Non‐linear stability in the Bénard problem for a double‐diffusive mixture in a porous medium

Abstract

SUMMARYThe linear and non-linear stability of a horizontal layer of a binary uid mixture in a porous medium heated and salted from below is studied, in the Oberbeck-Boussinesq-Darcy scheme, through the Lyapunov direct method. This is an interesting geophysical case because the salt gradient is stabilizing while heating from below provides a destabilizing e ect. The competing e ects make an instability analysis di cult. Unconditional non-linear exponential stability is found in the case where the normalized porosity is equal to one. For other values of a conditional stability theorem is proved. In both cases we demonstrate the optimum result that the linear and non-linear critical stability parameters are the same whenever the Principle of Exchange of Stabilities holds.

Effect of Vertical Magnetic Field on the Onset of Double Diffusive Convection in a Horizontal Porous Layer with Concentration Based Internal Heat Source

“…11, shows that for fixed values of , , and , ( for oscillatory convection), the solutal Rayleigh number exerts stability on the system. In the absence of and , this finding is in agreement with [25]. …”

“…Equation (43) corresponds to the result of [25] and [28]. Equation (42) gives the oscillatory neutral Rayleigh number; ( ) with critical oscillatory Rayleigh number, ( ) as…”

An operative method to obtain sharp nonlinear stability for systems with spatially dependent coefficients

“…To obtain sharp nonlinear stability bounds in the stability measure L 2 (U), (following an analogous argument to Lombardo et al (2001)), we multiply equation (4.5) by g and Dg, respectively, and equation (4.6) by j and Dj, respectively, and integrate over U to obtain 1 2…”

Energy Stability of Benard-Darcy Two-Component Convection of Maxwell Fluid

“…The quantities μ and K denote the viscosity of the fluid and permeability of the porous medium, g is the gravitation acceleration, k and k are the effective thermal diffusivity and solutal diffusivity of a Maxwell fluid saturated porous medium as defined in Lombardo et al (2001),  and   are the coefficients for thermal and solutal expansion, and 0 T and 0 C are the reference temperature and concentration, respectively.…”

“…is the velocity, P is the pressure, T is the temperature and C is the concentration, M is the ratio of heat capacities as defined by Lombardo et al (2001), and ε is the normalized porosity defined by M     where   is the porosity as in Lombardo et al (2001). The quantities μ and K denote the viscosity of the fluid and permeability of the porous medium, g is the gravitation acceleration, k and k are the effective thermal diffusivity and solutal diffusivity of a Maxwell fluid saturated porous medium as defined in Lombardo et al (2001),  and   are the coefficients for thermal and solutal expansion, and 0 T and 0 C are the reference temperature and concentration, respectively.…”