Analysis Of The Brinkman-Darcy Equation

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Consider the parallel flow of two viscous fluids in an infinite, fully saturated, uniform, homogeneous and isotropic two porous media with the Darcy's coefficients and . The statically stable situation was considered, so the upper fluid is assumed the lighter (vapor) while the lower one is assumed to be the heaver (liquid). The two fluids are incompressible and have uniform densities and viscosities for the liquid and for the vapor. The interface between the two fluids is assumed to be well defined and is initially flat to form the plane y = 0. Also, we consider that the two fluids are streaming with uniform horizontal velocities and throughout the two superposed porous media. The subscripts (1) and (2) refer to the lower and upper fluid, respectively. The acceleration due to gravity …show more content…

(4)
The balance of linear momentum for the viscous fluid through porous media according to Brinkman-Darcy equation is

, (5)
The basic assumptions that lead to the Brinkman-Darcy equation were illustrated by Rajagopal [23], and can be summarize in the following points:
1- The porous medium is a solid and thus the balance of linear momentum of the porous medium can be ignored.
2- The interactive force between the fluid and the porous medium is due to the frictional forces only and this force proportional to the flow velocity which represents by the term , where is the Darcy's coefficient, is the fluid viscosity and is the permeability of the porous medium.
3- The frictional effects due to viscosity were taken into account by the term , where is the effective viscosity of the fluid that flowing through the porous medium and is the porosity of the medium.
4- The flow is unsteady and sufficiently fast so that the inertial nonlinearities can not be ignored, thus the term needs to be retain.
According to the previous assumptions the balance of linear momentum can be written as in Eq.(5). Also, we want to confirm on the following

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