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.

Application of aggregate coarse rockfill in hydraulic structures is increasing due to their special properties and is imperative due to its importance and their specific characteristics. Accurate investigation of the flow properties through coarse materials in order to determine hydraulic parameters is one of the most important necessities considering high application of such media in civil works. On the other hands, there is less works about transitional flow through coarse materials. In order to study the flow behavior through granular materials, a packed column test was built. Some experiments have done on three samples of rounded particles with average size of 10.348, 12.127 and 17.785 mm and seepage flow and upstream and downstream heads were recorded in a certain intervals. By using some calculating and depicting some graphs, their behaviors have evaluated. Using the relationship of flow velocity-hydraulic gradient, it was fund that such relationship is non-linear and this is a strong criteria confirming non-DARCY flow. The evaluating of results of this study by statistical indexes and its comparison by Ergun (1952), Kovacs (1981) and Sidiropoulou et al (2007) relationships showed that the relation of Sidiropoulou et al (2007) is more accurate than the other equations. The result of Reynolds number-friction factor curves showed that with increasing the grain size diameter, friction factor is decreased and Reynolds number is increased

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.

Rockfill dams are usually used for affecting the flood and reducing the peak discharge. The materials used in these dams will usually make the flow turbulent, therefore DARCY's law can not be used for this kind of dams and the equations based on the non-DARCY flow must be applied. Non-DARCY flow equations are showed in two general ways including:i=aVN and i=rV+sV2 .One of the main parameters for flow simulationthrough this type of media is N, which can be specified empirically using experimental data. N is a parameter for flow turbulence and varies from 1 (laminar flow) to 2 (fully turbulent flow). One of the problems with some of non-DARCY equations (e.g. Ergun) is that they can not compute α and N directly. In this research using the Ergun equation and mathematical analysis, a few equations have been derived and presented for computing N and α and finally a new equation has been presented for the equivalent hydraulic conductivity (Ke) coefficients in porous media. A standard example was used for the verification of the resulting equations. Some values were assumed for the soil properties and the value of N, a were computed using the proposed equations in this research study. N was also specified using a non-linear regression analysis and was compared with the corresponding values obtained from the proposed equations. Comparison of the both sets of results showed that the computed N values are relatively precise especially for the boundaries (N=1 and 2).

DARCY's law is valid for laminar flow and its application is not recommended for turbulent flow. The turbulent flow equations are derived based on hydrodynamics principals with introducing turbulence effects. The purpose of this research is to derive the relationship between discharge and hydraulic potential in radial flow in unconfined aquifers under steady state. The basis of the relationship is fractional derivative approach of the velocity flow related to the hydraulic gradient. The previous study data was used for the verification of the derived equation. The profile of the water level was computed and plotted for different porous media with non-DARCY flow in unconfined aquifer for different discharges based on the extracted equation and compared with the results of the DARCY's law. The results showed with increasing the discharge values the highest and lowest increament of water level were 10 and 1. 48% related to DARCY flow respectively. These values for increasing of hydraulic conductivity were 19 and 9. 8% respectively. Overall, the difference between the DARCY and non DARCY flow in water level estimation increased with increasing discharge and hydraulic conductivity as well as the increase of both parameters, which water level was higher in DARCY flow than non DARCY flow in all water profile profiles.

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.