Impact of temperature-dependent viscosity and permeable boundaries on thermal instability in a Darcy–Brinkman ferrofluid layer

This study investigates the onset of thermal instability in a horizontal Darcy–Brinkman ferrofluid layer with temperature-dependent viscosity, bounded by permeable horizontal surfaces. The variation of viscosity with temperature is described using linear, exponential, and inverse-linear relations. The governing equations are nondimensionalized and reduced to an eigenvalue problem, which is solved using the single-term Galerkin method to determine the critical Rayleigh number for the onset of stationary convection. The incorporation of permeable boundaries, together with different viscosity relations in a ferrofluid-saturated Brinkman porous medium, provides a novel extension of the classical framework and broadens its applicability in various scientific and engineering contexts. The results reveal that the viscosity variation parameter stabilizes the system under linear and exponential laws, but has a destabilizing effect in the inverse-linear case across all boundary conditions. A transition in permeable boundaries from free to rigid enhances the stability of the system by delaying the onset of convection. Additionally, the magnetization parameters M1 and M3 both promote instability; however, their effects on convection cell size differ: M1 favors smaller convection cells, while M3 promotes larger ones. The permeability of the porous medium also supports the development of larger cells and reduces the critical Rayleigh number. Previous results are consistently recovered as limiting cases, confirming the validity of the present analysis. The study provides deeper insight into ferrofluid convection with relevance to aerospace, porous thermal management, and magnetically driven heat transfer applications.