.4). Recall that the action of the compressibility effects is only defined on the edges Γ between the points of the primal topology. In the equations of motion, this action is formulated as the gradient of a scalar, in this case the pressure. The only conditions imposed on these coefficients are as follows: 1 ≥ 0; χT

μ ≥ 0.

[6.17]

The two coefficients μ and χT allow us to fully separate the compression effects of the medium from the shear-rotation effects. These coefficients do not have the same units; to remedy this, we need to introduce time, or more precisely the time constant dt at which the evolution of the system is observed. Thus, the quantity dt/χT has the dimensions of a viscosity. The notion of time is effectively essential for understanding phenomena involving the propagation of longitudinal waves through a medium. For example, in a fluid, the pressure changes will be larger if the characteristic time scale of the phenomenon is small. Jumping into a swimming pool from a height of 1 m produces different effects on the body than jumping from 10 m. Accordingly, at large time scales, water can be considered to be incompressible, even though the fact that sound waves propagate at cl = 1, 500 m/s demonstrates that the phenomenon is in fact compressible. The coefficients μ and χT are intrinsic and only depend on the nature of the medium; they need to be measured by a suitable experimental methodology. The

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coefficient μ can be measured by considering slow shearing between two walls, e.g. by rheometry for a fluid. The compressibility coefficient can be determined extremely accurately in both fluids and solids from the propagation of longitudinal waves by measuring their celerity:  c=

1 . ρ χT

[6.18]

This expression is universally valid – it holds for fluids (gases or liquids) and solids alike. We simply need to know the celerity of the longitudinal waves ct = c to deduce the isothermal compressibility coefficient χT . In an adiabatic gas, there is a pressure contribution arising from the temperature variations. In this case, the isentropic compressibility coefficient χS = χT /γ is used instead. Table 6.1 lists the values of the viscosity μ and the compressibility coefficient χT calculated from the longitudinal celerity of the medium for a selection of fluid and solid media. These values are simply illustrative and may be subject to non-negligible variations. Material

ρ

cl

μ 5 × 10

χT 10

1.4 × 10

μ χT −11

Glass

2,500 5,300

Steel

7,500 5,600 8.1 × 1010 4.25 × 10−12

Aluminium 2,700 6,320 2.8 × 1010 9.27 × 10−12 1.14 × 10−7

Rubber

3,510

50

106

Cork

256

450

2.7 × 106

Air Water

1.18

347 1.85 × 10

1,000 1,480

10−3

−5

0.7 0.344 0.256 1.14 × 10−1

1.92 × 10−8 5.184 × 10−2 9.89 × 10−6 1.829 × 10−10 4.44 × 10−10 4.44 × 10−13

Table 6.1. Material properties calculated as the average of the values cited in the literature, which vary significantly. The quantity ρ is the density, cl is the longitudinal celerity, μ is the shear modulus or viscosity and χT is the isothermal compressibility coefficient. The product μ χT represents a characteristic relaxation time for the shear stress

Consider the evolution of a fluid or solid medium over time. We shall consider the motion of the medium in terms of the local velocity; the displacement is deduced by integrating with respect to time along the trajectory of a particle within the medium. Note that the clearest distinction between fluids and solids is in fact the shear-rotation viscosity μ rather than the compressibility coefficient. This confirms that the relaxation time τ of the shear stresses in a fluid is very small. We can estimate the order of magnitude by noting that τ /μχT is of the order of one; this gives values of dt = τ ≈ 10−12 s. In these conditions, a fluid cannot accumulate the shear stresses, which are relaxed almost instantaneously, and the stress ω o retains its previous value.

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The conclusion of this brief analysis is that the quantity: ct = cl



μ χT dt

[6.19]

is of order one when dt is of order one in solids, and of the same order as the relaxation time of the shear stresses in fluids. Note also that equation [6.15] is satisfied by each of the materials listed in Table 6.1: 0≤

c2t 1 ≤ . 2 cl 2

[6.20]

Clearly, interpreting these data can be tricky. An intuitive approach based on the Young’s modulus and the Poisson coefficient is not enough. The (incorrect) perception that rubber is incompressible arises from the mental image of an elastic band that is stretched, causing its cross-section to shrink. Large pieces of rubber deform much less in the transverse direction. Since our objective is to find a unified formulation for both fluids and solids – an entirely natural goal, given that they are governed by the same mechanics – we need a unified formulation for the boundary conditions as well as the conservation equations themselves. 6.4. Boundary conditions The concepts of boundary conditions and well-posed boundary value problems diverge significantly from the classical perspective, where the mathematical model is built on partial differential equations and boundary conditions are imposed on the variable. In discrete mechanics, the only constraints imposed on the variable derive from the equations themselves, together with any physically necessary restrictions. For example, the incompressibility of a fluid is not modeled by assuming a priori that ∇ · V = 0; instead, the properties of the medium are chosen in such a way that this relation holds as χT → 0. The effect of these properties, namely the relation ∇ · V = 0, is not imposed separately, which is problematic anyway from the perspective of causality. Similarly, the system of equations [3.38] is not accompanied by any separate boundary conditions on the variable; instead, the stresses are chosen to reflect the boundary conditions in the operators of the equations of motion. The ratio of the transverse and longitudinal celerities plays an important role in determining the behavior of materials under stress as a function of their environment. For instance, an extension phenomenon of a material that is very large in the lateral direction only involves the longitudinal celerity of the compression/extension waves. If the sample is smaller laterally, any longitudinal perturbation is accompanied by a reduction of the transverse section and a wave with celerity ct . In continuum

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mechanics, the ratio ct /cl only depends on the Poisson coefficient. Equation [6.19] is the analogous relation for a discrete medium. This dependency on the environment must be modeled by boundary conditions to describe the cohesion of matter when the system has a free surface, e.g. with the atmosphere. More precisely, any sample that is subject to an extension will experience a change in volume, typically an increase reflected by a positive total divergence. The behavior of the volume of a material depends on two local and directional physical properties: the viscosity μ and the ratio dt/χT . The compressibility coefficient is defined at a point and the gradient is calculated on the edge Γ, whereas the primal curl is defined on the contour as a circulation, which is then projected onto Γ by the dual curl. Within the medium, whether fluid or solid, the local distributions of the pressure stress p and the rotation stress ω are balanced by any forces per unit volume or imposed stresses. At the boundaries of the domain, the mechanical equilibrium must be defined specifically to reflect the absence of matter, e.g. as a fully rigid wall, some other material with different properties, a fluid or a vacuum. The boundary conditions, therefore, describe the integrity properties of the material that cause it to resist an extension or a compression; the medium is typically not incompressible and an extension can lead to an increase in volume, which must therefore be accompanied by a reduction of the density, since the divergence is positive. It is worth emphasizing that the formulation presented here is based on a dynamic approach where the principal variable is the velocity. The displacement is obtained naturally by integrating with respect to time. Similarly, the equations of discrete mechanics allow us to calculate not only the displacement, but also the pressure stress, represented by po , and the rotation stress ω o . These stresses are deduced from the accumulation process that observes the evolution of the system between times to and to + dt. The boundary conditions can be implemented by this same dynamic process by examining the behavior of the medium within its environment. For example, a displacement in the longitudinal direction should produce a displacement in the transverse direction unless the material is held fixed within a rigid cavity or the medium extends far in the lateral direction. In the latter two cases, the longitudinal celerity alone governs any displacements in the longitudinal direction. If the medium needs to interact directly with another medium, we must define external conditions. This must also be handled in a manner that is consistent with the formulation of the discrete equations. The boundary conditions are not presented as auxiliary conditions as in the classical approach (e.g. Neumann, Dirichlet, etc.); instead, they are fully integrated into the conservation equations. In discrete mechanics, the condition satisfied at an interface between two fluids with dynamic viscosities μ1 and μ2 may be expressed as follows, as well as the

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requirement that the velocities and pressure be continuous on either side of the interface: μ1 ∇ × V1 × n = μ2 ∇ × V2 × n,

[6.21]

where n is the normal vector of the interface Σ between the two fluids. Since the two vectors ∇ × V and n are orthogonal, their vector product has the same direction as the tangent vector t of the edge Γ. In fact, this boundary condition is just a formality, since the dual curl operator in the equations of motion already guarantees that it will be satisfied. As such, equation [6.21] is little more than a remnant of the formalism used for a continuum. In discrete mechanics, the velocity is just an accumulator of the acceleration; applying boundary conditions to the velocity itself doesn’t make sense. As we have seen, the boundary conditions are replaced by conditions imposed on the stress. Since the objective of this chapter is simply to validate the discrete model against various constitutive equations, only the shear-rotation stresses will be considered explicitly. The choice of the density ρv on the interface Σ between the two media is uniquely determined by the density at the vertices of the edge Γ. We can rewrite the condition on the rotation-shear stress as follows: c2t1 ∇ × V1 × n = c2t2 ∇ × V2 × n,

[6.22]

where c2t = ν = μ/ρv . Note that the equality of stresses is dynamic, as suggested by the condition [6.22], and not static, like the approach of continuum mechanics. The density does not appear anywhere in the equations of motion. Only the celerities cl and ct influence the equilibrium between the accumulators and deviators of the stress. For interactions between fluids and solids, the coupling condition on the stresses can be expressed as follows in terms of the velocity of the fluid and the displacement of the solid: νs ∇ × U × n = νf ∇ × V × n,

[6.23]

where νs is the shear modulus divided by the density and νf is the kinematic viscosity of the fluid. Note that this condition is associated with a plane; here, it is formulated in the direction of the normal vector to the (x, y)-plane. The analogous condition from continuum mechanics is expressed in terms of derivatives relative to the normal of the interface in the (x, y)-plane. In a monolithic approach, the condition needs to be formulated in terms of the velocity V = U/dt. It cannot, however, be expressed directly in an instantaneous

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form, since the stress within the solid is determined by the accumulation process of the vector potential ψ o . We can formulate it as follows: ∇ × ψ o × n = νf ∇ × V × n.

[6.24]

Since the shear stresses do not accumulate over large time scales in the fluid, the vector accumulator ψ o is eliminated. This removes the dimension of time from the corresponding term in the equations of motion for this medium. After imposing the condition [6.24], we need to verify that it is satisfied autonomously at every point on the solid–fluid interface to obtain a model that is both monolithic and implicit. This is effectively ensured by the dual curl operator ∇ × (ψ o − ν ∇ × V). This operator projects the vectors ν ∇ × V, which are orthogonal to each face of the primal topology, onto the edge Γ, where the acceleration, the velocity and the displacement are defined. It always acts on vectors in three-dimensional space by definition, regardless of whether the primal topology itself has one, two or three dimensions. 6.5. Rigid motion The study of rigid motion – flows with constant velocities and rigid rotations – is the key to understanding the behavior of a medium in general. Uniform translational flows are straightforward when working within a Galilean frame of reference; they are simply neglected by the equations of motion that describe the evolution of the acceleration. The case of a uniform rotational flow is slightly less obvious, since the mechanical equilibrium requires a term deriving from a scalar potential for the centripetal force in order to balance out the centrifugal force. A second issue is that the curl is constant throughout the medium, generating a non-zero dissipation term in μ (∇ × V)2 , even though there is no shearing in the flow. Continuum mechanics resolves this problem by balancing this term against the second invariant of the gradient tensor of the velocity, I2 . This approach to solving the obstacle, which ultimately arises from the constraints and conditions applied to the Navier–Stokes equations, creates new problems in turn. In fact, the behavior of rigidly rotating flows or solids raises questions that are much more general in scope, including purely theoretical questions about the solution of an open problem, namely the applicability of Stokes’ theorem at infinity. Here, we shall examine how long an observer must wait for the flow to appear to take on a steady state, i.e. with velocity V · eθ = Ω r. Strictly speaking, the time required is infinite, but we can, nonetheless, establish a characteristic time describing the settling of the velocity field in a finite medium.

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Consider a cylindrical vessel with vertical axis Oz filled with water of density ρ = 1,000 and viscosity μ = 10−3 . Initially at rest, the water begins to rotate under the effect of the surface of the vessel, which has radius R = 1 and is rotating at velocity V0 . We shall of course assume that the height of the cylinder along the Oz-axis is sufficiently large that the Eckman layers do not influence the motion of the fluid, which is therefore governed exclusively by the radial viscous effects. The solution of the problem can be found easily be solving the equations of motion with the approximation of incompressible motion. The method of separation of variables finds a general solution in terms of the difference vθ (r, t) = V · eθ − Ω r, obtained by superposition. This solution can be stated as follows: ∞

r

r  + b n Y1 α n e−ν a n J 1 αn vθ (r, t) = R R n=0

α2 n R2

t

,

[6.25]

where the coefficients αn are deduced from the boundary conditions using the orthogonality properties of the Bessel functions J and Y . We can now apply the initial conditions, namely the condition that the fluid is immobile at t = 0: −

∞

r

r  V0 r + b n Y1 α n . a n J1 α n = R R R n=0

[6.26]

This determines the coefficients an and bn . The solution V·eθ (r, t) follows the development of the dynamic boundary layer in the neighborhood of the solid wall and tracks its thickness over time. We can assume that the solution is accurately described by a single term at large time scales, and we can also bound the term αn /R2 above by one. This gives us: ε = V · eθ − V0

r ≈ e−ν t . R

[6.27]

For example, if the difference is ε = 10−4 , then, since ν = 10−6 , we find a settling time of t ≈ 107 s = 115 days! If the radius of the cylinder is made larger, the time required for the medium to settle approaches the time constant of an apparent solid with the characteristic value defined in Table 6.1 of τ = 1/μχT ≈ 1012 s. In a medium that is assumed to be solid, the viscosity μ – which is just the shear modulus in our approach – is 12 orders of magnitude larger than in a fluid, and the

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stresses are accumulated. Thus, the solid begins to rotate almost instantaneously when the cylinder itself starts turning. At a fundamental level, there are no differences between the behaviors of fluids and solids – they simply have different properties. From the theoretical perspective, the equations of motion are the same for both media. They can be stated as follows: dV = −∇ dt



o

dt ω po − ∇·V +∇× −ν∇×V . ρv ρv χT ρv

[6.28]

The term ω o can therefore be kept, including for fluid motions, since the viscosity is typically low and, more importantly, because phenomena that cause the curl to vary over time unfold extremely quickly in comparison to the phenomena described above. Finally, in practice, the quantity ω o will typically remain negligible throughout the evolution of the flow. The paradox of dissipation described above is fully resolved, since the dissipation of mechanical energy is no longer described by the term μ (∇ × V)2 , but rather the  term μ ∇ × V − ω o · ∇ × V. At large time scales, this term is zero, as is the dissipation of a rigid flow. To behave like a solid, the medium must accumulate the shear stresses in the vector potential ω o , which in this case is equal to μ ∇ × V; see [CAL 15a] (equation [5.40]). This observation naturally has significant ramifications on how we define the behavior of a medium. For example, a mountain glacier follows the same laws as a fluid at time scales of several months but behaves like a solid when subjected to faster stresses. To describe the rheology of complex media, which are typically relatively viscous, we might need to take advantage of the formulation proposed here. Similarly, it may be worth pondering whether the motion of fluids at extremely low velocities in the neighborhood of the wall could be viewed as solid-like behavior, with molecules and atoms arranged into an organized structure. 6.6. Validation of the model on examples The following examples are not intended to provide a demonstration of complex applications to fluid–structure interactions, which have already been published elsewhere [BOR 14, BOR 16]. Our objective is to give a few extremely simple examples, e.g. for which exact solutions of the mechanical equations are known. Nonetheless, the results obtained with the model presented here illustrate the validity of the approach in various situations featuring both fluids and solids. 6.6.1. Simple example of a monolithic fluid–structure interaction We shall study one of the simplest cases of an interaction that displays the behavior of two media, one viscous and the other elastic. This test case has a very

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simple analytic solution that clearly illustrates the behavior of both media within the discrete framework [3.38]. The domain has height h = 1 and is separated by an interface Σ located at height h/2; the lower wall is held at rest and the upper surface is initially set to V0 = 1. Let us begin by considering the purely viscous case of two fluids with kinematic viscosities ν1 = 1 and ν2 = 4. The solution obtained from the system [3.38] converges very rapidly to the steady solution, taking the form of two line segments that satisfy the boundary conditions as well as the condition ν1 ∇ × V1 = ν2 ∇ × V2 at the interface. Given these assumptions, the velocity at the interface is equal to Vi = 0.2. The one-dimensional solution does not depend on the choice of spatial approximation and the error is zero up to the machine error. Note that the condition at the interface is implicitly guaranteed by the operator ∇ × (ν ∇ × V). The curl is constant in each medium, with values ∇ × V1 = −1.6 and ∇ × V2 = −0.4, respectively. Since this problem does not have any compressibility terms, only the independent viscous terms appear in the discrete equations of motion: ⎧ dV   ⎪ = ∇ × ψ o − dt c2t ∇ × V ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ψ = ψ o − dt c2t ∇ × V ⎪ ⎪ ⎪ ⎪ V = Vo + γ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ U = Uo + V dt

[6.29]

Suppose now that the lower part of the domain is an elastic solid with celerity c2t = ν = 4. The upper part is occupied by a fluid with viscosity ν = 1. The vector potential ψ o accumulates the shear stresses within the solid, so that the stresses at the interface in the fluid are effectively transmitted and stored in the solid. The solution converges very rapidly to a velocity field that is strictly zero in the solid, with a linear velocity profile satisfying the relevant conditions at y = h, and zero velocity at the interface Σ. The vector equation of the system [3.38] is satisfied identically with ψ o = ν ∇ × V, where V is the velocity of the fluid and ψ o = 2. Thus, the exact solution does not depend on the spatial approximation. Figure 6.3 shows the evolution of the velocity at the interface Σ over time; the velocity attenuates relatively quickly to zero. In the steady state, the velocity field is zero within the solid and linear within the fluid. The figure also shows the displacement U of the solid once the steady state is attained.

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Although the fluid continues to move indefinitely under the shearing action, the displacement of the elastic solid rapidly settles to a steady state. The fact that there is no interpolation at the interface of the two media enables us to obtain the exact solution. This very simple example allows us to understand each of the mechanisms of equation [3.38], validating both unsteady and steady fluid–structure interactions.

Figure 6.3. Study of the fluid–structure interaction between a viscous fluid and an elastic solid; the viscosity of the fluid is equal to ν = 1 and the shear modulus of the solid is equal to ν = 4. The velocity of the interface over time is shown on the left, the velocity V within the domain in the steady-state regime is shown in the center, and the displacement of the solid U as a function of the y-coordinate is shown on the right.

In continuum mechanics, the theoretical solution of this problem can be obtained by considering the two media separately and imposing boundary conditions on the interface. The equations of the fluid and solid are as follows in the steady incompressible regime without inertial effects: ⎧    ⎨ ∇ · μf ∇V + ∇t V = 0 ⎩

∇ × (μs ∇ × U) = 0

[6.30]

If the properties μf and μs are constant, these equations reduce to Laplacians. The first can be deduced from the Navier–Stokes equations, and the second from the Navier–Lamé equations. The conditions at the interface are straightforward. For the fluid, the velocity is zero at y = h/2 and equal to V0 at y = h; for the solid, the displacement is zero at y = 0 and the stress is required to be equal to the stress of the fluid at the interface y = h. The velocity is, of course, 0 within the solid domain. The solution is also very straightforward: v(y) = V · ex = (2 y/h − 1) and u(y) = U · ex = μf /μs (2 y/h). As expected, the solution for the velocity v(y) does not depend on the viscosity, whereas the displacement depends on the ratio μf /μs . In this simple problem, the solution provided by discrete mechanics is, of course, the same as the solution from continuum mechanics. One of the advantages of the discrete approach is that the equations of motion are the same for every medium; formulating the equations of motion in terms of the acceleration enables us to simply

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view the velocity and the displacement as accumulators of the operators ∇ · V and ∇ × V. 6.6.2. Mechanical equilibrium of sloshing The system [3.38] can, of course, represent acoustic waves with celerity c but also various other phenomena, depending on the external actions that are introduced into the system: gravity, capillary forces, rotation, etc. In the presence of a constant and uniform gravitational force, multiple types of gravity waves can develop and persist over large time scales. This is, for example, the case with solitary waves, ocean waves and so on. The evolution of both media can be described succinctly by the equations of motion as follows: γ = −∇φ + ∇ × ψ + g.

[6.31]

The potentials φ and ψ associate the mechanical equilibrium potentials φo , ψ o with their respective deviators. The acceleration of each medium is non-zero; the evolutions are described in terms of the local velocity over time. In an equilibrium state, equation [6.31] may be stated as follows when the velocity is zero (or for uniform motion): 0 = −∇φo + ∇ × ψ o + g.

[6.32]

Figure 6.4. Mechanical equilibrium in a cavity filled with water and air, viewed as elastic solids; the phases are shown on the left, the scalar potential po (Δp = 617) is shown in the middle, and the vector potential ω o (max = 85.23) is shown on the right. The components of the velocity are zero, and of course so is the divergence of the velocity. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Consider the case of a solid medium contained by a cavity with rigid walls. The left part of Figure 6.4 shows the solid, bordered by three rigid walls, and another less dense solid medium lying above its upper surface. The pressure field in the middle part of this figure shows that the walls apply a pressure to the solid that is practically identical

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to the hydrostatic pressure of a fluid within a cavity. However, the free surface between the lower solid and the upper solid is not isobaric, and the other pressure values are coupled to the rigid walls of the cavity. In this situation, the stress state is determined by the two quantities po and ω o , which balance out the gravity field ρ g. The rotation stress field is shown in the right-hand graph in Figure 6.4. Equation [6.32] is satisfied exactly by these stress fields. Let us consider the same physical system again after changing the nature of each medium. The solids are now fluid at the initial time. Since air and water have very different densities, an oscillating motion arises under the effect of gravity. The nature of the medium can be changed simply by adjusting the factor αt of the system [6.29]; if αt = 0, the accumulation of the rotation stresses is eliminated and the medium behaves like a fluid. Figure 6.5 clearly shows the solid–fluid transition at the initial time t = 0. After this point, the pressure field is the only stress that balances out gravity. Inertia plays a significant role in this equilibrium, so the velocities oscillate periodically without attenuation, since the two fluids are assumed to be non-viscous.

Figure 6.5. Sloshing of water and air in a square cavity; evolution of the free surface on the left boundary of the cavity over time. At times t < 0, both media are assumed to behave like elastic solids. At times t ≥ 0, the media are assumed to behave like fluids

This example illustrates the flexibility of the discrete-mechanical formulation, where the behavior of both solids and fluids is governed by the same equations of motion. Here, the nature of the medium was assumed to change over time, but this is far from the only possible application of this approach. We could easily allow a complex physical system consisting of multiple domains to evolve by modifying the nature of its subdomains, for example by adjusting the rheology, and by changing their properties, for example as a function of the temperature. The factors αl and αt , which weight the equilibrium stresses po and ω o , as well as the coefficients χT and μ, cover an extremely broad spectrum of behaviors. In particular, if μ is a

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function of ∇ × V, we can model nonlinear rheological behaviors. Similarly, if χT depends on the pressure, we can generate shock waves and model strongly compressible flows. 6.6.3. Beam under extension This section gives a few examples of mechanical strains applied to a solid beam that will allow us to verify that the proposed model is suitable for describing the behavior of materials. An example that might seem extremely simple is the extension (or compression) of a beam that is stretched under an imposed displacement in the longitudinal direction. If we compress a material with large dimensions in the lateral direction, e.g. a plate or a board, the matter cannot escape laterally, and the compression unfolds similarly to the compression of matter within a rigid cavity. If there are no lateral forces on the material (e.g. a beam surrounded by air at atmospheric pressure), the extension of the sample is accompanied by a relative reduction in the lateral dimensions. If the volume is conserved throughout the transformation, the Poisson coefficient is equal to σ = 0.5 and the material is said to be “incompressible”. This is an abuse of language; for example, rubber is not inherently incompressible. But in any case, it is more compressible than steel, which has a Poisson coefficient of σ = 0.3, as can be seen by comparing the isothermal compressibility coefficients of each material. In the general case, an arbitrary applied mechanical strain may be accompanied by a variation in the volume, and hence the density, if the mass is conserved by the phenomenon. Note that significant variations in the density can in turn modify the other properties of the material, rendering the problem nonlinear. Large displacements can also create variations in the properties, which must be updated at each observation of the equilibrium state by solving the equations of motion without any residual terms. As an example, let us consider the longitudinal stretching of a beam of length L0 = 1 m and width l0 = 0.2 m made from a material with compressibility coefficient χT = 10−6 and shear modulus μ = 106 . The displacement is imposed by setting the velocity to V0 ± 10−3 m s−1 at the two endpoints of the beam. The Poisson coefficient is chosen to have value σ = 0.1 in the linear domain, which gives a transverse velocity of Vt = 10−4 m s−1 . Slipping conditions are imposed on the vertical borders of the beam shown in Figure 6.6 to ensure that the retraction unfolds uniformly, which significantly improves the accuracy of the calculations. Figure 6.6 shows the initial state of the mesh, as well as the final state obtained after 200 s of extension. As the width of the domain decreases, more and more stress is required to stretch it further. For example, if the deformations are large, the Poisson coefficient of the beam becomes σ = 0.034/0.433 = 0.078. The volume of the beam increases from V0 = 1 · 0.2 = 0.2 to V1 = 1.433 · 0.166 = 0.2379. The divergence is positive and constant, with a value of ∇ · V = 6.03 × 10−3 .

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Figure 6.6. Beam subjected to a continuous extension at a horizontal velocity of V0 ±10−3 m s-1 at the two vertical boundaries for a period of t = 200 s. After the extension phase is complete, the velocities are 0 and the stress state (po , ω o ) remains constant. The shear modulus is equal to μ = 106 and the compressibility coefficient is equal to χT = 10-6 . The final pressure of po = −158,446 Pa is homogeneous, and the rotation stress ω o is zero. The ALE mesh is deformed at each time step, and Δt = 1 s. The bottom part of the figure shows the streamlines during the extension phase. The final length of the beam is L = 1.433 m, and the final width is l = 0.166 m

Figure 6.7 shows the influence of the lateral boundary conditions, i.e. the environment of the beam. These conditions play an extremely important role in the behavior of solids under an applied mechanical strain phrased in terms of a displacement or a stress. The Young’s modulus and the Poisson coefficient alone are not sufficient to determine the behavior of the solid.

/ Figure 6.7. Beam under a continuous extension obtained by imposing a horizontal velocity of V0 ±10−3 m s−1 at the two vertical boundaries for a period of t = 200 s. The curve shows the evolution of the pressure −p as a function of the ratio of the compressibility coefficients χsT /χbT , where χbT = 10−6 is the bulk compressibility and χsT is the surface compressibility

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The compression of a solid under an imposed displacement is analogous to the extension considered above. In the case where the lateral width of the solid is of the same order of magnitude as its length, compression creates a lateral dilation. Typically, the average pressure is positive, the volume decreases and the divergence is negative. These extension and compression experiments illustrate the concept of cohesion of matter, at least how this concept is described by our physical model. For a fluid, the presentation of the problem is slightly different, since the material takes on the shape of its container and the environment is always present. For a solid, the internal resistance to deformations makes the boundary conditions less relevant. The role played by the ratio of the longitudinal and transverse celerities is extremely important, including for solids at equilibrium. When an extension is applied to a sample, the deformation travels in the transverse direction at the celerity of sound. The approach developed here gives a description of each phenomenon in terms of the relevant physical quantities, such as the compressibility or the celerities. 6.6.4. Multimaterial compression To extend the analysis given above, we shall consider the compression of two materials within a non-deformable enclosure. In this case, only the longitudinal compressibility is relevant, and no lateral displacement can occur.

Figure 6.8. Compression of two materials in the x-direction at a velocity of V0 = −10−3 m s−1 for a period of t = 8 s. After the extension phase is complete, the velocities are zero and the stress state (po , ω o ) is subsequently constant. The shear modulus of both materials is equal to μ = 106 and the compressibility coefficient is χT = 10−6 for the right material and χT = 10−2 for the left material. The final pressure is po ≈ 140 Pa and is homogeneous throughout both materials. The rotation stress ω o is zero. The ALE mesh is deformed after each time step Δt = 10−2 s. The initial length of the system is 0.2 m and the final length is 0.12 m. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Figure 6.8 shows the results of a compression obtained by imposing a displacement at constant velocity V0 = −10−3 m s−1 in the longitudinal direction ex on the right-

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hand material. The two materials have different compressibility coefficients; χT = 10−2 for the left material and χT = 10−6 for the right material. The shear modulus is assumed to be negligible. At each stage of the process, the displacement is calculated from the local velocities and the ALE mesh is updated accordingly. As we might have suspected, the final pressure is constant after the process terminates at time t = 8 s, with po ≈ 140 Pa. The pressure is uniform and equal in both materials. For simplicity, the densities are assumed to be constant. However, for large deformations, we would need to account for the pressure, as well as any evolutions in the compressibility as a function of the pressure.

6.6.5. Planar shearing We shall now consider a simple example of shearing, with a constant imposed velocity of V0 = 10−2 m s−1 over 40 s on the outer surface of a ring and an imposed displacement of zero on the inner face. The shear modulus of the material of the ring is μ = 106 , with a compressibility coefficient of χT = 10−6 , which does not affect the results. Again, if the deformations were large, we would need to account for any variations in the properties as a function of the shear rate; here, they are simply neglected.

Figure 6.9. Solid ring subjected to stress by imposing a velocity of V0 = 10−2 m s−1 on the inner cylinder over a period t = 40 s. After the rotation phase is complete, the velocities are zero and the stress state (po , ω o ) is subsequently constant. The shear modulus is equal to μ = 106 and the compressibility coefficient is equal to χT = 10−6 . The final pressure is zero and the rotation stress ω o = −4 · 107 Pa is homogeneous throughout the material

Figure 6.9 shows the results of the planar cylindrical shearing; the ALE mesh is deformed after each time step Δt = 1 s. As expected, the pressure is uniform and equal to the initial pressure; similarly, the rotation stress is uniform throughout the

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ring and equal to ω o = −4 · 107 Pa. In this example, like the two previous examples, the motion is ideal; in other words, an applied strain of a certain type does not generate other types of strains. Accordingly, the pressure stress po and the rotation stress ω o are fully separate. In practice, it can be difficult to distinguish between the two effects; for example, if a beam is subjected to an extension, there will be shearing at the couplings. 6.6.6. Flexing beam In the case of a flexing beam, the pressure effects cannot be completely neglected during the displacement of the beam, e.g. when the beam is subjected to an external force, such as gravity for the problem considered in this section. We shall consider a beam lying on two supports that bends under its own weight until it reaches an equilibrium position that depends on the properties of the material – primarily the Young’s modulus, the density, gravity and the moment of inertia of the beam. According to a simplified theory of beams, we can calculate the bending moment from these properties and explicitly determine the deflection by integrating a fourth-order equation. This simplified theory only considers the mean fiber, making the assumption that there are compression and extension zones on either side of this fiber. However, the pressure stress is neglected in the final equilibrium form of the beam. Although the shear stress is indeed predominant in this problem, the pressure stress is nonetheless present and varies along the curvilinear path of the mean fiber. Let us now apply the discrete model to describe the bending of a thin plank of length L = 1 m, width l = 0.02 m and density ρ = 1 kg m−3 . Initially, the beam is assumed to be horizontal; gravity then pulls it downward at a constant velocity. The beam is supported at two points, causing the middle section of the beam to bend slightly more over time. The entire beam experiences shearing, and the upper and lower parts experience a compression and an extension, respectively. The behavior of this system is entirely governed by the equations of [6.29]; from its initial position, the beam is dragged downward by the force ρ g, and the medium accelerates. The plank crosses its mechanical equilibrium position, and the inertial terms in the equations of motion cause the plank to rebound, oscillating briefly before stabilizing. The behavior of the system is determined by the transfer of mechanical energy between the acceleration, the pressure term ∇p, the rotation term ∇ × ω and the gravity force ρ g over time. Figure 6.10 shows the mechanical equilibrium state of the flexion of the beam. Observe that the middle part of the beam may be either compressed or extended depending on its position, unlike the solution proposed by the simplified beam

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theory. The shearing remains predominant in the mechanical equilibrium state, and the distribution of the stress ω o is nearly constant. Since the acceleration is zero and the pressure gradient is very small, the mechanical equilibrium is approximately determined by the equation ∇ω o + ρ g = 0.

Figure 6.10. Beam acted upon by gravity, at mechanical equilibrium. The pressure field (top) shows the compression in the upper part and the extension in the lower part of the beam. The bottom part of the figure shows the component of the field ω o that is orthogonal to the plane. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

6.6.7. Settling of a block under gravity This problem considers the behavior of a two-dimensional block of a soft gelatinous material. The gel settles under gravity but ultimately stabilizes at an equilibrium position with solid-like behavior. As a brief prelude, consider the two limiting cases of the problem – fluid-like behavior and quasi-rigid behavior. In the first case, only the pressure balances the gravity, but the boundary conditions need to satisfy the underlying principle of the reaction, namely that the pressures on the wall and in the fluid are locally equal. In the second case, the equilibrium is fully determined by the shearing, and the pressure is zero. These two limiting behaviors, which can be fully described simply by choosing the properties χT and μ accordingly, are represented by the following equations: ⎧ o ⎪ ⎨ −∇φ + g = 0 ⎪ ⎩

[6.33] ∇×ψ +g =0 o

Figure 6.11 shows the results obtained in both cases; on the left, the scalar potential is stratified like a fluid in a cavity; on the right, the potential field ψ o is shown, which is orthogonal to the plane. This nicely demonstrates the duality between the gradient and the primal curl defined at the very start of this book.

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Figure 6.11. Quasi-non-deformable solid square cavity of height h = 0.1 at mechanical equilibrium, with g = −10 and ρ = 1,000. On the left, for χT = 10−12 and μ = 0, the solution is p = ± 500, ω = 0; on the right, for χT = 0 and μ = 1012 , the solution is p = 0 and ω = ±500. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

We can now consider the main problem of this section: a gel with density ρ =1,000 kg m−3 and arbitrarily chosen properties, χT = 10−4 and μ = 104 . The initial height of the block is h = 0.1 m and gravity is g = −10 m s−2 .

Figure 6.12. Gelatinous block of initial height h = 0.1 at mechanical equilibrium with g = −10 and ρ = 1,000. The bottom wall is allowed to slip, V · n = 0. The properties of the medium are χT = 10−4 and μ = 104 , and the extrema of the solution are given by po = [−91, +494] for the pressure (left) and ω o = ± 417 for the rotation stress (right). For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The initial conditions are as shown in Figure 6.12. The gel initially has the shape of a square block, and the initial stresses are zero, po = 0 and ω o = 0. The settling of the block under gravity can be calculated from the system of equations [6.29]; the pressure increases at the center of the lower wall, which is allowed to slip, while shear stresses simultaneously begin to appear in the upper corners of the block. This shifts the center of gravity downward as the block settles. The inertial terms cause the gelatinous block to pass through its equilibrium position, then oscillate around it.

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Every component of this equilibrium – inertia, pressure and rotation stresses, gravitational force – contributes to the acceleration of the medium as well as its subsequent deceleration, ultimately determining the behavior of the final solid. The pressure field po and the rotation stress field ω o are shown in Figure 6.12 after the velocities become zero. The equilibrium satisfies the solid statics equations [6.7] up to the machine error. If the shape of the object was fixed beforehand, we could calculate the equilibrium by solving equation [6.29] directly while keeping the velocities at the boundary of the domain equal to zero, or alternatively by choosing large values of χT and μ. The solution of this problem can then be found in a single step, starting from the rest state. Finally, the divergence and the curl are used to deduce the equilibrium potentials instantaneously. 6.6.8. Mechanical equilibrium of a solid object We can calculate the equilibrium state of an arbitrarily shaped object from the equations of motion [6.29] in a single step if we assume that the shape of its outer surface is fixed. If the geometry of the object needs to be modified, the same equations enable us to calculate the dynamics of the motion, and finally the static equilibrium state. In three spatial dimensions, the primal topology is constructed from the elementary objects shown in Figure 1.4 – the edges Γ and a primal planar surface constructed from three edges. By assembling these elementary objects, we can construct polyhedral cells, such as tetrahedra, hexahedra or other cells with arbitrarily many faces. The two privileged directions, the direction of the edge Γ and the normal vector of the face, can be arbitrary in general for polyhedral cells, which makes the virtual medium represented by the primal mesh isotropic. In particular, the vector potential ψ is only defined for each face as a vector that is normal to the face.

Figure 6.13. The Stanford bunny. On the left, the pressure field of liquid water inside a non-deformable bunny-shaped container (po = ρ g z, ω o = 0). The center and right-hand images show the pressure po and the component in a plane orthogonal to y of the vector potential ω o in a solid bunny at mechanical equilibrium. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The example shown in Figure 6.13 is a popular test model commonly used by the imaging community, known as the Stanford bunny. After allowing gravity to act on

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the bunny, we shall calculate the mechanical stresses that prevent it from collapsing under its own weight in this section. The volume of the bunny is meshed by tetrahedra that align with another triangular surface mesh. The figures were produced with a tetrahedron count of 1.4 × 105 , but a comparison with a finer mesh (1.4 × 106 cells) would allow us to verify that the solution has converged adequately. The left image plots the pressure field of a liquid inside a bunny-shaped cavity; the isobaric surfaces are orthogonal to gravity. The middle and right images in Figure 6.13 show the equilibrium stresses po and ω o when the object is viewed as a solid. These two fields locally balance out the gravitational force exactly. The same procedure would allow us to extend the calculation to any type of applied stress regardless of the medium – even fluids. 6.6.9. Extension to other constitutive laws Even if the rheology of the medium is more complex, e.g. viscoelastic fluids, nonlinear viscosity laws, viscoplastic fluids, time-dependent properties and so on, we should still be able represent its behavior under various types of applied stress. In some cases, the shear-rotation stresses may only be partially accumulated. We can describe viscoelastic behavior by weighting the accumulation term of ψ o by an accumulation factor 0 ≤ αt ≤ 1. Fluids with thresholds can also easily be represented by specifying a value of ψ o = ψc below which the medium behaves like an elastic solid. Many of the difficulties that are typically encountered in rheologies with nonlinear viscosities are no longer an issue with this model. In discrete mechanics, the concepts of viscosity and shear-rotation are exclusively associated with the faces of the primal topology, where the stress may be expressed in the form ν ∇ × V in fluids and dt ν ∇ × V in solids. As an example, let us examine the interaction of an incompressible viscous Newtonian fluid and a neo-Hookean elastic solid. The stress tensor of an incompressible isotropic hyperelastic material is as follows in the neo-Hookean model: σs = −p I + μs B,

[6.34]

where B = F Ft is the left Cauchy–Green deformation tensor. In two spatial dimensions, the Cayley–Hamilton theorem can be used to show that the Mooney–Rivlin model of a hyperelastic material is equivalent to the neo-Hookean model. We shall study a problem that was published by Sugiyama in 2011 [SUG 11]. Consider an elastic band with an applied shear stress generated by the periodic flow of an incompressible Newtonian fluid. The flow is laminar and periodic in x. Given that there are no compression terms, we can solve the problem in one spatial dimension along the y-axis for y ∈ [0, 1]. Suppose that the upper interface follows the periodic motion V (t) = V0 sin(ω t), where V0 = 1 and ω = π. The velocity of the lower

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surface is kept at zero. The solid occupies the lower part of the domain, and the fluid occupies the upper part of the domain; the position of the interface is y = 1/2. The theoretical solution found by Sugiyama was obtained by separating the spatial variable y from the time variable t; a homogeneous solution is found by considering a basis of Fourier functions on the interval y ∈ [0, 1] and exponential functions on the time interval in each of the fluid and solid domains separately. The sequence of Fourier coefficients can be determined from the coupling at the interface by requiring the velocity and the stress to be continuous. We can find a solution V (y, t) directly from the equations of discrete mechanics [3.38] simply by imposing the relevant conditions at y = 0 and y = 1. The coupling conditions at the interface, namely the continuity of the velocity and the stress, are implicitly guaranteed to hold by the dual curl operator. The notion of a 2D or 3D space does not exist in discrete mechanics. Instead, the operators define the orientations of the normal and tangent directions within a three-dimensional space. Despite this, the hypotheses of this example enable us to solve along a single spatial dimension. The time step is chosen to be δt = 10−4 to ensure good overall levels of accuracy; by comparing against the theoretical analytic solution, it can be shown that the numerical solution is second order in space and time. Figure 6.14 plots the velocity and the displacement of the interface Σ over time. The velocity of the upper wall is also shown. The solution establishes itself very quickly. After just a few periods, the velocity becomes fully periodic. The velocity profiles are shown until t = 10. The displacement of the solid over time may be deduced from the relation U = Uo + V dt, where dt represents both the differential element and the time increment δt = dt. Note that the displacement is strongly out of phase with the velocity of the interface. A selection of the velocity profiles in the y-direction are shown in Figure 6.15 once the periodic regime is fully established. The results converge to second order in space and time. Given the absolute accuracy (of the order of 10−4 s) obtained using a coarse mesh (n = 32), we can conclude that there is no observable error between the theoretical solution and the numerical solution. One advantage of the fluid–structure interaction for a neo-Hookean model described by Sugiyama is that it has a theoretical solution. This allows us to compare the numerical solutions that we obtain more precisely, but also allows us to develop new concepts, as we did for discrete mechanics in this section. Sugiyama obtained a first-order error in the L2 and L∞ norms, whereas the model [3.38] achieves second-order results with much lower absolute errors. This improvement is ultimately attributable to the separation of the properties at the interface, as well as the fact that no interpolation is performed, despite a fully monolithic and implicit treatment of the fluid–structure interaction.

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Figure 6.14. Study of a periodic fluid–structure interaction between a viscous fluid and an elastic solid; the viscosity of the fluid is ν = 1 and the shear modulus of the solid is ν = 4. The velocity of the fluid at the upper wall is shown in black, the velocity of the interface Σ is shown in red, and the displacement over time of the solid U at the interface is shown in blue. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Figure 6.15. Study of a periodic fluid–structure interaction between a viscous fluid and an elastic solid; the viscosity of the fluid is ν = 1 and the shear modulus of the solid is ν = 4. The figures show the velocity profiles as a function of y at times t = 10, t = 10.5 and t = 10.8. The solid line shows the theoretical solution, and the points show the spatial approximation obtained with 32 cells for y ∈ [0,1]

Fluid–structure interactions on 2D or 3D geometries with a moving interface can of course also be solved using the system [3.38]. However, without an analytic solution for comparison, there is little benefit in doing so, since the errors of the various methodologies accumulate over each step of the process. Other, more complex constitutive laws can also be modeled. However, although specific research in this domain is interesting in its own right, this does not offer any additional validation of the discrete model. The complete separation of the laws of motion and

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the constitutive and state laws in discrete mechanics will likely enable us to apply any kind of law to problems such as this.

6.7. Toward a unification of solid and fluid mechanics The formulation presented in this book is a continuation of the theory of discrete media [CAL 15a]. Its objective is to unify the equations, boundary conditions and numerical processing of all mechanical problems involving solids or fluids. Although this was also the stated objective of continuum mechanics, we are forced to recognize that the continuum approach has been unsuccessful; the various differences studied in these pages show the extent of the work that still remains. To cite just one example, the variable of the formulation depends on the medium – the velocity for fluids and the displacement or the stress for solids. Many other obstacles make fluid–structure interactions complex and often inconsistent. Likewise, the dynamic behavior of solids involving waves seems irreconcilable with how static stresses are calculated. The most significant achievements of the discrete approach are as follows: – discrete mechanics proposes a unique formulation of the equations of motion in terms of the velocity to represent the motion of both fluids and solids; – the velocity variable, the displacement and the stresses (po , ω o ) are calculated simultaneously and accumulated by simple differential operators; – the accumulation process for the stress holds for large displacements and large deformations; – the boundary conditions are consistently incorporated directly within the formulation itself; – the discrete generalization of the fluid statics law holds in every medium. This theory describes the motion and displacements of solids and fluids consistently, but the scope of the proposed description also extends to the dynamic behavior of these materials and the propagation of waves within them. The number of coefficients expressing the dependency between the stress and the displacement is strictly minimized by eliminating the unnecessary formalism of tensors. As a result, no other stresses such as compatibility conditions are required to calculate the displacement as a function of the deformation. Similarly, the concept of material frame-indifference, although perfectly valid, now becomes unnecessary to impose formally, since the stresses are naturally invariant under rotations. The elementary topology described in Figure 1.4 can be used to formulate any type of anisotropic material by defining μ appropriately on each face. The mechanical coefficients χT and μ defined at points and on faces can naturally be allowed to vary

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over space, or even depend on the stress itself; this will allow us to reproduce even complex rheological behaviors. The model proposes a macroscopic representation of the cohesion of matter arising from molecular and atomic bonds using only the intrinsic properties of the medium, the viscosity and the celerity of waves. Thus, we can apply shearing in a fluid indefinitely without modifying the applied force, whereas solids resist the shearing effects much more strongly (around 1012 times as strongly), thereby accumulating the stress. This key aspect of the model opens a path to studying limiting behaviors such as the rheology of fluids with extremely low velocities. Possibly the most important result of discrete mechanics for fluids and solids is the formal Hodge–Helmholtz decomposition of the equations of motion. The decomposition into irrotational and solenoidal components enables us to understand the mechanisms governing the equilibrium of a medium, and the divergence and curl of the velocity can be used to deduce the stresses, namely the equilibrium pressure po and the rotation stress ω o .

7 Multiphysical Extensions

Many problems with academic or industrial applications have already been successfully handled with the discrete mechanical model. All of the simulations in this book were performed with Aquilon®, a software program based on the concepts of discrete mechanics. The examples presented below are emblematic of certain key aspects of mechanics – turbulence, incompressible flows, thermoconvection and a fluid–structure interaction. The deflection of light by the sun, a problem that has been solved by general relativity, is instead approached from the perspective of discrete mechanics; the exactness of the results suggests that the scope of the discrete equations of motion may exceed that of classical mechanics. 7.1. Deflection of light The objective of this section is to show that the equations of motion of discrete mechanics are capable of reproducing the first major success of relativity – an accurate prediction of the deflection of light by the sun. Although the relativistic approach draws from differential geometry and geodesics, discrete mechanics simply establishes a system of equations of motion in a local frame of reference. The solutions of these equations fully reproduce all usual phenomena – fluid flows, stresses and displacements in solids, wave propagation, etc. One of the terms that appears in the discrete equations of motion can be assimilated with the curvature of space-time, which leads us to ask: can discrete mechanics be viewed as a relativistic theory? An affirmative answer to this question, even an incomplete one, would inspire hope that we might be able to achieve some consistency between the various approaches of mechanics. Discrete mechanics is based on two fundamental principles: the equivalence principle and Hodge–Helmholtz decomposition postulating the existence of two potentials (a vector potential and a scalar potential of the acceleration).

Discrete Mechanics: Concepts and Applications, First Edition. Jean-Paul Caltagirone. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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7.1.1. Description of the physical phenomenon Since Galileo’s time, various experiments have been performed to investigate the action of gravity on two different types of mass that appear to be accelerated identically, regardless of the internal structure or composition of the object being accelerated – this is the equality between gravitational mass and inertial mass (also known as the Weak Equivalence Principle, or WEP). Einstein would later extend the WEP into a stronger equivalence principle that relates to the fact that the velocity is always bounded by the celerity of light in a vacuum in special relativity. Today, the weak equivalence principle has been repeatedly verified by an impressive array of experiments. The equivalency is quantified by the Eötvös ratio η = 2 |γ1 − γ2 | / |γ1 + γ2 |, where γ1 and γ2 are the accelerations of the two masses. The acceleration can be measured independently from any frame of reference to extremely high accuracy. The modern view is that the WEP holds exactly, with an Eötvös ratio of η < 10−14 ; the article by Will [WIL 09] presents the various experiments that have been conducted over the last century. Other experiments led by France and the United States are also currently underway. It is hoped that they will achieve even greater levels of precision (10−15 or 10−18 ) to confirm (or refute) the exactness of the Weak Equivalence Principle (WEP). The equivalence principle allows us to rewrite the fundamental law of dynamics established by Newton in his Principia papers [NEW 90] in the following form: γ = g.

[7.1]

This equality might seem self-evident, since an isolated observer cannot distinguish the effect of gravity from their own specific acceleration. The equality had been known since Galileo, but Newton formulated his second law as m γ = F even though gravity was the most important force at the time. This law would prove to be somewhat problematic in the field of electromagnetism, but would be adopted by the field of dynamics nonetheless. The next question concerns the underlying meaning of the force per unit mass g. Is it exclusive to gravitational forces, or can we view g as the sum of the forces acting upon any particle of matter? The distinction is irrelevant anyway if gravity is the only force exerted by a body of mass M , viewed as a point, on a particle with (possibly zero) mass m. In this case, the acceleration due to gravity is equal to −G M/r2 , derived from the potential G M/r, where G is the universal gravitational constant and r is the distance between the particle and the body of mass M . If the gravitational

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force exerted by a body reduced to a point mass M is small, we can consider the Taylor expansion at G M/r c2 30.

Figure 7.6. Turbulent channel with Reτ = 590; average velocity profiles in reduced coordinates u+ = f(y+ ), deduced from the various contributions to [DEN 11]. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Starting from these average velocity profiles, we can characterize other zones in terms of the operators of discrete mechanics, including the dual curl of the primal

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curl r = −∇ × ∇ × V; this vector points along the edge Γ with unit vector t. The vector r is therefore collinear with the component V = W · t, where W is the velocity in a Galilean frame of reference. In practice, V can be viewed either as a vector or as a scalar defined on the edge Γ. This concept generalizes the ideas introduced by Harlow and Welch [HAR 65] for Cartesian meshes and arbitrary generalized polyhedral topologies.

Figure 7.7. Turbulent channel with Reτ = 590 [DEN 11], r = f(y+ ) = −∇ × ∇× V in logarithmic coordinates. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Figure 7.7 plots the variations of r = −∇ × ∇ × V as a function of the wall coordinates y + . We can distinguish four zones separated by the values y + = δv , y + = δl , y + = δn as follows: – y + < δv , a layer where only the viscosity effects are present. The value of r = −∇ × ∇ × V is of order one. It is difficult to truly characterize the behavior of this function from this example since the mesh is relatively coarse. But in general, the law of the velocity profile is known to be almost linear, u+ ≈ y + ; – δv < y + < δl , a zone that can be calculated very precisely by simulations, where the viscosity effects remain dominant but the turbulence is no longer negligible. This zone is characterized by a minimum of the function r·t at y + = δr = 9. While bearing in mind the relative imprecision in the definition of the zones δv , δl and δn , we can expect this value of δr = 9 to hold more generally; this will need to be confirmed by analyzing other flows with different turbulent Reynolds numbers Reτ ; – δl < y + < δn , a zone where the turbulence is not yet homogeneous, with some remaining viscosity effects, and where δl = 30 is approximately equal to the inverse

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of the curvature of r. On either side of this boundary, the curvature of r appears to be constant; – y + > δn , a region sometimes called the kernel where the turbulence is fully developed. Above δn = 200, the value of r · t is constant. Since the curvature of the velocity profile is small in this region, direct simulations are less accurate and depend on the degree of convergence of the turbulent statistics. The choice of values of δv , δl and δn will of course affect the solution, but contributions based on very different numerical methodologies (research codes, commercial software, etc.) nonetheless find very similar solutions. We shall choose the following boundaries between the regions described above: δv = 2, δl = 35 and δn = 200. These values allow us to reproduce the average velocity profile of the contributor unit − 2 very accurately. However, the variations of these boundaries are not so good at representing some of the other average profiles. At this stage, it would be difficult to fix the boundaries more precisely without having better convergence toward a unique solution. Once fixed, they can be viewed as universal constants. To gain a better understanding of the behavior of the solution in the turbulent kernel, we can plot the function 1/r = 1/∇ × ∇ × V as a function of y + in linear coordinates. Figure 7.8 shows that this function is practically constant in the center of the channel. Of course, this is the region where the mesh is coarsest and the numerical errors are largest. It is also more difficult to ensure the convergence of the turbulent statistics in this region.

Figure 7.8. Turbulent channel with Reτ = 590. The function 1/r = f(y+ ) = 1/∇ × ∇× V is plotted across the width of the channel, reproduced from [DEN 11]. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

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To analyze the behavior of the mean flow more finely, let us compare the velocity profile of the contribution unit − 2 against a parabolic profile. A regression of the profile u+ = f (y) gives us the curvature κ, which is constant. Figure 7.9 shows a comparison with the parabolic profile; the graph of the function u+ = f (y) on half of the channel shows that the function is strictly of order two.

Figure 7.9. Turbulent channel with Reτ = 590, results of unit-2 [DEN 11]. The average velocity profile u+ = f(y) is shown on the left, together with a parabolic profile for comparison. On the right, the same profile is plotted in logarithmic coordinates in the left half of the channel

The same procedure can be extended to each of the other contributions – only the curvature itself depends on the individual results. Table 7.1 lists the values of the curvature κ for each contribution. Contribution Curvature κ unit-1

-5.35

unit-2

-5.21

unit-3

-5.17

unit-4

-5.25

unit-5

-4.89

unit-6

-5.31

unit-7

-5.24

MKM99

-6.6

Table 7.1. Curvature in the zone y+ > δn calculated by model-free simulations [DEN 11] and a direct simulation [MOS 99]

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The table also shows the curvature found by the direct simulation by Moser et al. [MOS 99]. It is noticeably higher than the values of the model-free simulations from [DEN 11]. At this stage, the most important conclusion is that the turbulent kernel can indeed be associated with an average velocity profile with constant curvature κ. To determine more accurate universal values for the boundaries δv , δl and δn , we need an extremely accurate set of databases. Although the mean velocity profile itself seems very smooth, the values of the first- and second-order operators vary strongly, including in the direct simulations. The databases that are currently available are already sufficient to begin interpreting and analyzing the average profiles to establish a discrete mechanical model of turbulence. 7.2.4. Model of the turbulence in discrete mechanics One of the fundamental hypotheses of discrete mechanics is that the pressure effects are separable from the viscosity effects. Another key assumption is that the sum of all forces per unit mass can be expressed as two potentials, a scalar potential for the pressure stress and a vector potential for the shear-rotation stress γ = −∇φo + ∇ × ψ o , where po = ρv φo is the mechanical equilibrium pressure and ω o = ρv ψ o is the equilibrium rotation stress. The equilibrium pressure and shear-rotation stresses are accumulators of the pressure and viscosity forces. Recall the discrete equations established earlier: ⎧ o

 2  ∂V 1  2  1 p ⎪ 2 ⎪ − dt cl ∇ · V + ∇ |V| − ∇ × |V| n = −∇ ⎪ ⎪ ⎪ ∂t 2 2 ρv ⎪ ⎪ ⎪ ⎪ −∇ × (ν ∇ × V) + g ⎪ ⎨ ⎪ ⎪ p = po − dt ρv c2l ∇ · V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρ = ρo − dt ρ ∇ · V v

[7.23]

where ρv is the density associated with the inertial or gravitational mass, defined as a constant on the edge Γ, whereas ρ is the density defined on the points of the primal mesh. The system [7.23] is autonomous and does not need to be accompanied by any state equations; only the physical parameters themselves must be known. It does not matter how these parameters are determined: from tables, correlations, databases or by any other means. These quantities are local and instantaneous and can depend on the variables themselves. After the system has been solved, the quantities (p, ρ) are advected at the velocity of the fluid. This step is best accomplished by a Lagrangian transport calculation. The term in |V|2 from equation [7.23] can be incorporated into

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the gradient, corresponding to the dynamic pressure, and associated with the accumulation pressure po to give the Bernoulli pressure. The formulation of the system [7.23] is the most compatible with the original discrete perspective. The underlying idea is that the evolution of the physical system is observed as it passes from one mechanical equilibrium state to another. Interpreting the system in terms of motion in a Galilean frame does not change the equilibria themselves, which are entirely governed by the Lagrangian form of equations of motion. Note that the system [7.23] represents every kind of flow, mechanical behavior and wave, including shock waves. Solving this system via a direct simulation (DNS) gives all of the physical information that we need – not just the velocity field or the displacement field, but also the instantaneous pressure and shear stresses. In particular, the system represents turbulent flows if the spatial and temporal scales are chosen accordingly. The spatial scale characterizing a turbulent flow becomes smaller as the Reynolds number of the flow increases. Interactions between different scales of turbulence generate an apparent diffusion that could potentially be used to define a turbulent kinematic viscosity νdm in the context of an ad hoc model. Note that the fluid is assumed to be Newtonian and the turbulent dynamic shear viscosity μdm is not an intrinsic property but depends on the characteristics of the flow. This idea of apparent viscosity can be incorporated into the discrete model. Unlike the turbulent viscosity μt of a statistical model, the apparent viscosity is not defined in terms of an equilibrium between production and dissipation. The production does not appear anywhere directly; for example, it cannot be found in the Navier–Stokes equations. The viscosity represented by the quantity νdm is not an intrinsic property of the fluid, but it is nonetheless a physical quantity. Knowledge of this quantity can be exploited to model the turbulence realistically. The objective of this section is to apply the concepts of discrete mechanics in order to establish a model of turbulent flows phrased exclusively in terms of the averages of specific quantities, such as the velocity or the pressure stress. 7.2.5. Application to a flow in a channel with Reτ = 590 Historically, one of the most emblematic problems ever studied since the original experiments by Reynolds is the flow within a channel, which becomes turbulent at Reynolds numbers of the order of Re ≈ 2, 300. Many experimental studies, DNS, LES and even RANSE statistical models, have examined this problem, resulting in extremely accurate databases. The problem setup is straightforward. A flow with a constant Reynolds number is considered within a channel with rectangular cross-section, driven by a fixed pressure

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difference δp in the x-direction. Since the length of the channel is assumed to be large relative to its hydraulic diameter, the average pressure gradient ∇p is constant along the channel. To simulate this problem, periodic boundary conditions are typically assumed in the x- and z-directions. Each zone of the flow in the channel is modeled directly from equation [7.23]. The conclusions of our model will be fully consistent with the original dimensionless analysis that was performed to describe the behavior of this flow. But for now, these equations are already fully representative of any laminar or turbulent flow; the analysis that we shall perform below is based on the equilibrium between the pressure and the curl of the velocity of the equations. In particular, between any two layers of the fluid (1) and (2), the following equality holds: −ν1 (∇ × V)1 = −ν2 (∇ × V)2 .

[7.24]

This condition is intrinsically satisfied. It replaces the equality of shear stresses described by the Cauchy tensor. It also holds for two immiscible fluids in problems with multiphase flows. Since we are only interested in finding the steady average flow, the partial derivative of the velocity with respect to time is zero. The two inertial terms are also zero. This property is of course not satisfied in direct simulations. The equations of motion characterizing established turbulent motion in a channel with constant cross-section may therefore be expressed as follows in terms of average values: −∇po − ∇ × (μdm ∇ × V) = 0.

[7.25]

The Hodge–Helmholtz theorem defines the gradient of a scalar function and the curl of a vector quantity as being orthogonal. The two fields ∇p and ∇ × ω are equal and orthogonal, and so they must both necessarily be equal to a constant. Since the average pressure field is only a function of x and the shear field is only a function of y, equation [7.25] could in principle be reduced to partial derivatives along x and y. However, we shall avoid this reduction, since these operators have a physical meaning that disappears if we replace ∇ × V by dV/dy, which is just part of the component of the curl. This zoneof the flow is defined by the reduced coordinates y + = y uτ /ν < δv , where uτ = τw /ρ is the shear velocity and ν is the kinematic viscosity. The reduced velocity is classically defined by u+ = u/uτ . The characteristic length δv of this zone can thus be determined by studying the turbulent boundary layer of a planar sheet, where it is typically equal to y + ≈ 7.

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Within this layer, the viscous effects are dominant and any remaining velocity is maintained by the outer layers. This is a Couette flow generated by the adjacent fluid, known as the buffer zone. However, in a planar Couette flow, the curl is constant and the pressure gradient is zero. This is not the case for the average gradient ∇po . The only way to recover Couette flow in this zone is to take ∇ × ω o = ∇po and μdm = μ; equation [7.25] then becomes: ∇ × V = 0,

[7.26]

i.e. u+ (y + ) = V · ex = y + . This recovers the classical linear law, except that the slope is imposed by the external flow. From a physical point of view, the imposed pressure in any given cross-section is determined solely by the constant gradient ∇po . This is especially true if the thickness δl is very low. Without the term in ∇ × V, the gradient in the viscous sublayer cannot be eliminated, and the law u+ = f (y + )+ is not linear but quadratic. Hence, the shear stress is steady and accumulates in ω o according to the law ω o (y + ) = y + ez . In boundary layer theories, these two regions – the buffer zone and the logarithmic zone – are separate. This is not the case here, in discrete mechanics. The buffer zone is not just a coupling zone but also plays an essential role in the equilibrium between the pressure and the viscous stresses. These two effects are of the same order and modeling them can be particularly tricky. We shall assume that the flow is unsteady in each zone, allowing the accumulated viscous stresses to be destroyed, with αt = 0. Consequently, the flow is generated by the constant pressure gradient ∇po = δ. The equations of motion now become: −∇po − ∇ × (μdm ∇ × V) = 0,

[7.27]

which gives: d dy +

μdm

du+ dy +

= δ.

[7.28]

Since δ is a constant, we must have μdm = y +2 , since the product y + · du+ /dy + is constant, as established by dimensionless analysis and corroborated by direct simulations. The solution of the above equations with μdm = y +2 is: u+ = δ Ln(y + ) −

a + b. y+

[7.29]

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The logarithmic law is recovered as expected, but there is also an additional term in 1/y + that increases in significance as y + approaches δv and becomes negligible as y + approaches δl ≈ 140 at the end of the logarithmic zone. The constants a and b can be deduced from the profiles calculated by DNS, but they are no longer relevant anyway after equation [7.27] is integrated directly. The coupling between the different zones is ensured by the implicit condition [7.24]. Hence, the apparent viscosity μdm is equal to μ in the viscous sublayer, increases slowly in the buffer zone, and increases more rapidly in the logarithmic zone. These variations are fully determined by the apparent viscosity in these two zones, μdm = y +2 . In the kernel of the flow, after y + = δl ≈ 140, the turbulence is fully developed and the diffusion of momentum is maximal; in other words, μdm attains its highest values in this zone. In this region, the flow is mainly characterized by the pressure gradient. To satisfy equation [7.27], we must invoke the results of direct simulations, as well as the physical fact that this zone has a high mixing rate. The average velocity profiles can be very precisely represented by a second-order law that takes into account the symmetry about the axis of the channel. This law can be expressed in the form +2 u+ = u+ /κ, where κ is the absolute value of the curvature of the velocity m − y profile; this is a constant that depends on the turbulent Reynolds number Reτ . This gives the value μdm = −δ/2 κ = μm of the viscosity. The apparent viscosity is constant for a given turbulent Reynolds number. Let us briefly present the results obtained by the discrete approach [CAL 15a] for the flow in a channel with Reτ = 590. Consider a 3D channel with rectangular cross-section whose lateral dimensions in the x- and z-directions are larger than the vertical dimension in the y-direction. No-slip conditions are imposed on the horizontal walls; periodic boundary conditions are applied in the x-direction. A constant pressure gradient δp ex is maintained throughout the volume to fix the value of the average velocity V0 = V0 · ex . The dimensions and the velocity V0 are chosen in such a way that the turbulent Reynolds number Reτ is 590. Dimensionless quantities are then derived from the vertical dimension and the local velocity based on this value to deduce the quantities y + and u+ . This configuration has been extensively studied since the early 1980s by performing direct simulations or LESs; the results are broadly consistent at low Reynolds numbers like these. We can cite the work of Kim et al. [KIM 87] and Moser et al. [MOS 99] as examples. More recently, other authors have published refined comparisons that were reviewed in a test case organized by Denaro in 2011. The results were updated in 2014 and are available in the literature [DEN 11]. Our objective here is to develop a steady turbulence model for the averaged parameters that is fully based on the discrete mechanical approach and which may be

276

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applied to the discrete equations arising from this model. Even if the Navier–Stokes equations lead to identical results when the physical properties are constant, they differ significantly from the discrete equations, meaning that a new model of turbulence is needed for the discrete approach. The concept of rotation effectively lies at the heart of the discrete theory, which describes viscous effects solely in terms of the vector potential ω rather than a stress tensor. Many of the approaches currently used to model the turbulence appear to conflate the diffusion of momentum and the viscous dissipation. These are two different effects, and most models from statistical modeling and LES are in fact based on the dissipation. In discrete mechanics, the transfer of momentum within the fluid is governed by the dual curl and the shear-rotation stress ω through the term ∇ × (ω o − μ ∇ × V). The quantity ω o represents the accumulation stress of the diffusion effects associated with μ, the shear-rotation stress, which is the asymptotic value of the instantaneous effects. For solids or fluids with very small time constants, this term can be written as dt μe , where μe is the elastic shear modulus. In fact, this time constant corresponds to the period dt between two observations of the physical system. In turbulence, this characteristic time scale becomes smaller as the Reynolds number becomes larger. The shear stresses are diffused by vortices. This creates an apparent diffusion phenomenon as a result of mixing. We can define viscosity arising from this turbulence, but it represents a completely separate concept from the viscous dissipation, which does not appear directly in the equations of motion. It is neither a turbulent viscosity, nor a subgrid-scale viscosity. We shall call this viscosity μdm , and it will of course depend on the local conditions of the flow. The simulation was performed in a 2D domain with periodic conditions in x and a mesh with 100 cells in the x-direction and 64 cells in the y-direction. The initial conditions on (p, V) were set to zero and the pressure gradient was chosen in such a way that Reτ = 590. The steady solution was calculated up to the machine error. A one-dimensional model was also developed and the results are identical; the steady version of the problem that is periodic in x effectively reduces to a one-dimensional equation along y. The problem is modeled implicitly by layers using the term ∇×(ω o −μdm ∇×V), which already appears in the equations of motion, based on the various zones of the flow: – the laminar boundary sublayer for y + < δ1 ; – the buffer zone for δ1 < y + < δ2 ; – the logarithmic zone for δ2 < y + < δ3 ; – the fully turbulent zone for y + > δ3 .

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The values of the boundaries δ1 , δ2 , δ3 are defined from values in the literature that are consistent with previously conducted direct simulations. A strictly linear law u+ = f (y + ) can only be obtained for the accumulation stress ω o , since the constant gradient maintained within the laminar boundary sublayer is not compatible with a linear law. However, the thickness δ1 is very small and the pressure is the same as in the buffer zone. In practice, the term ω o balances out the effect of the pressure gradient and the velocity law is perfectly linear in the boundary sublayer. The physical explanation for this is as follows: within the laminar boundary sublayer, the flow is steady, and μ has attained its asymptotic limit, known as the molecular viscosity. After this point, unsteady exchanges between vortices cause a small part of the rotation stresses to accumulate, but anything that accumulates is then immediately diffused by these same exchanges. This leads to larger viscosity values, μdm > μ. Figure 7.10 shows the average velocity u+ = f (y) in physical coordinates and u = f (y + ) in logarithmic coordinates. The distinct zones of the flow are clearly distinguishable. The results are compared against the results found by Germano, reproduced from [DEN 11]. The objective of this section is to use physical concepts to establish a robust and reliable averaged model with as few parameters as possible that can be determined from direct simulations. Figure 7.11 plots the values of u+ (y + ) over time obtained by the discrete model against the results of direct simulations recorded in the reference [DEN 11]; the points issued to the model accurately reproduce the evolution in dimensionless coordinates. The dispersion of the results of the direct simulations can be attributed to methodological differences in the authors’ approaches, as well as refinements of the mesh near the walls. +

Figure 7.10. Turbulent channel with Reτ = 590, comparison between the results of DNS (unit-2) [DEN 11] (solid line) and the discrete approach [CAL 15a] (dots). The profile u+ = f(y+ ) of the average velocity is shown in reduced coordinates on the left and logarithmic coordinates on the right. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

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Figure 7.11. Turbulent channel with Reτ = 590. Results of the TDM model (dots) compared to direct simulations reproduced from [DEN 11] (solid lines). For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

Figure 7.12 gathers together the results for the reduced velocity μdm /μ as a function of the reduced position y + in physical coordinates and logarithmic coordinates. In the central zone, the ratio is essentially constant. The same is true in the viscous boundary sublayer, where the ratio is equal to one. Between these two zones, the evolution is quadratic.

Figure 7.12. Turbulent channel with Reτ = 590. The reduced turbulent viscosity μdm /μ is shown for the TDM model (dots) and for the DNS by [DEN 11] (solid lines). For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

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The simulations by Moser et al. [MOS 99] are one of the possible references for this turbulent flow problem in the literature. Figure 7.13 plots the reduced velocity as a function of the reduced position in wall units, showing that these reference results are highly consistent with the results of the discrete model.

Figure 7.13. Turbulent channel with Reτ = 590, results from MKM99 [MOS 99] (solid line) and from the TDM model (dots). The average velocity profile u+ = f(y+ ) is shown in wall units on the left and in logarithmic coordinates on the right. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

We do not impose or even consider any laws on the velocity profiles, since they can be implicitly reproduced by choosing the viscosity μdm accordingly. So, which parameters do we need to model the flow in a turbulent channel, and hence to model an arbitrary problem? In the viscous boundary sublayer, the viscosity is given by μ, the molecular viscosity, but the thickness δv of this layer needs to be known. The intermediate layer requires us to know another thickness to determine the end of the logarithmic zone, and the turbulence zone presented above requires us to know κ, the curvature of the velocity profile in this zone. Therefore, we at the very least need the three parameters δv , δl and δn , as well as κ itself. This model also needs to satisfy various constraints. In particular, it must naturally reduce to Poiseuille’s solution for a laminar flow in the degenerate case. In this case, the curvature κ is given by κ = −1/2 μ, and the viscous sublayer and the intermediate zone vanish; the laminar flow occupies the whole channel. Locally, the viscosity μdm can be calculated from the equilibrium between the pressure effects and the shear effects as follows in discrete mechanics: μdm =

∇p , ∇×∇×V

[7.30]

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where the denominator is implicitly understood as ∇d × ∇p × V; the term ∇d × is the dual curl and ∇p × is the primal curl. This is in fact just the curvature of the velocity profile. The two vector quantities in equation [7.30] both have the same direction as the unit vector t of the edge Γ. It therefore makes sense to define the scalar μdm on this same edge. Statistical models produce much more fragmented results involving the introduction of so-called universal constants. Furthermore, models based on Boussinesq’s hypothesis and the concept of turbulent viscosity μt typically only hold in specific conditions – high Reynolds numbers, low Reynolds numbers, etc. ⎧ μdm ⎪ y + < δv , =1 ⎪ ⎪ μ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ μdm ⎪ ⎪ ⎪ δv < y + < δl , = 1 + 5 (y + − δv )2 ⎪ ⎪ μ ⎨ ⎪ μdm ⎪ ⎪ δl < y + < δn , = μm − (y + − δn )2 ⎪ ⎪ μ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ μdm δ ⎪ ⎪ ⎩ y + > δn , = μm = − μ 2κ

u+ = y + u+ = δ Ln(y + ) −

a +b y+

u+ = δ Ln(y + ) −

a +b y+

u+ = u+ m−

[7.31]

y +2 κ

The system [7.31] summarizes the law of the wall in each zone as a function of the reduced position. These zones are defined by the distances δv = 2, δl = 35, δn = 200 and μm = −δ/2 κ. 7.3. The lid-driven cavity problem with Re = 5,000 Incompressible flows are an important category of mechanical problems in which the constraint ∇ · V = 0 must be satisfied throughout the duration of the simulation. This constraint is a property of the motion rather than an intrinsic property of the medium, which has well-defined physical characteristics that in particular include a specific compressibility χT or celerity. As we have seen, the divergence appears in the term dt c2l ∇ · V; for a given celerity, motions with low time constants are fundamentally compressible in nature. For example, even though water is a very low-compressibility medium, sound waves propagate through water at a celerity that is just four times larger than the celerity of air.

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As well as compressible flows, the formulation of the system [3.38] enables us to simulate incompressible motion by choosing high values for the grouped term r = dt c2l (of the order of r ≈ 1/∇ · V∗ , where ∇ · V∗ is the desired divergence). The discrete equations of motion are just as good at representing flows in realistic media as flows that are viewed as incompressible. They do not require specific adjustments in either case. Naturally, any simulation that aims to fully represent the physics of a phenomenon must be performed with a time increment dt that is compatible with the time constants of this phenomenon. The lid-driven cavity problem with a sufficiently high Reynolds number is ideal for testing the validity of the inertial terms of the equations of motion. In the general case, the solenoidal and irrotational terms of the inertia do not have straightforward expressions. This is the perfect opportunity to present concrete results to the many numerically inclined mathematicians who are understandably strongly invested in the omnipresent Navier–Stokes equations. The Navier–Stokes equations have proven their worth in the vast majority of problems that have been studied for centuries. The objective here is not to obtain results that differ from the Navier–Stokes equations, but simply to verify that the solutions are the same up to any errors in the numerical methodology. We shall consider the problem of a lid-driven cavity with a Reynolds number of Re = 5, 000. Figure 7.14 shows the streamlines after convergence. Even though we are using a different physical model, based on the equations of discrete mechanics instead of the usual Navier–Stokes equations, which handles the pressure as a Bernoulli pressure and writes the inertial term as −∇ × (φi n) + ∇ (φi ), the solution obtained is very close to previous results found for this Reynolds number.

Figure 7.14. Lid-driven cavity with Re = 5,000: streamlines calculated by a simulation on a Cartesian mesh with 2562 cells. The vortices have amplitudes ψmax = 0.1211 and ψmin = −0.003138, as well as positions xmax = 0.5152, ymax = 0.5353 and xmin = 0.8024, ymin = 0.07180, respectively

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The results found by Bruneau [BRU 06] for a Reynolds number of Re = 5, 000 and a steady flow are listed in Table 7.2. This quantitative comparison considers the amplitude and the position of the vortices generated by the flow detaching from the walls of the cavity; the reference results are reproduced to an extremely good level of accuracy despite the differences in the numerical methodology and the approach based on the discrete formulation instead of the Navier–Stokes equations. A convergence study of this configuration found a spatial rate of convergence of 2 for the velocity and the pressure. The time steps calculated the Bernoulli pressure, from which the pressure itself was then deduced. Ref.

ψmax

xmax

ymax

ψmin

xmin

ymin

This study 2562

0.1219

0.5153

0.5352

−3.086 · 10−3

0.8040

0.07310

Bruneau et al. 20482 0.12197 0.515465 0.53516 −3.0706 · 10−3 0.80566 0.073242

Table 7.2. Comparison of the results obtained by continuum mechanics [BRU 06] and the results obtained from the discrete formulation presented here with Re = 5,000 for a Chebyshev mesh of 2562 cells

Discrete mechanics does not call into question the results obtained by the Navier–Stokes equations. As the numbers show, the same solutions are obtained at least within the various inevitable numerical errors. Still, there are many differences between the continuum and discrete models. One example is the principle of conservation of mass, which systematically accompanies the Navier–Stokes equations in continuum mechanics. The discrete equations of motion do not need to invoke this equation, but the mass is, nonetheless, strictly conserved by any compressible or incompressible motion – the discrete equations are autonomous and do not require any supplementary constitutive laws or state equations. In practice, conservation of mass is implicitly incorporated into the equations of motion via the term dt c2l ∇ · V. Showing that the discrete equations yield exactly the same solutions for this problem with a Reynolds number of Re = 5, 000 is not our only objective. We shall also take the opportunity to exhibit the various components of the discrete equations of motion. In the example presented here, the density is constant, ρ = 1. We therefore have that p = φ and ω = ψ; the equations of motion are given by:

2 ∂V |V|2 |V| −∇ × (ν ∇ × V) . [7.32] −∇× n = − ∇ φo − dtc2 ∇ · V + ∂t 2 2 The flow is steady, ∂V/∂t = 0, and incompressible, ∇ · V = 0. Moreover, the accumulation factor is αt = 0 in a Newtonian fluid, which gives us ψ = ψ o =

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−ν ∇ × V, the instantaneous shear stress. There is also one other equality between the gradient of the potential φ and the curl of the shear potential ψ:



1 1 −∇ φo + |V|2 + ∇ × ψ o + |V|2 n = 0, 2 2

[7.33]

where φoB = φo + |V|2 /2 is the Bernoulli potential. As before for the Bernoulli pressure, the duality-of-action postulate is applied to the inertia to define the rotation-shear stress. The two vector fields in equation [7.33] are orthogonal. Other effects, such as the gravitational acceleration or the capillary acceleration, can also be perfectly incorporated into the discrete formulation in a consistent form. Neglecting any uniform translational or rotational motion, the variations of the velocity over time are induced by the imbalance between the two terms of this equation. This example of a simple flow demonstrates the value provided by the discrete formulation over a continuum-type approach. When the properties are constant, the solutions are strictly the same, regardless of whether the problem involves analytically solving the Navier–Stokes equations or simulating a complex flow whose solution must be computed numerically. The added value of the discrete formulation arises from the physical understanding that it provides of these scalar and vector potentials, allowing us to fundamentally view the equations of discrete mechanics as a Hodge– Helmholtz extractor of the components of the acceleration. 7.4. Natural convection into the non-Boussinesq approximation The natural convection induced by a horizontal temperature gradient caused by the action of gravity in a closed cavity results in a noticeable modification of the average pressure in the cavity when the temperature differences are large; the same phenomenon can also be observed with lower temperature differences, including when Boussinesq’s approximation is supposedly applicable. This problem is impossible to simulate (without sophisticated trickery) if the formulation does not have a continuous-memory medium. In Boussinesq’s approximation, the average pressure is not a useful quantity; only the constraint ∇ · V = 0 is applied. If a low Mach number is assumed, we need to handle the conservation of mass explicitly, for example as described by Paolucci [PAO 82]. Given that the box is closed and the total mass is conserved, we must impose an average thermodynamic pressure that reflects this conservation; accordingly, the low Mach model assumes that the thermodynamic pressure p(t) is decorrelated from the other pressures, including the driving pressure. The local volumetric density satisfying the state law is therefore corrected accordingly. The divergence of the velocity is strictly zero and does not take into account any of the significant temperature variations within the cavity. This model is clearly inapplicable if the box is partially open.

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Consider the physical problem of a 2D square cavity with a temperature difference of δT = 720K applied to the two vertical walls. The two horizontal walls are assumed to be adiabatic. We are only interested in finding the steady regime, which is known to exist for Ra = 106 . The initial conditions are given by T0 = 600 K, and the fluid inside the cavity is air, viewed as an ideal gas with pressure p0 = 101, 325 Pa. Since the evolution of the density as a function of T and p is not linear, there is no reason for the pressure to remain constant over time in the cavity; its value depends on the temperature field and the local density field. Average pressure

p0

Average temperature T0

101,325 600

Average density

ρ0

0.588414

Ideal gas constant

r

287

Ratio of specific heats γ

1.4

Prantdl’s number

Pr

Thermal conductivity λ Specific heat Dynamic viscosity

cp

0.71 0.041800782 1004.5

μ0 2.95456 × 10−5

Table 7.3. Characteristics of the test problem for non-Boussinesq convection

The initial conditions of the test case are summarized in Table 7.3. Rayleigh’s number and the dimensionless parameter ε characterizing the temperature difference are given by: ⎧ ν ρ20 g δT L3 ⎪ ⎪ Ra = ⎪ ⎪ ⎨ a μ20 T0 ⎪ ⎪ ⎪ T − Tc ⎪ ⎩ ε= h Th + Tc

[7.34]

The hot and cold temperatures that must be imposed on the vertical walls of the cavity are therefore Th = T0 (1 + ε) , Tc = T0 (1 − ) , δT = 2T0 . For Tm = 600K and ε = 0.6, we have that δT = 2 · 0.6 · 600 = 720 K, and so the length of the sides of the cavity must be equal to L = 0.067066263247 m. The Mach number is of the order of M = 7 × 10−3 . Traditionally, the incompressible model is considered to be applicable whenever M < 0.2. This test case was originally studied in order to determine the differences between the low

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Mach and incompressible solutions given identical Rayleigh numbers. The results were relatively disappointing; very few differences could be observed. The same problem was then studied numerically, with around 20 contributors performing simulations with different numerical methodologies. The author’s own contributions [CAL 00] for this test case are presented as tables to facilitate a quantitative comparison. The principal characteristics of the solution are given by a Nusselt number of N u = 8.8604 and a ratio of the average pressure at convergence to the initial pressure of p/p0 = 0.85635, for a Rayleigh number of 106 . The principal fields of the results are shown in Figure 7.15.

Figure 7.15. Top row: temperature field, pressure field (min = −13,690, max = −13,364 Pa) and density field (min = 0.307, max = 1.27 kg m−3 ). Bottom row: divergence field (min = −7.66, max = 9.46), horizontal velocity field (min = −0.106, max = 0.175) and vertical velocity field (min = −0.213, max = 0.274). For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The discrete model based on the system of equations [2.33] was applied directly to the problem. The principal characteristics of the model are summarized by the following properties: – no approximations or additional hypotheses; – the total mass of the cavity is conserved to within the machine error throughout the entire calculation without needing to be artificially updated, even though the local divergence is strongly different from zero; – all relevant phenomena – wave propagation, advection, diffusion – are taken into account and reproduced by the model; – the model is valid at any time constant (waves, flows, diffusion) without requiring modification.

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The temperature field is not very different from the low Mach solution, but the pressure and density fields differ significantly. As expected, the divergence of the velocity is of order 1, rather than 0 like in the low Mach model. In the steady regime, the fluid alternates between heating and cooling near the vertical walls. This phenomenon shows how density varies with temperature, which is significant due to the extent of the imposed temperature difference. The results show that the divergence is far from zero. The constraint ∇ · V = 0 does not describe the conservation of mass, but rather the conservation of volume. Clearly, the approximation of incompressibility does not make sense if there are density variations arising from temperature gradients, concentration gradients, etc. The greatest advantage of the discrete mechanical model relates to the conservation of mass within the cavity. In this problem, the mass is conserved up to the machine error. The density of course varies as a function of the other variables, in particular the temperature and the pressure, but its evolution is fully determined by the relation ρ = ρo − dt ρo ∇ · V. Since the divergence of the velocity is obtained from the equations of motion, the total mass is implicitly conserved in a manner that is entirely consistent with the rest of the formulation. The models developed for Boussinesq or low Mach approximations do not implicitly conserve the mass, even when the pressure and temperature differences are small. This shortcoming arises from the fact that state equations are used to calculate the density from the pressure and the temperature. If the thermophysical coefficients need to be fixed, then the local equilibrium hypothesis on which the state equations are based is redundant and inconsistent with the equations of motion and the conservation of mass. We cannot simply add the state equations to the system just because we have too many variables. Furthermore, a low Mach number is not sufficient to guarantee that the low Mach model, let alone the incompressible model, is applicable to the problem. The equivalence principle eliminates ρ from the equations of motion for every term including an acceleration, but this does not prevent us from considering situations such as the problem presented here, where the density is allowed to vary continuously. The density is calculated from the equations of conservation of mass and introduced into the equations of motion as a parameter in the term po /ρv . 7.5. Fluid–structure interaction This section considers a simple example of a fluid–structure interaction in a square cavity of size h containing a fluid of viscosity νf . A solid cylinder whose axis of rotation goes through the point with coordinates x0 and y0 is held fixed within the cavity.

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A rotation is imposed on the solid domain by solving equation [4.74] for a fixed value of Ωo and a very large shear modulus νs = dtc2t :     dV = −∇ φo − dt c2l ∇ · V + ∇ × Ωo − dt c2t ∇ × V + g. dt

[7.35]

Unlike techniques that penalize the velocity inside the circle, which generate pressure discontinuities at the interface Σ, the formulation [7.35] allows us to solve the problem without any numerical difficulties, giving a continuous pressure field throughout the domain and at the interface. Figure 7.16 presents the Bernoulli pressure fields and the horizontal and vertical components of the velocity. Note that the Bernoulli pressure field in the solid is influenced by the pressure in the fluid. As expected, the velocity field in the solid cylinder is that of a rigid rotational motion.

Figure 7.16. Fluid–structure interaction; the imposed rate of rotation is equal to Ωo = 50, the size of the square is h = 0.1, the circle has radius R = 0.02 and position x0 = 0.04, y0 = 0.4. The viscosity of the fluid is νf = 1. The Bernoulli pressure field is shown on the left, the horizontal velocity is shown in the center and the vertical velocity is shown on the right. The divergence is of the order of the machine error and the curl in the circle is exactly equal to Ω = 2 Ωo = 100. The velocity takes values in the interval ±1. For a color version of this figure, see www.iste.co.uk/caltagirone/mechanics.zip

The energy generated by the dissipation arising from the compression and viscous phenomena in the domain can be calculated as follows:  t Ed =

0 Ωf

c2l

2

(∇ · V) dt +

 t 0 Ωf

2

ν (∇ × V) dt.

[7.36]

Since the flow is incompressible, only the viscous dissipation is present. We can also compute the circulation V · t around the closed contour formed by the outer walls of the domain. Since the curl is conserved, the circulation can also be deduced by performing a volume integral.

References

[ALH 03] A L -H ADHRAMI A., E LLIOTT L., I NGHAM D., “A new model for viscous dissipation in porous media across a range of permeability values”, Transport in Porous Media, Kluwer Academic Publishers, vol. 53, no. 1, pp. 117–122, 2003. [ANG 12] A NGOT P., C ALTAGIRONE J.-P., FABRIE P., “A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems”, Applied Mathematics Letters, vol. 25, pp. 1681–1688, 2012. [ANG 13] A NGOT P., C ALTAGIRONE J.-P., FABRIE P., “Fast discrete Helmholtz-Hodge decomposition in bounded domains”, Applied Mathematics Letters, vol. 26, pp. 445–451, 2013. [BEA 67] B EAVERS G., J OSEPH D., “Boundary conditions at a naturally permeable wall”, Journal of Fluid Mechanics, vol. 30, pp. 197–207, 1967. [BEL 89] B ELL J., C OLELLA P., G LAZ H., “A second order projection method for the incompressible Navier-Stokes equations”, J. Comput. Phys., vol. 85, pp. 257–283, 1989. [BEN 02] B ENZI M., “Preconditioning techniques for large linear systems: A survey”, J. Comput. Phys., vol. 182, pp. 418–477, 2002. [BEN 05] B ENZI M., G OLUB G., L IESEN J., “Numerical solution of saddle point problems”, Acta Numerica, vol. 14, pp. 1–137, 2005. [BHA 14] B HATIA H., PASCUCCI V., B REMER P., “The natural Helmholtz-Hodge decomposition for open-boundary flow analysis”, IEEE Transactions on Visualization and Computer Graphics, vol. 20, no. 11, pp. 1566–1578, 2014. [BOR 14] B ORDÈRE S., C ALTAGIRONE J.-P., “A unifying model for fluid flow and elastic solid deformation: A novel approach for fluid structure interaction”, Journal of Fluids and Structures, vol. 51, pp. 344–353, 2014. [BOR 16] B ORDÈRE S., C ALTAGIRONE J.-P., “A multi-physics and multi-time scale approach for modeling fuid-solid interaction and heat transfer”, Computers and Structures, vol. 164, pp. 38–52, 2016.

Discrete Mechanics: Concepts and Applications, First Edition. Jean-Paul Caltagirone. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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[BOT 98] B OTELLA O., P EYRET R., “Benchmark spectral results on the lid-driven cavity flow”, Computers & Fluids, vol. 27, pp. 421–433, 1998. [BRI 49] B RINKMAN H.C., “A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles”, Flow, Turbulence and Combustion, vol. 1, p. 27, 1949. [BRU 06] B RUNEAU C., S AAD M., “The lid-driven cavity problem revisited”, Computers & Fluids, vol. 35, pp. 326–348, 2006. [CAI 14] C AI M., N ONAKA Y., B ELL J., et al., “Efficient variable-coefficient finite-volume Stokes solvers”, Communications in Computational Physics, vol. 16, pp. 1263–1297, 2014. [CAL 00] C ALTAGIRONE J.-P., “Modelling and simulation of natural convection: Flows with large temperature differences. A benchmark for low Mach numbers solvers”, Workshop/12th CFD Seminar, CEA Saclay, January 2000. [CAL 13] C ALTAGIRONE J.-P., Physique des écoulements continus, Springer-Verlag, 2013. [CAL 15a] C ALTAGIRONE J.-P., Discrete Mechanics, ISTE Ltd, London and John Wiley & Sons, New York, 2015. [CAL 15b] C ALTAGIRONE J.-P., “Équilibres statique et dynamique de l’ascension capillaire”, Available at: https://hal.archives-ouvertes.fr/hal-01200464, 2015. [CAL 15c] C ALTAGIRONE J.-P., V INCENT S., “A kinematics scalar projection method (KSP) for incompressible flows with variable density”, Open Journal of Fluid Dynamics, vol. 5, pp. 171–182, 2015. [CAL 16a] C ALTAGIRONE J.-P., “Déflexion gravitationnelle de la lumière en mécanique discrète”, Hal, Available at: https://hal.archives-ouvertes.fr/hal-01422632, 2016. [CAL 16b] C ALTAGIRONE J.-P., “Modélisation des effets capillaires en mécanique des milieux discrets”, Hal, Available at: https://hal.archives-ouvertes.fr/hal-01251670, 2016. [CAL 17] C ALTAGIRONE J.-P., “Décomposition complète de Hodge-Helmholtz en mécanique discrète”, Hal, Available at: https://hal.archives-ouvertes.fr/hal-01583505, 2017. [CHO 68] C HORIN A., “Numerical solution of the Navier-Stokes equations”, Math. Comput., vol. 22, pp. 745–762, 1968. [COI 07] C OIRIER J., Dunod, Paris, 2007.

NADOT-M ARTIN

C.,

Mécanique

des

milieux

continus,

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Index

A, B

C

acceleration absolute, 116 capillary, 50, 170, 177, 197 compression, 49 gravitational, 50 inertial, 50, 52 of gravity, 12, 250 viscous, 49, 50 accumulation of the compression effects, 42 of the rotation stress, 48 advection of a scalar, 88 of a vector, 88 of the density, 80 angle advancing contact, 197 contact, 175, 194 dynamic contact, 196 receding contact, 197 static contact, 196 anisotropic, 19, 20 attenuation of longitudinal waves, 48 of transverse waves, 48 axiom of discrete mechanics, 28 Boussinesq’s approximation, 283

calculation of the displacement, 217 celerity longitudinal, 35, 61, 221 of light, 12, 115, 251 transverse, 35, 61, 221 Clapeyron relation, 91 Clausius-Duhem inequality, 40, 218 coefficient barodiffusion, 83 compressibility, 219, 224 compression viscosity, 40 isothermal compressibility, 39 Lamé, 20 Lamé viscosity, 219 molecular diffusion, 80 Poisson, 39, 127, 219 shear viscosity, 40 thermal conductivity, 19, 99, 102 thermal diffusion, 83 thermal expansion, 219 viscosity, 224 volume viscosity, 40 component irrotational, 63, 118 solenoidal, 63, 118, 143 conditions boundary, 63, 225 compatibility, 126, 133 coupling, 63, 227

Discrete Mechanics: Concepts and Applications, First Edition. Jean-Paul Caltagirone. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

296

Discrete Mechanics

Dirichlet, 63 Fourier, 63 Neumann, 72 conservation of energy, 89 of flux, 89 of mass, 77 of momentum, 69 convection natural, 283 non-Boussinesq, 283 convergence in space, 147 in time, 147 coupling fluid-structure, 246 monolithic, 230 curvature contact, 178, 196, 204 discrete, 23 Gaussian, 23 mean, 23, 173 of space, 116 of the Universe, 259 principal, 23, 173 Riemann, 23 D Darcy, 95 deflection of light, 249 density intrinsic, 84 partial, 84 derivative time, 58, 72 total, 58 discrete horizon, 2 space-time, 1 dissipation mechanical, 94 of energy, 60 of longitudinal waves, 61 of the rotation stress, 131 of transverse waves, 61 of waves, 91, 131 dynamics of the vorticity, 74

E Eddington, 12, 52, 251, 257 effect Bénard-Marangoni, 175 capillary, 170 Marangoni, 175, 190 energy compression, 60 conservation, 89, 93 dissipated, 97 internal, 91, 95 kinetic, 60, 93, 94 mechanical, 91 thermal, 91 total, 93 enthalpy condensation, 111 fusion, 111 equation Brinkman’s, 162 discrete equations of motion, 56, 58 Ergun, 161 generalized Bernoulli, 151 Laplace, 187 Young-Dupré, 178 equations at a discontinuity, 65 at the interfaces, 169 conservation, 19, 124 constitutive, 19 Euler, 150 Navier-Lamé, 126 Navier-Stokes, 41, 127, 210 of discrete mechanics, 124 of motion, 174 state, 19 equilibrium local thermodynamic, 56 mechanical, 56, 94, 182 of a solid, 242 equivalence principle Einstein’s, 11 Galileo’s, 11, 250 Newton’s, 11 weak, 11, 31, 115, 250 Euler, 37

Index

expansion isenthalpic, 162 isothermal, 162 experiment Couette’s, 44 Michelson and Morlay, 33 Poiseuille’s, 49 F factor longitudinal attenuation, 214 transverse attenuation, 214 flow capillary ascension, 206 compressible, 155 Green-Taylor, 145 in a porous medium, 160 incompressible, 74, 124, 249 irrotational, 74 isentropic, 153 isothermal, 153 lid-driven cavity, 148, 280 perfect fluid, 150 planar Couette, 138 planar Poiseuille, 140 radial, 144 single-phase Poiseuille, 179 sonic, 117 steady, 124 supersonic, 117 thermoconvective, 249 turbulent, 264 two-phase, 169 two-phase gravitational, 204 two-phase lens, 205 two-phase Poiseuille, 179 fluid immiscible, 181 Newtonian, 45 viscoelastic, 46 flux conservation, 89, 90 heat, 19, 100

297

force Archimedean, 85 gravitational, 85 forms of energy enthalpy, 96, 99 entropy, 96 internal energy, 97 temperature, 97, 99 formulation conservative, 69, 156 curl and steam function, 68 curl and vector potential, 67 incompressible, 70 non-conservative, 69 Fourier, 91 fraction mass fraction, 84 molar, 84 frame of reference absolute, 116 local, 117 G, H general relativity, 1 Hadamard, 63 Helmholtz, 9 Hodge-Helmholtz decomposition, 9, 50, 75, 118, 158, 177, 211 Hugoniot, 37 hypothesis Boussinesq’s, 280 local equilibrium, 29 material frame-indifference, 101, 121, 214 objectivity, 101 of a continuous medium, 56 of a continuum, 79 Stokes’, 40, 77, 222, 262 tracer, 82 I, L invariance under rotation, 9 under translation, 9 law Biot-Savart, 119

298

Discrete Mechanics

constitutive, 19 Darcy’s, 47, 86, 160 Fick’s, 82 Forchheimer’s, 161 Fourier’s, 90, 99 Hooke’s, 20, 126, 209 Jurin’s, 175, 207 Laplace’s, 177 linear law of the wall, 268 logarithmic law of the wall, 268 Newton’s, 209 of gravity, 12, 250 Stokes’, 128, 130, 163, 219 viscoelastic constitutive, 243 line contact, 196 triple, 196, 198 M Mach, 37 material anisotropic, 101 frame-indifference, 10 hyperelastic, 243 isotropic, 101 neo-Hookean, 243 mechanics classical, 117 relativistic, 117 medium compressible, 128 continuous-memory, 46, 219 method kinematic projection, 70, 178 penalty-projection, 73 scalar projection, 70, 178 vector projection, 70 mixture binary, 86 multicomponent, 81 of miscible fluids, 86 model discrete multicomponent, 88 low-Mach, 283, 284 mixing length, 262 of gravity, 50 of the compression effects, 42

of the curvature, 170 of the rotation effects, 48 of the shear effects, 49 subgrid-scale, 276 turbulent statistical, 269 modulus compression, 64, 126, 228 Poisson, 77 shear, 64, 126, 227 Young’s, 39, 78, 127, 219 molar concentration, 84 volume, 84 monolithic approach, 65 monotonically decreasing of kinetics energy, 97 motion incompressible, 145 rigid, 228 rigid rotational, 136 translational, 136 uniform rotational, 228 uniform translational, 228 N, P Navier-Stokes, 129 Newton, 4, 31 Principia, 12, 250 number Eötvös, 12, 250 Mach, 284 Reynolds, 148, 266 turbulent Reynolds, 268 Weber, 179 phase function, 176 polar coordinates, 136 potential Bernoulli scalar, 61 equilibrium scalar, 42 scalar, 35, 93 scalar potential of the acceleration, 119 vector, 35, 93 vector potential of the acceleration, 119 power heating, 93 mechanical, 93 total, 93

Index

pressure Bernoulli, 142, 213 capillary, 198 hydrostatic, 234 mechanical equilibrium, 216 partial, 84 thermodynamic, 213 Prigogine, 80 principle Clausius-Duhem, 210 duality-of-action, 14 of accumulation of stresses, 13 propagation of light waves, 251 of linear waves, 152 of non-linear waves, 155 of polarizable waves, 154 of sound waves, 251 S simulation direct, 277 large-eddy, 262 statistical, 276 sloshing linear, 184 non-linear, 185 Sod, 155 shock tube, 155 space-time discrete, 1 Stefan problem, 102, 103 statics of fluids, 158, 216 of solids, 158, 216 strain compression, 237 flexing, 238 of a beam under extension, 235 planar shear, 238 under gravity, 240 stress compression, 46, 216 in a solid, 209 normal, 211 rotation, 46 shear, 47, 272

299

shear-rotation, 211, 214 turbulent, 276 stretching of space-time, 165 surface tension, 175, 190 per unit mass, 175 T tensor Cauchy, 126, 211 conductivity, 19 elasticity, 39, 220 Green-Lagrange, 126 permeability, 19 Ricci, 257 Riemann, 23 second invariant, 134 term capillary, 172 inertial, 141, 196 scalar accumulation, 213 vector accumulation, 213 theorem divergence, 6 fundamental theorem of analysis, 6 Green-Ostrogradski, 7 Hodge-Helmholtz, 273 Hugoniot’s, 165 Kelvin’s, 74 kinetic energy, 94 Lagrange’s, 74 Rolle’s, 7 Stokes’, 6 theory kinetic theory of gases, 222 of general relativity, 12, 52, 251, 259 of special relativity, 32 topology Cartesian, 133, 173 dual, 5, 56, 120 hybrid, 100 polygonal, 100 polyhedral, 100 primal, 5, 56, 102 random, 100 structured, 142, 179 unstructured, 143, 179 Voronoi, 100, 174

300

Discrete Mechanics

transformation Galilean, 21 Lorentz, 12, 20, 116, 251 Truesdell, 10, 121 turbulence discrete approach, 262 fully developed, 275 homogeneous isotropic, 266 V vector binormal, 170 curvature, 174 Darboux, 170 displacement, 213 Lamb, 55, 136 velocity, 212 velocity barycentric, 82, 83 bounded, 116 capillary, 192

correction, 72 Eulerian, 58 Lagrangian, 58 predicted, 71 relative, 116 superluminal, 259 viscosity compression, 77, 136 shear, 133 turbulent, 262 W, Y wave acoustic, 181 longitudinal, 48 shock, 155 transverse, 48 wetting partial, 175, 195, 200 perfect, 175 Young, 36

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