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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 938624, 13 pages
http://dx.doi.org/10.1155/2012/938624
Research Article

Two-Dimensional and Axisymmetric Unsteady Flows due to Normally Expanding or Contracting Parallel Plates

1Young Researchers Club, Islamic Azad University, Central Tehran Branch, Tehran, Iran
2Mechanical Engineering Department, Islamic Azad University, Central Tehran Branch, Tehran, Iran

Received 17 December 2011; Accepted 22 February 2012

Academic Editor: Md. Sazzad Chowdhury

Copyright © 2012 Saeed Dinarvand and Abed Moradi. 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 of a viscous incompressible fluid between two parallel plates due to the normal motion of the plates for two cases, the two-dimensional flow case and the axisymmetric flow case, is investigated. The governing nonlinear equations and their associated boundary conditions are transformed into a highly non-linear ordinary differential equation. The series solution of the problem is obtained by utilizing the homotopy perturbation method (HPM). Graphical results are presented to investigate the influence of the squeeze number on the velocity, skin friction, and pressure gradient. The validity of our solutions is verified by the numerical results obtained by shooting method, coupled with Runge-Kutta scheme.

1. Introduction

Most of the scientific problems and phenomena are modeled by nonlinear ordinary or partial differential equations. In recent years, many powerful methods have been developed to construct explicit analytical solution of nonlinear differential equations. Among them, two analytical methods have drawn special attention, namely, the homotopy perturbation method (HPM) [1, 2] and homotopy analysis method (HAM) [36]. The essential idea in these methods is to introduce a homotopy parameter, say 𝑝, which takes the value from 0 to 1. For 𝑝=0, the system of equations takes a simplified form which readily admits a particularly simple solution. When 𝑝 is gradually increased to 1, the system goes through a sequence of deformations, the solution of each of which is close to that at the previous stage of deformation. Eventually at 𝑝=1, the system takes the original forms of equation, and the final stage of deformation gives the desired solution.

We know that all perturbation methods require small parameter in nonlinear equation, and the approximate solutions of equation containing this parameter are expressed as series expansions in the small parameter. Selection of small parameter requires a special skill. A proper choice of small parameter gives acceptable results, while an improper choice may result in incorrect solutions. The homotopy perturbation method, which is a coupling of the traditional perturbation method and homotopy in topology, does not require a small parameter in equation modeling phenomena. In recent years, the HPM has been successfully employed to solve many types of linear and nonlinear problems such as the quadratic Riccati differential equation [7], the axisymmetric flow over a stretching sheet [8], the fractional Fokker-Planck equations [9], the magnetohydrodynamic flow over a nonlinear stretching sheet [10], the thin film flow of a fourth grade fluid down a vertical cylinder [11], the fractional diffusion equation with absorbent term and external force [12], Burgers equation with finite transport memory [13], the system of Fredholm integral equations [14], the generalized Burger and Burger-Fisher equations [15], the wave and nonlinear diffusion equations [16], the flow through slowly expanding or contracting porous walls [17], the torsional flow of third-grade fluid [18], Emden-Fowler equations [19], and the long porous slider [20]. All of these successful applications verified the validity, effectiveness, and flexibility of the HPM.

The problem of unsteady squeezing of a viscous incompressible fluid between two parallel plates in motion normal to their own surfaces independent of each other and arbitrary with respect to time is a fundamental type of unsteady flow which is met frequently in many hydrodynamical machines and apparatus. Some practical examples of squeezing flow include polymer processing, compression, and injection molding. In addition, the lubrication system can also be modeled by squeezing flows. Stefan [21] published a classical paper on squeezing flow by using lubrication approximation. In 1886, Reynolds [22] obtained a solution for elliptic plates, and Archibald [23] studied this problem for rectangular plates. The theoretical and experimental studies of squeezing flows have been conducted by many researchers [2435]. Earlier studies of squeezing flow are based on Reynolds equation. The inadequacy of Reynolds equation in the analysis of porous thrust bearings and squeeze films involving high velocity has been demonstrated by Jackson [34], Ishizawa [35], and others. The general study of the problem with full Navier-Stokes equations involves extensive numerical study requiring more computer time and larger memory. However, many of the important features of this problem can be grasped by prescribing the relative velocity of the plates suitably. If the relative normal velocity is proportional to (1𝛼𝑡)1/2, where 𝑡 is the time and 𝛼 a constant of dimension [𝑇1] which characterizes unsteadiness, then the unsteady Navier-Stokes equations admit similarity solution.

With the above discussion in mind, the purpose of the present paper is to examine analytically the problem of unsteady flows due to normally expanding or contracting parallel plates. The governing equations here are highly nonlinear coupled differential equations, which are solved by using the homotopy perturbation method. In this way, the paper has been organized as follows. In Section 2, the problem statement and mathematical formulation are presented. In Section 3, we extend the application of the HPM to construct the approximate solution for the governing equations. Section 4 contains the results and discussion. The conclusions are summarized in Section 5.

2. Flow Development and Mathematical Formulation

Let the position of the two plates be at 𝑧=±(1𝛼𝑡)1/2, where is the position at time 𝑡=0 as shown in Figure 1. We assume that the length 1 (in the two-dimensional case) or the diameter 𝐷 (in the axisymmetric case) are much larger than the gap width 2𝑧 at any time such that the end effects can be neglected. When 𝛼 is positive, the two plates are squeezed until they touch at 𝑡=1/𝛼. When 𝛼 is negative, the two plates are separated. Let 𝑢, 𝑣, and 𝑤 be the velocity components in the 𝑥, 𝑦, and 𝑧 directions, respectively. For two-dimensional flow, Wang introduced the following transforms [36]: 𝑢=𝛼𝑥[]2(1𝛼𝑡)𝑓(𝜂),𝑤=𝛼2(1𝛼𝑡)1/2𝑓(𝜂),(2.1) where𝑧𝜂=(1𝛼𝑡)1/2.(2.2) Substituting (2.1) into the unsteady two-dimensional Navier-Stokes equations yields nonlinear ordinary differential equation in form: 𝑓+𝑆𝜂𝑓3𝑓𝑓𝑓+𝑓𝑓=0,(2.3) where𝑆=𝛼2/2𝜈(squeeze number) is the nondimensional parameter. The flow is characterized by this parameter. The boundary conditions are such that on the plates the lateral velocities are zero and the normal velocity is equal to the velocity of the plate, that is,𝑓(0)=0,𝑓(0)=0,𝑓(1)=1,𝑓(1)=0.(2.4) Similarly, the Wang’s transforms [36] for axisymmetric flow are 𝑢=𝛼𝑥[]4(1𝛼𝑡)𝑓(𝜂),𝑣=𝛼𝑦[]𝑓4(1𝛼𝑡)(𝜂),𝑤=𝛼2(1𝛼𝑡)1/2𝑓(𝜂).(2.5) Using transforms (2.5), unsteady axisymmetric Navier-Stokes equations reduce to𝑓+𝑆𝜂𝑓3𝑓+𝑓𝑓=0,(2.6) subject to the boundary conditions (2.4).

938624.fig.001
Figure 1: Schematic diagram of the problem.

Consequently, we should solve the nonlinear ordinary differential equation𝑓+𝑆𝜂𝑓3𝑓𝛽𝑓𝑓+𝑓𝑓=0,(2.7) where𝛽=0,Axisymmetric,1,Two-dimensional,(2.8) and subject to boundary conditions (2.4).

3. Solution by Homotopy Perturbation Method

3.1. Basic Idea

Now, for convenience, consider the following general nonlinear differential equation𝐴(𝑢)𝑓(𝑟)=0,𝑟Ω,(3.1) with boundary conditions𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟Γ,(3.2) where 𝐴 is a general differential operator, 𝐵 is a boundary operator, 𝑓(𝑟) is a known analytic function, and Γ is the boundary of the domain Ω.

The operator 𝐴 can, generally speaking, be divided into two parts 𝐿 and 𝑁, where 𝐿 is linear and 𝑁 is nonlinear; therefore (3.1) can be written as𝐿(𝑢)+𝑁(𝑢)𝑓(𝑟)=0.(3.3) By using homotopy technique, one can construct a homotopy 𝑣(𝑟,𝑝)Ω×[0,1] which satisfies homotopy equation:𝐻𝐿𝑢(𝑣,𝑝)=(1𝑝)(𝑣)𝐿0[𝐴]+𝑝(𝑣)𝑓(𝑟)=0,(3.4) or𝐻𝑢(𝑣,𝑝)=𝐿(𝑣)𝐿0𝑢+𝑝𝐿0[𝑁]+𝑝(𝑣)𝑓(𝑟)=0,(3.5) where 𝑃[0,1] is an embedding parameter and 𝑢0 is the initial approximation of (3.1) which satisfies the boundary conditions. Clearly, we have𝐻𝑢(𝑣,0)=𝐿(𝑣)𝐿0=0,𝐻(𝑣,1)=𝐴(𝑣)𝑓(𝑟)=0.(3.6) The changing process of 𝑝 from zero to unity is just that of 𝑣(𝑟,𝑝) changing from 𝑢0(𝑟) to 𝑢(𝑟). This is called deformation, and, also, 𝐿(𝑣)𝐿(𝑢0) and 𝐴(𝑣)𝑓(𝑟) are called homotopic in topology. If the embedding parameter 𝑝(0𝑝1) is considered as a small parameter, applying the classical perturbation technique, we can naturally assume that the solution of (3.4) and (3.5) can be given as a power series in 𝑝, that is,𝑣=𝑣0+𝑝𝑣1+𝑝2𝑣2+,(3.7) and setting 𝑝=1 results in the approximate solution of (3.8) as𝑢=lim𝑝1𝑣=𝑣0+𝑣1+𝑣2+.(3.8)

The convergence of series (3.8) has been proved by He in his paper [37]. It is worth to note that the major advantage of He’s homotopy perturbation method is that the perturbation equation can be freely constructed in many ways (therefore is problem dependent) by homotopy in topology and the initial approximation can also be freely selected. Moreover, the construction of the homotopy for the perturb problem plays very important role for obtaining desired accuracy.

3.2. Guidelines for Choosing Homotopy Equation

In a homotopy equation, what we are mainly concerned about are the auxiliary linear operator 𝐿 and the initial approximation 𝑢0. Once one chooses these parts, the homotopy equation is completely determined, because the remaining part is actually the original equation (see (3.12)), and we have less freedom to change it. Here we discuss some general rules that should be noted in choosing 𝐿 and 𝑢0.

3.2.1. Discussion on Auxiliary Linear Operator 𝐿

According to the steps of the homotopy perturbation method, 𝐿 should be as follows.

(i) Easy to Handle
We mean that it must be chosen in such a way that one has no difficulty in subsequently solving systems of resulting equations [38]. It should be noted that this condition does not restrict 𝐿 to be linear. In scarce cases, as was done by He in [37] to solve the Lighthill equation, a nonlinear choice of 𝐿 may be more suitable. But, it is strongly recommended for beginners to take a linear operator as 𝐿.

(ii) Closely Related to the Original Equation
Strictly speaking, in constructing 𝐿, it is better to use some part of the original equation [39]. We can see the effectiveness of this view in [40] where Chowdhury and Hashim have gained very good results with technically choosing the 𝐿 part.

3.2.2. Discussion on Initial Approximation 𝑢0

There is no unique universal technique for choosing the initial approximation in iterative methods, but from previous works done on HPM [41, 42] and our own experiences, we can conclude the following facts.

(i) It Should Be Obtained from the Original Equation
For example, it can be chosen to be the solution to some part of the original equation, or it can be chosen from initial/boundary conditions.

(ii) It Should Reduce Complexity of the Resulting Equations
Although this condition only can be checked after solving some of the first few equations of the resulting system, these are the criteria that have been used by many authors when they encountered different choices as an initial approximation.

3.3. Application for Unsteady Flows due to Normally Expanding or Contracting Parallel Plates

To investigate the explicit and totally analytic solutions of present problem by using HPM, we first define homotopy 𝑣(𝜂,𝑝)Ω×[0,1] for (2.7) which satisfies𝐿𝑓(1𝑝)(𝑣)𝐿0𝑣+𝑝𝐼𝑉+𝑆𝜂𝑣3𝑣𝛽𝑣𝑣+𝑣𝑣=0,(3.9) where 𝐿 is linear operators as follows:𝑑𝐿(𝑣)=4𝑣𝑑𝜂4.(3.10) We choose𝑓01(𝜂)=23𝜂𝜂3,(3.11) as initial approximation of 𝑓(𝜂), which satisfy the boundary conditions (2.4). Assume that the solution of equation (3.9) has the form𝑣(𝜂)=𝑣0(𝜂)+𝑝𝑣1(𝜂)+𝑝2𝑣2(𝜂)+,(3.12) where 𝑣𝑖(𝜂),𝑖=1,2,3, are functions yet to be determined. Substituting (3.12) into (3.9) and equating the terms with identical powers of 𝑝, we have 𝑝0𝑣0𝑓0𝑣=0,0(0)=0,𝑣0𝑣(0)=0,0(1)=1,𝑣0𝑝(1)=0,1𝑣1+𝑓0+𝑆𝜂𝑣03𝑣0𝛽𝑣0𝑣0+𝑣0𝑣0𝑣=0,1(0)=0,𝑣1𝑣(0)=0,1(1)=0,𝑣1𝑝(1)=0,2𝑣2+𝑆𝜂𝑣13𝑣1𝑣𝛽1𝑣0+𝑣0𝑣1+𝑣1𝑣0+𝑣0𝑣1𝑣=0,2(0)=0,𝑣2𝑣(0)=0,2(1)=0,𝑣2𝑝(1)=0,3𝑣3+𝑆𝜂𝑣23𝑣2𝑣𝛽2𝑣0+𝑣1𝑣1+𝑣0𝑣2+𝑣2𝑣0+𝑣1𝑣1+𝑣0𝑣2𝑣=0,3(0)=0,𝑣3𝑣(0)=0,3(1)=0,𝑣3𝑝(1)=0,4𝑣4+𝑆𝜂𝑣33𝑣3𝑣𝛽3𝑣0+𝑣2𝑣1+𝑣1𝑣2+𝑣0𝑣3+𝑣3𝑣0+𝑣2𝑣1+𝑣1𝑣2+𝑣0𝑣3𝑣=0,4(0)=0,𝑣4𝑣(0)=0,4(1)=0,𝑣4𝑝(1)=0,5𝑣5+𝑆𝜂𝑣43𝑣4𝑣𝛽4𝑣0+𝑣3𝑣1+𝑣2𝑣2+𝑣1𝑣3+𝑣0𝑣4+𝑣4𝑣0+𝑣3𝑣1+𝑣2𝑣2+𝑣1𝑣3+𝑣0𝑣4𝑣=0,5(0)=0,𝑣5𝑣(0)=0,5(1)=0,𝑣5𝑝(1)=0,6𝑣6+𝑆𝜂𝑣53𝑣5𝑣𝛽5𝑣0+𝑣4𝑣1+𝑣3𝑣2+𝑣2𝑣3+𝑣1𝑣4+𝑣0𝑣5+𝑣5𝑣0+𝑣4𝑣1+𝑣3𝑣2+𝑣2𝑣3+𝑣1𝑣4+𝑣0𝑣5𝑣=0,6(0)=0,𝑣6𝑣(0)=0,6(1)=0,𝑣6(1)=0.(3.13) Using the Mathematica package, the solutions of system (3.13) may be written as follows:𝑣01(𝜂)=23𝜂𝜂3,𝑣1𝑆(𝜂)=560(37+15𝛽)𝜂(73+33𝛽)𝜂3+(35+21𝛽)𝜂5(1+3𝛽)𝜂7,𝑣2𝑆(𝜂)=155232001025640+415800𝛽1025640415800𝛽153060𝑆126789𝑆𝛽25875𝑆𝛽2𝜂2023560+914760𝛽2023560914760𝛽349010𝑆325392𝑆𝛽73998𝑆𝛽2𝜂3+970200+582120𝛽970200582120𝛽227304𝑆281358𝑆𝛽79002𝑆𝛽2𝜂527720+83160𝛽+2772083160𝛽20196𝑆92268𝑆𝛽40392𝑆𝛽2𝜂7+10780𝑆8085𝑆𝛽10395𝑆𝛽2𝜂9378𝑆+1428𝑆𝛽882𝑆𝛽2𝜂11,(3.14) Other terms are too long for presentation. According to the HPM, we can conclude that𝑓(𝜂)=lim𝑝1𝑣(𝜂)=𝑣0(𝜂)+𝑣1(𝜂)+𝑣2(𝜂)+𝑣3(𝜂)+𝑣4(𝜂)+𝑣5(𝜂)+𝑣6(𝜂).(3.15)

4. Results and Discussion

The fourth-order ordinary differential equation (3.2), with the boundary conditions (3.3), is solved numerically using shooting method, coupled with Runge-Kutta scheme. Our main concern is the various values of 𝑓(𝜂) and 𝑓(𝜂). These quantities describe the flow behaviour. For several values of 𝑆, the function 𝑓(𝜂) obtained by the different order of approximation for the axisymmetric and two-dimensional cases are compared with the numerical results in Tables 1 and 2, respectively. We can see a very good agreement between the purely analytic results of the HAM and numerical results. The variation of 𝑓(𝜂) with the change in the positive values of 𝑆 for the two-dimensional case is plotted in Figure 2. Figure 3 shows the influence of negative 𝑆 on 𝑓(𝜂) for the axisymmetric case. Note that for the large negative values of 𝑆, the results of similarity analysis are not reliable. 𝑓(1) gives skin friction, and 𝑓(1) represents pressure gradient. 𝑓(1) and 𝑓(1) as functions of 𝑆 are illustrated in Figures 4 and 5, respectively.

tab1
Table 1: The analytic results of 𝑓(𝜂) at different terms of approximation compared with the numerical results (RK4) for the axisymmetric case.
tab2
Table 2: The analytic results of 𝑓(𝜂) at different terms of approximation compared with the numerical results (RK4) for the two-dimensional case.
938624.fig.002
Figure 2: The influence of positive 𝑆 on 𝑓(𝜂) for the two-dimensional case.
938624.fig.003
Figure 3: The influence of negative 𝑆 on𝑓(𝜂)for the axisymmetric case.
938624.fig.004
Figure 4: The skin friction (𝑓(1)) for the axisymmetric and two-dimensional cases.
938624.fig.005
Figure 5: The pressure gradient (𝑓(1)) for the axisymmetric and two-dimensional cases.

5. Conclusions

In this paper, the unsteady axisymmetric and two-dimensional squeezing flows between two parallel plates are studied using the homotopy perturbation method (HPM). Graphical results and tables are presented to investigate the influence of the squeeze number on the velocity, skin friction, and pressure gradient. Here, the results are compared with the numerical solution obtained using shooting method, coupled with Runge-Kutta scheme. The obtained solutions, in comparison with the numerical solutions, demonstrate remarkable accuracy. This method provides an analytical approximate solution without any assumption of linearization. This character is very important for equations with strong nonlinearities which could be extremely sensitive to small changes in parameters. In this regard the homotopy perturbation method is found to be a very useful analytic technique to get highly accurate and purely analytic solution to such kind of nonlinear problems.

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