Onset of Bénard–Marangoni convection in a micropolar fluid

This investigation reports on a stability analysis of the quiescent state within a horizontal layer of a micropolar fluid. The horizontal boundaries are considered rigid–rigid, rigid–free or free–free. Thermal boundary conditions of the Neumann type are applied on the boundaries of the system. The critical Rayleigh and Marangoni numbers for the onset of supercritical convection of micropolar are predicted analytically on the basis of the parallel flow approximation. The onset of motion is found to depend on the materials parameters K, B  , λλ and the micro-rotation boundary condition n. Furthermore, a linear stability analysis is conducted yielding numerically the critical Rayleigh and Marangoni numbers for the onset of motion from the rest state. A good agreement is observed between the analytical model and the numerical simulations.