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In this work, we consider the flow through composite porous layers of variable permeability, with the middle layer representing a porous core bounded by two Darcy layers. Brinkman’s equation is valid in the middle layer and has been reduced to an Airy’s inhomogeneous differential equation. Solution is obtained in terms of Airy’s functions and the Nield-Kuznetsov function.

Fluid flow through and over porous layers has been receiving increasing interest in the porous media literature for over half a century, due to the importance of this type of flow in industrial and natural situations including lubrication problems, heating and cooling system design, groundwater flow, and the movement of oil and gas in earth layers [

Many excellent reviews are available in the literature which has been centred on the problem of flow through and over porous layers of constant permeability [

It is worth noting that there exist a large number of functions that can be used to model the variable permeability and result either in an Airy’s equation or in a different special differential equation. In the current work we will introduce a permeability function that is suitable for describing permeability variations in a Brinkman layer bounded two Darcy layers of variable permeability. This will be used in the analysis of the problem of flow through a variable permeability Brinkman porous channel bounded on either side by a variable permeability Darcy layer. The Darcy layers are terminated on their outer sides by solid, impermeable walls. This problem is representative of flow in a porous channel with a porous core that is of different porosity and permeability than its bounding porous lining.

A main objective of this undertaking is to study the effects of thin porous Darcy layers on the variable permeability flow in a Brinkman layer. In order to accomplish this work, we choose a Brinkman permeability function that reduces Brinkman’s equation to the well-known inhomogeneous Airy’s differential equation [

Consider the flow configuration in

In setting up the above flow problem, we make the following assumptions that are essential for the current work.

1) In the lower Darcy regiment, permeability is an increasing function of y. It starts at zero on the lower macroscopic wall and reaches a maximum,

2) In the upper Darcy regiment, permeability is a decreasing function of y. It starts at its maximum,

3)

4) All permeability functions are assumed continuous. At each interface, the permeability of the lower channel is equal to permeability of the upper channel. However, the rates of change of Darcy permeability are not necessarily equal at the interfaces.

5) At each interface, we assume velocity continuity and shear stress continuity.

6) Flow is driven by the same constant pressure gradient

7) Solutions below will depend on

8) We will choose

Equations governing the flow in the three regions in

For

For

For

where in

Boundary conditions associated with the above flow are as follows.

Darcy’s Equations (1) and (3) are algebraic equations from which we can determine the velocities once the pressure gradient and viscosity are given, and the permeability distributions (i.e.

Once

Solution to Brinkman’s equation is given in terms of Airy’s functions. These are computed in this work using Maple’s built-in functions.

In this work we consider the variable permeability distribution in the Brinkman layer to be given by the following expression that satisfies

Using (6) in (2) reduces Equation (2) to the form

where

Now, letting

and

then

and

Equation (7) then becomes:

Equation (14) is Airy’s inhomogeneous equation, which admits the following general solution for

where

Equation (15) takes the following form in terms of the original variable y:

and the following form in terms of the original velocity

It is convenient at this stage to introduce the following dimensionless variables with respect to a characteristic length M, in which the quantities identified by an asterisk (*) are dimensionless:

Dropping the asterisk (*), we obtain the following dimensionless equations:

Permeability distribution in Brinkman’s layer:

Velocity distribution in Brinkman’s layer:

Shear stress distribution in Brinkman’s layer:

where prime notation denotes differentiation with respect to the respective arguments.

Velocity at the interfaces between layers:

where in:

Solution to Brinkman’s equation, obtained above is predicated upon

The dimensionless forms of linear, quadratic and exponential permeability distributions and associated velocity distributions for the Darcy layers, together with the shear stress terms, summarized in

Dependence of permeability profiles on the thickness of the porous layers is illustrated in

Tables 5-10 document the values of velocities and shear stresses at the interfaces between the porous layers, and list values of parameters involved in velocity computations. It should be emphasized here that some of the computed values of velocity and shear stresses become inaccurate or extremely large for small values Da, hence not listed in this work. This may be attributed to inaccuracy in computations and approximations of Airy’s functions when Da is small (that is, when Da < 0.001).

Graphs illustrating linear, quadratic, and exponential permeability profiles are illustrated in Figures 2(a)-(c). These figures show the relatives shapes of the permeability distribution in each of the layers, and the decreasing permeability in the middle layer. How the permeability distributions affects the velocity profiles across the layers is illustrated in Figures 3(a)-(e). These figures show regions of expected increase and decrease in the velocity across the layers in a manner that is reflective of the increase and decrease in the permeability profiles.

In this work we considered flow through composite porous layers of variable permeability. The problem considered is that of a porous core the flow through is governed by Brinkman’s equation for variable permeability media, while the core is bounded by two Darcy layers of variable permeability. Various types of variable Darcy

Permeability Distribution | Velocity Distribution | Shear Stress Term |
---|---|---|

Permeability at the Interface | Velocity at the Interface | Shear Stress Term at the Interface |
---|---|---|

Lower Layer | D | y | |||
---|---|---|---|---|---|

0.1 | 1 | 1 | Linear: | ||

Middle Layer | L | y | |||

0.9 | |||||

Upper Layer | H | y | |||

1 | 0.1 | 0.1 | Linear: |

Lower Layer | D | y | |||
---|---|---|---|---|---|

0.4 | 1 | 1 | Linear: | ||

Middle Layer | L | y | |||

0.6 | |||||

Upper Layer | H | y | |||

1 | 0.1 | 0.1 | Linear: |

1 | 0.1 | 0.01 | |
---|---|---|---|

0.1 | |||

0.01 | ` | ||

0.001 |

1 | 0.1 | 0.01 | |
---|---|---|---|

0.1 | |||

0.01 | |||

0.001 |

1 | 0.1 | 0.01 | |
---|---|---|---|

0.1 | |||

0.01 | |||

0.001 |

1 | 0.1 | 0.01 | |
---|---|---|---|

0.1 | |||

0.01 | |||

0.001 |

1 | 0.1 | 0.01 | |
---|---|---|---|

0.1 | |||

0.01 | |||

0.001 |

1 | 0.1 | 0.01 | |
---|---|---|---|

0.1 | |||

0.01 | |||

0.001 |

permeability have been considered and solution to flow through the Brinkman layer is cast in terms of Airy’s and the Nield-Koznetsov functions.

M. S. Abu Zaytoon,T. L. Alderson,M. H. Hamdan, (2016) Flow through a Variable Permeability Brinkman Porous Core. Journal of Applied Mathematics and Physics,04,766-778. doi: 10.4236/jamp.2016.44087