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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 257903, 11 pages
http://dx.doi.org/10.1155/2011/257903
Research Article

Analytical Method in Solving Flow of Viscoelastic Fluid in a Porous Converging Channel

1Department of Mechanical Engineering, Islamic Azad University, Nowshahr-Branch, Nowshahr, Iran
2Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran
3Computer Engineering Department, Amir-Kabir University of Technology, Tehran, Iran

Received 19 December 2010; Accepted 24 March 2011

Academic Editor: Mónica Clapp

Copyright © 2011 M. Esmaeilpour 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

An analytical method, called homotopy perturbation method (HPM), is used to compute an approximation to the solution of the nonlinear differential equation governing the problem of two-dimensional and steady flow of a second-grade fluid in a converging channel. The table and figures are presented for influencing various parameters on the velocity field. The results compare well with those obtained by the numerical method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

1. Introduction

In recent years, there has been a considerable interest in the channel flow of non-Newtonian fluid because of its various applications in different fields of engineering. The problem of laminar flow of a viscous fluid in a parallel-walled channel was first studied by Berman [1] for two-dimensional case and the case of very cross-flow Reynolds number. In 1963, Rosenhead [2] explained this phenomenon more adequately by the theory of converging and diverging channel with a permeable wall and suction or injection at the wall whose magnitude is inversely proportional to the distance along the wall from the origin of the channel. Terrill [3] presented the new form for channel flow and considered suction at one wall and injection at the other wall where the rates of this injection and suction were equal. Some other researchers continued the solution of this phenomenon for different cases [47]. In 2001, Baris restated Terrill’s problem by introducing a second-grade fluid and presented the effect of the elasticity of the fluid on the velocity distribution for different values of Reynolds number and cross-flow Reynolds number [8].

In all of these cases, these problems do not admit analytical solution, so these equations should be solved using special techniques. In recent years, much attention has been devoted to the newly developed methods to construct an analytic solution of equation; such methods include the Adomian decomposition method [912], the homotopy perturbation method (HPM) [1324], and the variational iteration method (VIM) [2531]. HPM is the most effective and convenient one for both linear and nonlinear equations. This method does not depend on a small parameter. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter 𝑝[0,1], which is considered as a “small parameter.” HPM has been shown to effectively, easily, and accurately solve a large class of linear and nonlinear problems with components converging rapidly to accurate solutions. HPM was first proposed by He [1419] and was successfully applied to various engineering problems.

The organization of this paper is as follows: in Section 2, the mathematical framework of the homotopy perturbation method is illustrated. To present a clear overview of this method, Section 3 contains the description of the laminar flow of a second-grade viscoelastic fluid in a porous converging channel of total opening of 60. In Section 4, this analytical method is utilized to solve the nonlinear equation governing the described problem. Finally, some results are provided.

2. Fundamentals of the Homotopy Perturbation Method

To illustrate the homotopy perturbation method (HPM) for solving nonlinear differential equations, He [14, 15] considered the following non-linear differential equation:𝐴(𝑢)𝑓(𝑟)=0,𝑟Ω,(2.1) with the boundary condition of𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟Γ,(2.2) where 𝐴 is a general differential operator, 𝐵 is a boundary operator, 𝑓(𝑟) is a known analytic function, Γ is the boundary of the domain Ω, and 𝜕()/𝜕𝑛 denotes differentiation along the normal vector drawn outwards from Ω. The operator 𝐴 can generally be divided into two parts 𝐿 and 𝑁, where 𝐿 is linear and 𝑁 is nonlinear. Therefore, (2.1) can be rewritten as follows:𝐿(𝑢)+𝑁(𝑢)𝑓(𝑟)=0,𝑟Ω.(2.3)

Homotopy perturbation structure is shown as follows:𝐻𝐿𝑢(𝑣,𝑝)=(1𝑝)(𝑣)𝐿0[𝐴]+𝑝(𝑣)𝑓(𝑟)=0,(2.4) where[]𝑣(𝑟,𝑝)Ω×0,1𝑅.(2.5)

Equation (2.4) is equivalent to𝐻𝑢(𝑣,𝑝)=𝐿(𝑣)𝐿0𝑢+𝑝𝐿0[𝑁]+𝑝(𝑣)𝑓(𝑟)=0.(2.6)

So, we have𝐻𝑢(𝑣,0)=𝐿(𝑣)𝐿0=0,𝐻(𝑣,1)=𝐴(𝑣)𝑓(𝑟)=0.(2.7)

The changing process of 𝑝from zero to unity is just that of 𝐻(𝑣,𝑝) from 𝐿(𝑣)𝐿(𝑢0) to 𝐴(𝑣)𝑓(𝑟). In topology, this is called deformation and 𝐿(𝑣)𝐿(𝑢0) and 𝐿(𝑣)𝐿(𝑢0) are called homotopic. According to the homotopy perturbation method, the parameter 𝑝 is used as a small parameter, and the solution of (2.4) can be expressed as a series in 𝑝 in the form of𝑣=𝑣0+𝑝𝑣1+𝑝2𝑣2+.(2.8)

When 𝑝1, (2.4) corresponds to the original one and (2.5) and (2.8) become the approximate solution of (2.3), that is,𝑢=lim𝑝1𝑣=𝑣0+𝑣1+𝑣2+.(2.9) The convergence of the series in (2.9) is discussed by He in [14, 15].

3. Problem Statement and Mathematical Formulation

In this section, we state the problem and present the governing equation. The phenomenon of two-dimensional non-Newtonian viscoelastic fluid flow in a converging channel whose wall has suction and injection is considered. The rates of these suction and injection in two walls are equal and vary in inverse proportion to the distance along the wall from the center of the channel. We assume that the wall which has suction is located in 𝜃=+𝛼 and the other wall which has injection of fluid is in 𝜃=𝛼 (Figure 1). According to Figure 1, 𝑢(𝑟,𝜃) and 𝜈(𝑟) are velocity components in the directions of 𝑟 and 𝜃, respectively.

257903.fig.001
Figure 1: Schematic diagram of the physical system.

The governing equations for this problem arecontinuity equation:𝑉=0.(3.1)Momentum equation:𝜌(𝑉𝑉)=𝑇,(3.2) where 𝑉 is the velocity, 𝜌 is the density, and 𝑇 is The Cauchy stress tensor [5, 9] and it is presented as follows:𝑇=𝑝0𝐼+𝐴1𝐴+𝛽2𝐴21,𝛽=𝛼1=𝛼2.(3.3)

Terrill and Roy and Nayak presented velocity field in the form of [3, 6]𝑈𝑢(𝑟,𝜃)=0𝑟0𝑟𝑉𝑓(𝜃),𝑣(𝑟)=0𝑟0𝑟,(3.4) where 𝑉0 is the velocities of suction and injection at 𝜃=+𝛼 and 𝜃=𝛼, respectively, at the typical length of 𝑟0. Also, 𝑈0 is the magnitude of the velocity in the direction of 𝑟 at the center line of the channel.

The boundary conditions for this problem are𝑢𝑢𝑟𝑢(𝑟,𝛼)=0,(𝑟,+𝛼)=0,0||𝑈,0=0||.(3.5) Using (3.1)–(3.5), we will have𝜕𝑃=𝜕𝜃2𝜇𝑈0𝑟0𝑟2𝑓+2𝛽𝑟20𝑈20𝑟4𝑓𝑓𝜕𝑃=𝜕𝑟𝜌𝑟20𝑟3𝑉20+𝑈20𝑓2𝑈0𝑉0𝑓+𝜈𝑈0𝑟0𝑓+𝛽𝑟20𝑟58𝑉208𝑈20𝑓2+4𝑈0𝑉0𝑓2𝑈20𝑓24𝑈20𝑓𝑓+𝑈0𝑉0𝑓,(3.6) where, in these equations, prime denotes derivation with respect to 𝜃 and 𝜈 is the kinematic viscosity. Using (3.6) and eliminating the pressure term, (3.1) and (3.2) reduce to the following ordinary differential equation or similarity equation:𝑓𝑅𝑓+2Re𝑓𝑓+4𝑓+𝑁16Re𝑓𝑓+4𝑅𝑓4Re𝑓𝑓+𝑅𝑓𝐼𝑉=0,(3.7) where 𝑅, Re, and 𝑁are the cross-flow Reynolds number, Reynolds number, and elastic number, respectively.

The boundary conditions for this equation are𝑓(𝛼)=0,𝑓(0)=1,𝑓(+𝛼)=0.(3.8)

Equation (3.7) with the boundary conditions (3.8) is solved numerically by Bariş [9] for different cases of 𝑅, Re, and 𝑁. In this paper, we reconsider the governing equation and solve it by He’s homotopy perturbation method.

4. HPM Solutions for Velocity and Temperature Fields

In this section, we will solve (3.7) using HPM. So, we construct the following homotopy for these equations:𝐻𝑓(𝑓,𝑝)=(1𝑝)𝑓0𝑓+𝑝𝑅𝑓+2𝑅𝑒𝑓𝑓+4𝑓+𝑁16𝑅𝑒𝑓𝑓+4𝑅𝑓4𝑅𝑒𝑓𝑓+𝑅𝑓𝐼𝑉=0.(4.1)

According to the definition of the HPM, we consider function 𝑓(𝜃) as follows:𝑓=𝑓0+𝑝𝑓1+𝑝2𝑓2+𝑝3𝑓3+.(4.2) Assuming 𝑓0=0 substituting 𝑓(𝜃) from (4.2) into (4.1), and using some simplification and rearranging based on powers of 𝑝-terms, we have 𝑝0𝑓0𝑓=0,(4.3)0(𝛼)=0,𝑓0(0)=1,𝑓0𝑝(+𝛼)=0,(4.4)1𝑓1𝑅𝑓0+4𝑁𝑅𝑓0+2Re𝑓0𝑓0+𝑁𝑅𝑓IV0+4𝑓04𝑁Re𝑓0𝑓016𝑁Re𝑓0𝑓0𝑓=0,(4.5)1(𝛼)=0,𝑓1(0)=0,𝑓1𝑝(+𝛼)=0,(4.6)2𝑓2+4𝑓116𝑁Re𝑓1𝑓0+2Re𝑓1𝑓016𝑁Re𝑓0𝑓1𝑅𝑓14𝑁Re𝑓0𝑓1+4𝑁𝑅𝑓14𝑁Re𝑓1𝑓0+2Re𝑓0𝑓1+𝑁𝑅𝑓IV1𝑓=0,(4.7)2(𝛼)=0,𝑓2(0)=0,𝑓2(+𝛼)=0,(4.8)

By using Maple software, we solve (4.3), (4.5), and (4.7) with considering boundary conditions (4.4), (4.6), and (4.8). We have obtained the 4-term approximations to 𝑓(𝜃), but for lack of space, only the first 3 terms produced are given below𝑓0(𝜃)=36𝜃2𝜋2𝑓1,1(𝜃)=114.048𝜋4𝜃6+3.840𝜋2𝜃4+56.400𝜋2𝜃30.01866𝜃2𝑓1.56666𝜃,2(𝜃)=240.8693𝜋6𝜃10263.9396𝜋6𝜃815.6408𝜋4𝜃8331.8253𝜋4𝜃7+0.2821𝜋2𝜃6+25.5052𝜋4𝜃6+22.1088𝜋2𝜃5+87.8688𝜋4𝜃5+65.5788𝜋2𝜃411.3040𝜋2𝜃30.001991𝜃40.363466𝜃31.835365𝜃2+0.247672𝜃,(4.9)

According to the definition of the HPM in Section 2 ((2.8) and (2.9)), we consider 4-order solution of 𝑓(𝜃) as following:𝑓(𝜃)=4𝑖=0𝑓𝑖(𝜃).(4.10)

In the case ofRe=3,𝑅=5, and 𝑁=0.015 (Table 1), functions 𝑓(𝜃) are𝑓(𝜃)=0.008022𝜃180.062658𝜃160.245778𝜃15+0.110238𝜃14+1.386855𝜃13+2.362529𝜃121.570382𝜃118.762153𝜃108.056363𝜃9+5.259996𝜃8+14.617810𝜃7+5.696890𝜃64.530889𝜃54.098633𝜃4+3.711309𝜃3+4.284551𝜃20.929478𝜃1.(4.11)

tab1
Table 1: The results of HPM and NM methods for Re=3, 𝑅=5, and 𝑁=0.015.

In the same manner, we have obtained the other cases for Re,𝑅, and 𝑁. The results are presented in Figures 2, 3, 4, and 5.

257903.fig.002
Figure 2: Comparison of HPM and the numerical method forRe=3, 𝑅=5, and 𝑁=0.015.
257903.fig.003
Figure 3: Velocity distribution for Re=0.5, 𝑅=2, and different 𝑁 using HPM.
257903.fig.004
Figure 4: Velocity distribution for Re=0.5, 𝑅=5, and different 𝑁 using HPM.
257903.fig.005
Figure 5: Velocity distribution for Re=5, 𝑅=2, and different 𝑁 using HPM.

5. Conclusion

In this paper, we have successfully developed HPM to obtain the solutions of non-Newtonian viscoelastic fluid flow in a porous channel with suction and injection in their walls. The governing equation is solved for different cases of Reynolds number,Re, the cross-flow Reynolds number, 𝑅, and elastic number, 𝑁. It is apparently seen that homotopy perturbation method is a very powerful and efficient technique for solving different kinds of problems arising in various fields of science and engineering and present a rapid convergence for the solutions.

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