Rayleigh Bénard-Poiseuille stabity problem for viscoplastic fluids
Résumé
The stability of Rayleigh-Bénard Poiseuille flow is investigated for a yield stress fluid. It is assumed that the rheological behavior is described by the Bingham model. The basic flow is characterized by a central plug zone in which the second invariant of the deviatoric stress tensor is less or equal to the Bingham number B, a dimensionless yield stress. The Bingham model assumes that inside this zone the material moves as a rigid solid and that outside this zone, it behaves as a viscous fluid. The aim of our study is to highlight the effect of the yield stress on the stability conditions. A major difficulty with this type of fluid is the possibility to have two phases: a “gel-like” behavior in which regions the stress is not determined and a fluid-like behavior where yielded. The linear stability analysis of this flow leads to propagating convective patterns, on the both sides of the plug zone, in the form of travelling waves. This analysis shows that the critical conditions, i.e., critical Rayleigh Rac and wave numbers, increase with B. Thus, increasing the Bingham number stabilizes the flow. New results concerning the evolution of the amplitude and of the plug zone are obtained with the weakly non linear analysis. This analysis allows us to take into account the non linearities due to, for example, the effective viscosity. It assumes that the perturbation remains very weak in amplitude and close to the linear conditions. It is shown that, at low values of Peclet number, the bifurcation is supercritical, similarly to the Newtonian case. At larger Peclet numbers, there is a sharp change from supercritical to subcritical bifurcation. This sharp change in behavior is a consequence of the large nonlinear viscosity. The weakly non-linear analysis only remains valid very close to Rac. In other words, the perturbed problem becomes rapidly fully non linear. Extension of the weakly non-linear method becomes impossible since the yield surface topology would change.