Yoshio Sone

Kinetic Theory and Fluid Dynamics

In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard‐sphere molecules

The Poiseuille and thermal transpiration flows of a rarefied gas between two parallel plates are investigated on the basis of the linearized Boltzmann equation for hard‐sphere molecules and diffuse reflection boundary condition. The velocity distribution functions of the gas molecules as well as the gas velocity and heat flow profiles and mass fluxes are obtained for the whole range of the Knudsen number with good accuracy by the numerical method recently developed by the authors.

Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules

Shear flow and thermal creep flow (flow induced by the temperature gradient along the boundary wall) of a rarefied gas over a plane wall are considered on the basis of the linearized Boltzmann equation for hard‐sphere molecules and diffuse reflection boundary condition. These fundamental rarefied gas dynamic problems, typical half‐space boundary‐value problems of the linearized Boltzmann equation, are analyzed numerically by the finite‐difference method developed recently by the authors, and the velocity distribution functions, as well as the macroscopic variables, are obtained with good accuracy. From the results, the shear and thermal creep slip coefficients and their associated Knudsen layers of a slightly rarefied gas flow past a body are derived. The results for the slip coefficients and Knudsen layers are compared with experimental data and various results by the Boltzmann–Krook–Welander (BKW) equation, the modified BKW equation, and a direct simulation method.

Kinetic Theory of Evaporation and Condensation –Hydrodynamic Equation and Slip Boundary Condition–

The steady behavior of a gas in contact with its condensed phase of arbitrary shape is investigated on the basis of kinetic theory. The Knudsen number of the system (the mean free path of the gas molecules divided by the characteristic length of the system) being assumed to be fairly small, the hydrodynamic equations for the macroscopic quantities, the velocity, temperature, and pressure, of the gas and their boundary conditions on the interface of the gas and its condensed phase are derived, and two simple examples (evaporation from a sphere, two-surface problem of evaporation and condensation) are worked out.

Temperature jump and Knudsen layer in a rarefied gas over a plane wall: Numerical analysis of the linearized Boltzmann equation for hard‐sphere molecules

A semi‐infinite expanse of a rarefied gas over a plane wall where there is a constant heat flow normal to the wall from infinity is considered. The behavior of the gas is analyzed numerically by a finite difference method on the basis of the standard linearized Boltzmann equation for hard‐sphere molecules with diffuse reflection at the wall. From the result the temperature jump coefficient and its associated Knudsen layer of a slightly rarefied gas flow around a body are derived.

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.

Kinetic Theory of Evaporation and Condensation

The behavior of a gas in contact with its condensed phase is considered on the basis of a relaxation model of the linearized Boltzmann equation. The temperature and density distributions of the gas in the Knudsen layer as well as so called slip boundary condition on the interface of the gas and its condensed phase are obtained.

Numerical analysis of gas flows condensing on its plane condensed phase on the basis of kinetic theory

Gas flows condensing on its plane condensed phase are considered on the basis of the Boltzmann–Krook–Welander equation. First, the formation and propagation of disturbances in an initially uniform gas blowing against its plane condensed phase are investigated under the conventional boundary condition on the condensed phase to supplement the authors’ previous work, where some solutions require more detailed computation. Then, with the aid of time‐dependent analysis, the behavior of steady condensation is clarified: The range of the variables at infinity and on the condensed phase that allows a steady solution is determined; the profiles of typical steady solutions are presented.

Flow Induced by Thermal Stress in Rarefied Gas

A new kind of mechanism which induces a flow around a solid body in a slightly rarefied gas is proposed. In order to demonstrate the flow induced by this mechanism the behavior of gas around a sphere with a constant temperature which is placed in an infinite expanse of gas at rest with a uniform temperature gradient is investigated on the basis of the asymptotic theory for a slightly rarefied gas. A flow with magnitude of the order of the Knudsen number squared is induced from the hotter to the colder region. The sphere is subject to a force in the direction of the given temperature gradient.

Asymptotic Theory of a Steady Flow of a Rarefied Gas Past Bodies for Small Knudsen Numbers

A survey is made of the asymptotic behavior for small Knudsen numbers of the time-independent solution of the boundary-value problem of the Boltzmann equation over a general domain. Included is the hydrodynamic system (hydrodynamic type equations and their slip boundary conditions) describing the asymptotic behavior, with several new results.