Development of a two-stage explicit numerical scheme for mixed convective flow in hybrid materials with variable properties

The accurate numerical simulation of unsteady convective transport phenomena is pivotal in understanding a wide spectrum of physical processes, including heat exchangers, chemical reactors, geothermal systems, and magnetohydrodynamic (MHD) flows. Among these, mixed convective flow arising from the combined influence of forced and free convection is particularly significant in engineering applications involving both buoyancy and externally driven flows. The complexity of such problems increases when fluid properties, such as viscosity and thermal conductivity, vary with temperature, introducing strong nonlinearity into the governing equations. This study proposes a two-stage explicit numerical scheme for solving time-dependent partial differential equations that govern such flows. The first stage modifies the exponential time integrator, while the second employs a classical Runge-Kutta formulation. A sixth-order compact scheme is used for spatial discretization, and second-order accuracy in time is achieved. The scheme is applied to simulate the mixed convective flow of a Newtonian fluid with variable properties in a porous medium under the influence of a transverse magnetic field. Simulation results reveal that increasing the Darcy number enhances the velocity profile while higher magnetic parameter values suppress it. Similarly, the Eckert number amplifies the temperature due to viscous dissipation, whereas the Prandtl number reduces thermal diffusion. For concentration profiles, both the Schmidt number and reaction rate parameter cause a significant decline, demonstrating their damping effect on solute transport. The local Sherwood number is shown to decrease with increases in both Schmidt number and reaction rate. The proposed scheme consistently yields lower $L_2$-norm errors compared to traditional Euler and second-order Runge-Kutta methods, with up to 18 % error reduction observed at smaller time step sizes. These results highlight the method's accuracy and computational efficiency, demonstrating its potential as a high-fidelity tool for simulating nonlinear convective flows with variable material properties in hybrid thermal systems.