Mixed-convection instability in a horizontal Brinkman porous layer saturated with a viscoelastic fluid under magnetic-field and thermal-radiation effects: Application to renewable energy systems

This study examines the influence of magnetic field and thermal radiation on the onset of mixed convection instability in a horizontal porous layer saturated by a viscoelastic Kelvin-Voigt fluid with a special focus on applications to renewable energy and geothermal science. Many geothermal reservoirs underground heat-storage units, and solar assisted porous heat exchangers involve complex fluids whose rheology depart from Newtonian behavior due to presence of polymers, suspended particles or bio-geochemical interaction. Understanding the stability of thermally driven flow in such environments is crucial for optimizing heat extraction, improving long-term reservoir performance, and preventing undesired thermal stratification. A linear stability analysis is conducted to determine the critical Reynolds number as a function of Hartmann number, Darcy number, Kelvin-Voigt parameter, radiative parameter, Richardson number and Prandtl number. The results show that increasing the Darcy number enhances the effective permeability to the porous matrix, strengthens viscous dissipation, and therefore stabilizes the flow by raising the critical threshold of instability. Similarly, a stronger magnetic field, represented by the Hartmann number, generates a Lorentz damping force that suppresses transverse perturbations, acting as a major MHD stabilizing mechanism. The radiation parameter also contributes to flow stabilization by increasing effective thermal diffusion and weakening temperature gradients. In addition, the Kelvin-Voigt viscoelastic parameter introduces a memory-driven elastic resistance that absorbs perturbations and significantly delays the onset of instability. The Prandtl number plays a critical role in modulating the competition between thermal diffusion, shear and viscoelasticity, producing a Reynolds-dependent dual effect at low wavenumbers, but an exclusively stabilizing effect in high wavenumber regimes. Conversely, the Richardson number, which measures the competition between buoyancy and shear, exhibits a destabilizing effect: higher buoyancy forces intensify natural convection motions, making the system more sensitive to disturbance and facilitating the transition to instability. Quantitatively, we find that an increase in Hartmann number raises the stability threshold by 25%, whereas changes in Darcy number and viscoelastic coefficient modify the threshold by 10%. Thermal radiation increases the threshold by 12%. Increasing the Richardson number decreases the critical threshold by approximately 25%, with a typical reduction ranging between 20% and 30% depending on wavenumber domain. Theses findings provide valuable insight for the design and optimization of geothermal heat exchangers, porous thermal-energy storage units, and bio-energy and renewable energy systems, where controlling thermal and hydrodynamic stability is essential for improved efficiency, enhanced heat transfer, and safe long-term operation.