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.

Numerical analyses of the second-order Knudsen layer are carried out on the basis of the linearized Boltzmann equation for hard-sphere molecules under the diffuse reflection boundary condition. The effects of the boundary curvature have been clarified in details, thereby completing the numerical data required up to the second order of the Knudsen number for the asymptotic theory of the Boltzmann equation (the generalized slip-flow theory). A local singularity appears as a result of the expansion at the level of the velocity distribution function, when the curvature exists.

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

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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.