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Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows

Published online by Cambridge University Press:  10 August 2012

Jason Nassios
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
John E. Sader*
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia Kavli Nanoscience Institute and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
*
Email address for correspondence: jsader@unimelb.edu.au

Abstract

Kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow at arbitrary degrees of rarefaction, with solutions of the Boltzmann equation requiring numerical methods in many cases of practical interest. Importantly, the near-continuum regime can be examined analytically using asymptotic techniques. These asymptotic analyses often assume steady flow, for which analytical slip models have been derived. Recently, developments in nanoscale fabrication have stimulated research into the study of oscillatory non-equilibrium flows, drawing into question the applicability of the steady flow assumption. In this article, we present a formal asymptotic analysis of the unsteady linearized Boltzmann–BGK equation, generalizing existing theory to the oscillatory (time-varying) case. We consider the near-continuum limit where the mean free path and oscillation frequency are small. The complete set of hydrodynamic equations and associated boundary conditions are derived for arbitrary Stokes number and to second order in the Knudsen number. The first-order steady boundary conditions for the velocity and temperature are found to be unaffected by oscillatory flow. In contrast, the second-order boundary conditions are modified relative to the steady case, except for the velocity component tangential to the solid wall. Application of this general asymptotic theory is explored for the oscillatory thermal creep problem, for which unsteady effects manifest themselves at leading order.

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Copyright © Cambridge University Press 2012

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