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Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows

Nassios, Jason and Sader, John E. (2012) Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows. Journal of Fluid Mechanics, 708 . pp. 197-249. ISSN 0022-1120. doi:10.1017/jfm.2012.302. https://resolver.caltech.edu/CaltechAUTHORS:20121030-101034564

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1017/jfm.2012.302DOIUNSPECIFIED
http://journals.cambridge.org/abstract_S0022112012003023PublisherUNSPECIFIED
ORCID:
AuthorORCID
Sader, John E.0000-0002-7096-0627
Additional Information:© 2012 Cambridge University Press. Received 12 January 2012; revised 2 May 2012; accepted 13 June 2012; first published online 10 August 2012. The authors acknowledge financial support from an Australian Postgraduate Award and the Australian Research Council Grants Scheme.
Funders:
Funding AgencyGrant Number
Australian Postgraduate AwardUNSPECIFIED
Australian Research Council Grants SchemeUNSPECIFIED
Subject Keywords:kinetic theory, MEMS/NEMS, non-continuum effects
DOI:10.1017/jfm.2012.302
Record Number:CaltechAUTHORS:20121030-101034564
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20121030-101034564
Official Citation:Jason Nassios and John E. Sader (2012). Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows. Journal of Fluid Mechanics, 708, pp 197-249 doi:10.1017/jfm.2012.302
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:35173
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:30 Oct 2012 18:59
Last Modified:09 Nov 2021 23:13

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