The linear and nonlinear stability of double diffusive convection with nonlinear Boussinesq approximation and vertical throughflow

Abstract

The objective of the present study is to investigate the onset of double-diffusive convection in a horizontal porous layer saturated with fluid. The governing formulation employs the nonlinear Boussinesq approximation, with flow dynamics described by Forchheimer’s extension of Darcy’s law to incorporate quadratic drag effects. Permeable boundary conditions are considered to account for realistic exchange at the interfaces. Stability analysis is carried out through two complementary approaches: linear and nonlinear stability analyses. The eigenvalue problem for linear stability is solved using the Chebyshev tau spectral method, while the nonlinear stability boundary is determined via an energy method combined with a shooting technique and fourth-order Runge–Kutta method. This dual framework allows evaluation of the critical thermal Rayleigh number in both linear ($R_L$) and nonlinear ($R_E$) theories, thereby identifying subcritical instability regimes. The results reveal several significant trends. An increase in the Forchheimer coefficient ($J$) enhances flow resistance and delays the onset of convection. Larger Péclet numbers ($Pe$) amplify advective effects and further elevate the stability threshold, underscoring their stabilizing role. Nonlinear buoyancy contributions, represented by the parameters $a_1$ and $b_1$, provide additional damping under certain parameter ranges and contribute to further stabilization. In contrast, the Lewis number ($Le$) is found to exert only a minor influence on the convective behavior of the system.