The effects of non-uniform internal heating on the convective instability of power-law fluid-saturating-porous layer

Abstract

The onset of convection in a porous plate with non-Newtonian saturating fluid is studied using linear stability theory. The horizontal plate is heated internally by a heat source with varying strength along its vertical axis, a situation relevant to certain geophysical applications. Two distinct heating profiles are considered: a linearly antisymmetric distribution in Case $A$ and a quadratically symmetric distribution in Case $B$. The non-Newtonian behavior of the saturating fluid is modeled using the power-law rheology in the Darcy–Forchheimer equation. An inclined throughflow is introduced at an angle $\xi$ to the horizontal direction. The eigenvalue problem is solved analytically using the Galerkin method of weighted residuals, and the results are verified numerically, via the 4th-order Runge–Kutta technique. The critical values of the internal Rayleigh number $R_{ic}$, wave number $k_c$, and angular frequency ${\omega }_c$ for transverse and longitudinal rolls are affected by Peclet number $Pe$, form drag coefficient $G$, power law index $n$, and non-uniformity coefficients $q_1$ and $q_2$. The results relative to $q_{1,2}\to 0$ converge well to those relevant to uniform internal heating presented in the literature, serving as a benchmark for the two methods. According to the findings, the effect of the quadratic form drag number $G$ causes the critical values of shear-thinning fluids to converge to those of Newtonian fluids for both scenarios of stationary and oscillatory convection. This phenomenon is not observed for shear-thickening behavior. In addition, the change in the non-uniformity coefficients $q_{1,2}$ had a significantly larger impact on the critical values for a shear-thickening fluid than for a shear-thinning one, particularly in Case $A$.