Weakly Non-linear Stability Analysis of Triple-Diffusive Convection in a Bi-viscous Bingham Fluid Layer with Cross-Diffusion Effects

The paper investigates the impact of cross-diffusion on triple-diffusive convection in a bi-viscous Bingham fluid layer. Non-linear stability analysis is performed, and the expression of the critical-Rayleigh-number is obtained, resulting in an analytical solution of the Ginzburg-Landau model (GLM). The coefficients in the GLM involve the scaled Rayleigh-number, the solutal Rayleigh-numbers, the solutal diffusivity rates, the bi-viscous Bingham fluid parameter, and the cross-diffusion parameters. The solutal Rayleigh-numbers, the solutal diffusivity rates, and the bi-viscous Bingham fluid parameter alone determine the critical-Rayleigh-number, which provides the condition for the stationary onset. The neutral curves for the stationary mode are examined. It is found that the solutal diffusivities and bi-viscous Bingham fluid parameter advance the onset of convection, whereas the solutal Rayleigh-numbers delay it. The Nusselt number, Nu, and the Sherwood numbers, \(Sh_{1}\) $S{h}_1$ and \(Sh_{2}\) $S{h}_2$ , determine the heat- and mass-transfer rates obtained for the convection system. We see that Nu, \(Sh_{1}\) $S{h}_1$ and \(Sh_{2}\) $S{h}_2$ increase with an increase in the values of the bi-viscous Bingham fluid parameter. Also, we observe that increase in the Prandtl number effect increases them, and the same is true of the solutal Rayleigh-numbers, whereas the opposite impact on Nu, \(Sh_{1}\) $S{h}_1$ and \(Sh_{2}\) $S{h}_2$ is seen for solutal diffusivities, Soret and cross-diffusion parameters. In general, we observe that mass-transfer is more than the heat-transfer ( \(Sh_{1}> Sh_{2} > Nu\) $S{h}_1>S{h}_2>Nu$ ) depending on the value of diffusivities.