The Asymptotic Behaviour of Oscillatory Flows in a Slightly Rarefied Gas: A Kinetic Theory Formulation
by Jason Nassios
Abstract: The Navier-Stokes equations and the no-slip boundary condition provide a rigorous mathematical description of the dynamics of many important flow phenomena. The validity of these equations is contingent on the continuum approximation, which is violated when operating at low gas densities or in miniaturised systems. In contrast, kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow at arbitrary degrees of rarefaction. Concomitant with the generality of this approach is its analytical intractability in many cases of practical interest. Importantly, the near-continuum regime has been examined analytically using a model equation and asymptotic techniques. This asymptotic analysis assumed steady flow, for which analytical slip models were derived. Recently, developments in nanoscale fabrication have stimulated research into oscillatory (time-varying) rarefied flows, drawing into question the applicability of the steady flow assumption.
In this talk, I will outline some key findings of a formal asymptotic analysis of the unsteady linearised Boltzmann-BGK equation for small mean free path, where interparticle collisions are modelled via a relaxation process. In the near-continuum limit where the characteristic oscillation frequency of the flow is much smaller than the collision frequency of gas particles, we generalise existing steady theory to the unsteady (oscillatory) case; this formally elucidates the effect of unsteadiness on all classical (steady) hydrodynamic equations, slip models and Knudsen layer corrections. In addition, we outline the key findings from a formal asymptotic analysis of the flow in the complementary high oscillation frequency limit. Critically, this system can be solved explicitly and in complete generality; in stark contrast to other asymptotic regimes, these explicit formulae eliminate the need to solve a differential equation for a body of arbitrary geometry. Implications of this theory are explored for two canonical thermally-driven flows, where oscillatory temperature fields along adjacent walls generate a rarefied flow.
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