Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows

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.

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

“…Critically, these formulae supplant the need to solve any additional differential equations for the bulk flow quantities. This explicit solvability property contrasts to slightly rarefied flows at low frequency (k 1 and θ 1), and ultra-rarefied flows at high frequency (k 1 and θ 1); see Sone (2007) and Nassios & Sader (2012) and Takata & Hattori (2012). In line with previous work, application of this theory is illustrated with a study of the oscillatory (time-varying) thermal creep flow generated between two adjacent walls.…”

“…In so doing, we elucidate the physical distinction between this class of highly oscillatory flows and the ultra-rarefied limit (k 1) investigated by Sone (2007). Our work also provides the complementary high frequency solution to the recent studies by Nassios & Sader (2012) and Takata & Hattori (2012), of slightly rarefied flows at low frequency (θ 1). These two analyses can be realized physically using a single oscillating device of fixed (small) size, operating at low and high frequency.…”

“…The approach of Sone (1969Sone ( , 1974 was revisited recently by Nassios & Sader (2012), who examined the linearized Boltzmann-BGK equation for oscillatory gas flows. In independent work, Takata & Hattori (2012) presented an analysis of the linearized Boltzmann equation for hard spheres, and the linearized Boltzmann-BGK equation.…”

“…The low frequency assumption was implicit in both cases. In Nassios & Sader (2012), θ is written in the form 4) where β = ωL 2 /ν Kin is the Stokes number and ν Kin is the kinematic viscosity. The frequency ratio θ is small when operating in the asymptotic limit of slight gas rarefaction k 1, for all values of inertia β.…”

“…Therefore, the hydrodynamic equations are modified by unsteadiness at O(k), whereas the slip boundary conditions are altered at O(k 2 ). Nassios & Sader (2012) explored the effect of the additional body force at O(k), by studying the oscillatory (time-varying) thermal creep problem, where temperature gradients are applied along two adjacent walls. Unsteady effects in the complementary ultra-rarefied limit Kn 1 have also been investigated using asymptotic methods, by Sone (2007).…”