The Boltzmann equation for a bounded medium I. General theory

Abstract

A general theoretical treatment is given of the linearized Boltzmann equation for flow in a bounded medium under conditions when the collision mean free path is of the order of the dimensions of the cross-section of the specimen. The approach given may be used for any type of particle; we consider gas molecules, electrons and phonons. Part 1 is concerned with elucidating various results for a specimen of general cross-section. Having obtained the Boltzmann equation for any type of internal scattering mechanism, we deal with the boundary conditions relating to the scatter of particles by the specimen surfaces. Employing a very general form for these, a uniqueness theorem is proved for the solution of our problem. Certain general symmetry properties of the solution are discussed, and transport considerations are dealt with. Finally, a frequently used approximate treatment of boundary scatter is placed on a firmer mathematical foundation. Part 2 is concerned with a detailed evaluation of the solution of the Boltzmann equation and boundary conditions for the flow between parallel plates. Previous treatments of this problem have assumed a ‘relaxation-time’ approximation for representing the effect of interparticle collisions in the medium, together with certain simplified boundary conditions. These two assumptions effectively remove the ‘coupling’ which should exist between the equations relating to the different particle modes, and thus greatly simplify the solution of the problem. We retain the complete Boltzmann equation, which is equivalent to a set of coupled first-order linear differential equations, and find its general solution containing various undetermined constants, which are then calculated via our general boundary conditions. This general solution is obtained as the sum of a complementary function and a particular integral. The former involves the eigenfunctions and eigenvalues of a modified collision operator, while the form taken by the latter depends on whether or not wave number (equivalent to momentum) is conserved in interparticle collisions. If wave number is not conserved, the particular integral is the solution of the Boltzmann equation for an infinite medium. The complementary function is then a combination of terms, varying exponentially with respect to distance, which correspond to a decrease in the neighbourhood of the boundary; these are qualitatively of the same form as when a relaxation time is employed. On the other hand, if wave number is conserved, the equation for an infinite medium possesses no solution and it is then found that the particular integral corresponds to a quadratic variation with respect to distance between the boundary surfaces. When the distance between these surfaces is sufficiently greater than the collision mean free path this quadratic variation is shown to differ from the usual 'viscous flow’ theory by terms which are of importance only in the neighbourhood of the boundary; these, together with the complementary function, give the boundary corrections to the usual theory. The combination of quadratic particular integral and exponential complementary function is shown to give rise to the possibility of a ‘Knudsen minimum’, which has so far been observed both in gases and in phonon flow in liquid helium. Throughout the paper a general anisotropic medium is assumed and we thus incidentally generalize the theory of viscous flow, previously considered only for an isotropic medium in the case of gas molecules. Finally, a consideration is given of the situation when a small proportion of collisions not conserving wave number occurs together with a very large proportion of collisions that do conserve it; this is relevant to the effect of impurities and other momentum destroying mechanisms at low temperatures. The result for general separation of the boundaries is obtained, and it is found that if this is large enough, the total particle flow is similar to that occurring in the absence of wave number conserving processes. However, the flow variation on leaving the boundary is now accurately characterized by a relaxation length which is the geometrical mean of the relaxation lengths for the two types of collision process acting separately.