Flow through a Variable Permeability Brinkman Porous Core

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.


Introduction
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 [1]- [4].Much of the work in this field has been devoted to the derivation of appropriate conditions at the interface between a fluid and a porous layer, or at the interface between two composite porous layers [5]- [9].Various flow models have also been tested to find the most appropriate model to use in a given flow situation, and the most appropriate model that provides compatibility with the Navier-Stokes equations [2] [10] [11].
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 [1] [10]- [13].More recently, however, there has been increasing interest in the use of Brinkman's equation with variable permeability due to the usefulness of Brinkman's equation in modelling the flow in the transition layer between a Darcy porous medium and Navier-Stokes channel.In fact, this has been extensively analyzed by Nield and Kuznetsov [14] in their introduction of the variable per-meability transition layer.Their analysis introduced the use of Brinkman's equation to model the flow in which they chose a permeability function that reduced Brinkman's equation into an Airy's differential equation.
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 [15].We provide an analytical solution to the resulting Airy's inhomogeneous equation, and we provide computations using Maple's built-in functions to evaluate Airy's functions.

Problem Formulation
Consider the flow configuration in Figure 1.The flow domain is a channel composed of three porous layers, where the flow in the middle layer is governed by Brinkman's equation with variable permeability, and in the lower and upper layers with variable permeability Darcy law.The channel is bounded by solid, impermeable walls at 0 y = and y H = .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, max1 k , at the lower interface (y = D) of the Brinkman layer.2) In the upper Darcy regiment, permeability is a decreasing function of y.It starts at its maximum, max 2 k , at y = L (the upper interface with the Brinkman layer) and drops to zero on the upper macroscopic wall (y = H).
3) max1 max 2 k k > due to the choice of decreasing permeability distribution in the Brinkman regime.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 x p .7) Solutions below will depend on max1 k and max 2 k whose values will be determined from given permeability distributions in the Darcy regiments.For 0 y D < < : ( ) For D y L < < : ( ) For L y H < < : ( ) where in 1 v is the velocity in the lower Darcy layer, 2 v is the velocity in the upper Darcy layer, u is the ve- locity in the Brinkman layer, 0 x p < is the common driving pressure gradient, µ is the fluid viscosity and eff µ is the effective viscosity of the fluid in the porous medium associated with Brinkman's flow.Boundary conditions associated with the above flow are as follows.

Conditions at the Interfaces y = D and y
.

Solution Methodology
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. ( ) ( ) , k y k y ) are prescribed.With this knowledge, we can determine the velocity distributions, ( ) ( ) , v y v y , and calculate the velocity and permeability at each interface.The distribution ( ) k y is chosen as any increasing and differentiable function on 0 y D < < , with ( ) ( ) k D k = , and the distribution ( ) k y is chosen as any decreasing and differentiable function on L y H < < , with ( ) U v L = .These are the velocities at the lower and upper interfaces, respectively, that will be used as boundary conditions in the solution of Brinkman's equation.We will choose a Brinkman permeability function and solve the Brinkman equation for the velocity, ( ) u y , subject to known interfacial velocities, ( ) ( ) , and ( ) ( ) . 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.

Solution to Brinkman's Equation in the Middle Layer
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 Equation ( 7) then becomes: Equation ( 14) is Airy's inhomogeneous equation, which admits the following general solution for u [14] [15]: 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.

Results and Analysis
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 Table 1, above, are used to generate Table 2, which lists the values of permeability, velocity and shear stress term at the lower and upper interfaces between layers in terms of the permeability at the lower interface max1 k , and that at the upper interface max 2 k for all chosen permeability distributions.Similarly, the velocity at the lower interface is 1 max1 for all chosen permeability distributions.These values are independent of the dimensionless thickness of each layer.The shear stress terms at the interfaces, on the other hand, are dependent on the dimensionless permeability values at the interfaces and the dimensionless thicknesses of the porous layers.
Dependence of permeability profiles on the thickness of the porous layers is illustrated in Table 3 by taking D = 0.1, L = 0.9 and H = 1 for a thick middle layer, and in Table 4 with D = 0.4, L = 0.6 and H = 1 for a thin middle layer.Permeability distributions in the lower and upper bounding Darcy layers are given in Table 3 and Table 4, as functions of y, and permeability distribution in the middle, Brinkman layer is calculated by the dimensionless expression of equation ( 20) and given in Table 3 and Table 4 for chosen values of max1 k and max 2 k .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.

Conclusion
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 Table 1.Dimensionless permeability, velocity, and shear stress terms for darcy layers.

Permeability Distribution
Velocity Distribution Shear Stress Term

8 )
We will choose L D ≠ so that the Brinkman permeability remains finite (for the function chosen in this work).
Airy's functions of the first and second kind, and ( ) i N Y is the Nield-Kuzn- etsov function, defined by[15]

Table 4 .
Dependence of permeability profiles on the thickness of the porous layers: thin layer.

Table 6 .Table 7 .
Values of dimensionless 1 Γ , 2 Γ for D = 1/3, L = 2/3 and different values of max1 k Values of dimensionless 1 c and 2 c for D = 1/3, L = 2/3 and different values of max1 k the interfaces for linear permeability functions, D = 1/3, L = 2 permeability functions in the Darcy layers.We illustrate the dependence of the solution of Brinkman's equation on max1 k and max 2 k by choosing linear, quadratic and exponential permeability functions in the Darcy layers, whose dimensionless forms together with the velocity distribution and shear stress terms (the first derivative of velocity functions), are summarized in Table1, below, after dropping the asterisks (*).

Table 2 .
Dimensionless permeability, velocity, and shear stress terms at the lower and upper interfaces.

Table 3 .
Dependence of permeability profiles on the thickness of the porous layers: thick layer.