Kinetic theory based nonlinear treatment for motions of a binary gas mixture involving slightly strong evaporation and condensation processes II — A system of macroscopic equations, the boundary conditions and the Knudsen-layer corrections
Kinetic theory based nonlinear treatment for motions of a binary gas mixture involving slightly strong evaporation and condensation processes II — A system of macroscopic equations, the boundary conditions and the Knudsen-layer corrections
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基于动力学理论的涉及弱蒸发和冷凝过程的二元气体混合物运动的非线性处理 II — 宏观方程组、边界条件和克努森层校正
DOI：
10.1007/bf00942812
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
Y. Onishi
中科院分区：
文献类型：
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作者：
Y. Onishi
基于边界扩散边界条件下的BGK型玻尔兹曼方程，对一般流域中可凝性和不可凝性气体二元混合物的稳态行为进行了非线性分析：（i）可凝性气体的量相当大，使得在某个稳态平衡参考状态下分子的平均自由程与系统的特征长度（即克努森数）相比较小。参考状态下的系统与统一相比较小； (ii) 系统与参考状态的偏差较小，但其幅度为努森数数量级； (iii)浓度比，即参考状态下不凝性气体的数密度与可凝性气体的数密度之比，较小，最多为克努森数的数量级。本研究是第一部分针对中等浓度比值的情况（本作者早期的工作）到较小浓度比值的情况的延伸。当前问题与第一部分问题的主要区别在于，不可凝性气体的数密度和分压的完全非线性行为不可避免地在凝结表面附近表现出来，因为其分子被驱向表面并通过与流向它的可凝性气体分子的碰撞而在那里积累。在分析这种完全非线性时，通过适当考虑，导出了以下宏观系统作为玻尔兹曼方程的渐近解： BGK 型适用于小努森数； (1) 控制流体动力学量的纳维-斯托克斯型宏观方程； (2) 它们在界面处的适当边界条件； (3) 应用于上述宏观系统界面附近的解的努森层修正； (4) 混合物及其组分气体的应力张量以及热能通量矢量的公式。导出的宏观系统使得在普通流体动力学水平上处理涉及蒸发和冷凝过程的二元气体混合物的各种流动问题成为可能，充分描述了混合物及其组分气体在浓度比的小值范围内的行为。当然，当应用于特定问题时，随着集中比增加到阶数统一的值，它所产生的结果会顺利地融入到第一部分中获得的结果中。
A nonlinear analysis for the steady behaviour of a binary mixture of a condensable and noncondensable gases in a general flow domain has been carried out based on the Boltzmann equation of BGK type subject to the diffusive boundary condition at the interface(s) under the following conditions: (i) the amount of the condensable gas is fairly large so that the mean free path of the molecules at a certain stationary equilibrium reference state is small compared with the characteristic length of the system, i.e., the Knudsen number of the system at the reference state is small compared with unity; (ii) the deviation of the system from the reference state is small, but its magnitude is of the order of the Knudsen number; and (iii) the concentration ratio, the ratio of the number density of the noncondensable gas to that of the condensable gas at the reference state, is small, being of the magnitude at most of the order of the Knudsen number. The present study is an extension of Part I for the case of moderate values of the concentration ratio (the earlier work by the present author) to the case of smaller values of the ratio. The main difference of the present problem to the problem of Part I lies in the fact that thefull nonlinear behaviourof the number density and partial pressure of the noncondensable gas inevitably manifests itself near the surface of condensation owing to its molecules being driven toward the surface and accumulating there by the collisions with the molecules of the condensable gas flowing toward it.With proper account being taken in the analysis of thisfull nonlinearity, the following macroscopic system has been derived as an asymptotic solution to the Boltzmann equation of BGK type for small Knudsen numbers; (1) the macroscopic equations of Navier-Stokes type governing the fluid dynamic quantities; (2) the appropriate boundary conditions for them at the interface(s); (3) the Knudsen-layer corrections to be applied to the solutions to the above macroscopic system near the interface(s); and (4) the formulas for the stress tensors and heat and energy flux vectors of the mixture and its component gases. The derived macroscopic system makes possible at the level of ordinary fluid dynamics the treatment of various flow problems of a binary gas mixture involving evaporation and condensation processes, giving an adequate description of the behaviour of the mixture and its component gases for a range of small values of the concentration ratio. It produces, of course, when applied to specific problems, results merging smoothly into those obtained from Part I as the concentration ratio increases to values of order unity.