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

The onset of thermal convection of a Boussinesq fluid 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 around the z axis. The fluid layer is sheared by moving the lower and upper boundaries parallel to themselves with constant velocity -U and U respectively. The temperature of the lower boundary is higher than the temperature of the upper boundary. Due to the physical situation, the basic state of the system has a linear temperature profile and a two dimensional Ekman-Couette flow. The Orr-Sommerfeld type equations for the perturbations of the vertical velocity, temperature and vorticity, are formulated in terms of the six parameters (the wave number, the angle of the steady convective rolls, and the Rayleigh, Taylor, Reynolds and Prandtl numbers) that govern the system under investigation. The linear stability equations are solved by using the Tau-Chebyshev spectral numerical method, taking into account no slip boundary conditions. We present, for a fluid with Prandtl number equal to 0.7 and at Taylor numbers up to 200, the neutral stability curves for transverse, longitudinal and oblique convective rolls.