Nonlinear stability of the motionless state of thermosolutal Rivlin–Ericksen fluid in porous medium

Energy method is used to study the nonlinear stability of the motionless state of thermosolutal Rivlin–Ericksen fluid in porous medium for stress-free boundaries. By defining energy functionals we will show that for τ = ( E ′ P C ) / ( E P T ) ≤ 1 , α ˆ = ( C / R ) ≥ 1 the motionless state is always stable and for τ ≤ 1 , α ˆ < 1 the sufficient and necessary conditions for stability coincide, where P C , P T , C and R are the Schmidt number, Prandtl number, Rayleigh number for solute and heat, respectively, E ′ and E are two constants related to porosity of porous medium. Unlike the energy-decay rate in previous works concerning the nonlinear stability of Bénard problem for the same boundaries, this quantity in present work is completely independent of mode numbers.