The onset of Darcy–Forchheimer convection in an inclined porous medium with the temperature dependence of viscosity and anisotropic thermal diffusivity

This study investigates how the impact of an internal heat source and anisotropic thermal diffusivity influences convective motion in a horizontally inclined porous medium saturated with fluid, assuming an adiabatic bottom boundary and an isothermal top boundary. Here, the momentum equation is formulated using the Darcy–Forchheimer model and assumes that the viscosity is a linear function of the temperature. Linear and nonlinear stability analyses have been conducted to determine the stability by calculating the critical Rayleigh number and the associated wavenumber. For nonlinear stability, conditional stability has been established for all initial data with higher-order norms, showing that the upper L squared L 2 $L^2$ theory is insufficient to achieve similar stability, where upper L squared L 2 $L^2$ represents Hilbert space over the domain omega ω $\omega$ . It has also been shown that the conditional stability bound can be attained by restricting the internal heat source parameter Q using higher-order norms. The Chebyshev tau-QZ algorithm is used to solve the generalised eigenvalue problem, which is obtained by employing the normal mode technique on the disturbance equations. The effect of various dimensionless parameters, such as anisotropic thermal diffusivity parameters left parenthesis xi right parenthesis ( ξ ) $(\xi )$ , Rayleigh number ( Ra ), angle of inclination ( phi ϕ $\phi$ ) and viscosity parameter left parenthesis upper Gamma right parenthesis ( Γ ) $(\varGamma )$ , on the stability of the medium is investigated, and streamlines are also shown by contour plots to depict the flow pattern. The results show that the system is stable when xi greater than 1 ξ > 1 $\xi \gt 1$ and unstable when xi less than 1