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 initial-or 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 f e , 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 |f e − f e0 |/f e0 , where f e0 is the stationary Maxwellian with the representative density and temperature in the gas.(1) |f e − f e0 |/f e0 = 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) |f e − f e0 |/f e0 = o(Kn) : Linear system (the Stokes set).(2) |f e − f e0 |/f e0 = O(1) with | ξ i f e dξ|/ |ξ i |f e dξ = O(Kn) (ξ i : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity v i of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) 2 must be retained in the equations (non-Navier-Stokes effect). The thermal creep [1] in the boundary condition must be taken into account.(3) |f e − f e0 |/f e0 = O(1) with | ξ i f e dξ|/ |ξ i |f e dξ = 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 T w and δT w 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 ...

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.

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.

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.

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 initial-or 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 f e , 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 |f e − f e0 |/f e0 , where f e0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |f e − f e0 |/f e0 = 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) |f e − f e0 |/f e0 = o(Kn) : Linear system (the Stokes set). (2) |f e − f e0 |/f e0 = O(1) with | ξ i f e dξ|/ |ξ i |f e dξ = O(Kn) (ξ i : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity v i of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) 2 must be retained in the equations (non-Navier-Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |f e − f e0 |/f e0 = O(1) with | ξ i f e dξ|/ |ξ i |f e dξ = 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 T w and δT w be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified wit...