Rayleigh-Marangoni-Bénard instability in an Oldroyd-B fluid layer overlying a highly porous layer with a deformable surface

Abstract

This paper describes the linear stability analysis of Rayleigh–Marangoni–Bénard convection with a deformable surface in a fluid overlying a highly porous layer. Using the Chebyshev tau-QZ method, we investigate the oscillatory mode of both the Rayleigh and Marangoni instabilities for non-Newtonian fluids. The numerical results indicate that a deformable upper surface destabilizes the system. Moreover, the Marangoni instabilities in long-wave branches first decrease and then increase with increasing depth ratio, while the opposite effects are observed in short-wave branches. The Rayleigh instabilities decrease monotonically as the depth ratio increases. For certain reference parameters, the system becomes more stable as the Biot number and Galileo number increase. A greater strain retardation time and a smaller stress relaxation time lead to more stable Marangoni convection in long-wave branches, whereas these viscoelastic times have the opposite effects in short-wave branches. The influence of viscoelasticity on the Rayleigh instability is also investigated. Finally, the coupling mode of the two instabilities is studied in detail. Variations in the Marangoni number influence long- and short-wave branches differently. Interestingly, there is an interval of the Marangoni number in short-wave branches for which no critical Rayleigh number exists.