Vladimir Khristinich

Divergence Form of the Boltzmann Kinetic Equations

Faculty of Mathematics and Mechanics St Petersburg State University, Russia



    Various forms of the kinetic equations being equivalent from the mathematical point of view are convenient in devising and grounding of computational methods in rarefied gas dynamics. It must be noted that this various forms can be derived from definite phenomenological hypothesis on which direct simulation method of rarefied gas flow is founded. The purpose of the work is to derive and construct the system of kinetic equations which is equivalent to the Boltzmann equation. Equations of the system are well known models of mathematical physics and different methods are applicable for their solution. In deriving of the system of divergence equations the processes of convectional transfer and particle collisions in gas are separated and a construction of statistical models become more easy. In the Bird method this simplification has been achieved due to an assumption about the decomposition of both processes into two steps - moving the particles and computing collisions appropriate to a small time interval. To derive the system of kinetic equations in divergence form a force field F, which effect is equivalent to a collision integral in the Boltzmann equation is introduced. This in contrast to the Vlasov equation. In divergence form of kinetic equations self - consistent field F is introduced according to the relation between collision integral J(f,f) and divergence in velocity space (u) - div, of vector flux (1/m)(Ff)

1/m div(F(r,u,t) f(r,u,t)) = - J(r,u,t).

   In the report this approach is generalized for the case of gas mixture. The models of field F are constructed for solving the problems of spatially homogeneous gas relaxation and plane steady shock. One of the interest points is transformation of the equation from the divergence form to integral one. The integration is realized along the trajectories which are formed by the use of the field F. The integral form of the divergence equations represents another variant of a description of rarefied gas flow when the assumptions of Boltzmann equation take place. The iteration scheme for the divergence form of kinetic equation is not traditional. One must solve motion equations for particles in the field F on every step of this procedure. Use of the divergence form of kinetic equations allows to derive net analog of kinetic equation, thus for the phase sell we get the balance law on which the derivation of Boltzmann equation is based. The difference equations based on the net analog will satisfy conservation equations.

Filippov B.V., Khristinich V.B. Divergence form of the kinetic equations of rarefied gas dynamics. // Dynamical processes in gases and solids. Ser. Physical mechanics, issue 4, Leningrad State University, 1980, p. 7 - 18.

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