Advanced Hydrodynamics Problems in Earth Sciences 3031230493, 9783031230493

Table of contents :
Contents
Mitigation of Rayleigh–Taylor Convection in a Porous Medium by Initial Periodic Fluctuations
1 Introduction
2 Description of the Problem and Mathematical Model
3 Numerical Modeling and Quantities
4 Results
5 Conclusion
References
Viscose Mutual Involvement of Moving Contacting Liquids
1 Introduction
2 Problem Statement
3 Conclusions
Reference
The Study of a Drop Collision with an Obstacle
1 Introduction
2 Numerical Modelling Two-Phase Air–Liquid System
2.1 Mathematical Model
3 Results of Numerical Simulation
3.1 3D Simulation of Drop Spreading After Impact on Plane Surface
3.2 Initial Stage of Impact Drops on Solid Surface
3.3 A Drop Spreading
3.4 The Tightening of Lamella
3.5 Universal Dependencies of Droplet Spreading
4 Conclusions
Appendix: Simulation of Changing in the Interface Level of a Two-Layer Oil–Water System Rotating in a Cylinder
References
Methods of Cavitation Flows Investigation
References
Modeling of a Ring Wave System Source Using Satellite Surveillance
1 Introduction
2 Characteristics of the Wave System
3 Model of the Wave Source
4 Finding a Solution
5 Calculation and Modeling of the Wave System
6 Conclusion
References
Static Form and Quasi-stationary Evaporation of Groundwater in Soil Cracks and Human Biofluid Between Slides
1 Introduction
2 Mathematical Model of Problem
Reference
Effect of Barrier Discharge Plasma Treatment on Winter Wheat Seed Germination
1 Introduction
2 Motivation
3 Experiment Setup
4 Results and Discussion
5 Conclusion
References
On the Treatment of Seeds with Cold Plasma to Improve Germination Processes
1 Introduction
2 Germination Processes
3 Bлияниe низкoтeмпepaтypнoй плaзмы
3.1 Particle Bombardment
3.2 UV Radiation
3.3 The Presence of Chemical Radicals
3.4 Exposure to Electromagnetic Fields
4 Inhibition of Fungal Diseases of Winter Wheat
5 Coaxial Barrier Reactor for Seed Treatment
6 Generation of a High-Frequency Discharge in Air Flow to Improve the Parameters of Germination of Onion Seeds
7 Hazelnut Surface Treatment with Low-Temperature Plasma in an Upward Flow of Air
8 Deactivation of Surface Microorganisms and Increased Germination of Chickpea Seeds
9 Conclusion
References
Non-potential Waves on the Surface of Stratified Liquid
1 Introduction
2 Problem Statement
3 Finite Amplitude Waves
4 Operability of the Received Expressions
5 The Solution for Stratified Liquid
References
Scattering Short Hydroacoustics Wave in the Presence of Long Surface Waves
1 Introduction
2 Problem Statement
3 Results
4 Discussion
4.1 Corrections to the Scattering Coefficient GR0
4.2 Calculation of Corrections g2 overlineβ2 and g3 overlineβ3
5 Conclusion
References
On the Determination of Diffuse Reflectance of PTFE
1 Introduction
2 General Information
3 Reflection Coefficient
4 ePTFE Features
5 Conclusions
References
Features of Admixture Dynamics in the Fluid Depth for Complex Vortex with Free Surface
1 Experimental Base
2 Programs for Processing Video Files with Recordings of the Flow Pattern
3 Processing of Patterns of Distribution of Marking Impurity
4 Conclusions
References
Features of Using Perturbation Theory to Study Convective Diffusion
1 Relevance of the Topic’s Research
2 Analysis of Publications on the Topic of Research
3 Unsolved Parts of the General Problem
4 Outcomes
5 Conclusion
References
Stratification and Segregation Under Laminar Convection
1 Introduction
2 Problem Statement and Mathematical Model
3 The Results of Numerical Simulation
3.1 Vertical Stratification Induced by Gravitational Convection
3.2 The Dependences of the Temperature and Concentration Stratification on the Determining Dimensionless Parameters
3.3 Changing the Direction of Density Stratification of a Two-Layer “Water–Air” System Under Zero Gravity
4 Conclusions
References
Arctic Air Intrusions and Changes in 7Be Fluxes from the Atmosphere to the Earth's Surface
1 Introduction
2 Materials and Methods
3 Results and Discussion
4 Conclusion
References
On Simulation of the Physical and Chemical Complex for Processing Hydrogen Sulfide in the Black Sea
1 Introduction
2 Hydrogen Sulfide as a Deposit of Useful Chemicals
3 Underwater Physical and Chemical Complex—PCC
4 Modeling PCC with a Capacity of 300 kW
5 Some Economic Estimates of the Project
6 Conclusions
References
Analysis of Microbiological and Sanitary-Hygienic Indicators of Water from an Underground Source
1 Conclusion
References
Joint Influence of Arctic Ice Dynamics and Magnetic Field Diffusion on Magnetohydrodynamic Processes in Geoenvironments
1 Introduction
2 Basic Shallow Water Equations
3 Small Perturbations
4 Waves Caused by Vibrations of a Flat Wall
5 Results
6 Conclusion
References
Approximation of the Soil Temperature by Piecewise-Continuous Functions
1 Introduction
2 Characteristics of Research Objects
3 Methods of Mathematical Calculations
3.1 Approximation Technique
3.2 Calculation Results Without Dividing the Data into Periods
3.3 Results of Calculations When Dividing Data Arrays
4 Conclusions
References

Citation preview

Earth and Environmental Sciences Library

Tatiana Chaplina Editor

Advanced Hydrodynamics Problems in Earth Sciences

Earth and Environmental Sciences Library Series Editors Abdelazim M. Negm, Faculty of Engineering, Zagazig University, Zagazig, Egypt Tatiana Chaplina, Antalya, Turkey

Earth and Environmental Sciences Library (EESL) is a multidisciplinary book series focusing on innovative approaches and solid reviews to strengthen the role of the Earth and Environmental Sciences communities, while also providing sound guidance for stakeholders, decision-makers, policymakers, international organizations, and NGOs. Topics of interest include oceanography, the marine environment, atmospheric sciences, hydrology and soil sciences, geophysics and geology, agriculture, environmental pollution, remote sensing, climate change, water resources, and natural resources management. In pursuit of these topics, the Earth Sciences and Environmental Sciences communities are invited to share their knowledge and expertise in the form of edited books, monographs, and conference proceedings.

Tatiana Chaplina Editor

Advanced Hydrodynamics Problems in Earth Sciences

Editor Tatiana Chaplina Alicante, Spain

ISSN 2730-6674 ISSN 2730-6682 (electronic) Earth and Environmental Sciences Library ISBN 978-3-031-23049-3 ISBN 978-3-031-23050-9 (eBook) https://doi.org/10.1007/978-3-031-23050-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Mitigation of Rayleigh–Taylor Convection in a Porous Medium by Initial Periodic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. B. Soboleva

1

Viscose Mutual Involvement of Moving Contacting Liquids . . . . . . . . . . . A. V. Kistovich

11

The Study of a Drop Collision with an Obstacle . . . . . . . . . . . . . . . . . . . . . . A. I. Fedyushkin and A. N. Rozhkov

21

Methods of Cavitation Flows Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Yu. Kravtsova

43

Modeling of a Ring Wave System Source Using Satellite Surveillance . . . S. A. Kumakshev

51

Static Form and Quasi-stationary Evaporation of Groundwater in Soil Cracks and Human Biofluid Between Slides . . . . . . . . . . . . . . . . . . . A. V. Kistovich Effect of Barrier Discharge Plasma Treatment on Winter Wheat Seed Germination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ivan I. Pashkov, Nikolay A. Sharapov, Sergey E. Malanichev, Fedor A. Shishkin, and Sofia V. Zhelezova On the Treatment of Seeds with Cold Plasma to Improve Germination Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ivan I. Pashkov and Mikhail A. Kotov

63

73

83

Non-potential Waves on the Surface of Stratified Liquid . . . . . . . . . . . . . . . 101 A. V. Kistovich Scattering Short Hydroacoustics Wave in the Presence of Long Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A. S. Zapevalov

v

vi

Contents

On the Determination of Diffuse Reflectance of PTFE . . . . . . . . . . . . . . . . . 123 Mikhail A. Kotov Features of Admixture Dynamics in the Fluid Depth for Complex Vortex with Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 V. P. Pakhnenko Features of Using Perturbation Theory to Study Convective Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Kouadio Kouadio Julien Stratification and Segregation Under Laminar Convection . . . . . . . . . . . . 153 A. I. Fedyushkin Arctic Air Intrusions and Changes in 7 Be Fluxes from the Atmosphere to the Earth’s Surface . . . . . . . . . . . . . . . . . . . . . . . . . 171 A. V. Kholoptsev and G. F. Batrakov On Simulation of the Physical and Chemical Complex for Processing Hydrogen Sulfide in the Black Sea . . . . . . . . . . . . . . . . . . . . . 183 O. A. Saprykin and V. N. Nosov Analysis of Microbiological and Sanitary-Hygienic Indicators of Water from an Underground Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 S. I. Fonova and A. V. Epitashvili Joint Influence of Arctic Ice Dynamics and Magnetic Field Diffusion on Magnetohydrodynamic Processes in Geoenvironments . . . . 199 S. I. Peregudin, S. E. Kholodova, and K. M. Cherkay Approximation of the Soil Temperature by Piecewise-Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 L. J. Lapina, D. A. Kaverin, A. V. Pastuhov, and M. K. Chebanova

Mitigation of Rayleigh–Taylor Convection in a Porous Medium by Initial Periodic Fluctuations E. B. Soboleva

Abstract Numerical simulation of gravity-driven convection in a porous medium in application to geological problems is conducted. A stability of two-layered fluid system with the lower layer being a pure fluid and the upper layer being the fluid with dissolved admixture is analyzed. A growth of instability is triggered by initial periodic density fluctuations at the interface. Peculiarities of convective flows and mass transfer depending on the wave length of disturbances are investigated. As obtained, if the wave length of disturbances is of the order of average double width of convective fingers originated under random fluctuations, those disturbances can lead to mitigating of convection. In this case, the prescribed motion cannot develop freely and transit to stochastic mode, that leads to its deceleration and a decrease in convective mixing. Keywords Porous medium · Dissolved admixture · Rayleigh–Taylor convection · Convective mixing · Periodic disturbances · Numerical simulation

1 Introduction Natural or gravity-driven convection plays an important role in underground fluid dynamics. Subsurface formations contain water and other fluids because of natural conditions and human activities. If the density of fluid is variable, its state can be unstable in the gravity field. Gravity-driven flows of fluids at the density depending on the amount of dissolved admixture is classified as haline or density-driven convection. Onset and peculiarities of density-driven convection in a porous medium are of wide interest with respect, for example, to the long-term geological storage of carbon dioxide [3, 4]. Dissolution of carbon dioxide into resident brines leads to one-sided descending convective flows which promote geological trapping. In other E. B. Soboleva (B) Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia e-mail: [email protected] URL: http://ipmnet.ru/~soboleva/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina (ed.), Advanced Hydrodynamics Problems in Earth Sciences, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-23050-9_1

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cases, geological conditions can be reduced to the Rayleigh–Taylor problem which is about an instability of two-layered fluid system in the gravity field. In those cases, small disturbances develop into two-sided convection with descending and ascending plumes. This problem inside a porous medium is considered by different authors. In [2], linear stability analysis and nonlinear simulations have been carried out to study the Rayleigh–Taylor instability in homogeneous porous media; the problem is complicated by allowing vertical flow displacements. In [5], fluid mixing induced by Rayleigh–Taylor convective dissolution in a domain with top and bottom confinement is investigated. In [6], stochastic Rayleigh–Taylor convection in a fluid pair with extremely large viscosity contrast is studied. However, Sabet et al. try in [6] to explain the observed effect of fingering asymmetry by analyzing the vorticity field that is not convincing. This effect can be explained clearly based on analyzing the velocity of finger tips [11]. As discussed in the case of one-sided convection, the onset of convective motions of miscible fluids depends on fluctuations of physical parameters, particularly fluctuations of admixture concentration at the source as well porosity or permeability fluctuations [1]. In [10, 12], a role of initial random fluctuations of density at the interface in the Rayleigh–Taylor problem for viscosity-uniform and viscosity-variable fluid systems is studied numerically. It was obtained for fluids at uniform viscosity that the time of convection onset increases roughly in fifteen times if the amplitude of fluctuations decreases in six orders of magnitude and the effect of fluctuations is visible even in stochastic mode of convection [12]. In the present work, we continue to study the effect of density fluctuations on the onset and development of Rayleigh–Taylor convection. However, on the contrary to [12], we set here initial periodic fluctuations at the interface. We consider different ratios of the wave length of periodic fluctuations to the average wave length of convective motions induced by random fluctuations. Our study demonstrate how one can mitigate Rayleigh–Taylor convection in a pair of miscible fluids.

2 Description of the Problem and Mathematical Model A rectangular porous domain containing one-component pure fluid in its lower part and the same fluid with a dissolved admixture in its upper part is considered. The porosity φ and permeability k of porous medium are assumed to be constant. The pure fluid/solution interface is horizontal. Initially, the system is motionless and in hydrostatic equilibrium. The density of upper fluid, ρb , is bigger than the density of lower fluid, ρ0 , therefore the fluid system is unstable in the gravity field. Fluid is not able to pass through domain boundaries whereas a little amount of admixture can diffuse through vertical boundaries to satisfy the diffusion equation. Initially, small sinusoidal fluctuations of density are set at the interface. Starting from the initial instant to some time, only admixture diffusion from the upper part of domain into its lower part occurs. During this time period, the interface smooths out and becomes a

Mitigation of Rayleigh–Taylor Convection in a Porous Medium …

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transition zone with the height h c which increases in time. Next, the transition zone is broken and convective motions start to develop. We employ the continuity equation, Darcy equation and convection-diffusion equation to describe flows and mixing in the fluid system. The equations are transformed to the dimensionless form. The scales are: characteristic height H , velocity D/H , time H 2 /D, density (ρb − ρ0 ), and pressure (ρb − ρ0 )g H . The quantities D and g are the diffusion coefficient and mass force acceleration. The basic equations are written as follows [10, 12]:

φ

∇·u=0

(1)

u = −Ra φ(∇Π − Se)

(2)

∂S + u · ∇ S = ∇ · (φ∇S) ∂t

(3)

Here, u, Π = (P − Pin )/((ρb − ρ0 )g H ), S = (ρ − ρ0 )/(ρb − ρ0 ) are the dimensionless Darcy velocity, pressure and density. The quantity Pin is the initial pressure in hydrostatic conditions. The unit vector e is co-directional with the gravity force. The Darcy velocity u relates with the fluid velocity v by the relation: u = φv. The Cartesian coordinate system with (x, y) being the coordinates of point is used. We have e = (0, −1). The dimensionless parameter in Eq. (2) is the Rayleigh–Darcy number (ρb − ρ0 )g H k (4) Ra = φμD Here, μ is the viscosity coefficient. We assume that D and μ are constants. The calculation domain is at 0 ≤ x ≤ h x , −0.5h y ≤ y ≤ 0.5h y , where h x and h y are the dimensionless width and height. Initial conditions for the density are S(x, y) = 0, 0 ≤ x ≤ h x , −0.5h y ≤ y < 0

(5)

S(x, y) = 1, 0 ≤ x ≤ h x , 0 < y ≤ 0.5h y

(6)

The interface is located at y = 0. Initial conditions for the density at the interface are specified with small periodic fluctuations as follows:   x S(x, 0) = 0.5(1 + s), s = σsin 2π λp Here, σ and λ p are the amplitude and wave length of fluctuations; σ  1.

(7)

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3 Numerical Modeling and Quantities The basic equations (1)–(3) with boundary and initial conditions are solved numerically with the use of the author’s 2D solver which has been employed for simulating density-driven convection under different conditions during several years [7–10, 12, 13]. The solver is based on a finite-difference method with the use of SYMPLE-type algorithm. The details of the numerical method can be found in [9]. In computations, the maximal fluid velocity | v |max over the calculation domain is analyzed. We obtain | v |max = 0 at the initial stage of mixing when only admixture diffusion occurs. Increasing | v |max indicates the convection development. The average density S(y) at some level with the coordinate y is calculated as follows:  hx 1 S(x, y)d x (8) S(y) = hx 0 The value of S(y) varies in the range: [0, 1]. We suppose that the lower boundary of mixing zone is located at y∗ and defined by S(y∗ ) = 0.01. The upper boundary of mixing zone located at y ∗ is supposed to be defined by S(y ∗ ) = 0.99. The height of mixing zone is h c = y ∗ − y∗ . Changing in h c with time shows the rate of mixing. Another quantity which describes fluid mixing is the parameter of admixture inhomogeneity G calculated at y = 0. From the definition, G = Smax (x, 0) − Smin (x, 0)

(9)

Here, Smax (x, 0) and Smin (x, 0) are the maximal and minimal values of density S(x, 0) at the level of initial location of interface (y = 0). Initially, G is close to zero. If the fluctuations Eq. (7) are set, G = σ. When convection develops, G increases up to the limiting value G = 1. One should set the Rayleigh–Darcy number Ra  1 to observe intensive convection in numerical simulations. We have chosen Ra = 2.67 × 103 in [12] and use the same value in the present study to compare results. The calculation domain is 10 × 6, the 4000 × 2000 grid is uniform, the time step is t = 1.0 × 10−7 . We carried out calculations at the constant amplitude of fluctuations, σ = 10−5 , but at the variable wave length of fluctuations λ p .

4 Results Rayleigh–Taylor convection triggered by random fluctuations has been simulated numerically in [12]. As exhibited, when the diffusion zone between fluid layers is broken, convection begins from quasi-periodic motion characterized by the average width of convective fingers λ. Later, the transition to stochastic flows occurs. The fields of density illustrating the convection structure at different times are shown

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Fig. 1 Density field in the case of random fluctuations with σ = 10−5 at the times t = 4.8 × 10−4 (a), 1.2 × 10−3 (b), 2.2 × 10−3 (c). Here, h c is the height of mixing zone

in Fig. 1. Only a fragment of calculation domain is exhibited, the density increases from 0 to 1 with color changing from white to dark blue. One can match the convective fingers in the beginning of quasi-periodic mode with the wave of disturbances characterized by the wave length λr . One wave contains two fingers moving up and down, therefore λ considered in [12] relates with λr by the relation: λr = 2λ. The value of λ remains about the same in calculations with a constant amplitude σ; calculations differ from each other in a series of random numbers determining initial fluctuations. However, λ varies with σ. It is found that λ = 3.28 × 10−2 at σ = 10−5 (see Fig. 5a in [12]). The quasi-periodic motion is resulted from a selforganization of weak stochastic motions induced by random fluctuations therefore the length λr can be treated as an inherent characteristic of fluid system. We will call λr the eigen wave length of convection. Let’s introduce the ratio of wave lengths L = λ p /λr = λ p /2λ. In our calculations, we varied L in a wide range from L  1 to L  1. However, the most interesting results are obtained if L is of the order of unity. The calculations with L = 0.75 (1), 1.0 (2), 2.0 (3) are shown in Fig. 2. For comparison, the case of ran-

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Fig. 2 Maximal fluid velocity | v |max (a), height of mixing zone h c (b), and parameter of admixture inhomogeneity G (c) for the cases with periodic fluctuations at σ = 10−5 and L = 0.75 (1), 1.0 (2), 2.0 (3), and with random fluctuations at σ = 10−5 (4), σ = 0 (5)

dom fluctuations with the amplitude σ = 10−5 (4) is displayed as well. Note, that the computer number format occupying 64 bits in computer memory is used. This format gives from 15 to 17 significant decimal digits precision. This restriction leads to roundoff errors. Note also, because exact differential equations are replaced with approximate finite-difference equations in numerical simulations, truncation errors arise. The roundoff and truncation errors affect numerical solutions as fluctuations which can be called the noise fluctuations. They trigger convection if initial fluctuations are not set explicitly (σ = 0). We have conducted the calculation at σ = 0 and obtained the limiting solution which is demonstrated in Fig. 2 by curves 5. As clear in Fig. 2a, the maximal velocity | v |max in cases (1)–(3) starts to increase from 0 at roughly the same time t ≈ 2.5 × 10−4 as in the case (4). We see that the beginning of convection induced by periodic and random fluctuations is very similar. However, the curves (1)–(3) reach the local maxima and further move down indicating the deceleration of convective motions. This unexpected result can be explained as follows. As discussed in [12], when quasi-periodic motion re-orders into stochastic motion, some convective fingers move faster than others and some of them merge with each other. In the case of periodic fluctuations, all fingers are identical and competition between them cannot lead to some of them to be faster than others. Among the identical fingers, it is also impossible to distinguish some of them which should merge together. In order for the periodic motion to transit to stochastic one, random fluctuations ensuring a bit of difference between fingers are necessary. Later, the velocity | v |max after some time period starts to grow and

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this is observed at t ≈ 1.8 × 10−3 (1), t ≈ 2.3 × 10−3 (2) and, probably, will be observed at t > 4.0 × 10−3 (3). One can assume, this growth is determined by noise fluctuations. An increase in the height of mixing zone h c with time is shown in Fig. 2b. When the fluid velocity decelerates, spreading of mixing zone decelerates as well. After t ≈ 7 × 10−4 , h c (2) at L = 1.0 lags far behind h c (4), which corresponds with the case of random fluctuations, and, after t ≈ 2.4 × 10−3 , lags even behind h c (5), which corresponds with the case of noise fluctuations. We see that the behavior of h c (3) at L = 2 looks similar to that of (2), however, a lag between curves (3) and (4) is observed later and is less. The curve (1) with L = 0.75 is close to (5) controlled by noise fluctuations. The Fig. 2b shows that periodic fluctuations can mitigate convection and decrease the mixing zone significantly. Periodic fluctuations are able to suppress an effect of noise fluctuations as, during a long time period, the mixing zone remains smaller than it would be at σ = 0. The parameter of admixture inhomogeneity G depending on time is exhibited in Fig. 2c. The curves (1)–(3) rise from 0 corresponding with a homogeneous diffusion zone between fluid layers when convection has not observed yet. Then, curves (1)– (3) reach their local maxima at times nearly equal to the times of local maxima of | v |max in Fig. 2a. After that, they move down indicating a decrease of G which is associated with an admixture dissipation due to diffusion. In this time period, convective motions decelerate and play a minor role. Later, when the velocity | v |max increases and convective flows transfer more admixture, G (1), (2) increases again. The value of G (3) will be seemed to increase after t = 4.0 × 10−3 when | v |max is grow. In the case of random fluctuations, the parameter of inhomogeneity increases fast and then vibrates near G ≈ 0.9 whereas, in the case of periodic fluctuations, G is not able to hold this value. The density field of convection induced by periodic fluctuations is shown in Fig. 3; a fragment of calculation domain is demonstrated. We see that patterns in Figs. 1a and 3a depict descending and ascending fingers, which are roughly similar to each other. Convection induced by random fluctuations transits to stochastic mode in Fig. 1b, c, however, as discussed above, periodic fluctuations prevent to do that. For this reason, convective flows in Fig. 3b, c decelerate and remain periodic, whereas the inhomogeneity of density is smoothed with time by diffusion.

5 Conclusion Numerical simulation of Rayleigh–Taylor convection in a porous medium was carried out. An instability is caused by initial sinusoidal fluctuations of density at the interface between fluid layers. The periodic fluctuations lead to forming convection with periodic structure; descending and ascending convective fingers occur. As obtained, if the wave length of fluctuations is of the order of eigen wave length of convection, these fluctuations are able to mitigate flows and mixing. To develop further and transit to stochastic mode, convective fingers should be a bit different from

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Fig. 3 Density field in the case of periodic fluctuations with σ = 10−5 and L = 1.0 at the times t = 4.3 × 10−4 (a), 1.2 × 10−3 (b), 2.2 × 10−3 (c). Here, h c is the height of mixing zone

each other in order to competition between them would lead to some fingers being faster than others or able to merge together. However, periodic fluctuations give rise to identical fingers and prevent their transition to chaos. Our study shows the way how one can mitigate convective flows and mixing. For this purpose, one should form initial periodic fluctuations at the interface which will suppress an effect of random fluctuations in the system. Acknowledgements This work has been supported by the Russian Science Foundation (Grant No. 21-11-00126).

References 1. Bestehorn M, Firoozabadi A (2012) Effect of fluctuations on the onset of density-driven convection in porous media. Phys Fluids 24:114102. https://doi.org/10.1063/1.4767467 2. Elgahawy Y, Azaiez J (2020) Rayleigh-Taylor instability in porous media under sinusoidal time-dependent flow displacements. AIP Adv 10:075308. https://doi.org/10.1063/5.0018914 3. Emami-Meybodi H, Hassanzadeh H, Green ChP, Ennis-King J (2015) Convective dissolution of CO2 in saline aquifers: Progress in modeling and experiments. Int J Greenh Gas Control 40:238–266. https://doi.org/10.1016/j.ijggc.2015.04.003 4. Huppert HE, Neufeld JA (2014) The fluid mechanics of carbon dioxide sequestration. Annu Rev Fluid Mech 46:255–272. https://doi.org/10.1146/ANNUREV-FLUID-011212-140627 5. Paoli M, Giurgiu V, Zonta F, Soldati A (2019) Universal behavior of scalar dissipation rate in confined porous media. Phys Rev Fluids 4:101501(R). https://doi.org/10.1103/PhysRevFluids. 4.101501 6. Sabet N, Hassanzadeh H, De Wit A, Abedi J (2021) Scalings of Rayleigh-Taylor instability at large viscosity contrasts in porous media. Phys Rev Lett 126:094501. https://doi.org/10.1103/ PhysRevLett.126.094501

Mitigation of Rayleigh–Taylor Convection in a Porous Medium …

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7. Soboleva E (2017) Numerical simulation of haline convection in geothermal reservoirs. J Phys: Conf Ser 891:012105. https://doi.org/10.1088/1742-6596/891/1/012105 8. Soboleva EB (2018) Density-driven convection in an inhomogeneous geothermal reservoir. Int J Heat Mass Transf 127(part C):784–798. https://doi.org/10.1016/j.ijheatmasstransfer.2018. 08.019 9. Soboleva EB (2019) A method for numerical simulation of haline convective flows in porous media applied to geology. Comput Math Math Phys 59(11):1893–1903. https://doi.org/10. 1134/S0965542519110101 10. Soboleva EB (2021) Onset of Rayleigh-Taylor convection in a porous medium. Fluid Dyn 56(2):200–210. https://doi.org/10.1134/S0015462821020105 11. Soboleva E (2022) Comment on “Scalings of Rayleigh-Taylor instability at large viscosity contrasts in porous media”. arXiv:2203.16249 [physics.flu-dyn], https://doi.org/10.48550/arXiv. 2203.16249 12. Soboleva EB (2022) Effect of finite fluctuations on development of Rayleigh-Taylor instability in a porous medium. Theor Math Phys 211(2):724–734. https://doi.org/10.1134/ S0040577922050129 13. Soboleva EB, Tsypkin GG (2016) Regimes of haline convection during the evaporation of groundwater containing a dissolved admixture. Fluid Dyn 51(3):364–371. https://doi.org/10. 1134/S001546281603008X

Viscose Mutual Involvement of Moving Contacting Liquids A. V. Kistovich

Abstract The presented article deals with the problem of introducing characteristic scales of the velocity field during oil product spills on the water surface. The model problem is solved in the case of a flat contact boundary between an oil product, water and water. The coefficients of mutual involvement of each other in the movement of the mentioned media are determined. Expressions for characteristic dynamic viscous scales are also given. Quantitative estimates are given for spills of specific grades of oil and mineral oils. Keywords Spreading · Involvement coefficient · Dynamic and kinematic viscosities · Velocity field

1 Introduction The rapid development of oil production and oil transportation over the past century has led to the problem of pollution of the World Ocean with petroleum products as a result of accidents of oil tankers with subsequent spill of their contents on the water surface. The need to obtain a quick expert assessment of the propagation speed and size of such spills, their dynamics and other characteristic features, led to the formulation of the scientific problem of describing oil spills taking into account the physicochemical properties of the contacting media. The creation of adequate mathematical models in this case, however, as in many other hydrodynamic problems, requires preliminary calculations to determine the characteristic scales of flows, both in oil and in water, including a quantitative assessment of the coefficients of mutual involvement of these media in motion as a result of the spill. In this paper, an approach to this problem is formulated based on solving the model problem of mutual involvement of media in the general movement when a viscous A. V. Kistovich (B) A. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina (ed.), Advanced Hydrodynamics Problems in Earth Sciences, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-23050-9_2

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tangential contact occurs between them. Quantitative estimates of the required characteristics for a whole set of grades of crude oil and mineral oils are given, which allows us to get closer to answering questions about the parameters of spills in natural (oil) and laboratory (mineral oils) conditions. Expressions for the dynamic scales of spills and the coefficients of mutual involvement in the movement of contacting media are explicitly presented.

2 Problem Statement One dimensional model problem of mutual involvement of moving contacting liquids is considered. The scheme of the problem is shown on the Fig. 1. Let two media, i-th and j-th, characterized by densities and dynamic viscosities ρi , ηi and ρ j , η j correspondingly, are moving in arias z > 0 and z < 0 along the x-axis with constant in the space velocities vi0 and v j0 . At the initial time moment they are brought into contact with each other on the plane z = 0. It is needed to find the velocities distributions vi (z, t) and v j (z, t) in the time and space when t > 0. The governing equations have the forms ∂ vi, j /∂ t − νi, j ∂ 2 vi, j /∂ z 2 = 0,

(1)

where νi, j = ηi, j /ρi, j are the corresponding kinematic viscosities of the media. The initial conditions have the forms  vi, j (z, t)t = 0 = vi0, j0 = consti, j ,

(2)

kinematic boundary condition is the equality of velocity fields on the contact plane z=0 Fig. 1 Scheme of the involvement problem

Viscose Mutual Involvement of Moving Contacting Liquids

 vi (z, t)|z = 0 = v j (z, t)z = 0 ,

13

t > 0,

(3)

and dynamic boundary condition is the equality of viscose tangent strengths on the mentioned plane  ηi ∂ vi /∂ z|z = 0 = η j ∂ v j /∂ z z = 0 ,

t > 0.

(4)

Also the conditions at infinity must be satisfied vi (z, t)|z = + ∞ = vi0 ,

 v j (z, t)z = − ∞ = v j0 , t > 0.

(5)

It is clear in advance that a physically meaningful solution of the equation of the form f t − ν f zz = 0 is given by the expression z f = t α ϕα (ξ), ξ = √ , 2 νt

α ≥ 0.

(6)

In this case, the function ϕα must satisfy the equation ϕ  α + 2ξ ϕ  α − 4αϕ α = 0,

(7)

where superscript means the derivative with respect to ξ. The solution of (7) has the form [1] ϕα = ξe− ξ

 2

    3 3 A(α)M α + 1, , ξ2 + B(α)U α + 1, , ξ2 , 2 2

(8)

where M and U are Kummer’s functions. The properties of the Kummer functions are such that in order to satisfy the conditions at infinities (5) it is necessary to put α = 0. Thus, the solutions of Eqs. (1) are given in the form   |z| , t > 0, vi, j (z, t) = Ai, j + Bi, j erf √ 2 νi, j t

(9)

√  2 x where erf(x) = 2π 0 e−y dy is the error function. Substituting solutions (9) into the initial (2) and boundary (3), (4) conditions leads to the final result       |z| |z| 1 vi0 ki j + erf √ + v j0 erfc √ vi (z, t) = 1 + ki j 2 νi t 2 νi t       (10) |z| |z| 1 + vi0 erfc √ , v j0 k ji + erf √ v j (z, t) = 1 + k ji 2 νjt 2 νjt

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A. V. Kistovich

where erfc(x) = 1 − erf(x), and the coefficient ki j is defined by expression  ki j =

ρi ηi = k −1 ji . ρjηj

(11)

The presentation of the expressions (10) in the form  √ ρjηj |z| 1 , κi j = =√ √ √ 2 νi t 1 + ki j ρ j η j + ρi ηi   √

ρi ηi |z| 1 , κ ji = v j (z, t) = v j0 + vi0 − v j0 κ ji erfc √ =√ √ 2 νjt 1 + k ji ρ j η j + ρi ηi (12)

vi (z, t) = vi0 + v j0 − vi0 κi j erfc



allows us to consider the value κi j as the coefficient of involvement of the i-th fluid by the relative motion of the j-th fluid. All further calculations were done for crude and mineral oils, which characteristics are presented in Tables 1 and 2. The dynamic viscosity of mineral oils at other temperatures is determined by formula η = η0 exp(−0.03 · T ),

(13)

where T is oil’s temperature in centigrade. Below, in Figs. 2, 3, 4, 5, 6 and 7, the coefficients of air and water involvement by various grades of crude and mineral oils are shown. The numbers of the corresponding grades of crude or mineral oil are shown on the horizontal axis of Figs. 2, 3, 4, 5, 6 and 7. Figures 8, 9, 10 and 11 show graphs of the vertical velocity distribution with the involvement of air and water by oil No. 12 and oil No. 2. The air and water velocities were initially assumed to be zero (vi0 = 0), and the crude and mineral oil velocities were assumed to be one (v j0 = 1). Calculations were carried out according to (10). The axes in Figs. 8, 9 and 10 are presented in non-dimensional values.

3 Conclusions The calculation results shown in Figs. 2, 4 and 5 indicate extremely small values of the coefficient of air involvement of crude oil and mineral oil. At the same time, the velocity distribution in the air, crude oil and mineral oil (Figs. 8 and 9) is such that the braking of oil spill on the air can be neglected. Thus, when creating a spreading model, the air medium should not be included in its composition at all, so as not to complicate calculations.

Viscose Mutual Involvement of Moving Contacting Liquids

15

Table 1 Physical characteristics of crude oils Density T = 20 °C ρo , g/sm3

Dynamic viscosity g ηo , sm·s

Kinematic viscosity

1. Romashkinskoe

0.862

0.123

0.1422

2. Tuymazinskoe

0.852

0.06

0.0707

3. Mukhanovskoe

0.84

0.065

0.0765

4. Trekhozernoye

0.848

0.083

0.0975

5. Teterevo-Mortyminskoe

0.825

0.034

0.041

6. Pravdinskoe

0.854

0.092

0.1076

7. Salymskoe

0.826

0.038

0.0454

8. Yuzhno-Balykskoye

0.868

0.144

0.1658

9. Mamontovskoe

0.878

0.189

0.2151

10 Ust-Balykskoye

0.874

0.153

0.1748

11. Lyantorskoe

0.887

0.143

0.1614

12. ZapadnoSurgutskoe

0.885

0.368

0.4160

13. Kholmogorskoe

0.860

0.067

0.0783

14. Pokachevskoe

0.865

0.048

0.0552

15. Megionskoe

0.850

0.066

0.0782

Oil fields

νo ,

sm2 s

16. Sovietskoe

0.852

0.052

0.0613

17. Samotlorskoe

0.851

0.042

0.0494

18. Varioganskoe

0.832

0.036

0.0437

19. Pervomaiskoe

0.844

0.036

0.043

Table 2 Physical characteristics of mineral oils DensityT = 0 °C ρo , g/sm3

Dynamic viscosity g ηo , sm·s

Kinematic viscosity

1. Olive

0.92

2.34

2.54

2. Peanut

0.92

2.22

2.41

3. Cotton

0.92

1.936

2.1

4. Sunflower

0.92

1.751

1.9

5. Soy

0.92

1.52

1.65

6. Linen

0.93

1.36

1.46

Mineral oils

νo ,

sm2 s

At the same time, the results presented in Figs. 2, 6 and 7, as well as in Figs. 10 and 11 show that the coefficients of water involvement by crude oil or mineral oils are not so small and the water flow arising under the spill must be taken into account when creating an adequate mathematical model of the process. In this case, it is necessary

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A. V. Kistovich

Fig. 2 The coefficient of air involvement by various grades of crude (a) and various grades of crude oil by air (b)

Fig. 3 The coefficient of water involvement by various grades of crude oil (a) and various grades of crude oil by water (b)

to take into account the temperatures of the contacting media and the dynamics of heat exchange. The obtained theoretical results (10)–(12) indicate that the coefficients of mutual involvement of media are determined not by a simple ratio of their kinematic (or dynamic) viscosities, as is often assumed in studies, but have a significantly different form (12) that meets all the requirements of the physical meaning of this value.

Viscose Mutual Involvement of Moving Contacting Liquids

17

Fig. 4 The coefficient of air entrainment by various grades of mineral oil (a) and mineral oils by air (b) at temperature T = 0 °C

Fig. 5 The coefficient of air entrainment by various grades of mineral oil (a) and mineral oils by air (b) at temperature T = 20 °C

As follows from (10) and (12), the characteristic scale of involvement is dynamic in nature, the influence of which increases with time as the spill area increases. The work was carried out with the financial support of the project of the Russian Federation represented by the Ministry of Education and Science of Russia № 07515-2020-802.

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A. V. Kistovich

Fig. 6 Coefficient of water involvement by various grades of mineral oil (a) and mineral oils by water (b) at temperature T = 0 °C

Fig. 7 Coefficient of water involvement by various grades of mineral oil (a) and mineral oils by water (b) at temperature T = 20 °C

Viscose Mutual Involvement of Moving Contacting Liquids Fig. 8 Velocity distribution when air is involved by crude oil

Fig. 9 Velocity distribution when air is involved by mineral oil

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Fig. 10 Velocity distribution when water is involved by crude oil

Fig. 11 Velocity distribution when water is involved by mineral oil

Reference 1. Abramowitz M, Stegun I (1964) A handbook of mathematical functions. NBS. Applied mathematics series, vol 55

The Study of a Drop Collision with an Obstacle A. I. Fedyushkin and A. N. Rozhkov

Abstract The paper considers the dynamics of the spreading liquid droplets after impact on a solid surface. The dynamics of a drop falling on a solid surface is shown using numerical simulation based on the solution of 2D and 3D Navier–Stokes equations for a two-layer liquid–gas system. The results of numerical simulation are compared with experimental data. Keywords Numerical simulation · Navier–Stokes equations · Two-phase liquid system · Droplet · Spreading

Nomenclature di d d m , d max dt F g g h H k ls n p

Impact drop diameter (m) Rim diameter (m) Maximum rim diameter (m) Target diameter (m) Force vector (N) Acceleration vector of the Earth’s gravity (m/s2 ) Acceleration of the Earth’s gravity (m/s2 ) Height of interface (mm) Height of calculation region (m) Surface curvature (m− 1 ) Point on interface Unit normal vector (m) Pressure (Pa)

A. I. Fedyushkin (B) · A. N. Rozhkov Ishlinsky Institute for Problems in Mechanics RAS, IPMech RAS, 101(1) Prospect Vernadskogo, Moscow 119526, Russia e-mail: [email protected] A. N. Rozhkov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina (ed.), Advanced Hydrodynamics Problems in Earth Sciences, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-23050-9_3

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22

r, z R u u1 u2 vi t x, y, z Rei Wei

A. I. Fedyushkin and A. N. Rozhkov

Radial and axial coordinates (m) Radius of calculation region (m) Vector of fluid velocity (m/s) Radial component of fluid velocity(m/s) Axial component of fluid velocity(m/s) Impact velocity (m/s) Time (s) Cartesian coordinates (m) Impact Reynolds number Impact Weber number

Greek Symbols βm ε γ θ μ ρ ταij ζ1 , ζ2

Dimensionless diameter of the maximum spreading Function of the volume phase fraction Surface tension (N/m) Wetting angle (degree) Dynamic viscosity (Pa s) Density (kg/m3 ) Viscous stress tensor (Pa) Radii of curvature (m)

Subscripts α i m, max s

Phase index Impact Maximum Point

Superscript T

Transposed

The Study of a Drop Collision with an Obstacle

23

1 Introduction The study of droplet spreading when falling on a solid surface has both fundamental scientific significances for the study of the laws of multiphase hydrodynamics, and applied significance in various physical processes, for example, in metallurgy, energy, micro-technologies, in the electronic, nuclear and aviation industries, medicine and health care, in the processes of cooling, irrigation, fire extinguishing, in jet and 3D printing and in other areas. A lot of papers devoted to the study of droplet dynamics, for example, works, [1–8]. Reviews of works devoted to the study of hydrodynamics during impact and spreading of droplets can be found in [6–8]. This paper presents the results of mathematical modeling of the spreading of water droplets falling on a solid surface based on the numerical solution of the Navier–Stokes equations for two-phase air–liquid systems. The simulation results are compared with experimental data:

2 Numerical Modelling Two-Phase Air–Liquid System 2.1 Mathematical Model Simulation of the process drop collision with obstacles was compared with experiment which scheme shown in Fig. 1 [1, 9–12]. Numerical simulation is based on the numerical solution of three- and two-dimensional axisymmetric Navier–Stokes equations for two immiscible liquids [13].    ∂(ρu) + ∇ · (ρuu) = −∇ p + ∇ · μ ∇u + ∇uT + ρ g + F. ∂t

(1)

For two-phase air–liquid system the only one momentum equation is solved in all calculation domain, and the resulting velocity field is shared among the phases. The momentum Eq. (1) is dependent on the function of the volume fractions ε (0 ≤ ε ≤ 1) of liquid phase through the properties: density ρ = ερair + (1 − ε)ρliquid and viscosity μ = εμair + (1 − ε)μliquid , where the values with the index ‘air’ refer to air, and with the index ‘liquid’ refer to liquid. The volume fraction of liquid ε was determined +∇ · (ρεu) = 0. from the solution of the transfer equation: ∂ρε ∂t For the two-dimensional case of a two-phase incompressible air–liquid system the Navier–Stokes equation (3) in a cylindrical coordinate system for an axisymmetric approximation without taking into account the circumferential velocity can be written as follows: u1 ∂u 2 ∂u 1 + + = 0, ∂ x1 x1 ∂ x2

(2)

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A. I. Fedyushkin and A. N. Rozhkov

Fig. 1 Drop impact on small disc-like target

  ∂(ρu 1 ) ∂(ρu 1 ) ∂(ρu 1 ) ∂p 1 ∂ ∂u 1 x1 μ + u1 + u2 =− + ∂t ∂ x1 ∂ x2 ∂ x1 x1 ∂ x1 ∂ x1   ∂u 2 u1 ∂ μ − μ 2 + F1 , + ∂ x2 ∂ x2 x1   ∂(ρu 2 ) ∂(ρu 2 ) ∂(ρu 2 ) ∂p 1 ∂ ∂u 2 x1 μ + u1 + u2 =− + ∂t ∂ x1 ∂ x2 ∂ x2 x1 ∂ x1 ∂ x1   ∂u 2 ∂ μ − ρg + F2 , + ∂ x2 ∂ x2

(3)

(4)

where t is the time, x1 ≡ r, x2 ≡ z are the radial and axial coordinates, u 1 , u 2 are the radial and axial components of the velocity vector u(u1 ,u2 ), p is the pressure, ρ is the density, μ is the dynamic viscosity coefficient, and F1 , F2 are the radial and axial components of external force F(F1 , F2 ), acting only in very narrow zone along the air–liquid interface, g is axial component of the gravity acceleration vector g. To describe the fluid flow of a two-phase air–liquid system in the approximation of axial symmetry, a model based on 2D system of Navier–Stokes Eqs. (2)–(4) was used, which made it possible to determine the velocities for liquid and air. Density

The Study of a Drop Collision with an Obstacle

25

and viscosity were determined through the function of the volume fraction of the liquid ε as ρ = ερ air +(1 − ε)ρ liquid , μ = εμair + (1 − ε)μliquid . The function of the volume fraction of the liquid ε was determined from the solution of the transfer +u 1 ∂∂εx1 +u 2 ∂∂εx2 = 0. equation: ∂ε ∂t The following boundary conditions were determinate: at the remote external boundaries—flow symmetry conditions, on a solid wall, there were boundary conditions no slip with a wetting angle of θ , at the air–liquid interface were determined from the equilibrium condition of surface tension forces and pressure forces [13]: ( p1 − p2 + γ k)ni = (τ1ij − τ2ij )nj

(5)

where γ is the surface tension coefficient assumed to be constant; p1 , p2 are the liquid and air pressures; k = 1/ζ1 + 1/ζ2 is the surface curvature, where ζ1 , ζ2 are the radii of curvature of the air–liquid interface; n is the unit normal vector at the interface directed from the liquid into the air; ταij is the viscous stress tensor and phase index α denotes: α = 1—liquid, α = 2—air. Condition (5) is written for a constant the surface tension coefficient γ . In the case of variability of surface tension (for example, these are cases of heat- or concentration-capillary convection) necessary additionally to take into account the changing of the surface tension force to the right side of the tangential along the interface (l), and append the term ∂γ ∂l projection of expression (5). Changing the interphase boundary shape of the air–liquid interface was performed using the volume of fluid method (VOF). The interface was defined using an implicit VOF method with Euler iterations. To increase the accuracy of calculating the location of the interface, a piecewise linear geo-reconstruction approach was used [14]. The interphase boundary was determined from the solution of the system of Eqs. (2)−(5) taking into account surface forces by the method of continuous surface forces (CSF) [15]. The CSF method allows the condition (5) at the interface to be taken into account through an additional local bulk force F in the right part of the momentum transfer Eqs. (4)−(5) for 2D case. The force F acts only in a very narrow zone enclosed along an interface line of width h (Fig. 2). For every point ls of interface lines l, at

h striving to zero, the force F can be written, for example, as F(ls ) = γ k(ls )n(ls ) that was shown by [15], where k(ls ) is the interface curvature at a point ls , and n(ls ) is the normal to the site at the interface point ls (Fig. 2). The CSF method makes it possible to eliminate the singularity in the case of turning the radius of curvature of the interface to zero and improve the accuracy of calculations when calculating forces at the interface in the form F = 2γρk∇ε/(ρair + ρliquid ) [15]. For numerically solve the Navier–Stokes equations the conservative control volume method [16] was used with an adaptive grid. To verify this VOF-CSF mathematical model and determine the accuracy of reproducing changes in the shape of the interface between two liquids, the simulation results were compared with spreading experiments [1, 4] and droplet coalescence (Fedyushkin and Rozhkov 2014, [5]). In addition, the dynamic problem of changing the interface level of a two-layer system in time, rotating rapidly in a cylinder, was solved and compared with experimental data published in the work [17]. When modeling

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A. I. Fedyushkin and A. N. Rozhkov

Fig. 2 The scheme of numerical method for definition of interface [15]

this test problem, the presence of a circumferential velocity was taken into account and the system of Eqs. (2)–(4) was supplemented by the equation of its momentum transfer. Comparison of the simulation results of the test problem with experimental data [17] showed good accuracy and are given in the Appendix.

3 Results of Numerical Simulation The calculation areas and grids for 3D (a) and 2D (b, c, d) models are shown in Fig. 3. The dimensions of 2D regions were height H and width R, and in the case of a 3D model, the geometry of the calculation area was a truncated cone with height H with radii of the lower circle R and upper H/2. To exclude the influence of external boundaries, the dimensions of the calculated areas (H and R) were much larger than the initial diameter of the drop (d i ), for example, in the 2D model 5 < R/d i < 25. The sizes of the “target” on which the drop falls were different, but always larger than the diameter of the drop d t > d i , the same as in the experiments. Some series of calculations, for example, to determine the effect of viscosity and wetting angle, were performed for a “target”—horizontal bottom flat of all calculation region with size of d t /2 = R. Non-uniform and dynamic grids were used in 2D and 3D models. For the 3D model, along with hexagonal and triangular grids, polygonal grids were used to reduce the calculation time (Fig. 3a). Dynamic grids were used to reduce the calculation time, for example, in the case of 2D shown in Fig. 3d, the sizes of the calculation area H and R varied over time according to changes the shape of the drop. In the mathematical model, it is assumed that a drop at the moment of contact with an obstacle has a spherical shape with a diameter of d i and a velocity of vi . A drop after impact on solid surface deformation into a thin lamella which can elongate, fragment and change into a new drop due to surface tension forces (Fig. 6, 8, 9).

The Study of a Drop Collision with an Obstacle

27

(a)

(b)

(c)

(d)

Fig. 3 Schemes of computational regions with grids for 3D (a) and 2D (b, c) models

3.1 3D Simulation of Drop Spreading After Impact on Plane Surface In Fig. 4 experimental data on the spreading of a water droplet and the results of 3D modeling of the dynamics of a drop of water (vi = 3.87 m/s, d i = 1 mm) falling on a solid hydrophobic surface showing the formation of a lamella, an edge jet and the separation of secondary droplets aproximately after t = 1 ms of impact on a solid

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A. I. Fedyushkin and A. N. Rozhkov

a) side view

b) top view

Fig. 4 Spreading of water drop (vi = 3.87 m/s, d i = 1 mm) aproximately after t = 1 ms of impact on a solid surface a side view, b top view, left photos—experiment, right—results of 3D modeling: isolines of the water fraction, the rightmost picture—isolines of the velocity module

surface are presented. In Fig. 4a are shown side view of a drop: in left it is photo— experiment and right 3D modeling result as cross section of drop by plane y = 0. In Fig. 4b are shown top view of a drop: left photo—experiment and right—results of 3D simulation isolines of the water fraction and the rightmost picture—isolines of the velocity module from which the contour of a spreading drop where an edge jet is visible for the dimensionless time τ = t/(d i /vi ) approximately equal to 2 (Fig. 4). Depending on the time of change of the maximum spreading (Rm = d m /2) drops with a diameter d i = 1 mm falling on a solid surface at a velocity vi = 3.87 m/s to the origin of coordinates are shown in Fig. 5. Lines 1 and 2 show the change on time of the maximum spreading distances on the plane x = 0 in both directions from the origin of coordinates along the y direction, lines 3 and 4 are the dependencies of Rm on time on the plane y = 0 along both directions from the origin of coordinates along the x direction. Line 5 in Fig. 5 shows the change in the maximum thickness of the spreading drop (film)—the height of the spreading drop on the z axis. These dependencies show the features of the spreading of the drop, the time and asymmetry degree of the drop spreading. In the work [1] it was shown that these dependences for dimensionless variables are universal and weakly depends on the properties of the liquid (Fig. 11, 12 and 13).

3.2 Initial Stage of Impact Drops on Solid Surface The results of experiments and mathematical modeling have shown that the initial stage of interaction of a falling drop with a solid surface is short-term and non-trivial. In Fig. 6 the dynamics of a falling water drop in the vicinity of the contact of the drop with the target and the formation of an initial jet and a lamella with an edge jet at the initial moments of interaction with a solid surface are shown. Firstly, when a drop falls at high velocity and collides with a solid surface, it is possible to capture air (in the form of a bubble or a mini torus) between the drop and the surface [18]. Secondly, in initial moment of contact drop with the surface, a very thin and, as a result, very fast initial jet is formed (Fig. 6a–d), and then a lamella with an edge jet is formed (Fig. 6e), which schematically was depicted also in Fig. 1. In Fig. 6f the

The Study of a Drop Collision with an Obstacle

29

Fig. 5 The dependences of the change in the maximum spreading distance of the drop (Rm ) on time along the main coordinate planes: lines 1, 2 are along the plane x = 0; lines 3, 4 are along the plane y = 0; line 5 is the maximum thickness of the spreading drop (lamella) along the z axis)

results of the formation of an edge jet at the initial moment of impact of a drop on a hydrophobic solid surface with a wetting angle of θ = 1800 (t = 6.8 10–2 ms), which show the separation of the jet from the surface in contrast to the case of the jet flow on a hydrophilic surface (Fig. 6a–e) are shown.

3.3 A Drop Spreading The dynamics of the spreading of a falling drop depends not only on its diameter and rate, but also on the properties of the surface on which it falls and on which it spreads. In Fig. 7 the numerical calculated dependences of the maximum spreading diameter of the lamella d m on time are shown. In Fig. 7a the spreading diameter for a frictionless surface with θ = 180° for small drop size (d i = 2.67 mm, vi = 3.87 m/cek, Wei = 550) and for large drop (d i = 3.89 mm, vi = 3.84 m/s, Wei = 792) are presented. In Fig. 7b effect of droplet spreading the (d i = 3.89 mm, vi = 3.84 m/s) on the surface with and without friction for wetting angel θ = 135° was shown. Numerical simulation of the spreading of a falling drop (d i = 3.89 mm, vi = 3.87 m/s) on surfaces with different wetting angles was carried out. In Fig. 7c the dependences of the droplet spreading diameter for surfaces with and without friction for different wetting angles (θ = 0, 90, 135 and 180°) are shown. In the paper [9], the specificity and difference of the dynamics of the spreading of “large” and “small” drops was shown, therefore, mathematical modeling of the

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A. I. Fedyushkin and A. N. Rozhkov

(a) The initial moment of impact of a drop on the surface (t=7 10-3 ms)

(b) Enlarged view of the formation of the initial jet at the initial moment of impact of a drop on a solid surface (t=7 10-3 ms)

(c) The initial moment of impact of a drop on the surface (t=6.8 10-2 ms)

(d) Enlarged view of the formation of the initial jet at the initial moment of impact of a drop on a solid surface (t=3.3 10-2 ms)

(e) Enlarged view of the formation of the lamella and the edge jet at impact of the drop on a solid surface ( t=0.1ms)

(f) Enlarged view of the formation of the initial jet at the initial moment of impact of a drop on a hydrophobic solid surface with wetting angle θ = 1800 (t = 6.8 10-2 ms) Fig. 6 Formation of an initial jet and a lamella rim with an edge jet in initial moments of impact of a drop on a solid surface (d i = 2.67 mm, vi = 3.87 m/s)

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b)

a)

c)

Fig. 7 The dependence of the diameter of the spreading of the drop d m on time. a spreading of small drops (d i = 2.67 mm) and large drops (d i = 3.89 mm), b spreading on surfaces with and without friction, c spreading of drops for surfaces with and without friction for different wetting angles (θ = 0, 90, 135 and 180°)

spreading of a “large” drop (d i = 3.89 mm, vi = 3.84 m/s) after its impact on the target (d t = 4 mm) was carried out, case of relatively large  corresponding to the Weber (Wei = 792) and Reynolds Rei = vi μdi ρ = 1.5 · 103 numbers. The results of numerical simulation are presented in Fig. 8 shows the dynamics of changing the shape of a drop of the distribution of water fractions during the fall and spreading of a drop on a solid surface (d i = 3.89 mm, vi = 3.83 m/s, Wei = 792, Rei = 1.5 103 ) for different time moments in range from t = 0.007 ms until t = 2.23 ms. The results of experiments and mathematical modeling have shown that over time, due to the instability of the liquid–air interface, small droplets break off from the lamella, as shown in Figs. 1 and 8. Mathematical modeling for this case was conducted up to the time t = 10 ms. After reaching the maximum spreading of the dm droplet, the remainder of the lamella due to the tension forces begins to contract (t = 5.5 ms) back to the target, collapses and breaks away from the target in the vertical direction (t = 8.5 ms), and later falls down on the target in the second time (Fig. 9).

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Fig. 8 Spreading of water drop (d i = 3.89 mm, vi = 3.84 m/s, Wei = 792, Rei = 1.5 103 ) on a solid disk target for time from t = 0.007 ms until t = 2.23 ms

3.4 The Tightening of Lamella When the droplet spreads (at high Reynolds and Weber numbers), there comes a moment when the forces of viscosity and surface tension balance the forces of inertia (the expansion rate of the lamella slows down and the diameter of the edge jet of the lamella reaches the maximum value of d m ), the expansion of the lamella stops and it begins to tighten under the prevailing surface tension forces. In Fig. 9 shows a continuation of the fall drop process shown in Fig. 8 of the collapse of the lamella formed during fall a drop on a solid surface (this is (d i = 3.89 mm, vi = 3.84 m/cek, We = 792). In Fig. 9 the dynamics of tightening lamellar, starting from the time when the returning edge jet of the lamella has already reached the edge of the “target”, i.e. when the diameter of the lamella is less than the diameter of the “target” (d t = 4mm) are shown. The time from the beginning of the drop falling (touching the surface) is indicated in the figures. This case was considered without taking into account gravity (g = 0), therefore, after the collapse of the lamella, the newly formed drop rushes up and leaves the calculated region. In Fig. 10 the dependences of the dimensionless diameter of the spreading droplet βm = d m /d i on the Weber number We in comparison with experimental data are shown. In Fig. 10 by line 1—numerical results, line 2—experimental data, line 3— correlation expression for the maximum spreading of the droplet βm = (Wei /20)1/2 are presented. The results in Fig. 10 show a fairly good agreement with the experiment of the presented data, up to the value of the Weber number equal to 600.

The Study of a Drop Collision with an Obstacle t=5.96 ms

33 t=7.40 ms

dt=4mm t=7.61 ms

t=8.40 ms

t=9.67 ms

t=11.7 ms

t=1.32 ms

t=16.2 ms

Fig. 9 The collapsing of the lamella formed during fall a drop on a hard surface (this is a continuation of the fall drop process shown in Fig. 8)

3.5 Universal Dependencies of Droplet Spreading In paper Rozhkov et al. [1] was shown that the spreading of the droplet is universal with respect to the dimensionless velocity V = v/vi and the dimensionless flow rate Q = q/(πd i 2 vi /6), as a function of the dimensionless coordinate Y = r/d i and the dimensionless time T = t/(d i /vi ), respectively; t-time, r is the radial coordinate (Fig. 1), vi , the local velocity, and q is the local flow velocity, is defined as the amount

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Fig. 10 Dependences of the dimensionless diameter of the maximum spreading of the droplet β m = d m /d i on the Weber number We (line 1—numerical results, 2—experimental data, line 3— approximation β m = (Wei /20)1/2 )

of liquid flowing through a circular contour of radius r per unit time, that is, q = 2πrhv, where h is the local thickness of the film. In Fig. 11. Influence of conditions (vi , d i liquid density ρ and surface tension γ ) shown in Fig. 11. All curves can be approximated by universal functions V (τ, Y ) and Q(τ, Y ), characterizing the outflow of the liquid film from the point source in origin coordinates. The results of the dependences of the dimensionless velocity V = v/vi and the dimensionless flow Q = q/(π d i 2 vi /6) on the dimensionless coordinate Y = r/d i and the dimensionless time τ = t/(d i /vi ) show that all curves form surfaces with a certain accuracy. These surfaces V(τ, Y) and Q(τ,Y) are a universal approximation and are shown in Fig. 12a, b. The results of numerical simulation in Figs. 11 and 12 presented by the lines. Figure 13 shows the distribution surface of the dimensionless film thickness H = h/d i depending on the dimensionless coordinate Y and the dimensionless time τ. In Fig. 14 shown the dependences of droplet spreading diameters on time (a)—in dimensionless variables d i = 2.79 mm, vi = 3.40 m/s, Wei = 448, (b)—in dimensional variables d i = 2.67 mm, vi = 3.87 m/s, Wei = 550 (“Small” drop) and d i = 3.89 mm, vi = 3.84 m/s, Wei = 792 (“Large” drop). In Fig. 14, the experimental data are indicated by circles (O) [1]. Solid lines (E) are theoretical predictions based on universal outflow functions calculated on the basis of experimental data [1]. Dotted lines (N) are theoretical predictions based on universal expiration functions calculated based on numerical simulation data.

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vi=3.87 m/s, di=1÷4 mm, ρ=1000 kg/m, γ=0.0726 N/m

vi=3.87 m/s, di=1÷4 mm, ρ=1000 kg/m, γ=0

vi=3.87 m/s, di=1÷4 mm, ρ=70 kg/m, γ=0

Fig. 11 Universality of the flow in the impacting drop (it is resulted on numerical data): the dimensionless velocity V = v/vi and dimensionless flow rate Q = q/(πd i 2 vi /6), as functions of dimensionless coordinate Y = r/d i and dimensionless time = τt/(d i /vi ), respectively; t is the time, r, the radial coordinate (Fig. 1), v, the local velocity, and q, the local flow rate defined as the amount of liquid that flows through a circular contour of radius r per unit time, i.e., q = 2πrhv with h the local film thickness

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Fig. 12 Surfaces of universal functions (lines—calculation) a V (τ, Y ), b Q(τ, Y )

a)

b)

4 Conclusions Experimental and numerical results confirming universal patterns of droplet spreading on a solid surface are shown. The dynamics of a drop falling on a solid surface is shown using numerical simulation based on the solution of 2D and 3D Navier–Stokes equations for a two-layer liquid–gas system. The results of numerical modeling are generally in good agreement with experimental data and confirm the reliability of the developed approach. The breakup of the splash (lamella) into droplets is caused by the Savart-PlateauRayleigh capillary instability of the rim with the formation of thickened (droplets)

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Fig. 13 The surface of a universal function of dimensionless lamella thickness H = h/d i as a function n of dimensionless time τ and dimensionless coordinate Y

a)

b)

Fig. 14 The dependences diameter of the spreading of droplets on time a dimensionless variables, b dimensional variables

and thinned (bridges) sections in the rim. The thickened and thinned sections have different inertia during radial deceleration of the rim, which enhances the growth of liquid fingers in the rim and, ultimately, the separation of secondary droplets.

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Universal dependencies are found for all drops falling on a solid surface. The flow structure for a low-viscosity droplet in a collision with an obstacle is universal and does not depend on the impact parameters and properties of the liquid. When a drop falls on a solid surface, the maximum spreading diameter of the drop increases with increasing hydrophilicity of the surface was numerically shown. These studies allow to trace the physical features of fragmentation of oral and bronchial fluids and their transformation into a large number of tiny droplets that spread infection in the air. The results of the study showed that the spread of viral infection can occur due to the disintegration of droplets containing viruses when hitting solid obstacles in real natural conditions (in-situ). Acknowledgements This work was supported by the Government program (contract # AAAAA20-120011690131-7) and was funded by RFBR, project number20-04-60128.

Appendix: Simulation of Changing in the Interface Level of a Two-Layer Oil–Water System Rotating in a Cylinder The purpose of this test problem is to determine the possibilities of reproducing the unsteady effects of changing the shape of the interface of two-layer liquid systems in a cylinder by numerical simulation. The simulation is based on the numerical solution of the Navier–Stokes Eqs. (2)–(5) with defining interface by VOF method. The numerical simulation results were compared with experimental data [17] (Fig. 15). Fig. 15 Scheme of the experimental setup (left) and changing the shape of the interface between silicone oil and water (right) for different time moments (νso = 100 mm2 /s, Vw /Vso = 1.0, Rew = 74.9). (The drawing is taken from paper of [17]

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Test Problem Statement The calculation area is a vertical cylindrical vessel filled with immiscible liquids (silicone oil and water) which are at rest in the initial moment. There is a layer of silicone oil above the water. Oil and water have similar densities, but the viscosity of oil is 10 times greater than that of water: the viscosity of μoil = 0.0103(kg/m s), the density of ρoil = 935 (kg/m3 ) is higher than that of water ρw = 1030 (kg/m3 ), μw =0.00103 (kg/m s). The cylindrical vessel is covered with a lid from above. The cylinder suddenly begins rotation around its axis at a constant angular velocity. The diameter of the vessel is 2R = 46 mm, and the height is H so + H w = 120 mm, respectively (Fig. 16).  1/2 -Reynolds Dimensionless parameters are defined as follows: Reω = R ων number, RV = VVsow —ratio of volumes occupied by silicone oil and water. This paper presents the results of calculations for Rv = 1. To model this test problem, a twodimensional axisymmetric model (2)–(4) was used with an additional equation for the transfer of the circumferential velocity momentum. For numerically solve the Navier–Stokes equations the conservative control volume method [16] with VOF method for defining of interface was used. It should be noted that the using of the VOF method imposes additional restrictions on the values of spatial and especially time steps. For the calculation, quadrangular grids with a different number of cells with a thickening near the interface were used. Numerical Simulation Results When the cylinder filled with oil and water is instantaneously brought into rotational motion, due to the difference in viscosity of oil and water and as a consequence of different scales of dynamic times, a change in the interface will occur. At the initial moment, silicone oil (as more viscous) will begin to displace water at the cylinder wall, and give way to water on the cylinder axis. Changes in the magnitude and direction of the pressure forces near the walls of the cylinder leads to a change in the direction of the vector of pressure forces in the water at the axis of the cylinder, where the acting forces add up, and therefore the maximum deviation of the interface from its initial equilibrium position can be expected on the axis of the cylinder. In Fig. 17, the blue color shows the area occupied by water, and the red color shows the area occupied by silicone oil at different time moments when the cylinder starts rotating at a speed of ω = 10.6 rad/s (Re = 74.9). The maximum displacement h of the interface of liquids in the calculations is observed on the axis and for the parameters Re = 74.9, ω = 10.6 rad/s and Rv = 1 at the fifth second, after which, oscillating in time, it slowly levels out. Figure 18 shows the dependence of the maximum vertical displacement h of the interface on the initial plane horizontal surface. The graphs in Fig. 18 show a good correspondence of the calculations with the experiment [15, 19]. Calculations were carried out until of the physical time equal to 60 s. Animated images of interface behavior and fluid flow were obtained. The fluid flow is complex with multi-vortex and oscillatory, with thin boundary layers at the walls of the cylinder

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Fig. 16 Cylindrical calculation area and grid

Fig. 17 The shape of the oil–water interface at different time moments (Re = 74.9, ω = 10.6 rad/s)

and near the interface. Due to the different properties of liquids, the flows structures of silicone oil and water are different. This can be seen, for example, in Fig. 19, where fields of density, velocity, stream function, tracks and isobars are shown for the time t = 5 s. This figure shows a thin flow structure near the interface, a different flow structure in water and oil, as well as a different pressure distribution in silicone oil and water.

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Fig. 18 Time variation of the vertical displacement h of the silicone oil–water interface (Re = 74.9)

Density

Velocity magnitude

Stream Function

Tracks

Pressure

Fig. 19 The distributions of density, velocity, stream function, tracks and total pressure at the moment (time = 5 s) of the maximum deviation of the interface from the horizontal position for Re = 74.9

Conclusions Benchmark of mathematical model using a VOF-CSF method for the numerical simulation unsteady shape of the interphase boundary for silicon oil–water systems are performed. Data on the structure of the flow caused by rotation, the maximum altitude of lifting of interface and its shape are presented. A comparison with the experiment data of the results of numerical simulation of a non-stationary change of the shape of the interface between two liquids showed good accuracy of the mathematical model and the calculation method.

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References 1. Rozhkov A, Prunet-Foch B, Vignes-Adler M (2004) Dynamics of a Liquid Lamella Resulting from the Impact of a Water Drop on a Small Target. Proc R Society London Ser A 460(2049):2681–2704 2. Rozhkov A, Prunet-Foch B, Vignes-Adler M (2002) Impact of water drops on small targets. Phys Fluids 14(10):3485–3504 3. Rozhkov A, Prunet-Foch B, Vignes-Adler M (2010) Impact of drops of surfactant solutions on small targets. Proc R Soc A 466:2897–2916. First published online 14 April 2010. https://doi. org/10.1098/rspa.2010.0015 4. Fedyushkin AI, Rozhkov AN (2018) Spreading of drops at impact on solid surfaces. In: Proceedings of International conference actual problems of applied mathematics, informatics and mechanics. Research publications, Voronezh, pp 966–977. (In Russian) 5. Fedyushkin AI, Rozhkov AN (2020) A coalescence of the droplets. IOP Conf Ser: Mater Sci Eng 927(012055):1–6 6. Rozhkov AN (2005) Dynamics and breakup of viscoelastic liquids (a review). Fluid Dyn 40(6):835–853 7. Yarin AL, Roisman IV, Tropea C (2017) Collision phenomena in liquids and solids. Cambridge University Press, UK, Cambridge 8. Yarin AL, Weiss DA (1995) Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity. J Fluid Mech 283:141–173 9. Rozhkov AN, Prunet-Foch B, Vignes-Adler M (2006) Dynamics and disintegration of drops of polymeric liquids. J Nonnewton Fluid Mech 134(1–3):44–55 10. Vernay C, Ramos L, Ligoure C (2015) Free radially expanding liquid sheet in air: time-and space-resolved measurement of the thickness field. J Fluid Mech 764:428–444 11. Vernay C (2015) Destabilization of Liquid Sheets of Dilute Emulsions, Soft Condensed Matter [cond-mat.soft]. Université de Montpellier, English, HAL Id: tel-01254934. Accessed April 4, 2019, from https://tel.archives-ouvertes.fr/tel-01254934 12. Villermaux E, Bossa B (2011) Drop fragmentation on impact. J Fluid Mech 668:412–435 13. Landau LD, Lifshitz EM (1987) Fluid mechanics, vol 6, 2nd edn. Pergamon Press, Oxford 14. Youngs DL, Morton KW, Baines MJ (1982) Time-dependent multi-material flow with large fluid distortion. In: Morton KW, Baines MJ (eds) Numerical methods for fluid dynamics. Academic Press, pp 273–285. Accessed from https://www.researchgate.net/publication/ 249970655_Time-Dependent_Multi-material_Flow_with_Large_Fluid_Distortion?enrichId= rgreq-7f9a39787ef5a845cf3e5b340a101896-XXX&enrichSource=Y292ZXJQYWdlOzI0OT k3MDY1NTtBUzoxMzkxOTQwNjQxODMyOTZAMTQxMDE5Nzg1OTUxMw%3D% 3D&el=1_x_2&_esc=publicationCoverPdf 15. Brackbill J, Kothe D, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335–354 16. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere, p 214 17. Sugimoto T, Iguchi M (2002) Behavior of immiscible two liquid layers contained in cylindrical vessel suddenly set in rotation. ISIJ Int 42:338–343 18. Weiss DA, Yarin AL (1999) Single drop impact onto liquid films: neck distortion, jetting, tiny bubble entrainment, and crown formation. J Fluid Mech 385:229–254 19. Fedyushkin AI, Rozhkov AN (2021) Criterion of drop fragmentation at a collision with a solid target (numerical simulation and experiment). J Phys Conf Ser 2057(012129):1–5

Methods of Cavitation Flows Investigation A. Yu. Kravtsova

Abstract The review is devoted to the description of experimental methods for the study of hydrodynamic cavitation in laboratory conditions. At the moment, the optimal methods for studying cavitation flows are non-contact measurement methods. The high-speed visualization allows analyzing the evolution of cavitation bubbles and cavities simultaneously with the dynamics of the development of large-scale structures in the flow over quite long period of time. The particle image velocimetry (PIV) method serves to obtain correct quantitative values of components of average velocity and statistical moments of velocity up to and including the third order. Keywords Methods of cavitation flows investigation · Pitot tube · Visualization · Particle image velocimetry (PIV) · Tomo-PIV Cavitation occurs in areas of low pressure, when it becomes lower than the critical pressure of water vapor. Theoretically, cavitation was predicted by Leonhard Euler in the middle of the XVIII century, however, as a hydrodynamic phenomenon in technical systems, it was first experimentally detected at the end of the XIX century on propellers (Fig. 1) and subsequently widely studied in the units of various scales and purposes by a large number of measurement methods. All methods of studying cavitation flows can be divided into direct (contact and non-contact) and indirect ones [3]. Indirectly, the presence of cavitation in the fluid flow is inferred from its effect on structural elements, for example, from the erosion of parts. Charles Parsons was the first to show the relationship between the resulting cavitation and the erosion of the ship’s propeller blades [2] when studying cavitation flows in the created small cavitation tube in England in 1895. He also authored the first photograph of a cavitation cavity. In addition, the occurrence of cavitation in the flow may be indirectly inferred from the appearance of noise and the distribution of pressures measured along the boundary of the cavitating streamlined solid. These A. Yu. Kravtsova (B) S.S. Kutateladze Institute of Thermophysics SB RAS, Novosibirsk, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina (ed.), Advanced Hydrodynamics Problems in Earth Sciences, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-23050-9_4

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a

b

Fig. 1 a Daring boat [1], b boat’s propeller [2]

cavitation properties, in combination with visual observations and photography, were widely used in the study of the flow of hydrofoils [4, 5], in confusor-diffuser working channels [4, 5], hydraulic pumps [6] and turbines [7, 8] on cavitation pipes of England, USA, Germany, and Russia in the period from 1910 to 1940. A direct method of studying cavitation flows is visualization. German scientist Reichardt [9, 10] in the 1940s of the twentieth century studied cavitation by observing streamlines in the flow, visualized using small gas bubbles [11]. An approximate expression was found for the cavity elongation and for the drag coefficient of the solid. However, photography remains the main experimental method of research. The first method of obtaining images is photographing with a fairly short exposure. This method provides clear images. The second one is taking a photo with a long exposure. It is used to determine the average shapes of cavitation areas. In addition, a photo-expansion onto a moving photographic film is used. Based on the photographic data, Birkhoff and Sarantonello [12] came to the conclusion about the ellipsoid shape of the cavity formed behind the supercavitating body. Epstein and Logvinovich [13] proposed to call the distance between the point of formation of a vapor–gas cavity in a liquid and the complete disappearance of the cavity the length of the cavity. They also established the dependence of the cavity size on the cavitation number. Logvinovich in 1954 formulated the principle of “independence of expansion” of the cavity, where of is writted by his colleague Buivol V.N. in the book “Thin cavities in in perturbed flows” [14]. Based on the results of numerous experimental works, Epstein in 1948 [13] derived an approximate formula for determining the drag coefficient of a cavitating disk, which was subsequently refined by Plesset and Schaeffer [15]. Knapp et al. in 1974 [3] proposed the first classification of hydrodynamic cavitation reflecting the emerging large-scale cavitation structures and their properties. However, in recent decades, owing to the active development of technology, it has become possible to register instantaneous images of cavities with a sampling frequency of up to 100 kHz or more (Fig. 2) for quite a long period of time, which enables the analysis of both the type of cavities that arise and the evolution of cavitation bubbles and cavities simultaneously with the dynamics of the development of large-scale structures in the flow. Brandner in [16] described

Methods of Cavitation Flows Investigation

a

45

b

Fig. 2 Instantaneous images of a cavity near hydrofoil with a different of a sampling rate a 1.5 kHz, b 20 kHz [22]

the so-called “drip-streak” cavitation, which is characterized by the separation of the fluid flow from the solid boundary of a streamlined body with the formation of triangular cavities that do not intersect with each other, having a smooth interphase boundary. The existence of this type of cavitation was shown in [17]. The author of [16] demonstrated that an attached cavity with a smooth surface at the initial stages of development can be divided into some structures elongated in the longitudinal direction; this type of cavitation flow was called divots. The time evolution of cavities, as well as the causes of its instability, have been considered by many authors. In the period from 1969 to 1986, Maltsev et al. [17–19] proved that the main cause of the flow instability is the reverse jets formed in the back of the cavity due to a large pressure gradient. In [20], two different types of instability of the cavity were distinguished. The first type occurs in the case of formation of quite thick cavity on the streamlined solid, while the entrainment of a single cavitation cloud is observed downstream; the second takes place at quite thin cavity, while the cavity does not separate entirely, as in the previous case, but breaks up into small cavitation clouds. In 2016, the authors of [21] experimentally discovered the third—transverse—type of the cavity instability and systematized the resulting instabilities based on high-speed visualization data. Measuring the quantitative characteristics of the cavitation flow is an extremely difficult and time-consuming task. From the middle of the XX century, Pitot tubes or thermoanemometers have been used to measure the flow velocity at a point [23]. To assess the nature of changes in sound and dynamic pressure, a hydrophone is installed in the working channel. Pressure fluctuations in a liquid measured with its help serve to obtain near-real spectra of cavitation noise, including discrete components. This method measures sound and ultrasonic frequencies ranging from tens of hertz to hundreds of kilohertz. The high sensitivity of hydrophones allows avoiding preamplifiers that reduces signal distortion. Sensors of rapidly alternating pressures mounted in the streamlined solid serve to measure the pressure on its surface during the development and collapse of the cavity. As is known, the contact measurement methods, nevertheless, introduce insignificant disturbances into the cavitation flow, thereby not providing fully correct information about the structure of the cavity, its

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size and location. Therefore, the study of cavitation flows requires the simultaneous use of contact and non-contact measurement methods. In 1965, Ellis [24] proposed to highlight the flow with a laser beam and shoot the resulting picture on a photocell. In this method, the laser beam almost contacts the surface of the object in the area of the cavity appearance. Due to cavitation, light scattering occurs. This scattering can be detected using a sensitive photocell mounted at an angle of 90 degrees to the laser beam. The field (panoramic) method of particle image velocimetry (PIV) in a standard configuration for the study of cavitation flows was first applied by Tassin et al. [25]. A schematic plan view of the test section with the planar PIV set-up is shown in Fig. 3. This work investigated the hydrodynamics of cavitating flows in the vicinity of the surface of a model streamlined solid of the NACA63A012 series. The flow near the boundaries of the individual formed cavitation bubbles and near the incipient cavities was considered in detail. It was shown that tracer particles added to the working fluid were not active centers of cavitation bubbles formation. The boundary layer can be visualized by the PIV method. Ensembles of average velocities were calculated from a small number of images, and thus the possibility of calculating velocity vectors near individual bubbles, during the origin and development of a cavity was shown. The authors of later papers increased the sample size from which the components of the average and turbulent pulsations of the flow velocity were calculated. Thus, 72 double images were used in [26], 760 images in [27], and 1000 images in [28]. Only in 2014 the authors of [29] revealed the minimum required number of double images for the correct determination of the quantitative characteristics of the cavitation flow, including the average velocity, as well as the statistical moments of velocity up to and including the third order for the particle image velocimetry (PIV) method. Image processing algorithms play an important role in determining the velocity characteristics of the cavitation flow in the PIV method [29]. At the first stage, the instantaneous velocity field is calculated using an iterative cross-correlation algorithm with continuous displacement and deformation of elementary calculation cells. The overlap of the computational domains is 50, 75%. The sub-pixel interpolation of the cross-correlation peak is carried out at three points using a one-dimensional approximation by the Gaussian function. In order to have a relatively large dynamic range, the initial size of the elementary computational domain is 64 × 64 pixels [30]. The obtained vector fields of instantaneous velocity are subjected to several stages of vector filtering. At the first stage, a filter with a signal-to-noise ratio of 2 is applied. Next, an adaptive median filter proposed by Westerville and Scarano in 2005 was applied for the area of 7 × 7 pixels [31]. At the third stage, the procedure of cluster validation of vectors with a threshold value of 50, proposed by a team of authors of the Institute of Thermophysics SB RAS, is employed [29]. The above filtering algorithms allow obtaining an instantaneous velocity field without deliberately "false" vectors. However, when calculating the average velocity field by averaging over the ensemble, due to the unsteadiness of cavities and their spatial heterogeneity velocity vectors may appear within the cavity. The issue of the correctness of such vectors was considered in [29]. Based on statistical analysis, the authors proposed a criterion for estimating the number of correct/incorrect vectors N for each calculated cell in

Methods of Cavitation Flows Investigation

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Fig. 3 A schematic plan view of the test section with the planar PIV set-up

Fig. 4 A velocity field of a cavitating hydrofoil obtained by PIV method [32]

the PIV method: 30 < N < 80 for average flow velocity, 160 < N < 300 for turbulent flow velocity pulsations, and 600 < N < 1000 for mixed velocity components. Thus, the correctness of the results of measuring the quantitative characteristics of the velocity of cavitation flows by digital tracer visualization was shown by the authors of [29] and confirmed by numerous experimental results. An example of the velocity field obtained by planar PIV method is shown in the Fig. 4. The PIV method in stereo configuration, which allows obtaining a threedimensional velocity field, was used by Choi and Cessio [33] in 2007 to study the effect of a vortex core on the dynamics and noise radiation of cavitation bubbles originating in a Venturi nozzle. In 2015, Felli and co-authors [34] for the first time applied the tomographic method of digital tracer imaging to study the propeller of a ship. The method serves to determine the quantitative characteristics of the velocity in

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the entire volume of the measuring area at once and is one of the promising methods for studying cavitation flows in various configurations. Acknowledgements The study was supported by the grant of the Russian Science Foundation (Project No. 19-79-10217).

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10. 11. 12. 13. 14. 15. 16. 17.

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Modeling of a Ring Wave System Source Using Satellite Surveillance S. A. Kumakshev

Abstract Natural (or man-made) phenomena sometimes occur in the Earth’s atmosphere and on the surface of the World Ocean, the development of which can produce a danger to people. In order to monitor such phenomena, it is necessary to provide satellite surveillance of the Earth’s surface from the spacecraft. The space experiment “Uragan” (Hurricane) is dedicated to this goal. It consists in photographing such phenomena from the ISS. However, after fixing the origin of such a phenomenon, it is necessary to predict its further development. To do this, it is necessary to propose a physical model and build a mathematical model of the phenomenon on its basis. In this article, based on a snapshot of the ring wave system near Darwin Island in the Pacific Ocean, such work has been carried out. Based on the results of multivariate calculations, it was possible to bring the calculated wave system closer to the one depicted in the photograph. This made it possible to verify the mathematical model and evaluate the physical parameters of the wave source. Keywords Ring wave system in the ocean · Modeling of natural phenomena · Space experiment “Uragan” (hurricane)

1 Introduction To prevent and mitigate the impact of catastrophic events, it is necessary not only to observe the Earth’s surface, but also to process incoming information promptly. For this, in turn, ready-made mathematical models are needed, according to which calculations can be carried out quickly on a personal computer in order to predict the development of such natural (or man-made) phenomena. From aboard the Russian segment of the International Space Station (ISS), astronauts conduct such visual monitoring of the Earth’s surface and record the phenomena that interest them on video and photo equipment. This is happening according to the program of the S. A. Kumakshev (B) Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo, 101-1, Moscow 119526, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina (ed.), Advanced Hydrodynamics Problems in Earth Sciences, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-23050-9_5

51

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space experiment “Hurricane”, in which, in particular, the task of operational classification and decoding of potentially dangerous natural and man-made catastrophic phenomena is set. Similar modern foreign programs are mainly aimed at automatic monitoring of natural phenomena from artificial Earth satellites. The mathematical software performs statistical processing of continuously incoming measurements. Unfortunately, at the moment, according to the Hurricane experiment, an archive is available that registers natural phenomena on one or more photographs that give only a general presentation. This imposes restrictions on the mathematical models being created. In particular, they should allow fast calculations on a personal computer, and the number of input parameters should not be large. Detecting and identification of various ocean wave systems is one of the important tasks in the Uragan experiment. Examples of such systems can be found in [1], where a ring wave system in the Atlantic Ocean was modeled. The analysis of the created model allowed us to conclude about the pulsed nature of the wave source (explosion). The model allowed us to estimate the power of the explosion. In this paper, we analyze a system of ring surface waves, also recorded in a photograph from the ISS taken over the Pacific Ocean near Darwin Island [2]. Apparently, the source of the waves in this case has a different character. Modeling has shown that the formation of these waves is most likely as a result of time-harmonic impact, rather than pulsed. A seismic impact, or, for example, a pulsating geyser, is suitable as a source of waves. As a result of mathematical modeling, it was possible to estimate the physical parameters of such a source.

2 Characteristics of the Wave System 1000 km west of Ecuador in the Pacific Ocean is the Galapagos Archipelago, in the north of which is Darwin Island. On May 2, 2006, a photograph was taken near it from the ISS of the ring wave system (Fig. 1). For the scale in figure the segment AB, 1 km long, is depicted. Some quantitative estimates can be made from the photograph. It is clearly seen that, unlike the case [1], the wavelengths in the wave system are approximately equal to each other. The source of this system is localized and does not have a large size. It can be assumed that the frequency of the source is constant, and the intensity has a harmonic nature. Thus, it is assumed that we are dealing with a seismic type source or, possibly, a pulsating geyser. The segment AB allows you to make a number of geometric observations. Let’s first consider the zone closest to the source. It stands out because there are no waves. Let’s estimate its radius d0 at 90 m. Next comes the transition zone. It is characterized by the fact that the waves in it are not quite clearly expressed, since they are only being formed. Let’s take its radius dw equal to 180 m. And finally, consider the far (relative to the source) zone. From 14 to 16 ring waves can be observed in this zone.

Modeling of a Ring Wave System Source Using Satellite Surveillance

53

Fig. 1 A photograph of the ring ocean wave system near Darwin Island. The length of the segment AB is 1 km

The width of this zone is assumed to be equal to 860 − 180 = 680 m. Now we can calculate the necessary parameter of the registered ring waves—the length. It is put equal to λ = 680/15 ∼ 45 m.

3 Model of the Wave Source Taking into account the previously established characteristics of the wave system, it is possible to make assumptions about its physical model. It is most likely that a harmonic source is located at the bottom of a liquid layer with a constant depth H0 and unlimited horizontally. Judging by the photo, although it is limited in size, it may not be a point. This source excites waves with a length of λ = 680/15 ∼ 45 m and a constant frequency σ . We introduce a Cartesian rectangular coordinate system O x1 x2 z as follows. Let’s take the source of waves as the center (point O). The axis Oz is directed vertically upwards. We will direct the axis O x1 along the segment AB. The plane O x1 x2 will be horizontal and coincide with the free undisturbed surface of the liquid. We have the bottom equation in the form z = −H0 . The presence of a source leads to the presence of a vertical velocity W (x, t) at a moment t in time for liquid particles in a layer with a horizontal coordinate  x = (x1 , x2 ). Obviously that |x| = introduced earlier, we have the ratio

x12 + x22 . Taking into account the frequency

W (x, t) = W (x) cosσ t,

(1)

where the magnitude W (x) depends on the source. We introduce some characteristics of the source: l as the characteristic horizontal length of the source and W0 as the maximum intensity of the source. Obviously, intensity has the dimension of velocity. Then you can write

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S. A. Kumakshev

W (x) =

W0 l 3 . [|x|2 + l 2 ]3/2

(2)

In the area of Darwin Island, the characteristic depth varies in the interval 5 ≤ H0 ≤ 10 (in m), so for our simulation we can put the constant depth of the ocean H0 . Comparing the estimates made of the wavelength and depth of the ocean, you can see that λ  H0 . This allows the use of the long wave approximation. Modeling will be carried out within the framework of linear theory. We assume that the pressure at the undisturbed level of the free surface of the liquid is zero, that is, when z = 0 p = 0. Within the framework of the linear theory of long waves [3, 4], we will carry out a mathematical formalization of the accepted physical model. Since we observe surface waves in the photo, we will take the elevation of the free surface of the liquid η(x, t) as the main value. Let’s denote the component of the fluid velocity v1 (x, t) along the axis O x1 and v2 (x, t) along the axis O x2 . Then, taking into account the assumptions made above, we have a system of equations connecting the functions η, v1 , v2 ∂η ∂v1 = −g , ∂t ∂ x1

∂η ∂v2 = −g , ∂t ∂ x2

  ∂v1 ∂η ∂v2 + W (x, t). = −H0 + ∂t ∂ x1 ∂ x2 (3)

Here g is the acceleration of gravity. Thus, for the elevation of the free surface we have the equation  2  ∂ W (x, t) ∂ 2η ∂ 2η 2 ∂ η = C0 + 2 + , 2 2 ∂t ∂t ∂ x1 ∂ x2

(4)

 where C0 = g H0 . Taking into account the Eq. (1) introduced earlier for the vertical velocity, it is possible to rewrite (4) in the form   ∂ 2η 1 ∂ 2η σ ∂ 2η + 2 − 2 2 = Re −i 2 W (x) , ∂ x12 ∂ x2 C0 ∂t C0

(5)

Only the waves leaving the source at |x| → ∞ are suitable for us. This requirement allows us to single out a single solution η(x, t). We will look for it in the following form η(x, t) = Re[η(x) eiσ t ].

(6)

Then, substituting (6) into (5), one can obtain a mathematical model (with a condition at infinity) of the wave system under consideration for finding η(x)

Modeling of a Ring Wave System Source Using Satellite Surveillance

∂ 2 η(x) ∂ 2 η(x) σ2 σ + + 2 η(x) = −i 2 W (x), 2 2 ∂ x1 ∂ x2 C0 C0

−∞ < x1 , x2 < ∞.

55

(7)

4 Finding a Solution To find the elevation of the free surface according to the mathematical model (7), we use the two-dimensional Fourier transform by x = (x1 , x2 ). We will look for a solution to Eq. (7) in the form η(x) = Fk [Fx−1 [η(x)]].

(8)

Here Fk is the direct Fourier transform 1 Fk [ f (k)](x) = 2π

 R2

f (ξ )ei(x,k) dk.

(9)

and Fx−1 is the inverse Fourier transform Fx−1 [W (x)](k)

1 = 2π

 R2

W (x)e−i(k,x) d x,

(k, x) = k1 x1 + k2 x2 ,

(10)

Integration is performed in the range from −∞ to ∞ by two variables x1 and x2 . The given form η(x, t) (8) leads to an equation that will be needed in the future ∂ 2 η(x) ∂ 2 η(x) + = Fk [−|k|2 Fx−1 [η(x)]]. ∂ x12 ∂ x22

(11)

The function W (x) from the right side of Eq. (7) is represented as W (x) = Fk [Fx−1 [W (x)]],

(12)

Returning to the mathematical model (7), taking into account Eqs. (8), (11) and (12), it has the following form now  −|k|2 +

 σ2 σ Fx−1 [η(x)] = −i 2 Fx−1 [W (x)]. C02 C0

(13)

Let’s find what Fx−1 [W (x)] is equal to. To do this, substituting (2) into (10), we get Fx−1 [W (x)](k)

W0 l 3 = 2π

 R2

e−i(k,x) d x, (x 2 + l 2 )3/2

(14)

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This equation, by entering the polar coordinates x1 = |x|cosψ, x2 = |x|sinψ and k1 = |k|cosϕ, k2 = |k|sinϕ, we write as Fx−1 [W (x)](|k|, ψ)

 2π  |x|d|x| W0 l 3 ∞ = e−i|k| |x|cos(ψ−ϕ) dψ = 2π 0 (|x|2 + l 2 )3/2 0  W0 l 3 ∞ |x|J0 (|k||x|)d|x| , 2π 0 (|x|2 + l 2 )3/2

where J0 (.) is the zero-order Bessel function [3]. We get after calculating the last integral: Fx−1 [W (x)](|k|) = W0 l 2 e−|x||k| .

(15)

Substituting the resulting expression into our mathematical model in the form (7) we get Fx−1 [η(x)] = i

σ W0 l 2 e−l|k| 2 . C02 |k|2 − σ 2

(16)

C0

Now, substituting (16) into (8) taking into account (9), the following integral representation can be obtained for the function σ W0 l 2 η(x) = i 2πC02



e−l|k| R2

|k|2 −

σ2 C02

ei(x,k) dk.

(17)

We introduce a dimensionless integration variable κ = (κ1 , κ2 ) as follows: k1 = κ1 /l, k1 = κ2 /l, then the integral representation (17) will take the form η(x) = i

σ W0 l 2 2πC02

 R2

e−|κ| x ei ( l ,κ ) dκ, 2 2 |κ| − s

here s=

lσ , C0

(18)

and when moving to the polar coordinates x1 = |x|cosψ, x2 = |x|sinψ,κ1 = ρcosϕ, κ2 = ρsinϕ, we get η(x) = i

σ W0 l 2 2πC02

 0

∞

e−ρ ρdρ 2 ρ − s2



2π

ei 0

|x| l ρcos(ϕ−ψ)

dϕ.

Modeling of a Ring Wave System Source Using Satellite Surveillance

57

To get the integral representation for η(x) in the final form, it is necessary to calculate the integral by ϕ: η(x) = i

σ W0 l C02

2



∞

e−ρ J0



|x| ρ l

ρ dρ.

ρ2 − s2

0

Here, at a point ρ = s, the function under the integral has a non-integrable singularity. In the framework of the theory of functions of a complex variable, we connect two segments on the real axis (0 ≤ ρ ≤ s − ε and s + ε ≤ ρ < +∞) in the upper half—plane of the complex ρ by a semicircle with a center at a point ρ = s and a radius ε, while 0 < ε 0, then (6) says that the curvature at the lower boundary of the phases is greater than the curvature at the upper boundary: K − > K + . Thus, in the general case it is impossible to put the edge angles θ± equal to the edge angle of a free drop of biofluid on a flat surface, since this contradicts the relation (6). As a result of the research done, the system (3) has been reduced to a single boundary condition, which, after substituting into it (5), has the form. α K − − ρ g z − α K | = 0

(7)

To solve the problem, it is necessary to determine the curvature of K . Let’s define the free surface of the liquid by the equation  : r = f (z), as shown in Fig. 2. A point on a free surface with coordinates (r, z) is given by a radius vector r drawn from the origin (0, 0), so that. r = f (z) er + zez where er , ez are orts of a cylindrical system. Fig. 2 To calculation of the surface curvature

(8)

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A. V. Kistovich

Using ort differentiation rules. ∂ er ∂ eφ ∂ ez ∂ ei ∂ eφ ∂ ez ∂ er = 0, = eφ , = 0, = −er , = = 0, =0 ∂r ∂φ ∂r ∂φ ∂r ∂φ ∂z (9) Let’s determine the necessary values. ∂r ∂r = f eφ , r z ≡ = f  er + ez ∂φ ∂z (10) ∂2 r ∂2 r ∂2 r  = f ≡ = − f e , r = r ≡ e , r ≡ = f  e r φ z z φ φ zz r ∂ φ2 ∂φ∂z ∂ z2

rφ ≡ r φφ

In the ratios (10), the strokes denote differentiation by z. Using the obtained relations, we will find the elements of the first and second quadratic forms of the surface. 

E = r φ · r φ = f 2 , F = r φ · r z = 0, G = r z · r z = 1 + f 2     r φφ r φ r z rφ z rφ rz f2 L=√ = −√ , M=√ =0 EG − F2 EG − F2 EG − F2     r zz r φ r z f f   N=√ =√ , EG − F2 = f 2 1 + f 2 EG − F2 EG − F2

(11)

The relations (11) allow us to determine the curvature K K . 

K =

E N − 2F M + G L f f  − 1 − f 2 =  3/2 2 EG − F f 1 + f 2

(12)

Substituting (12) into (7) leads to the equation. 

α K − − ρgz − α

f f  − 1 − f 2  3/2 = 0 f 1 + f 2

(13)

determining the shape of the free surface f (z). The exact solution of Eq. (13) is practically unattainable. For this reason, the method of obtaining an approximate solution is used. The essence of the method is based on the fact that when the gap H between the plates is small compared to the radii r± , the shape of the free surface f (z) allows representation in the form. f (z) = ϕ(z) + ψ(z)

(14)

where ϕ(z) is the shape of the free surface in disregard of the action of gravity, ψ(z) is the correction due to gravitational effects. But before proceeding with the

Static Form and Quasi-stationary Evaporation of Groundwater in Soil …

67

implementation of this approach, it is useful to bring Eq. (13) to a dimensionless form. In the absence of gravity, the shape of the surface is symmetric with respect to the z = H/2 plane, and therefore in the zero approximation r− = r+ = r∗ , where r∗ is an unknown quantity, such that H/r∗