In this paper, we employ the finite element method based on non-uniform rational B-splines function approximation to solve the nonlinear Brinkman-Forcheimer-DARCY equation in a simply connected and bounded Lipschitz domain Ω. We provide both theoretical and numerical studies of the Dirichlet boundary problem. Utilizing a stream function formulation, we demonstrate the well-posedness of the weak form. Furthermore, we approximate the velocity and pressure fields by linearizing the nonlinear terms, resulting in an algebraic system. This Non-uniform rational B-splines method is more effective in terms of the exact representation of the geometry and the good approximation of the solution compared to the virtual element method. To validate the effectiveness of the non-uniform rational B-splines Finite Element Method, we conduct numerical simulations of fluid flow in porous media.

An attempt is made for the study of steady two-dimensional flow of a viscous and incompressible fluid striking at some angle of incidence on a stretching sheet. Fluid is considered in the porous media obeying DARCY law, in the presence of radiation effect. Rosseland approximation is use to model the radiative heat transfer. The stream function splits into a Hiemenz and a tangential component. Using similarity variables, the governing partial differential equations are transformed into a set of three non-dimensional ordinary differential equations. These equations are then solved numerically using fifth order Runge-Kutta Fehlberg method with shooting technique. In the present reported work the effects of striking angle, radiation parameter, porosity parameter and the Prandtl number on flow and heat transfer characteristics have been discussed. Variations of above discussed parameters with the stretching sheet parameter have been graphically presented.

The onset of DARCY– Benard penetrative convection in a liquid saturated porous layer of high permeability of practical importance is investigated by employing the Brinkman– Forchheimer– Lapwood extended DARCY flow model with fluid viscosity different from effective viscosity. The lower boundary is taken to be rigid and isothermal and the upper surface is free and subject to the general thermal condition. The critical eigen values are obtained numerically, in general, using Galerkin method. The stability of the system is found to be dependent on the dimensionless internal heat source strength Ns, permeability parameter  e and the ratio of effective viscosity to fluid viscosity  . It is observed that the increase in the value of permeability parameter is to delay while increase in the value of internal heat source strength is to hasten the onset of convection in a fluid saturated porous layer.

The linear and nonlinear stability analysis of double diffusive reaction-convection in a sparsely packed anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated analytically. The linear analysis is based on the usual normal mode method and the nonlinear theory on the truncated representation of Fourier series method. The DARCY-Brinkman model is employed for the momentum equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically.The effect of DARCY number, Damkohler number, anisotropy parameters, Lewis number, and normalized porosity on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the effect of DARCY number and mechanical anisotropy parameter have destabilizing effect, while the thermal anisotropy parameter has stabilizing effect on the stationary, oscillatory and finite amplitude convection. The Damkohler number has destabilizing effect in the case of stationary mode, with stabilizing effect in the case of oscillatory and finite amplitude modes. Further, the transient behavior of the Nusselt and Sherwood numbers are investigated by solving the nonlinear system of ordinary differential equations numerically using the Runge-Kutta method.

In the present study, the onset of DARCY-Brinkman double diffusive convection in a Maxwell fluid-saturated anisotropic porous layer is studied analytically using stability analysis. The linear stability analysis is based on normal technique. The modified DARCY-Brinkmam Maxwell model is used for the momentum equation.The Rayleigh number for stationary, oscillatory and finite amplitude convection is obtained analytically. The effect of the stress relaxation parameter, solute Rayleigh number, DARCY number, DARCY-Prandtl number, Lewis number, mechanical and thermal anisotropy parameters, and normal porosity parameter on the stationary, oscillatory and finite amplitude convection is shown graphically. The nonlinear theory is based on the truncated representation of the Fourier series method and is used to find the heat and mass transfer. The transient behavior of the Nusselt and Sherwood numbers is obtained by solving the finite amplitude equations using the Runge-Kutta method.