Effect of a non-uniform basic temperature gradient on the onset of Bénard-Marangoni convection: Stationary and oscillatory analyses

The qualitative effect of a non-uniform basic temperature gradient on the stationary and oscillatory stability analyses of the onset of Bénard-Marangoni convective in a horizontal fluid layer is investigated numerically using the fourth order Runger-Kutta-Gillʹs method coupled with the iterative Broydenʹs method. The effects of crispation at a deformable upper free surface and the conductive effect of non-steady conditions within the fluid layer on onset are discussed. Results show that the effect of crispation (i.e., Crispation number C) is a clearly destabilizing factor, but the conductive effect (i.e., ai* of non-steady conditions within the fluid layer does play a stabilizing state. The system becomes more stable at larger values of the Prandtl number Pr, the Biot number Bi and the Bond number Bo. We obtain a set of characteristic curves in the (MC, R)-plane that satisfy the linear relation on the onset of the stationary and oscillatory Bénard-Marangoni convective instability.