Nonlinear stability of the Bingham Rayleigh-Benard Poiseuille flow
Résumé
A nonlinear stability analysis of the Rayleigh-Bé}nard 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 $(Re, Ra)$-plane where the perturbation energy decreases monotonically with time. For increasing values of Reynolds numbers, we show that the energy bound for Ra varies like $(1-\frac{Re}{Re_{EN}} )$, where $Re_{EN}$ is the energy stability limit of isothermal Poiseuille flow. It is also shown that $Re_{EN}\sim 120 \sqrt{B}$ when $ B \rightarrow \infty$.