This paper investigates the cross-diffusion effects subject to exponential variable boundary conditions on transient double-diffusive natural convection flow in an enclosure. The flow domain is a two-dimensional inclined trapezoidal cavity filled with a porous medium. The top wall is assumed to be insulated and permeable, while the enclosure's bottom wall is subject to exponential varying temperature and concentration. The prescribed temperature and concentration are different at the vertical walls. Conservation equations are used as the governing equations. The finite element Galerkin weighted residual method, in association with the Newton-Raphson scheme is employed to solve the system of coupled nondimensional equations. The numerical tests are confirmed with existing literature and are found to be in excellent agreement. The simulations results for stream functions, isotherms, and isoconcentrations are discussed for the various flow parameters. A sensitivity analysis using the response surface method suggests that the average Nusselt and Sherwood numbers are more sensitive to the cross-diffusion effects. It is further observed that the cross-diffusion terms stabilize the sensitivity to the angle of inclination.

Flow analysis of the Al 2 O 3-water nanofluid in a porous trapezoidal enclosure is deliberated in this research work. The enclosure is bounded by four differentially heated walls with the bottom wall considered as wavy. The viscous and Joule dissipation effects are taken into account. The governing equations of the flow are non-dimensionalised, and the Galerkin finite element method is employed to solve the resulting unsteady initial and boundary value problem using Taylor-Hood elements to avoid instabilities. The numerical scheme is validated with available literature for a simplified case of a problem, and the results are found to be in good agreement. The influence of Darcy number and nanoparticle volume fraction on the dimensionless streamlines and isotherms are plotted and illustrated in detail. The rate of heat transfers near the bottom wall in a porous medium is analysed through simulated data. It is found that the heat transfer rate increases in the presence of a porous domain. Further, a sensitivity analysis has been performed using Response Surface Methodology (RSM) to identify the pertinent parameters of the problem such as Hartmann number, Darcy number, and nanoparticles whose variation have an important influence on the average Nusselt number. The analysis reveals that the Darcy number, Hartmann number and the nanoparticle volume fraction are related to the average Nusselt number. Thus, the finding suggests that the porous medium has a significant role in enhancing the heat transfer in a flow of MHD nanofluid within a wavy trapezoidal enclosure.

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