Mixed Convection in a One-Component Fluid

Abstract

In chapter III, we have shown that one can study the stability of isothermal forced convection, and determine the critical Reynolds number by solving the Orr-Sommerfeld equation, or the more complex nonlinear problem. Critical Reynolds numbers of the order of 10³ are obtained. However it is possible to induce hydrodynamic instabilities at much smaller Reynolds numbers by adding a destabilizing force as e.g. an adverse temperature gradient (the fluid is heated from below). In chapter VII we already studied the case of non-isothermal forced convection, but the effect of a temperature gradient on density was ignored. On the contrary, the present chapter is devoted to the coupling between the Orr-Sommerfeld problem and the Rayleigh-Bénard instability. Thus it focuses on the study of “stratified” flow, and mainly to its stability. Such a study is far from being purely academic since such flows are of considerable interest in nature as e.g. in geophysical fluid dynamics. Let us mention the effect of vertical temperature gradients on the wind in the low atmosphere, or the effect of temperature variations in oceanography.