On linear stability of Rayleigh–Bénard Poiseuille flow of viscoplastic fluids

The present paper deals with the onset of the two-dimensional Rayleigh–Bénard convection for a plane channel flow of viscoplastic fluid. The influence of the yield stress on the instability and stability conditions characterized by the Rayleigh numbers denoted, respectively, RaL and RaE is investigated in the framework of linear analysis using modal and energetic approaches. The results show that the yield stress, represented by the Bingham number B, delays the onset of convection. For low values of the Reynolds number Re, the critical conditions RaL and RaE tend to be equal and the difference RaL−RaE increases with increasing Re, highlighting the non-normality of the linear operator. For Re<1 and large B (B≥O(10)), it is shown that the critical Rayleigh number increases as B2 and the critical wave number evolves according to B1/4.