The Effect Of Concentration Diffusion On Thermal Convection In A Flow Of The Rivlin-Ericksen Fluid In Porous Medium

We consider an infinite horizontal incompressible Rivlin-Ericksen layer in porous medium. The layer is heated and soluted from below. When its boundaries are stress-free, Sharma and Pal have studied the stability of motionless state of this system by the method of linear stability. The linearly stabilizing effect of solute concentration is proved there. However, as we know, the stability condition obtained by method of linear stability is only necessary condition for stability, while subcritical instability cannot be excluded. Xu applied nonlinear stability theory to bring out the stabilizing effect of solute concentration. Through defining a generalized energy functional Xu provided a sufficient condition for nonlinear stability for a parameterτ. In particular,forτ= (E'Sc)/(EPr)≤1 Xu's sufficient condition for nonlinear stability and thenecessary condition for stability obtained by linear theory coincide. Most researchers have studied the stability of the Rivlin-Ericksen layer for the case when its boundaries are stress-free, but few people studied the cases when its boundaries are rigid or mixed (one boundary is free and the other is rigid). Because it is impossible to obtain the analytic solution in the last two cases. But the critical number and the corresponding instability-curve (above the curve the basic flow is linearly instabie) are what we want to obtain by the method of linear stability. In order to compare the three cases, we will calculate them together and obtain the relation of critical number R_c of the thermal Rayleigh number and the solute Rayleighnumbers. The calculation results show that the concentration diffusion has inhibiting effect on the convection.