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The problem of convection in a fluid with temperature dependent viscosity and imposed shear flow, driven by pressure gradients and by a top moving wall, is studied for the case of poorly thermal conducting horizontal walls. Analytical expressions accounting for temperature dependent viscosity effects were obtained for the critical Rayleigh number and frequency of oscillation under a shallow water approximation for Poiseuille, Couette and returning primary flows. The results of this investigation contirbute and extend previous findings showing that the onset of convection can be achieved at smaller critical Rayleigh and wavenumbers. The results include approximations of weak and strong shear flows along with conditions for rigid-rigid and rigid-free boundaries. It was found that the imposed shear flow does not influence the critical wavenumber but it does increases the critical Rayleigh number. In this case convection sets in as oscillatory.

On the Linear Stability of Thermal Convection with Three Different Imposed Shear Flows