Abstract
In this work we consider coupled-parallel flow through a finite channel bounded below by a porous layer that is either finite or infinite in depth. The porous layer is one in which Darcy’s equation is valid under the assumption of variable permeability. A suitable permeability stratification function is derived in this work and the resulting variable velocity profile is analyzed. It will be shown that when an infinite porous layer is implemented, Darcy’s equation must be used with a constant permeability.
Highlights
Flow through and over porous layers is encountered in natural and industrial settings
While other forms of variable permeability are possible, we will determine the form of permeability in this work and choose a function that smoothly varies over the thickness of porous layer in such a way that it falls to zero on the solid boundary and reaches a maximum value, kmax, at the interface y = 0, where kmax is to be determined
Dependence of the dimensional permeability function on the porous layer depth is illustrated in Figure 2, which demonstrates that the permeability is zero at the solid wall (y = −D) and increases until it reaches its maximum, kmax, at the porous interface, y = 0
Summary
Flow through and over porous layers is encountered in natural and industrial settings. Neale and Nader [8] discussed the significance of Brinkman’s equation and suggested the use of Brinkman’s equation in coupled parallel flow instead of Darcy’s law, due to the low differential order of Darcy’s law and its incompatibility with the Navier-Stokes differential order They [8], suggested conditions of velocity and shear stress continuity at the interface, and obtained Beavers and Joseph’s results by using α 2 = μe , where μ is the μ viscosity of the fluid and μe is the effective viscosity, that is, the viscosity of the fluid saturating the porous medium. The two cases of finite and infinite porous layers are considered to illustrate characteristics of the flow, and to emphasize the conclusion that when an infinite Darcy layer is used the permeability is essentially constant
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