Asymptotic Theory for the Time-Dependent Behavior of a Slightly Rarefied Gas Over a Smooth Solid Boundary

Abstract

A time-evolution of a slightly rarefied monoatomic gas, namely a gas for small Knudsen numbers, which is perturbed slowly and slightly from a reference uniform equilibrium state at rest is investigated on the basis of the linearized Boltzmann equation. By a systematic asymptotic analysis, a set of fluid-dynamic-type equations and its boundary conditions that describe the gas behavior up to the second order of the Knudsen number are derived. The developed theory covers a general intermolecular potential and a gas-surface interaction. It is shown that (i) the compressibility of the gas manifests itself from the leading order in the energy equation and from the first order in the continuity equation; (ii) although the momentum equation is the Stokes equation, it contains a double Laplacian of the leading order flow velocity as a source term at the second order; (iii) a double Laplacian source term also appears in the energy equation at the second order; (iv) the slip and jump conditions are the same as those in the time-independent case up to the first order, and the difference occurs at the second order in the jump conditions as the terms of the divergence of flow velocity and of the Laplacian of temperature. Numerical values of all the slip and jump coefficients are obtained for a hard-sphere gas by the use of a symmetric relation developed recently.

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.…”

“…Under the assumption of small Knudsen number, inter-particle collisions dominate gas particle advection. This result is also true for slightly rarefied flows at low oscillation frequencies θ 1; see Nassios & Sader (2012) and Takata & Hattori (2012), where collisional effects also dominate inertia. However, at high oscillation frequency ratios (θ 1), collisional effects are negligible relative to inertia.…”

“…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.…”

“…In independent work, Takata & Hattori (2012) presented an analysis of the linearized Boltzmann equation for hard spheres, and the linearized Boltzmann-BGK equation. Both studies derived slip models and bulk flow hydrodynamic equations for flows at low frequency in a slightly rarefied gas, i.e.…”

“…The frequency ratio θ is small when operating in the asymptotic limit of slight gas rarefaction k 1, for all values of inertia β. Takata & Hattori (2012) express the frequency ratio as…”