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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

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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

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