Boundary value problems in kinetic theory of gases

Abstract

The state of a gas, flowing between two parallel plates, is analyzed from the viewpoint of kinetic theory. When the mean free path is infinitely greater than the distance between the plates, the exact solution shows that the distribution function is discontinuous in velocity. One must distinguish between molecules impinging on a plate and those leaving. We investigate the problem of finding a theory valid for arbitrary ratio of mean free path to plate distance, and of plate speed to sound speed. This is most easily achieved by splitting the distribution function into the above mentioned parts, and expanding each part in polynomials in velocity space, which are orthogonal over half the velocity range. In every approximation, exact account is given of (1) the microscopic boundary conditions, (2) the conservation laws, and (3) the behavior in the low pressure region. The method, which can be applied to the Boltzmann equation, is here developed for the kinetic model of Bhatnager, Gross, and Krook. Variational principles are stated by noting the similarity of the linearized version of this theory to the Milne equation of radiative transfer. For the nonlinear, high speed case, a new approach in the low pressure region is indicated. The relationship to alternative methods is discussed. When the distribution function is expanded in full-range orthogonal polynomials it is necessary to go to high order to obtain an adequate representation of the low pressure region, and of the boundary layer. Very simple half-range distribution functions yield an accurate description of the state of the gas.