Prandtl number dependence of convective fluids in tall laterally heated slots

Abstract

The influence of the Prandtl number on the stable stationary and periodic flows of the fluids contained in laterally heated slots under realistic conditions is analyzed by using continuation methods. Namely, non-slip boundaries, insulated top and bottom horizontal limits and perfectly conducting lateral sides are considered. The branches of solutions are computed by decreasing the Prandtl number, for four Rayleigh numbers. The dynamical behavior depends strongly on both parameters. For a Rayleigh number, ${\rm Ra}=10^3$ the steady flow remains stable in the wide range of Prandtl numbers computed. At ${\rm Ra}=10^4$ and ${O(10^5)}$ the first bifurcations are of Hopf type giving rise to a type of oscillations that affect the bulk of the fluid, alternating from a general circulation to multi-vortex solutions, or the boundary layer, respectively. However, it is found that, in any case, the location of the shear determines the type of the oscillations. Moreover, at ${\rm Ra}=10^4$ the critical multipliers at the secondary bifurcations on the main branch of POs are real, giving rise to different kinds of very bounded stable periodic states of different symmetries and periods. At ${\rm Ra}$ of order $10^5$ the instability of the periodic orbits gives rise directly to quasi-periodic flows.