Velocity and temperature mappings on numerical solutions of momentum and energy transport equations

Abstract

The Golub-Swinney and the Benard-Rayleigh experiments are classic physical systems used in the study of hydrodynamic instabilities. The Golub-Swinney set-up consists of two coaxial cylinders with a fluid between them. The inner cylinder is given rotational motion while the outer one is at rest. On the other hand, the Benard-Rayleigh experiment concerns two horizontal parallel plates with a fluid placed between them and a temperature difference placed across the plates. Eventually, both systems undergo transition to unstable states otherwise known as Taylor vortices and Benard convection cells. Both systems are described using the standard mass, momentum and energy transport equation from which it is rather difficult to show instability and chaotic behavior implicitly.
However in a Letter [Phys. Lett. A 191, 422 (1994)], it was shown that it is possible to derive the corrections to the streaming terms of the Boltzmann kinetic equation using projection and perturbation techniques. Here the second-order kinetic equation is used to obtain three corrections to the standard momentum and kinetic energy transport equations.