Simulation of the onset of convection in a porous medium layer saturated by a couple-stress nanofluid

Linear and nonlinear stability analyses for the onset of time-dependent convection in a horizontal layer of a porous medium saturated by a couple-stress non-Newtonian nanofluid, intercalated between two thermally insulated plates, are presented. Brinkman and Maxwell–Garnett formulations are adopted for nanoscale effects. A modified Darcy formulation that includes the time derivative term is used for the momentum equation. The nanofluid is assumed to be dilute and this enables the porous medium to be treated as a weakly heterogeneous medium with variation of thermal conductivity and viscosity, in the vertical direction. The general transport equations are solved with a Galerkin-type weighted residuals method. A perturbation method is deployed for the linear stability analysis and a Runge–Kutta–Gill quadrature scheme for the nonlinear analysis. The critical Rayleigh number, wave numbers for the stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory, and the nonlinear analysis is executed with minimal representation of the truncated Fourier series involving only two terms. The effect of various parameters on the stationary and oscillatory convection behavior is visualized. The effect of couple-stress parameter on the stationary and oscillatory convections is also shown graphically. It is found that the couple-stress parameter has a stabilizing effect on both the stationary and oscillatory convections. Transient Nusselt number and Sherwood number exhibit an oscillatory nature when time is small. However, at very large values of time, both Nusselt number and Sherwood number values approach their steady-state values. The study is relevant to the dynamics of biopolymers in solution in microfluidic devices and rheological nanoparticle methods in petroleum recovery.