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High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

Nassios, Jason and Sader, John E. (2013) High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation. Journal of Fluid Mechanics, 729 . pp. 1-46. ISSN 0022-1120. doi:10.1017/jfm.2013.281. https://resolver.caltech.edu/CaltechAUTHORS:20131108-150149059

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Abstract

The Boltzmann equation provides a rigorous theoretical framework to study dilute gas flows at arbitrary degrees of rarefaction. Asymptotic methods have been applied to steady flows, enabling the development of analytical formulae. For unsteady (oscillatory) flows, two important limits have been studied: (i) at low oscillation frequency and small mean free path, slip models have been derived; and (ii) at high oscillation frequency and large mean free path, the leading-order dynamics are free-molecular. In this article, the complementary case of small mean free path and high oscillation frequency is examined in detail. All walls are solid and of arbitrary smooth shape. We perform a matched asymptotic expansion of the unsteady linearized Boltzmann–BGK equation in the small parameter ν/ω, where ν is the collision frequency of gas particles and ω is the characteristic oscillation frequency of the flow. Critically, an algebraic expression is derived for the perturbed mass distribution function throughout the bulk of the gas away from any walls, at all orders in the frequency ratio ν/ω. This is supplemented by a boundary layer correction defined by a set of first-order differential equations. This system is solved explicitly and in complete generality. We thus provide analytical expressions up to first order in the frequency ratio, for the density, temperature, mean velocity and stress tensor of the gas, in terms of the temperature and mean velocity of the wall, and the applied body force. In stark contrast to other asymptotic regimes, these explicit formulae eliminate the need to solve a differential equation for a body of arbitrary geometry. To illustrate the utility of these results, we study the oscillatory thermal creep problem for which we find a tangential boundary layer flow arises at first order in the frequency ratio.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1017/jfm.2013.281 DOIArticle
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8954446PublisherArticle
ORCID:
AuthorORCID
Sader, John E.0000-0002-7096-0627
Additional Information:© 2013 Cambridge University Press. Received 17 December 2012; revised 24 April 2013; accepted 27 May 2013; first published online 19 July 2013. 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; micro-/nano-fluid dynamics; rarefied gas flow
DOI:10.1017/jfm.2013.281
Record Number:CaltechAUTHORS:20131108-150149059
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20131108-150149059
Official Citation:Jason Nassios and John E. Sader (2013). High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation. Journal of Fluid Mechanics, 729, pp 1-46 doi:10.1017/jfm.2013.281
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:42345
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:08 Nov 2013 23:11
Last Modified:10 Nov 2021 16:21

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