Linear instability analysis of the onset of thermal convection in an Ekman-Couette-flow

Abstract

The onset of thermal convection in a plane layer with rotation and shear is investigated. The boundaries of the layer are parallel to the x−y plane of the Cartesian coordinate system. The layer rotates at a constant angular velocity Ωz around the z axis, and is sheared by moving, along the x direction, the lower and upper boundaries parallel to themselves with constant velocity −Uw and Uw respectively. The temperature of the lower boundary is higher than the temperature of the upper boundary. The linear instability equations are solved by using the Tau-Chebyshev spectral numerical method. The critical parameters of the stationary, transverse and longitudinal convective rolls are presented. The relationship between the critical Rayleigh number and: (i) the Taylor number, (ii) the wavenumber vector, (iii) the Reynolds number, (iv) the kinetic energy norm and (v) the heat transfer at the sliding plates, is presented. We find the combination of the critical parameters that lead to the occurrence of two interesting situations, these are (a) a high value of the kinetic energy norm which promotes the formation of inertial longitudinal rolls, which are considered as hydrodynamic instabilities modified by the buoyancy force, and (b) a small value of the kinetic energy norm together with a high value of the heat transfer at the sliding plates. The latter situation, may be considered in the single-crystal growth from the melt processes.