Nonlinear stability of the Bingham Rayleigh–Bénard Poiseuille flow

A nonlinear stability analysis of the Rayleigh–Bénard Poiseuille flow is performed for a yield stress fluid. Because the topology of the yielded and unyielded regions in the perturbed flow is unknown, the energy method is used, combined with classical functional analytical inequalities. We determine the boundary of a region in the (Re,Ra)(Re,Ra)-plane where the perturbation energy decreases monotonically with time. For increasing values of Reynolds numbers, we show that the energy bound for RaRa varies like (1−(Re)/(ReEN))(1−(Re)/(ReEN)), where ReENReEN is the energy stability limit of isothermal Poiseuille flow. It is also shown that ReEN∼120B when B→∞B→∞.