Mathematics and Computing for Scientific Innovation Vol. 1

This study investigates the onset of continuous double-diffusive convection in a binary liquid contained in a two-dimensional rectangular channel filled with an anisotropic porous material. The channel is heated and salted from below, and the coupled effects of thermal and solutal transport are represented through the Soret and Dufour cross-diffusion terms. The momentum equation is formulated using the Lapwood-Brinkman approach with the Darcy model and without a time-derivative term, while the Boussinesq approximation is applied to describe density variations. Linear stability analysis is used to determine the critical conditions for the onset of convection. The governing equations are non-dimensionalised, expanded using Fourier series, and reduced to a sixth-order differential equation. The resulting characteristic relation is solved numerically using the Newton-Raphson method to obtain the critical Rayleigh number for selected values of the solute Rayleigh number, Darcy number, mechanical anisotropy parameter, diffusivity ratio, Soret parameter, and Dufour parameter. The graphical results indicate that increasing the Dufour parameter raises the critical Rayleigh number and therefore stabilises the system. The Soret parameter generally lowers the critical Rayleigh number, although small values may produce a stabilising influence. The diffusivity ratio stabilises the system at small solute Rayleigh numbers but becomes destabilising at larger values. These findings clarify how anisotropy and cross-diffusion jointly influence the linear stability threshold in the considered porous channel.