Convection in a horizontal fluid layer under an inclined temperature gradient for Prandtl numbers Pr > 1

Abstract

In this paper calculations are done of the flow instability of a shallow horizontal fluid layer under an inclined temperature gradient for Prandtl numbers of magnitudes Pr>1. In a previous paper (Ortiz-Pérez and Dávalos-Orozco, 2011 [17]), the range of values investigated were Pr⩽1. There, the magnitudes of the horizontal Rayleigh number investigated were far larger than those used in the literature and it was found the new oblique oscillatory mode Obo. In the present paper it is found that this oblique oscillatory mode still appears for Pr=2. Further, it is also shown that some modes become irrelevant when the Prandtl number is increased, and different modes become the first to go unstable. An important result is that above a large enough Prandtl number, the curves of criticality remain almost the same. Therefore, the limit Pr→∞ is employed to calculate useful analytical expressions for the longitudinal modes of criticality which cover almost all the range of horizontal Rayleigh numbers. In contrast to the authors previous results, in this paper it is shown that for Pr>1 the transversal oscillatory mode has a range, in the horizontal Rayleigh number, as the first unstable one which first increases and then decreases with Pr, but never disappears.