Abstract: The Rayleigh-Marangoni convective instability of a non-Newtonian fluid in a fluid-porous medium two-layer system was studied. The linear stability analysis and Chebyshev tau-QZ algorithm were used to obtain the numerical results, and the neutral curves of the critical Rayleigh number and the critical Marangoni number were obtained. The instability variation trend of the coupling modes of the critical Rayleigh number for different Marangoni numbers was analyzed, and the instability of the Rayleigh convection and Marangoni convection under the oscillatory convection was discussed. The results show that the Rayleigh convection and the Marangoni convection under the oscillatory convection become more unstable when the stress relaxation time increases, the instability of the two types of convection has the opposite effects when the strain retardation time increases. The two types of convections become more stable when the thermal conductivity of the upper surface (the corresponding Biot number) increases, and the Rayleigh convection instability mode converts with the increase of Biot number when the depth ratio changes. The coupling instability mode of the Rayleigh-Marangoni convection for a given depth ratio is obtained.

Key words: non-Newtonian fluid, hydrodynamic instability, Rayleigh-Marangoni convection, Chebyshev tau-QZ algorithm