The purpose of this study is to analyse double-diffusive convection in a fluid-saturated porous layer using an extended Darcy–Brinkman model with higher-order (bi-Laplacian) thermal and solutal diffusion and to determine linear and nonlinear stability thresholds.
The governing equations are nondimensionalised and linearised about the conduction state to obtain the perturbation equations. Linear instability analysis was performed, and a nonlinear energy stability analysis was developed to determine unconditional decay thresholds for perturbations. The instability and nonlinear thresholds are computed using two high-accuracy Chebyshev collocation methods (standard and boundary-fitted).
Brinkman viscous diffusion and higher-order thermal/solutal diffusion act predominantly as stabilising mechanisms: they increase the critical Rayleigh numbers, reshape the neutral curves and shift the stationary–oscillatory transition in parameter space. The nonlinear (energy) threshold is consistently lower than the linear threshold, identifying a conditional-stability interval RaE < Ra < RaL in which linear stability holds but unconditional nonlinear decay is not guaranteed by the present energy estimate. In the top-heavy solutal configuration, higher-order solutal diffusion provides the strongest suppression of solutal-driven fingering. Both numerical schemes exhibit spectral convergence; however, the boundary-fitted method achieves smaller residuals at the same truncation order, indicating improved accuracy and efficiency.
To the best of the authors’ knowledge, this is the first combined linear/energy stability study of thermosolutal convection in a Darcy–Brinkman porous layer with higher-order thermal and solutal diffusion, supported by spectrally accurate Chebyshev collocation schemes for the associated high-order eigenvalue problems.
