Channel Flow of a Binary Mixture of Rigid Spheres Described by the Linearized Boltzmann Equation and Driven by Temperature, Pressure, and Concentration Gradients

Abstract

An analytical version of the discrete-ordinates method (the ADO method) is used with recently established analytical expressions for the rigid-sphere scattering kernels in a study devoted to the flow of a binary gas mixture in a plane channel. In particular, concise and accurate solutions to basic flow problems in a plane channel driven by temperature, pressure, and concentration gradients and described by the linearized Boltzmann equation are established for the case of Maxwell boundary conditions for each of the two species. The velocity, heat-flow, and shear-stress profiles, as well as the mass-and heat-flow rates, are established for each species of particles, and numerical results are reported for two binary mixtures (Ne-Ar and He-Xe). Introduction.While the classical problems of Poiseuille flow and thermalcreep flow in a plane channel in the general field of rarefied gas dynamics [24,3,5,4] have been extensively studied for the case of a single-species gas (see, for example, [1,25,20,22,21,14,17] and the references therein), there are relatively few works (for example, [23, 16, 13]) devoted to these problems for gas mixtures. While [23] and [16] are based on the McCormack kinetic model [15], the work of Kosuge et al. [13] is carried out in terms of the linearized Boltzmann equation (LBE). It can be noted that the paper by Siewert and Valougeorgis [23] reports (in terms of the McCormack model) concise and accurate solutions to the problems of channel flow driven by pressure, temperature, and concentration gradients. While the approach used in [16], also based on the McCormack model, is purely numerical, that work does investigate flow in a two-dimensional channel. Most closely related to this work is [13], where purely numerical methods are used to establish some results for channel-flow problems based on the LBE.In this work, we develop and evaluate concise and accurate solutions for flow problems in a plane-parallel channel driven by pressure, temperature, and concentration gradients. We make use of an analytical discrete-ordinates method (ADO method, [2]), and we use (in the LBE) explicit forms of the rigid-sphere collision kernels for binary gas mixtures [12,6,8]. The developed solutions depend (aside from some normalizations) only on the mass and diameter ratios and the relative equilibrium concentration of the two species of particles. We allow a free choice of the accommodation coefficients for each species at the confining surfaces of the channel. Our

DSMC Simulation of Binary Rarefied Gas Flows between Parallel Plates and Comparison to Other Methods

“…The method uses two types of definition of the rarefaction degree depending on other approaches with which the results are compared. First, the method is compared to the linearized Boltzmann equation [7] then to the McCormack model [8].…”

“…where l is the dimensionless channel width used in the LBE approach [7]. As a result, the cross section of the the first component can be written by…”

Particular solutions of the linearized Boltzmann equation for a binary mixture of rigid spheres

“…As a final comment, we note that Eq. (5.1) has been used recently [12] to solve well the problem of Poiseuille flow in a plane channel, as described by the linearized Boltzmann equation for a mixture of rigidsphere gases.…”

Discrete unified gas kinetic scheme for flows of binary gas mixture based on the McCormack model

“…Based on model equations, several types of kinetic method have already been developed for gas mixture flows, such as the lattice Boltzmann method 28,29 , the discrete velocity methods (DVM) 30,31 , and the analytical version of the discrete ordinate method 32,33 . Besides, the unified gas kinetic scheme (UGKS) for binary gas mixtures of hard sphere molecules has been developed based on the Andries-Aoki-Perthame (AAP) kinetic model 34,35 .…”