FDNT-code.m from Analytical and numerical steady states for thermal convection in an inclined, slender, two-dimensional enclosure

The present study considers uni- and multi-cellular natural convection of air in an inclined, slender, two-dimensional, rectangular enclosure, with emphasis on the existence of multiple steady states. The enclosure is subjected to a uniform heat flux along the long sidewalls and adiabatic conditions at the short-end walls. A mathematical model, based on the two-dimensional conservation equations under laminar flow and steady-state conditions, along with the Boussinesq approximation, is first formulated using non-dimensional stream function, vorticity and temperature, and then solved analytically and numerically, for different inclination angles, Rayleigh numbers and aspect ratios. The analytical solution, derived via the parallel-flow approximation, is used to describe the fluid flow and heat transfer in the core region, while detailed flow and temperature distributions are computed by finite difference approximations. By considering the cases of variable inclination angles for a fixed aspect ratio and different aspect ratios in a horizontal enclosure, the results reveal the emergence of multiple steady-state solutions for supercritical Rayleigh numbers around the horizontal configuration. Close agreement between numerical and analytical results confirms the robustness of the approach and provides insights into convective heat transfer mechanisms, including the influence of aspect ratio on the evolution of the steady states in horizontal enclosures.