A new hydrodynamic boundary condition simulating the effect of rough boundaries on the onset of Rayleigh-Bénard convection

This paper introduces a new hydrodynamic boundary condition which enables the simulation of the effects caused by rough boundaries. The classical Rayleigh-Bénard stability analysis is performed here to investigate the onset of thermal convection in a parallel-plate channel with rough boundaries. The hydrodynamic boundary conditions are modified, from the classical treatment, in order to consider channel boundaries characterised by non-negligible roughness. This roughness is simulated as a shallow fluid saturated porous medium and the Saffman interface condition is thus employed to model the hydrodynamic boundary conditions. The normal mode method is employed and the obtained eigenvalue problem is solved numerically. The Principle of Exchange of Stabilities is proved and the critical values of the Rayleigh number and of the wave number are obtained.