Shigeru Takata

Inappropriateness of the heat‐conduction equation for description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation

It is shown analytically and numerically on the basis of the kinetic theory that the heat‐conduction equation is not suitable for describing the temperature field of a gas in the continuum limit around bodies at rest in a closed domain or in an infinite domain without flow at infinity, where the flow vanishes in this limit. The behavior of the temperature field is first discussed by asymptotic analysis of the time‐independent boundary‐value problem of the Boltzmann equation for small Knudsen numbers. Then, simple examples are studied numerically: as the Knudsen number of the system approaches zero, the temperature field obtained by the kinetic equation approaches that obtained by the asymptotic theory and not that of the heat‐conduction equation, although the velocity of the gas vanishes.

Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard‐sphere molecules

A slow uniform flow of a rarefied gas past a sphere is investigated on the basis of the linearized Boltzmann equation for hard‐sphere molecules and the diffuse reflection condition. With the aid of a similarity solution, the Boltzmann equation is reduced to two simultaneous integrodifferential equations with three independent variables, which are solved numerically. The collision integral is computed efficiently by the use of a numerical collision kernel [Phys. Fluids A 1, 363 (1989)]. The velocity distribution function of the gas molecules, which has discontinuity in the gas, the density, flow velocity, and temperature fields of the gas, and the drag on the sphere are obtained accurately for the whole range of the Knudsen number. In spite of slow flow, the temperature is nonuniform (thermal polarization). From the behavior of the velocity distribution function, the kinetic transition region is clearly seen to separate into the Knudsen layer and the S layer for small Knudsen numbers.

Numerical analysis of a supersonic rarefied gas flow past a flat plate

A uniform supersonic flow of a rarefied gas past a flat plate at zero angle of attack is considered, and the steady behavior of the gas around the plate is investigated numerically on the basis of the Boltzmann–Krook–Welander equation (or the so-called BGK model) and the diffuse reflection boundary condition. An accurate finite-difference analysis, which gives the correct description of the discontinuity of the velocity distribution function of the gas molecules occurring in the gas, is carried out, and the features of the flow field (the velocity distribution function and the macroscopic variables such as the density, temperature, and flow velocity of the gas), in particular, those around the leading and trailing edges, are clarified for a wide range of the Knudsen number. The drag acting on the plate and the energy transferred to it are also obtained accurately. In addition, on the basis of the results for small Knudsen numbers, the behavior of the gas around the leading edge of a semi-infinite plate is...

Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of a noncondensable gas

A vapor in a gap between two parallel plane surfaces of its condensed phase, on which evaporation or condensation may take place, is considered in the case where another gas that neither evaporates nor condenses on the surfaces (say, a noncondensable gas) is also contained in the gap. The steady flow of the vapor caused by evaporation on one surface and condensation on the other and the behavior of the noncondensable gas are investigated on the basis of kinetic theory. First, fundamental features of the flow field are clarified for small values of the Knudsen number (associated with vapor–vapor collisions) by a systematic asymptotic analysis of the Boltzmann equation. Then, the problem is analyzed numerically by means of the direct simulation Monte Carlo method, and the steady behavior of the vapor and of the noncondensable gas (e.g., the spatial distributions of the macroscopic quantities) is clarified for a wide range of the Knudsen number. In particular, it is shown that, in the limit as the Knudsen number tends to zero (the continuum limit with respect to the vapor), there are two different types of the limiting behavior depending on the amount of the noncondensable gas, and evaporation and condensation can take place only when the average density of the noncondensable gas is vanishingly small in comparison with that of the vapor.

Shock-wave structure for a binary gas mixture: finite-difference analysis of the Boltzmann equation for hard-sphere molecules

Abstract The structure of a normal shock wave for a binary mixture of hard-sphere gases is analyzed numerically on the basis of the Boltzmann equation by a finite-difference method. In the analysis, the complicated collision integrals are computed efficiently as well as accurately by means of the numerical kernel method, which is the generalization to the case of a binary mixture of the method devised by Ohwada in 1993 in the shock-structure analysis for a single-component gas. The transition from the upstream to the downstream uniform state is clarified not only for the macroscopic quantities but also for the velocity distribution functions.

Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer

Abstract Discontinuity of the velocity distribution function in steady flows of a rarefied gas around a convex body is discussed, and its examples in an evaporating flow from a cylindrical condensed phase, in a slow uniform flow past a sphere, and in a flow induced around a sphere in a gas with a temperature gradient are presented. The same type of discontinuity is shown to be absent in a gas around a concave body. The S layer in the bottom of the Knudsen layer (Y. Sone, Phys. Fluids 16, 1422 (1973)) is attributed to this discontinuity, and its influence on the Knudsen layer and the slip boundary condition is discussed, with a numerical example.

Numerical analysis of thermal-slip and diffusion-slip flows of a binary mixture of hard-sphere molecular gases

The thermal-slip (thermal-creep) and the diffusion-slip problems for a binary mixture of gases are investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. The problems are analyzed numerically by the finite-difference method incorporated with the numerical kernel method, which was first proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)] for a single-component gas. As a result, the behavior of the mixture is clarified accurately not only at the level of the macroscopic variables but also at the level of the velocity distribution function. In addition, accurate formulas of the thermal-slip and the diffusion-slip coefficients for arbitrary values of the concentration of a component gas are constructed by the use of the Chebyshev polynomial approximation.

A rarefied gas flow caused by a discontinuous wall temperature

A flow of a rarefied gas caused by a discontinuous wall temperature is investigated on the basis of kinetic theory in the following situation. The gas is confined in a two-dimensional square container, and the left and right halves of the wall of the container are kept at different uniform temperatures, so that the temperatures of the top and bottom walls are discontinuous at their respective middle points. External forces are assumed to be absent. The steady flow of the gas induced in the container by the effect of the discontinuities is analyzed numerically on the basis of the Bhatnagar–Gross–Krook model of the Boltzmann equation and the diffuse reflection boundary condition by means of an accurate finite-difference method. The features of the flow are clarified for a wide range of the Knudsen number. In particular, it is shown that, as the Knudsen number becomes small (i.e., as the system approaches the continuum limit), the maximum flow speed tends to approach a finite value, but the region with appre...

The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: Asymptotic analysis of the Boltzmann equation

A binary mixture of a vapor and a noncondensable gas around arbitrarily shaped condensed phases of the vapor is considered. Its steady behavior in the continuum limit (the limit where the Knudsen number vanishes) is investigated on the basis of kinetic theory in the case where the condensed phases are at rest, and the mixture is in a state at rest with a uniform pressure at infinity when an infinite domain is considered. A systematic asymptotic analysis of the Boltzmann equation with kinetic boundary condition is carried out for small Knudsen numbers, and the system of fluid-dynamic type equations and their appropriate boundary conditions that describes the behavior in the continuum limit is derived. The system shows that the flow of the mixture vanishes in the continuum limit, but the vanishing flow gives a finite effect on the behavior of the mixture in this limit. This is an example of the ghost effect discovered recently by Sone and coworkers [e.g., Y. Sone et al., Phys. Fluids 8, 628 and 3403 (1996); Y. Sone, in Rarefied Gas Dynamics, edited by C. Shen (Peking University Press, Beijing, 1997), p. 3]. It is shown that there are several new source factors of the ghost effect that are peculiar to a gas mixture, i.e., that originate from the nonuniformity of the concentration.

Two-surface problems of a multicomponent mixture of vapors and noncondensable gases in the continuum limit in the light of kinetic theory

The steady behavior of a multicomponent mixture of vapors and noncondensable gases between two parallel plane condensed phases for small Knudsen numbers, especially for the continuum limit (i.e., the limit as the Knudsen number vanishes), is investigated in the light of kinetic theory. By a systematic asymptotic analysis of the Boltzmann equation with kinetic boundary conditions, the flow due to evaporation and condensation on the condensed phases is shown to vanish in the continuum limit, and then the system of fluid-dynamic-type equations and their boundary conditions which describes the behavior in the limit is derived. On the basis of the system, it is shown that the vanishingly weak evaporation and condensation give a finite effect on the behavior of the mixture in the continuum limit. This is an example of the ghost effect discovered recently by Sone and co-workers [e.g., Y. Sone et al., Phys. Fluids 8, 628 and 3403 (1996); Y. Sone, in Rarefied Gas Dynamics, edited by C. Shen (Peking U.P., Beijing, 1997), p. 3]. Finally, for the case of a binary mixture of a vapor and a noncondensable gas, two typical problems, the simultaneous mass and heat transfer and the plane Couette flow, are considered to demonstrate the effect more concretely. The result is also compared with that obtained by the numerical analysis of the Boltzmann equation by the direct simulation Monte Carlo method.The steady behavior of a multicomponent mixture of vapors and noncondensable gases between two parallel plane condensed phases for small Knudsen numbers, especially for the continuum limit (i.e., the limit as the Knudsen number vanishes), is investigated in the light of kinetic theory. By a systematic asymptotic analysis of the Boltzmann equation with kinetic boundary conditions, the flow due to evaporation and condensation on the condensed phases is shown to vanish in the continuum limit, and then the system of fluid-dynamic-type equations and their boundary conditions which describes the behavior in the limit is derived. On the basis of the system, it is shown that the vanishingly weak evaporation and condensation give a finite effect on the behavior of the mixture in the continuum limit. This is an example of the ghost effect discovered recently by Sone and co-workers [e.g., Y. Sone et al., Phys. Fluids 8, 628 and 3403 (1996); Y. Sone, in Rarefied Gas Dynamics, edited by C. Shen (Peking U.P., Beijing, ...