The stability of buoyancy-driven rolls aligned with a shear flow when the temperature gradient is nonlinear

Abstract

The buoyancy driven instability of rolls aligned with a shear flow of a Boussinesq fluid is considered. It is supposed that the temperature varies not only with height but also in the direction of the flow thus allowing the vertical temperature profile to be nonlinear. Conditions are derived for the neglect of the horizontal temperature gradient in the perturbation equations. These are (i) that the vertical temperature gradient varies sufficiently slowly in the direction of the flow and (ii) that Pr h/d≫1, where Pr is the Prandtl number and h and d are the length scales for the vertical variation of velocity and temperature, respectively. Under these conditions, stability is governed by the equation for Bénard convection. The principle of exchange of stabilities is proved for an arbitrary temperature gradient profile, with isothermal, free-surface boundary conditions, and the equation is solved numerically and by an asymptotic method for a model thermal boundary layer. The analysis is then applied to an experimental study of spoke patterns, observed in the growth of electronic materials. This work also indicates that a previous investigation of the stability of stratified shear flow has a restricted range of validity.