Combined effect of helical force and rotation on the double convection of a binary viscoelastic fluid in a porous medium: Linear and weakly nonlinear analysis

Abstract

We used linear stability theory based on the normal mode decomposition technique and nonlinear stability theory based on the minimum representation of double Fourier series to study the criterion of appearance of the stationary convection and the oscillatory convection; and the rate of heat and mass transfer in a binary viscoelastic fluid mixture in a rotating porous medium under the effect of helical force. We have determined the analytical expression of the Rayleigh number of the system as a function of the dimensionless parameters. Expressions for heat and mass transfer rates are determined as a function of Nusselt and Sherwood number, respectively. The transient behaviors of the Nusselt number and the Sherwood number are studied by solving the finite amplitude equations using the Runge‐Kutta method. Then, the effect of each dimensionless parameter on the system is studied pointed out interesting results. According to the analysis of the different results obtained, it appears that in a porous medium saturated by a binary mixture of viscoelastic fluid in rotation, the Taylor‐Darcy number TaD$Ta_D$, the delay time parameter λ2 and the mass Rayleigh‐Darcy number RaCD$Ra_{CD}$ delay the onset of stationary and oscillatory convection. On the other hand, the helical force SD$S_{D}$, the relaxation time parameter λ1, the ratio of diffusivities τ and the number of Vadasz Va$Va$ accelerate the onset of stationary and oscillatory convection. In the case of the nonlinear stability study, at steady state, the helical force SD$S_{D}$ accelerates the rate of heat and mass transfer. On the other hand, the Taylor‐Darcy number TaD$Ta_D$, the diffusivity ratio τ and the mass Rayleigh‐Darcy number RaCD$Ra_{CD}$ retard the rate of heat and mass transfer. In the unsteady state, the helical force SD$S_{D}$, the relaxation time parameter λ1 and the mass Rayleigh‐Darcy number RaCD$Ra_{CD}$ accelerate the rate of heat and mass transfer. This is not the case for the Taylor‐Darcy number TaD$Ta_D$ and the delay time parameter λ2.