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Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 349803, 10 pages
http://dx.doi.org/10.1155/2011/349803
Research Article

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

1COMSATS Institute of Information Technology, Chakshazad, Islamabad 44000, Pakistan
2Islamia College Peshawar (Chartered University), Khyber Pakhtunkhwa, Peshawar 25120, Pakistan
3Malakand University, Khyber Pukhtonnkhawa, Chakdara 1800, Pakistan

Received 5 October 2010; Revised 14 December 2010; Accepted 22 February 2011

Academic Editor: JaromirΒ Horacek

Copyright Β© 2011 S. Islam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 πœ“(π‘Ÿ,𝑧)=π‘Ÿ2𝐹(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.

1. 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 also interesting from rheological perspective [10]. The fundamental analysis of the squeezing flow between two large parallel plates was carried out by Stefan [11] in which he developed an ad hoc asymptotic solution for a thin film of Newtonian fluid. Naduvinamani et al. in [12, 13] studied static and dynamic behavior of squeeze-film lubrication of narrow porous journal bearings with coupled stress fluid and a squeeze film lubrication of a short porous journal bearing with couple stress fluids, respectively.

The determination of squeeze flow characteristics has attracted the attention of several investigators due to its importance in the practical problems of improving the performance of hydraulic machine elements, food industry, chemical engineering, polymer processing, compression, and injection molding [14–20].

Differential transform method (DTM) is one of the numerical methods for differential equations. The basic idea of differential transformation was initially introduced by Zhou [21] in 1986. Its main application therein is to solve both linear and nonlinear initial value problems in electrical circuit analysis. This method constructs the solution in the form of a polynomial. It is different from the traditional higher-order Taylor series method. The DTM is an alternative procedure for getting Taylor series solution of the differential equations. This method reduces the size of computational domain and is easily applicable to a variety of problems. The method was applied to differential algebraic equations (DAEs) of index-1 by Ayaz [22]. Liu and Song [23] analyzed higher index differential algebraic equations using this technique where they showed that the method is effective in case of index-2 DAEs but not suitable for DAEs of index-3. Application of the two-dimensional differential transform method was studied by Ayaz [24] for partial differential equations. Comparison of this method with ADM was done by Hassan [25] to solve PDEs. The same author has used this method to solve higher-order initial value problems where he studied second- and third-order initial value problems to show the efficiency of the method. Islam et al. [26] used this method for the solution of special twelfth-order boundary value problems.

The study reported in this paper considers the axisymmetric squeezing flow of a viscous fluid between two large parallel plates in a porous medium separated by a small distance 2𝐻 and the plates approaching each other with a low constant velocity 𝑉, and the flow can be assumed to be quasisteady. We use DTM to find the approximate solutions of the modeled problem.

2. 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]: βˆ‡β‹…π‘’=0,(2.1)πœŒπ‘‘π‘’π‘‘π‘‘=βˆ‡β‹…π‘‡+π‘Ÿ,(2.2) where 𝑒=[π‘’π‘Ÿ(π‘Ÿ,𝑧),0,𝑒𝑧(π‘Ÿ,𝑧)] is the velocity vector, 𝑑/𝑑𝑑 denotes the material time derivative, 𝜌 is the constant density, 𝑇 is the Cauchy stress tensor where 𝑇=βˆ’π‘πΌ+πœ‡π΄1 in which 𝐴1=βˆ‡π‘’+(βˆ‡π‘’)𝑇 is the Rivlin-Ericksen tensor, 𝜌 is the pressure, πœ‡ is the viscosity, andπ‘Ÿ is the Darcy’s resistance given by the relation,π‘Ÿ=βˆ’πœ‡π‘’π‘‘π‘˜,(2.3) here, π‘˜ is the permeability andπœ‡ is the effective viscosity of the porous medium and ξ‚πœ‡π‘‘ is the Darcian velocity which is related to the fluid velocity ̃𝑒 by 𝑒𝑑=Μƒπ‘’βˆ…, (0<βˆ…<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 πœ‡ and the fluid viscosity πœ‡ are different. However, at the macrolevel we may take them equal, though this assumption does not hold at the microlevel. Following Naduvinamani et al. [12] and Breugem [29], (2.3) can now be written asπ‘Ÿ=βˆ’πœ‡βˆ…Μƒπ‘’π‘‘π‘˜.(2.4) Continuity equation (2.1) implies the existence of a stream function πœ“(π‘Ÿ,𝑧) such that π‘’π‘Ÿ1(π‘Ÿ,𝑧)=π‘Ÿπœ•πœ“πœ•π‘Ÿ,𝑒𝑧1(π‘Ÿ,𝑧)=βˆ’π‘Ÿπœ•πœ“.πœ•π‘Ÿ(2.5) Substitution of (2.5) together with (2.4) in (2.2) and elimination of pressure from the resulting equations using πœ•2𝑝/πœ•π‘₯πœ•π‘¦=πœ•2𝑝/πœ•π‘¦πœ•π‘₯ yield ξƒ¬πœ•ξ€·βˆ’πœŒπœ“,𝐸2ξ€Έπœ“/π‘Ÿξƒ­=πœ‡πœ•(π‘Ÿ,𝑧)π‘ŸπΈ4πœ‡πœ“βˆ’π‘˜πœ‡πΈ2πœ“π‘Ÿ,(2.6) where𝐸2=πœ•2/πœ•π‘Ÿ2βˆ’(1/π‘Ÿ)(πœ•/πœ•π‘Ÿ)+(πœ•2/πœ•π‘§2).

3. Basic Idea of Differential Transform Method (DTM)

If𝐹(𝑧) is a given function, its differential transform is defined as1𝐹(π‘Ÿ)=π‘‘π‘Ÿ!π‘ŸπΉ(𝑧)π‘‘π‘§π‘Ÿ||||𝑧=0.(3.1) The inverse transform of 𝐹(π‘Ÿ) is defined by𝐹(𝑧)=βˆžξ“π‘Ÿ=0π‘§π‘ŸπΉ(π‘Ÿ).(3.2) In actual application, the function𝐹(𝑧) is expressed by a finite series𝐹(𝑧)=π‘ξ“π‘Ÿ=0π‘§π‘ŸπΉ(π‘Ÿ).(3.3) Equation (3.2) implies that 𝐹(𝑧)=βˆžξ“π‘Ÿ=𝑁+1π‘§π‘ŸπΉ(π‘Ÿ)(3.4) is negligibly small.

The fundamental operations of the DTM are given in Table 1.

tab1
Table 1

4. Analysis of the Method

Consider a fourth-order boundary value problem𝐹(4)(𝑧)=𝐺(𝑧,𝐹),0<𝑧<𝐻,(4.1) with the following boundary conditions:𝐹(0)=𝛼0,𝐹(𝐻)=𝛼1,𝐹(1)(𝐻)=𝛼2,𝐹(2)(0)=𝛼3,(4.2) where 𝛼𝑖, 𝑖=0,1,2,3 are given values.

The differential transform of (4.1) is𝐹(π‘Ÿ+4)=𝐺(π‘Ÿ)∏4𝑖=1,(π‘Ÿ+𝑖)(4.3) where 𝐺(π‘Ÿ) is the differential transform of 𝐺(𝑧,𝐹).

The transformed boundary conditions (4.2) are given by𝐹(0)=𝛼0,π‘ξ“π‘Ÿ=0π»π‘ŸπΉ(π‘Ÿ)=𝛼1,π‘ξ“π‘Ÿ=0π‘Ÿπ»π‘ŸπΉ(π‘Ÿ)=𝛼2,𝛼𝐹(2)=32.(4.4) Using (4.3) and (4.4), values of 𝐹(𝑖), 𝑖=4,5,… are obtained which give the following series solution up to 𝑂(𝑧𝑁+1),𝐹(𝑧)=π‘ξ“π‘Ÿ=0π‘§π‘Ÿξ€·π‘§πΉ(π‘Ÿ)+𝑂𝑁+1ξ€Έ.(4.5)

5. 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 𝑑, so that it can be taken as constant, the flow is steady [30, 31]:𝐹(𝑖𝑣)(𝑧)βˆ’π‘š2πΉξ…žξ…ž(𝑧)+2𝑅𝐹(𝑧)πΉξ…žξ…žξ…ž(𝑍)=0,(5.1) with the following boundary conditions: 𝐹(0)=0,𝐹(2)𝑉(0)=0,𝐹(𝐻)=βˆ’2,𝐹(1)(𝐻)=0,(5.2) where βˆšπ‘˜/πœ‡, 𝑅=𝜌/πœ‡.

The transformed boundary conditions (5.2) are 𝐹(0)=0,𝐹(1)=π‘Ž,𝐹(2)=0,𝐹(3)=𝑏,(5.3) where π‘Ž and 𝑏 will be determined later.

The differential transform of (5.1) is𝐹(π‘Ÿ+4)=π‘Ÿ!ξƒ―(π‘Ÿ+4)!(π‘Ÿ+1)(π‘Ÿ+2)𝐹(π‘Ÿ+2)βˆ’2π‘…π‘Ÿξ“π‘˜=0(π‘˜+1)(π‘˜+2)(π‘˜+3)𝐹(π‘˜+3)ξƒ°.𝐹(π‘Ÿβˆ’π‘˜)(5.4) Using (5.3) and (5.4), we obtain the following values of𝐹(𝑖), 𝑖=1,2,3,…15:𝐹(2π‘˜)=0,forπ‘˜=0(1),1𝐹(5)=ξ€·1206π‘π‘š2ξ€Έ,βˆ’12π‘Žπ‘Re1𝐹(7)=84020π‘š2𝐹(5)βˆ’2Re6𝑏2+60π‘Ž,𝐹(5)1𝐹(9)=302442π‘š2𝐹(7)βˆ’2Re66𝑏𝐹(5)+210π‘Ž,𝐹(7)1𝐹(11)=792072π‘š2𝐹(9)βˆ’2Re60𝐹2(5)+216𝑏𝐹(7)+504π‘Ž,𝐹(9)1𝐹(13)=17160110π‘š2𝐹(11)βˆ’2Re270𝐹(5)𝐹(7)+510𝑏𝐹(9)+990π‘Ž,𝐹(11)1𝐹(15)=37260156π‘š2𝐹(13)βˆ’2Re210𝐹2(7)+564𝐹(5)𝐹(9)+996𝑏𝐹(11)+171π‘Ž.𝐹(13)(5.5) We fix 𝐻=1 and 𝑉=0.25. The permeability π‘˜=1 and the fluid parameters πœ‡=1, 𝜌=1 are taken without units as they appear in the value of π‘š and 𝑅 which have been nondimensionalized. For these values, the unknowns π‘Ž and 𝑏 are determined by the following system:πœ‡15ξ“π‘Ÿ=0π‘Ÿ(𝐻)π‘Ÿβˆ’1𝐹(π‘Ÿ)=0,15ξ“π‘Ÿ=0π‘Ÿ(𝐻)π‘ŸπΉ(π‘Ÿ)=βˆ’π‘‰2.(5.6) We find that π‘Ž=βˆ’0.183491427, 𝑏=0.054618193.

Now, using the inverse differential transform, the following approximate solution of 𝑂(𝑧16) is obtained:𝐹(𝑧)=15ξ“π‘˜=0π‘§π‘˜πΉ(π‘Ÿ)=βˆ’0.183491427𝑧+0.054618193𝑧3+0.003733107𝑧5+0.000144123𝑧7βˆ’3.226Γ—10βˆ’6𝑧9βˆ’7.452Γ—10βˆ’7𝑧11βˆ’2.701Γ—10βˆ’8𝑧13+1.975Γ—10βˆ’9𝑧15.(5.7) Residual of the solution is𝑑ℏ=4𝐹(𝑧)𝑑𝑧4βˆ’π‘‘2𝐹(𝑧)𝑑𝑧2𝑑+2𝐹(𝑧)3𝐹(𝑧)𝑑𝑧3.(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.

tab2
Table 2

In Table 3, residuals of the DTM solutions have been evaluated for various values of π‘˜, πœ‡, 𝜌, keeping 𝐻=1 and 𝑉=1 fixed.

tab3
Table 3

In Table 4, we fix π‘˜=1, πœ‡=1, 𝜌=1 and vary 𝐻, 𝑉. The reliability of the solution can be seen by looking into the residual at different mesh points.

tab4
Table 4

6. 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, π‘š, 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 π‘˜β‰€1, the variation in flow pattern is negligible but for higher values of π‘˜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.

349803.fig.001
Figure 1: Velocity profiles of the squeezing flow in a porous medium when the plates velocity 𝑉 is varied.
349803.fig.002
Figure 2: Velocity profiles of fluid flow when permeability πœ…, is increased for a fixedπœ‡.

Acknowledgment

The authors would like to thank the anonymous reviewers for their helpful and encouraging comments on earlier draft of this paper.

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