Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below

Abstract

Based on a modified Darcy–Brinkman–Oldroyd model, linear and nonlinear thermal stability analyses of a horizontal layer of an Oldroyd-B fluid in a porous medium heated from below were performed. By using the linear stability theory, the critical Rayleigh number, wave number, and frequency for stationary and oscillatory convections were determined. The effects of the viscoelastic parameters and the porous parameter on the critical Rayleigh number for oscillatory convection were analyzed. Based on the results of the linear stability analysis, a nonlinear stability analysis was also conducted. It is shown that the onset of stationary convection has the form of a supercritical and stable bifurcation independent of the viscoelastic parameters. However, the onset of oscillatory convection has the forms of supercritical or subcritical bifurcations. The nature of the oscillatory mode depends strongly on the viscoelastic parameters. The variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Although the critical Rayleigh number for stationary convection is independent of the viscoelastic parameters, the Nusselt number depends on the viscoelastic parameters of the fluids, which is different from that for the modified Darcy–Oldroyd model.