Linear Theory of Particulate Rayleigh-Bénard Stability

The stability threshold of the Rayleigh-Bénard problem is studied for a two-phase situation in which particles are introduced uniformly at the upper plate with a prescribed temperature and velocity. The particles exert a drag force on the fluid and exchange energy with it. These processes have the effect of enhancing the stability of the of the system. In other words, the critical Rayleigh number for the onset of convection increases in order that buoyancy can overcome the drag force imposed by the particles on the fluid. The critical Rayleigh number for the onset of convection is calculated numerically by solving the mass, momentum and energy equations for the fluid and the particulate phase under the point-particle approximation. The effect of the particles is explored by varying the number density, the mechanical and thermal Stokes numbers and ratio of the particle to fluid densities and heat capacities. Although the principle of exchange of stability is not applicable in this case, the numerical evidence shows that, at onset, the eigenvalue with the largest real part is purely real. This circumstance permits a simplified analytical solution based on a Fourier series expansion which is found to be close to the numerical results.