Parabolic temperature profile and second-order temperature jump of a slightly rarefied gas in an unsteady two-surface problem

Abstract

The behavior of a slightly rarefied monatomic gas between two parallel plates whose temperature grows slowly and linearly in time is investigated on the basis of the kinetic theory of gases. This problem is shown to be equivalent to a boundary-value problem of the steady linearized Boltzmann equation describing a rarefied gas subject to constant volumetric heating. The latter has been recently studied by Radtke, Hadjiconstantinou, Takata, and Aoki (RHTA) as a means of extracting the second-order temperature jump coefficient. This correspondence between the two problems gives a natural interpretation to the volumetric heating source and explains why the second-order temperature jump observed by RHTA is not covered by the general theory of slip flow for steady problems. A systematic asymptotic analysis of the time-dependent problem for small Knudsen numbers is carried out and the complete fluid-dynamic description, as well as the related half-space problems that determine the structure of the Knudsen layer and the coefficients of temperature jump, are obtained. Finally, a numerical solution is presented for both the Bhatnagar-Gross-Krook model and hard-sphere molecules. The jump coefficient is also calculated by the use of a symmetry relation; excellent agreement is found with the result of the numerical computation. The asymptotic solution and associated second-order jump coefficient obtained in the present paper agree well with the results by RHTA that are obtained by a low variance stochastic method.

Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows

“…A requirement of steady flow is zero wall-normal velocity (see Sone 1969). Takata et al (2012). It is reported there as β…”

“…Even so, other second-order components of the boundary conditions are modified by unsteady effects. In particular, we show that oscillatory (time-varying) heating results in a modification to the second-order temperature slip model that is equivalent to that presented in Takata et al (2012). Our analysis is supplemented by additional terms, which account for non-zero boundary curvature, wall-normal velocity and gas compressibility effects, as we shall discuss.…”

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

“…On the other hand, the appearance of τ H0 term has an impact that the related Knudsen-layer problem for φ (0) 6 has not occurred in the theory for steady flows. It is this problem that causes the second-order temperature jump associated with the parabolic temperature profile in [13][14][15].…”

“…[12]) by the general theory of slip flow [5,11,12] for steady problems. Then, Takata et al [15] showed that this problem with volumetric heating corresponds to the time-evolution problem caused by a change of wall temperature, identified the source of the observed temperature jump by a systematic asymptotic analysis, and concluded that this jump is a new one that is not covered by the theory for steady problems. This result suggests a possibility that there might be other new jump phenomena which do not appear (or degenerate) in steady problems.…”

“…The present work is motivated by recent findings in Refs. [13][14][15] on the second-order temperature jump.…”