Random vortex and expansion-rate model for Oberbeck-Boussinesq fluid flows

By using a formulation of a class of compressible viscous flows with a heat source via vorticity and expansion-rate, we study the Oberbeck-Boussinesq flows. To this end we establish a new integral representation for solutions of parabolic equations subject to certain boundary condition, which allows us to develop a random vortex method for certain compressible flows and to compute numerically solutions of their dynamical models. Numerical experiments are carried out, which not only capture detailed B\'enard convection but also are capable of providing additional information on the fluid density and the dynamics of expansion-rate of the flow.