Toshiyuki Doi

Analytical study of transonic flows of a gas condensing onto its plane condensed phase on the basis of kinetic theory

Abstract A uniform flow of a gas condensing onto its plane condensed phase (commonly known as the half-space problem of condensation) is considered. The problem is studied analytically on the basis of the Boltzmann equation when the flow is in a transonic region. The paper clarifies the analytical structure of the solution, especially the mechanism by which the range of the parameters (the flow speed, pressure, and temperature of the uniform flow blowing from infinity) where a steady solution exists changes abruptly (from a surface to a domain in the parameter space) when the flow speed passes the sonic speed, the correspondence of a family of supersonic solutions to a subsonic solution, etc. The solutions constructed analytically are compared with new numerical solutions near the sonic point.

Bifurcation of and ghost effect on the temperature field in the Bénard problem of a gas in the continuum limit

A gas in a time-independent state under a uniform weak gravity in a general domain is considered. The asymptotic behavior of the gas in the limit that the Knudsen number of the system tends to zero (or in the continuum limit) is investigated on the basis of the Boltzmann system for the case where the flow velocity vanishes in this limit, and the fluid-dynamic-type equations and their associated boundary conditions describing the behavior of the gas in the continuum limit are derived. The equations, different from the Navier–Stokes ones, contain thermal stress and infinitesimal velocity amplified by the inverse of the Knudsen number. The system is applied to analysis of the behavior of a gas between two parallel plane walls heated from below (Benard problem), and a bifurcated strongly distorted temperature field is found in infinitesimal velocity and gravity. This is an example showing that the Navier–Stokes system fails to describe the correct behavior of a gas in the continuum limit.

Ghost effect of infinitesimal curvature in the plane Couette flow of a gas in the continuum limit

The time-independent plane Couette flow of a gas in the continuum limit is studied on the basis of kinetic theory as the limit of the cylindrical Couette flow of a rarefied gas between two rotating coaxial circular cylinders when the mean free path and the curvature (or the inverse of the radius) of the inner cylinder tend to zero simultaneously, keeping the difference of the radii of the two cylinders fixed. The fluid-dynamic-type equations and their boundary conditions governing the limiting state are derived for arbitrary circumferential speeds of rotation of the cylinders and for arbitrary temperature difference of the two cylinders. The resulting equations depend on the relative speed of decay of the two parameters and contain a term due to the infinitesimal curvature of the cylinder, as well as non-Navier–Stokes stress terms, when the curvature decays not faster than some function (generally, the square) of the mean free path. The bifurcation analysis of the plane Couette flow of a linear profile is...

Numerical study of the influence of gravity on the heat conductivity on the basis of kinetic theory

The Boltzmann–Krook–Welander (or Bhatnagar–Gross–Krook) model of the Boltzmann equation is solved numerically for the heat transfer problem of a gas enclosed between two parallel, infinite plates kept at different temperatures, in the presence of a constant gravity field normal to the plates. At each point where the direct effect of the boundaries is negligible, a relation among the relevant local quantities (heat flux, temperature gradient, temperature, and density) holds even if the temperature varies over a length scale comparable to the mean free path. The ratio of the actual heat flux to the value predicted by the Fourier law is seen to be determined by the local Knudsen number and the local Froude number which are defined with the local mean free path, local characteristic length, and the magnitude of gravity. It is observed that the gravity produces an enhancement of the effective heat conductivity when the heat flux and the gravity field are parallel, while it produces an inhibition when both vectors are antiparallel. This deviation from the Fourier law, which vanishes in the absence of gravity, increases as the local Knudsen number increases and is more remarkable when the heat flux is parallel to the gravity field rather than otherwise. Comparison of the numerical data with an asymptotic analysis as well as with Pade approximants derived from it is also made.

Analytical study of bifurcation of a flow of a gas between coaxial circular cylinders with evaporation and condensation

Time-independent behavior of a gas between two coaxial circular cylinders made of the condensed phase of the gas, where the cylinders are rotating around their common axis and evaporation or condensation is taking place, is considered with special attention given to bifurcation of the flow. The problem is studied analytically for small values of the speeds of rotation of the cylinders and the Knudsen number on the basis of the Boltzmann equation, and the solution is obtained explicitly. The bifurcation of flow occurs even in a simple case where the gas is axially symmetric and uniform (or the flow field depends only on the radial coordinate). The comprehensive feature of the bifurcation of flow is clarified with the explicit forms of solutions and the bifurcation diagram.

Ghost effect and bifurcation in a gas between coaxial circular cylinders with different temperatures

A gas in a time-independent state between rotating coaxial circular cylinders with different temperatures is considered. The bifurcation of the field in the continuum limit is studied on the basis of the system of fluid-dynamic-type equations and their boundary conditions derived from the Boltzmann system [Phys. Fluids 8, 628 (1996)]. When the ratio of the temperatures of the two cylinders is not close to unity, the bifurcation occurs at infinitesimal speeds of rotation of the cylinders of the first order of the Knudsen number. The temperature field in the continuum limit is determined together with the infinitesimal velocity field, and a bifurcated temperature field, as well as an axially uniform and symmetric field, exists in the absence of the flow velocity (ghost effect). Further, thermal stress, as well as viscous stress, produces a finite effect on the bifurcated temperature field (a non-Navier–Stokes effect). For example, when the ratio of the temperatures of the two cylinders is ten or one tenth, ...

Plane Thermal Transpiration of a Rarefied Gas Between Two Walls of Maxwell-Type Boundaries With Different Accommodation Coefficients

Plane thermal transpiration of a rarefied gas between two walls of Maxwell-type boundaries with different accommodation coefficients is studied based on the linearized Boltzmann equation for a hard-sphere molecular gas. The Boltzmann equation is solved numerically using a finite difference method, in which the collision integral is evaluated by the numerical kernel method. The detailed numerical data, including the mass and heat flow rates of the gas, are provided over a wide range of the Knudsen number and the entire range of the accommodation coefficients. Unlike in the plane Poiseuille flow, the dependence of the mass flow rate on the accommodation coefficients shows different characteristics depending on the Knudsen number. When the Knudsen number is relatively large, the mass flow rate of the gas increases monotonically with the decrease in either of the accommodation coefficients like in Poiseuille flow. When the Knudsen number is small, in contrast, the mass flow rate does not vary monotonically but exhibits a minimum with the decrease in either of the accommodation coefficients. The mechanism of this phenomenon is discussed based on the flow field of the gas.

Bifurcation of a Flow of a Gas between Rotating Coaxial Circular Cylinders with Evaporation and Condensation

Bifurcation of a flow of a gas between two rotating coaxial circular cylinders made of the condensed phase of the gas is studied analytically on the basis of the asymptotic equations and boundary conditions derived from the Boltzmann system for small Knudsen numbers. The bifurcation relation and behavior of solution near the bifurcation point are obtained, and the effect of evaporation and condensation on the cylinders on the bifurcation is clarified.

Transient Couette flow of a rarefied gas between plane parallel walls with different surface properties

Transient Couette flow of a rarefied gas between plane parallel walls with different surface properties induced by a sudden start-up of one of the walls is studied based on kinetic theory. The linearized Boltzmann equation for a hard sphere molecular gas is analyzed under the assumptions that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary. The initial and boundary value problem is solved numerically by using a modified hybrid scheme of characteristic coordinate and finite difference methods, to describe the discontinuities in the velocity distribution function correctly. The time evolution of the flow and the approach to the final time-independent state are studied over a wide range of the mean free paths and the accommodation coefficient of the boundary. In the transient process, the shear force acting on the moving wall depends on which wall moves. In contrast, the shear force acting on the wall at rest is independent of which wall moves; this property is a conse...

Instability of the Plane Couette Flow by the Ghost Effect of Infinitesimal Curvature

Instability of the plane Couette flow with a small Mach number of a gas in the continuum limit is studied on the basis of Boltzmann equation as the limit of the cylindrical Couette flow of a rarefied gas between two rotating coaxial circular cylinders when the mean free path and the curvature (or the inverse of the radius) of the inner cylinder tend to zero simultaneously, keeping the difference of the radii of the two cylinders fixed. The resulting equations depend on the relative speed of decay of the two parameters and contain a term due to the infinitesimal curvature of the cylinder. Owing to the infinitesimal curvature, the flow can be unstable. The stability diagram of the flow, i.e., the relation among the wave numbers, frequency, damping (or amplifying) factor, and wall speeds, are obtained.