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$H$-theorem and boundary conditions for the linear R26 equations: application to flow past an evaporating droplet

Published online by Cambridge University Press:  05 August 2021

Anirudh S. Rana Open the ORCID record for Anirudh S. Rana [Opens in a new window] ,
Vinay Kumar Gupta Open the ORCID record for Vinay Kumar Gupta [Opens in a new window] ,
James E. Sprittles Open the ORCID record for James E. Sprittles [Opens in a new window]  and
Manuel Torrilhon Open the ORCID record for Manuel Torrilhon [Opens in a new window]
Anirudh S. Rana*
Affiliation:
Department of Mathematics, BITS Pilani, Pilani Campus, 333031 Rajasthan, India
Vinay Kumar Gupta
Affiliation:
Department of Mathematics, Indian Institute of Technology Indore, Indore 453552, India
James E. Sprittles
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
Manuel Torrilhon
Affiliation:
Department of Mathematics, RWTH Aachen University, D-52062 Aachen, Germany
*
†Email address for correspondence: anirudh.rana@pilani.bits-pilani.ac.in
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Abstract

Determining physically admissible boundary conditions for higher moments in an extended continuum model is recognised as a major obstacle. Boundary conditions for the regularised 26-moment (R26) equations obtained using Maxwell's accommodation model do exist in the literature; however, we show in this article that these boundary conditions violate the second law of thermodynamics and the Onsager reciprocity relations for certain boundary value problems, and, hence, are not physically admissible. We further prove that the linearised R26 (LR26) equations possess a proper $H$-theorem (second-law inequality) by determining a quadratic form without cross-product terms for the entropy density. The establishment of the $H$-theorem for the LR26 equations in turn leads to a complete set of boundary conditions that are physically admissible for all processes and comply with the Onsager reciprocity relations. As an application, the problem of a slow rarefied gas flow past a spherical droplet with and without evaporation is considered and solved analytically. The results are compared with the numerical solution of the linearised Boltzmann equation, experimental results from the literature and/or other macroscopic theories to show that the LR26 theory with the physically admissible boundary conditions provides an excellent prediction up to Knudsen number $\lesssim 1$ and, consequently, provides transpicuous insights into intriguing effects, such as thermal polarisation. In particular, the analytic results for the drag force obtained in the present work are in an excellent agreement with experimental results even for very large values of the Knudsen number.

JFM classification

Rarefied Gas Flow: Kinetic theory Compressible Flows: Gas dynamics Drops and Bubbles: Drops
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 924 , 10 October 2021 , A16
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Aoki, K. & Sone, Y. 1987 Temperature field induced around a sphere in a uniform flow of a rarefied gas. Phys. Fluids 30, 2286–2288.CrossRefGoogle Scholar
Bailey, C.L., Barber, R.W., Emerson, D.R., Lockerby, D.A. & Reese, J.M. 2005 A critical review of the drag force on a sphere in the transition flow regime. AIP. Conf. Proc. 762, 743–748.CrossRefGoogle Scholar
Beckmann, A.F., Rana, A.S., Torrilhon, M. & Struchtrup, H. 2018 Evaporation boundary conditions for the linear R13 equations based on the Onsager theory. Entropy 20, 680.CrossRefGoogle ScholarPubMed
Beresnev, S. & Chernyak, V. 1995 Thermophoresis of a spherical particle in a rarefied gas: numerical analysis based on the model kinetic equations. Phys. Fluids 7, 1743–1756.CrossRefGoogle Scholar
Bobylev, A.V. 1982 The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29–31.Google Scholar
Bobylev, A.V. 2018 Boltzmann equation and hydrodynamics beyond Navier–Stokes. Phil. Trans. R. Soc. A 376, 20170227.CrossRefGoogle ScholarPubMed
Cai, Z., Fan, Y. & Li, R. 2014 Globally hyperbolic regularization of Grad's moment system. Commun. Pure Appl. Maths 67, 464–518.CrossRefGoogle Scholar
Cai, Z. & Wang, Y. 2020 Regularized 13-moment equations for inverse power law models. J. Fluid Mech. 894, A12.CrossRefGoogle Scholar
Cercignani, C. 1975 Theory and Application of the Boltzmann Equation. Scottish Academic.Google Scholar
Cercignani, C. 2000 Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge University Press.Google Scholar
Chapman, S. & Cowling, T.G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Chernyak, V.G. & Margilevskiy, A.Y. 1989 The kinetic theory of heat and mass transfer from a spherical particle in a rarefied gas. Intl J. Heat Mass Transfer 32, 2127–2134.CrossRefGoogle Scholar
Claydon, R., Shrestha, A., Rana, A.S., Sprittles, J.E. & Lockerby, D.A. 2017 Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects. J. Fluid Mech. 833, R4.CrossRefGoogle Scholar
Goldberg, R. 1954 Slow flow of a rarefied gas past a spherical obstacle. PhD thesis, New York University.Google Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2, 331–407.CrossRefGoogle Scholar
de Groot, S.R. & Mazur, P. 1962 Non-Equilibrium Thermodynamics. North-Holland.Google Scholar
Gu, X.-J., Barber, R.W., John, B. & Emerson, D.R. 2019 Non-equilibrium effects on flow past a circular cylinder in the slip and early transition regime. J. Fluid Mech. 860, 654–681.CrossRefGoogle Scholar
Gu, X.J. & Emerson, D.R. 2007 A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225, 263–283.CrossRefGoogle Scholar
Gu, X.-J. & Emerson, D.R. 2009 A high-order moment approach for capturing non equilibrium phenomena in the transition regime. J. Fluid Mech. 636, 177–216.CrossRefGoogle Scholar
Gu, X.-J. & Emerson, D.R. 2014 Linearized-moment analysis of the temperature jump and temperature defect in the Knudsen layer of a rarefied gas. Phys. Rev. E 89, 063020.CrossRefGoogle ScholarPubMed
Gu, X.-J., Emerson, D.R. & Tang, G.-H. 2010 Analysis of the slip coefficient and defect velocity in the Knudsen layer of a rarefied gas using the linearized moment equations. Phys. Rev. E 81, 016313.CrossRefGoogle ScholarPubMed
Gupta, V.K. 2015 Mathematical modeling of rarefied gas mixtures. PhD thesis, RWTH Aachen University, Germany.Google Scholar
Gupta, V.K. 2020 Moment theories for a $d$-dimensional dilute granular gas of Maxwell molecules. J. Fluid Mech. 888, A12.CrossRefGoogle Scholar
Gupta, V.K. & Torrilhon, M. 2012 Automated Boltzmann collision integrals for moment equations. AIP Conf. Proc. 1501, 67–74.CrossRefGoogle Scholar
Kalempa, D. & Sharipov, F. 2020 Drag and thermophoresis on a sphere in a rarefied gas based on the Cercignani–Lampis model of gas–surface interaction. J. Fluid Mech. 900, A37.CrossRefGoogle Scholar
Karlin, I. 2018 Derivation of regularized Grad's moment system from kinetic equations: modes, ghosts and non-Markov fluxes. Phil. Trans. R. Soc. A 376, 20170230.CrossRefGoogle ScholarPubMed
Karlin, I.V., Gorban, A.N., Dukek, G. & Nonnenmacher, T.F. 1998 Dynamic correction to moment approximations. Phys. Rev. E 57, 1668–1672.CrossRefGoogle Scholar
Koellermeier, J., Schaerer, R.P. & Torrilhon, M. 2014 A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinet. Relat. Models 7, 531–549.CrossRefGoogle Scholar
Kremer, G.M. 2010 An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Maxwell, J.C. 1879 On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
Müller, I. 1985 Thermodynamics. Pitman.Google Scholar
Müller, I. & Ruggeri, T. 1998 Rational Extended Thermodynamics. Springer.CrossRefGoogle Scholar
Ohr, Y.G. 2001 Improvement of the Grad 13 moment method for strong shock waves. Phys. Fluids 13, 2105–2114.CrossRefGoogle Scholar
Onsager, L. 1931 a Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426.CrossRefGoogle Scholar
Onsager, L. 1931 b Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265–2279.CrossRefGoogle Scholar
Padrino, J.C., Sprittles, J.E. & Lockerby, D.A. 2019 Thermophoresis of a spherical particle: modelling through moment-based, macroscopic transport equations. J. Fluid Mech. 862, 312–347.CrossRefGoogle Scholar
Rana, A., Torrilhon, M. & Struchtrup, H. 2013 A robust numerical method for the R13 equations of rarefied gas dynamics: Application to lid driven cavity. J. Comput. Phys. 236, 169–186.CrossRefGoogle Scholar
Rana, A.S., Gupta, V.K. & Struchtrup, H. 2018 a Coupled constitutive relations: a second law based higher-order closure for hydrodynamics. Proc. R. Soc. A 474, 20180323.CrossRefGoogle ScholarPubMed
Rana, A.S., Lockerby, D.A. & Sprittles, J.E. 2018 b Evaporation-driven vapour microflows: analytical solutions from moment methods. J. Fluid Mech. 841, 962–988.CrossRefGoogle Scholar
Rana, A.S., Lockerby, D.A. & Sprittles, J.E. 2019 Lifetime of a nanodroplet: kinetic effects and regime transitions. Phys. Rev. Lett. 123, 154501.CrossRefGoogle ScholarPubMed
Rana, A.S. & Struchtrup, H. 2016 Thermodynamically admissible boundary conditions for the regularized 13 moment equations. Phys. Fluids 28, 027105.CrossRefGoogle Scholar
Ringhofer, C., Schmeiser, C. & Zwirchmayr, A. 2001 Moment methods for the semiconductor Boltzmann equation on bounded position domains. SIAM J. Numer. Anal. 39, 1078–1095.CrossRefGoogle Scholar
Sarna, N. & Torrilhon, M. 2018 a Entropy stable Hermite approximation of the linearised boltzmann equation for inflow and outflow boundaries. J. Comput. Phys. 369, 16–44.CrossRefGoogle Scholar
Sarna, N. & Torrilhon, M. 2018 b On stable wall boundary conditions for the Hermite discretization of the linearised Boltzmann equation. J. Stat. Phys. 170, 101–126.CrossRefGoogle Scholar
Sazhin, S. 2014 Droplets and Sprays. Springer.CrossRefGoogle Scholar
Sone, Y. 2002 Kinetic Theory and Fluid Dynamics. Birkhäuser.CrossRefGoogle Scholar
Sone, Y., Takata, S. & Wakabayashi, M. 1994 Numerical analysis of a rarefied gas flow past a volatile particle using the Boltzmann equation for hard-sphere molecules. Phys. Fluids 6, 1914–1928.CrossRefGoogle Scholar
Struchtrup, H. 2004 Stable transport equations for rarefied gases at high orders in the Knudsen number. Phys. Fluids 16, 3921–3934.CrossRefGoogle Scholar
Struchtrup, H. 2005 Macroscopic Transport Equations for Rarefied Gas Flows. Springer.CrossRefGoogle Scholar
Struchtrup, H., Beckmann, A., Rana, A.S. & Frezzotti, A. 2017 Evaporation boundary conditions for the R13 equations of rarefied gas dynamics. Phys. Fluids 29, 092004.CrossRefGoogle Scholar
Struchtrup, H. & Nadler, B. 2020 Are waves with negative spatial damping unstable? Wave Motion 97, 102612.CrossRefGoogle Scholar
Struchtrup, H. & Taheri, P. 2011 Macroscopic transport models for rarefied gas flows: a brief review. IMA J. Appl. Maths 76, 672–697.CrossRefGoogle Scholar
Struchtrup, H. & Torrilhon, M. 2003 Regularization of Grad's 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 2668–2680.CrossRefGoogle Scholar
Struchtrup, H. & Torrilhon, M. 2007 $H$ theorem, regularization, and boundary conditions for linearized 13 moment equations. Phys. Rev. Lett. 99, 014502.CrossRefGoogle ScholarPubMed
Su, W., Zhang, Y. & Wu, L. 2021 Multiscale simulation of molecular gas flows by the general synthetic iterative scheme. Comput. Meth. Appl. Mech. Engng 373, 113548.CrossRefGoogle Scholar
Taheri, P., Struchtrup, H. & Torrilhon, M. 2009 Couette and Poiseuille microflows: analytical solutions for regularized 13-moment equations. Phys. Fluids 21, 017102.CrossRefGoogle Scholar
Takata, S. 2009 Note on the relation between thermophoresis and slow uniform flow problems for a rarefied gas. Phys. Fluids 21, 112001.CrossRefGoogle Scholar
Takata, S., Aoki, K. & Sone, Y. 1994 Thermophoresis of a sphere with a uniform temperature: Numerical analysis of the Boltzmann equation for hard-sphere molecules. In Rarefied Gas Dynamics: Theory and Simulations (ed. B.D. Shizgal & D.P. Weaver), Progress in Astronautics and Aeronautics, vol. 159, pp. 626–639. AIAA.CrossRefGoogle Scholar
Takata, S. & Sone, Y. 1995 Flow induced around a sphere with a non-uniform surface temperature in a rarefied gas, with application to the drag and thermal force problems of a spherical particle with an arbitrary thermal conductivity. Eur. J. Mech. B/Fluids 14, 487–518.Google Scholar
Takata, S., Sone, Y. & Aoki, K. 1993 Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the boltzmann equation for hard-sphere molecules. Phys. Fluids A 5, 716–737.CrossRefGoogle Scholar
Tang, J., Li, Y., Eames, I., Chan, P. & Ridgway, G. 2006 Factors involved in the aerosol transmission of infection and control of ventilation in healthcare premises. J. Hosp. Infect. 64, 100–114.CrossRefGoogle ScholarPubMed
Torrilhon, M. 2010 Slow gas microflow past a sphere: analytical solution based on moment equations. Phys. Fluids 22, 072001.CrossRefGoogle Scholar
Torrilhon, M. 2016 Modeling nonequilibrium gas flow based on moment equations. Annu. Rev. Fluid Mech. 48, 429–458.CrossRefGoogle Scholar
Torrilhon, M. & Sarna, N. 2017 Hierarchical Boltzmann simulations and model error estimation. J. Comput. Phys. 342, 66–84.CrossRefGoogle Scholar
Torrilhon, M. & Struchtrup, H. 2004 Regularized 13-moment equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513, 171–198.CrossRefGoogle Scholar
Torrilhon, M. & Struchtrup, H. 2008 Boundary conditions for regularized 13-moment equations for micro-channel-flows. J. Comput. Phys. 227, 1982–2011.CrossRefGoogle Scholar
Westerkamp, A. & Torrilhon, M. 2019 Finite element methods for the linear regularized 13-moment equations describing slow rarefied gas flows. J. Comput. Phys. 389, 1–21.CrossRefGoogle Scholar
Wu, L., Zhang, J., Liu, H., Zhang, Y. & Reese, J.M. 2017 A fast iterative scheme for the linearized Boltzmann equation. J. Comput. Phys. 338, 431–451.CrossRefGoogle Scholar
Xie, X., Li, Y., Chwang, A.T., Ho, P.L. & Seto, W.H. 2007 How far droplets can move in indoor environments – revisiting the Wells evaporation-falling curve. Indoor Air 17, 211–225.CrossRefGoogle ScholarPubMed
Yamamoto, K. & Ishihara, Y. 1988 Thermophoresis of a spherical particle in a rarefied gas of a transition regime. Phys. Fluids 31, 3618–3624.CrossRefGoogle Scholar
Yang, W., Gu, X.-J., Wu, L., Emerson, D.R., Zhang, Y. & Tang, S. 2020 A hybrid approach to couple the discrete velocity method and Method of Moments for rarefied gas flows. J. Comput. Phys. 410, 109397.CrossRefGoogle Scholar
Zhu, L., Pi, X., Su, W., Li, Z.-H., Zhang, Y. & Wu, L. 2021 General synthetic iterative scheme for nonlinear gas kinetic simulation of multi-scale rarefied gas flows. J. Comput. Phys. 430, 110091.CrossRefGoogle Scholar
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