An Axisymmetric Squeezing Fluid Flow between the Two Infinite Parallel Plates in a Porous Medium Channel

The flow between two large parallel plates approaching each other symmetrically in a porous medium is studied. The Navier-Stokes equations have been transformed into an ordinary nonlinear differential equation using a transformation ψ r, z r2F z . Solution to the problem is obtained by using differential transform method DTM by varying different Newtonian fluid parameters and permeability of the porous medium. Result for the stream function is presented. Validity of the solutions is confirmed by evaluating the residual in each case, and the proposed scheme gives excellent and reliable results. The influence of different parameters on the flow has been discussed and presented through graphs.


Introduction
The study through a porous medium is an interesting and hot issue in these days, especially, with the introduction of the modified Darcy's Law 1 , in contrast to the classical Darcy's Law 2 and the Brinkman model 3 .The flow through a porous media has wide spread applications in engineering and science, such as ground water hydrology, petroleum engineering, reservoir engineering, chemical engineering, chemical reactors to agriculture irrigation and drainage and the recovery of crude oil from the pores of the reservoir rocks 4-9 .
Squeezing flows are common in moulding, food industry, and chemical engineering, and they have, therefore, been studied for a long time as researchers and scientists have sought to optimize processing operations to produce improved components.These flows are

Basic Equations and Problem Formulation
Employing the modified Darcy's Law, the two-dimensional flow of an incompressible axisymmetric homogenous fluid in a porous medium in the absence of body force is governed by the following equations 27, 28 : where u u r r, z , 0, u z r, z is the velocity vector, d/dt denotes the material time derivative, ρ is the constant density, T is the Cauchy stress tensor where T −pI μA 1 in which A 1 ∇u ∇u T is the Rivlin-Ericksen tensor, ρ is the pressure, μ is the viscosity, and r is the Darcy's resistance given by the relation, here, k is the permeability and μ is the effective viscosity of the porous medium and μ d is the Darcian velocity which is related to the fluid velocity u by u d u∅, 0 < ∅ < 1 being the porosity of the medium.This assumes the same velocity in each pore, and the fluid in the pores is averaged over the volume.In general, the effective viscosity μ and the fluid viscosity μ are different.However, at the macrolevel we may take them equal, though this assumption does not hold at the microlevel.Following Naduvinamani et where

Basic Idea of Differential Transform Method (DTM)
If F z is a given function, its differential transform is defined as The inverse transform of F r is defined by In actual application, the function F z is expressed by a finite series is negligibly small.The fundamental operations of the DTM are given in Table 1.

Analysis of the Method
Consider a fourth-order boundary value problem with the following boundary conditions: where α i , i 0, 1, 2, 3 are given values.The differential transform of 4.1 is where G r is the differential transform of G z, F .The transformed boundary conditions 4.2 are given by

Our Problem
Here, we consider an incompressible Newtonian fluid, squeezed between two large planar, parallel plates which are separated by a small distance 2H and moving towards each other with velocity V .For small values of the velocity V , the gap distance 2H between the plates changes slowly with time t, so that it can be taken as constant, the flow is steady 30, 31 : with the following boundary conditions: where k/μ, R ρ/μ.The transformed boundary conditions 5.2 are where a and b will be determined later.The differential transform of 5.1 is

5.5
We fix H 1 and V 0.25.The permeability k 1 and the fluid parameters μ 1, ρ 1 are taken without units as they appear in the value of m and R which have been nondimensionalized.For these values, the unknowns a and b are determined by the following system: We find that a −0.183491427, b 0.054618193.

5.7
Residual of the solution is

5.8
In Table 2, numerical results of the DTM solution are compared with the results of Mathematica NDSolve solution.Residuals of both the solutions are given for comparison.List Interpolation is used for the construction of approximating polynomial of the numerical solution by NDSolve.
In Table 3, residuals of the DTM solutions have been evaluated for various values of k, μ, ρ, keeping H 1 and V 1 fixed.
In Table 4, we fix k 1, μ 1, ρ 1 and vary H, V .The reliability of the solution can be seen by looking into the residual at different mesh points.

Conclusion
In this paper, we studied squeezing flow in a porous medium between two parallel plates.Flow pattern for various parameters of Newtonian fluids are derived.When the parameters H, m, and R are fixed as 1, and velocity of the plates is varied, different velocity profiles are obtained.It is clearly visible that when the velocity of the plates is increased the fluid velocity is increased Figure 1 .For k ≤ 1, the variation in flow pattern is negligible but for higher values of k it does matter as seen in Table 2, and Figure 2. It is also noticed that at lower velocity of the plates the flow rate is also lower.The results obtained by the application of DTM are reliable with high accuracy.Applicability of the method is simple and needs no restrictions of large and small parameters.It avoids the difficulties and massive computational work encountered in other numerical techniques such linearization, discretization and perturbation.Moreover this method has superiority over the Adomian decomposition method as it does not require calculation of Adomian polynomials in case of nonlinearity.Furthermore, the solution obtained by this method converges rapidly to analytical solution in case of integrable system.

Table 2
Using 4.3 and 4.4 , values of F i , i 4, 5, . . .are obtained which give the following series solution up to O z N 1 ,