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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.

Flow through and over porous layers is encountered in natural and industrial settings. These include the natural flow of ground water and flow oil through the porous bedrock, the flow of nutrients into plants and blood flow in animal tissues [

The above and many other applications, together with a literature review of what has been accomplished in this field, have been discussed in greater details by Vafai and Thiagarajah [

Vafai and Thiagarajah [

Coupled parallel flow gained interest following the experiments of Beavers and Joseph [

where

Saffman [

It is worth noting that Saffman’s modification (Equation (2)) decouples the tangential velocity in the channel from both the slip velocity and Darcy’s filtration velocity, but retains its dependence on the permeability and slip parameter. Ehrhardt [

Based on their volume averaging analysis, Ochoa-Tapia and Whitaker [

where

Detailed analysis of boundary conditions at the interface and the jump stress conditions have been provided by Chandesris and Jamet [

Neale and Nader [

stress continuity at the interface, and obtained Beavers and Joseph’s results by using

viscosity of the fluid and

In addition to the work of Neale and Nader [

In the above investigations, permeability has, in general, been considered constant (although in some of the models reported in Alazmi and Vafai [

Hamdan and Kamel [

Consider the flow of a viscous fluid through a straight channel of depth D bounded by a porous layer of thickness D, as shown in

The channel is bounded above by a solid wall at y = D, and the porous layer is terminated from below by a macroscopic boundary (solid wall) at y = −D. Flow through the configuration is driven by the same constant pressure gradient. Flow in the channel is governed by the Navier-Stokes equations, which take the form for the unidirectional flow at hand:

where

where

For inhomogeneous soil layers the following form of variable permeability has been used [

where

It is thus required to solve Equations (4) and (5) subject to the following conditions.

No-slip on solid boundaries:

No penetration (zero permeability) on solid boundaries:

Maximum penetrability at the interface:

Velocity continuity at the interface:

Shear stress continuity at the interface:

Equation (4) admits the solution

where

Using (11) in (13) yields

while using (7) and (14) in (13) yields

Solution (13) thus takes the form:

An expression for the shear stress is obtained from (16) as:

At y = 0, Equation (17) reduces to:

Equation (5) gives the following expression at y = 0:

which, together with (11), gives the following expression for velocity at the interface:

From (5) we also obtain

and

Condition (12), together with (18) and (22) yield

and upon using (20) in (23), we obtain

or

Equation (25) provides an expression for the slope of the permeability function at the interface.

Velocity profile in the channel takes the following form, obtained by using (20) in (16):

Using (10) and (25) in (26) yields the following expression, equivalent to (16), for the velocity profile in the channel:

In order to determine a permeability profile, we rely on Equation (25) which

Solution to (28) is obtained via separation of variables as:

In order to find the arbitrary constant

Permeability profile (29) thus takes the form

The value of

while the velocity at the interface,

Velocity profile in the channel, Equation (26), thus takes the following form, in light of (32)

while the velocity profile in the porous layer is obtained from (5) and (31) as

Shear stress in the channel and in the porous layer are given, respectively, by

Shear stress at the interface, y = 0, is obtained from (36) or (37) as

Solution presented by Equations (31) through (38) is expressed in dimensional form in order to illustrate the effects of viscosity, pressure gradient and channel depth on the flow variables. It is also important to express the solution in dimensionless form to better understand the role of dimensionless numbers (such as Reynolds number and dimensionless pressure gradient). To this end we present two forms of dimensionless results: one with respect to channel depth and one with respect to channel depth and a characteristic velocity.

The dimensionless form of the solution with respect to channel depth is obtained by defining the following dimensionless variables, where quantities with asterisks (*) are dimensionless,

Equation (31) through (38) take the following dimensionless forms, respectively, which do not involve a Reynolds number of the flow or a dimensionless pressure gradient:

The dimensionless form of the solution with respect to channel depth and characteristic velocity is obtained by defining the following dimensionless variables with respect to channel depth, D, and characteristic velocity,

Equation (31) through (38) take the following dimensionless forms, respectively, which involve a Reynolds number of the flow and a dimensionless pressure gradient:

Both dimensionless forms are equivalent as they produce the same permeability, however, the velocities and shear stresses are scaled differently. Results based on both dimensionless forms are discussed in the Results and Discussion section, below.

If in

Now, from Equations (5) and (58), we obtain

From (29) and condition (58) we obtain

Equation (60) implies that

Observation 1:

In the study of coupled parallel flow through a channel over a porous layer of infinite depth, the flow through which is governed by Darcy’s law with an assumed exponential permeability, the only allowable permeability is essentially constant.

The Darcian volumetric flow rate through the porous layer is denoted here by

Using the velocity

and an average Darcian velocity in the porous layer of

The Navier-Stokes volumetric flow rate through the channel is denoted here by

Using the velocity

Under Poiseuille conditions, Navier-Stokes flow through a channel bounded by solid walls at y = 0 and y = D, on which no-slip conditions are imposed, has the velocity profile

and a volumetric flow rate of

The ratio of the volumetric flow rates, Equations (65) to (67), is given by

This ratio is greater than unity, thus indicating a reduction in the flow rate due to the introduction of a porous matrix in the flow domain.

The dimensional permeability function has been determined as the smoothly varying function given by Equation (31). Dependence of the dimensional permeability function on the porous layer depth is illustrated in

Equations (40) and (49) render the same dimensionless permeability function, whose graph in

The dimensional velocity profiles in the channel (Equation (34)) and in the porous layer (Equation (35)) are illus trated graphically in Figures 3-5 for different values of D and different ratios

valently higher values of the driving pressure gradient for a fixed viscosity, results in increasing the velocity in both the porous layer and the channel, as well as increasing the velocity at the interface. It is worth noting that the chosen permeability distribution results in a smoothly varying velocity profiles for all values of pressure gradients, without the occurrence of a velocity slip at the interface.

As is well-known, a Navier-Stokes Poiseuille flow in a channel in the absence of a porous layer results in a parabolic velocity profile that is symmetric about mid-channel. In other words, the maximum value of velocity in the profile occurs at mid-channel. The introduction of a porous layer with a non-zero velocity at the interface results in a loss of symmetry of the parabolic velocity profile in the channel, and the attainment of a maximum velocity in the channel at a point lower than mid-channel. In fact, the location of maximum velocity in the

channel is independent of the ratio

Each of Figures 3-5, shows the increase in the velocity at the interface,

Values of dimensional velocity,

Values of dimensionless velocity,

D = 0.5 | |||

D = 1 | |||

D = 2 |

Re = 1 | |||

Re = 2 | |||

Re = 5 | |||

Re = 10 |

In this work, we implemented a variable permeability Darcy equation in the study of flow through a channel bounded by a porous layer. The form of variable permeability was derived to produce a continuously varying velocity distribution across the channel and layer without resorting to the concept of slip that arises when the Beavers and Joseph condition is used. The main conclusion arrived at in this work is that when the porous layer is of infinite depth then the permeability must essentially be constant. This implies that the current model formulation should mainly be used with finite depth porous layers.

M. S. AbuZaytoon,T. L.Alderson,M. H.Hamdan, (2016) Flow over a Darcy Porous Layer of Variable Permeability. Journal of Applied Mathematics and Physics,04,86-99. doi: 10.4236/jamp.2016.41013