The Steady Boltzmann and Navier-Stokes Equations

The paper discusses the similarities and the differences in the mathematical theories of the steady Boltzmann and incompressible Navier-Stokes equations posed in a bounded domain. First we discuss two different scaling limits in which solutions of the steady Boltzmann equation have an asymptotic behavior described by the steady Navier-Stokes Fourier system. Whether this system includes the viscous heating term depends on the ratio of the Froude number to the Mach number of the gas flow. While the steady Navier-Stokes equations with smooth divergence-free external force always have at least one smooth solutions, the Boltzmann equation with the same external force set in the torus, or in a bounded domain with specular reflection of gas molecules at the boundary may fail to have any solution, unless the force field is identically zero. Viscous heating seems to be of key importance in this situation. The nonexistence of any steady solution of the Boltzmann equation in this context seems related to the increase of temperature for the evolution problem, a phenomenon that we have established with the help of numerical simulations on the Boltzmann equation and the BGK model.Comment: 55 pages, 4 multiple figure

Temperature, pressure, and concentration jumps for a binary mixture of vapors on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation

The half-space problem of the temperature, pressure, and concentration jumps for a binary mixture of vapors is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the complete condensation condition. First, the problem is shown to be reduced to three elemental ones: the problem of the jumps caused by the net evaporation or condensation, that caused by the gradient of temperature, and that caused by the gradient of concentration. Then, the latter two are investigated numerically in the present contribution because the first problem has already been studied [Yasuda, Takata, and Aoki, Phys. Fluids 17, 047105 (2005)]. The numerical method is a finite-difference one, in which the complicated collision integrals are computed by the extension of the method proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)] to the case of a gas mixture. As a result, the behavior of the mixture is clarified not only at the level of the macroscopic quantities but also at the level of the velocity distribution function. In addition, accurate formulas of the temperature, pressure, and concentration jumps are constructed for arbitrary values of the concentration of the background reference state by the use of the Chebyshev polynomial approximation. The solution of the corresponding problem of a vapor-gas mixture and that of the temperature-jump problem on a simple solid wall are also obtained as special cases of the present problem

Development of an Automatic Dishwashing Robot System

Proceedings of The 2009 IEEE International Conference on Mechatronics and Automation, August 9 - 12, Changchun, Chin

Cylindrical Couette flow of a rarefied gas: Effect of a boundary condition on the inverted velocity profile.

The cylindrical Couette flow of a rarefied gas between a rotating inner cylinder and a stationary outer cylinder is investigated under the following two kinds of kinetic boundary conditions. One is the modified Maxwell-type boundary condition proposed by Dadzie and Méolans [J. Math. Phys. 45, 1804 (2004)] and the other is the Cercignani-Lampis condition, both of which have separate accommodation coefficients associated with the molecular velocity component normal to the boundary and with the tangential component. An asymptotic analysis of the Boltzmann equation for small Knudsen numbers and a numerical analysis of the Bhatnagar-Gross-Krook model equation for a wide range of the Knudsen number are performed to clarify the effect of each accommodation coefficient as well as of the boundary condition itself on the behavior of the gas, especially on the flow-velocity profile. As a result, the velocity-slip and temperature-jump conditions corresponding to the above kinetic boundary conditions are derived, which are necessary for the fluid-dynamic description of the problem for small Knudsen numbers. The parameter range for the onset of the velocity inversion phenomenon, which is related mainly to the decrease in the tangential momentum accommodation, is also obtained

キタイ ブンシ ウンドウ ロン ニ ヨル ヒ ヘイコウ コンゴウ キタイ ノ キョドウ ノ ケンキュウ

京都大学0048新制・課程博士博士(工学)甲第8950号工博第2041号新制||工||1205(附属図書館)UT51-2001-F280京都大学大学院工学研究科航空宇宙工学専攻(主査)教授 青木 一生, 教授 斧 髙一, 教授 川原 琢治学位規則第4条第1項該当Doctor of EngineeringKyoto UniversityDFA

Slow evaporation and condensation on a spherical droplet in the presence of a noncondensable gas

A spherical droplet is placed in a binary mixture composed of the vapor of the droplet and another gas which neither evaporates nor condenses (a noncondensable gas). The mixture is in an equilibrium state at rest at infinity. A slow steady flow of the vapor caused by weak evaporation or condensation, under the influence of the noncondensable gas, is investigated on the basis of a linearized model Boltzmann equation. Numerical analyses by means of a finite-difference method are carried out for a wide range of the Knudsen number (i.e., from a large to small droplet compared to the molecular mean free path). The numerical results, together with analytical solutions for small and large Knudsen numbers, clarify the behavior the mixture, i.e., the mass- and heat-flow rates from or onto the droplet as well as spatial distributions of the macroscopic quantities, in the entire range of gas rarefaction. The solution for the steady heat transfer problem between a solid sphere and a binary gas mixture is also obtained as a byproduct