Convection in a horizontal fluid layer under an inclined temperature gradient

Abstract

In this paper, we investigate the flow instability of a horizontal fluid layer under an inclined temperature gradient. The fluid layer is supposed to be of infinite extension, and the differentially heated lateral walls are very far away from the central region which is the subject of research. The layer is also inside two rigid, horizontal and parallel walls which are perpendicular to gravity and subjected to a vertical adverse temperature gradient. Calculations are done for Prandtl numbers Pr in the range from 0.026 to 1, which include materials from liquid metals to gases. By improving the Galerkin numerical method, new results and important extensions and corrections to the work of Nield [Int. J. Heat Fluid Flow 15, 157 (1994)] are obtained. It is found that inside the range 0.2 < Pr < 0.45, a new oblique oscillatory mode starts to appear and that it can be the first unstable one in a particular range of the horizontal Rayleigh number RH for all Prandtl numbers until Pr = 1. To the left, this mode separates from an oscillatory longitudinal even mode already found by Nield [Int. J. Heat Fluid Flow 15, 157 (1994)]. A new codimension two point is found when the Prandtl number is increased to Pr = 0.4886, where the curves of the oblique oscillatory mode and the even stationary longitudinal mode touch each other for the first time. Another new codimension two point starts to appear in the range 0.5 < Pr < 1.0 which corresponds to the crossing between the curves of the oblique oscillatory mode and the odd stationary longitudinal mode. As a consequence, another interesting result is that, to the right of this point, the odd stationary longitudinal mode is the first unstable one in a range of RH. The qualitative form of the curves of criticality changes notably between Pr = 0.026 and Pr = 1.0. Therefore, here we present a detailed description of this change by means of calculations for different Prandtl numbers inside that range.