Nonlinear Problems: Analytical and Computational Approach with Applications
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Hamid Khan, S. Islam, Javed Ali, Inayat Ali Shah, "Comparison of Different Analytic Solutions to Axisymmetric Squeezing Fluid Flow between Two Infinite Parallel Plates with Slip Boundary Conditions", Abstract and Applied Analysis, vol. 2012, Article ID 835268, 18 pages, 2012. https://doi.org/10.1155/2012/835268
Comparison of Different Analytic Solutions to Axisymmetric Squeezing Fluid Flow between Two Infinite Parallel Plates with Slip Boundary Conditions
Abstract
We investigate squeezing flow between two large parallel plates by transforming the basic governing equations of the first grade fluid to an ordinary nonlinear differential equation using the stream functions and and a transformation . The velocity profiles are investigated through various analytical techniques like Adomian decomposition method, new iterative method, homotopy perturbation, optimal homotopy asymptotic method, and differential transform method.
1. Introduction
The study of squeezing flows has widespread applications in chemical engineering, industrial engineering, mechanical engineering, biomechanics, and food industry. Valves and diarthrodial joints are also the examples for squeeze flows relevant in biology and bioengineering. The first application to squeeze flow problem was made by Stefan in 1874 [1]. The motion of a thin film of lubricant, squeezed flow between two stationary parallel plane surfaces were reported by Tichy and Winner [2] and Wang and Watson [3]. The theoretical and experimental studies of squeezing flows have been conducted by many researchers [4–11]. The mathematical studies of these flows are concerned primarily with the nonlinear partial differential equations which arise from the NavierStokes equations. These equations have no general solutions, and only a few exact solutions have been attained by confining some physical aspects of the original problem [12]. To solve these nonlinear differential equations, different perturbation and analytical techniques have been extensively used in fluid mechanics and engineering [13].
In the literature only a few papers deal with the comparison of different analytical methods. In this paper we study the squeezed flow between two large parallel plates with slip boundary conditions. The velocity profile is obtained using various analytical techniques like Adomian decomposition method (ADM), new iterative method (NIM), homotopy perturbation (HPM), optimal homotopy asymptotic method (OHAM), and differential transform method (DTM) [14–23]. The residual of each technique is computed and a comparison is made to assess the efficiency of the above techniques. We select DTM for analyzing the velocity profile under different flow parameters.
Squeezing flows are produced by vertical movements of boundaries or by applying external normal forces. Commonly two types of boundary conditions are employed. For a viscous fluid at a solid wall, it is generally accepted that the fluid velocity matches the velocity of the solid boundary, and it is known as no slip boundary condition. While the noslip condition is experimentally proven to be accurate for a number of macroscopic flows. Navier [24] proposed a general boundary condition that incorporates the possibility of fluid slip at a solid boundary. He assumed that the velocity at a solid surface is proportional to the shear rate at the surface, that is, where is the slip length or slip coefficient. If, the generally assumed noslip boundary condition is obtained, and if a finite constant, fluid is slip occurs at the wall. Its effect depends upon the length scale of the flow [25–28].
2. Basic Equation
We consider a steady axisymmetric flow where the velocity vector, is represented by In the absence of body forces, the NavierStokes equations are obtained for the first grade fluid by using equations of continuity and momentum, where is the constant density, is the pressure, is the viscosity, and is the vorticity vector.
Substituting (2.1) into (2.2), we get the following Continuity equation:component of N.S equation:component of N.S equation: where is the vorticity function.
Now we define a function which is known as generalized pressure, Using (2.6), (2.4) and (2.5) take the following form: We now introduce the stream functions, and obtain the following results: where , Eliminating , from (2.9), we obtain We consider viscous incompressible fluid, squeezed between two large planar and parallel plates, separated by a distance . The plates are moving towards each other with velocity . The surfaces of both plates are covered by special material with slip length (slip coefficient). For small values of the gape distance between the plates varies slowly with the time , so the flow can be taken as quasisteady [29–32]. See Figures 1, 2, 3, and 4.
The boundary conditions are as follows:
Now we use the transformation
By virtue of and (2.12), the compatibility equation (2.10) and the boundary conditions equation (2.11) become Now by introducing dimensionless parameters and dropping “*” for simplicity, the boundary value problem (2.13) become, with boundary conditions
3. Analytical Techniques
In this section, we give the basic idea of various analytical techniques and evaluate the velocity profile of our problem by considering . The residuals of all the techniques are computed and the results are displayed in Table 2.
3.1. Adomian Decomposition Method
According to [13, 14], we consider the differential equation where is the operator of the highest order derivative with respect to, , is the reminder of the linear term, and the nonlinear term is represented by. Operating on both sides of (3.1) we get the following: where, the constants can be determined by using initial or boundary conditions.
The unknown function can be expressed by an infinite series of the form where and , . The nonlinear term is decomposed by an infinite series of polynomial given by where are the socalled Adomian polynomials that can be determined by the formula
It has been observed that these polynomials can be constructed for a wide class of nonlinear functions.
The solution is approximated by the truncated series of order, that is, In our case, .
Now appling ADM on (2.15) and (2.16), for .
We obtain
Considering the Adomian 5thorder solution,
The boundary conditions at are used to get the following values of and .
By substituting these values our solution is
3.2. New Iterative Method
The basic idea of new iterative method [15, 16]. Consider the following nonlinear general differential equation: where is the operator of the highest order derivative with respect to, , is the reminder of the linear term, and the nonlinear term is represented by. Operating on both sides of (3.11) we get, where , are constants to be determined by using initial or boundary conditions.
The unknown function can be expressed by an infinite series of the form where and , The nonlinear term is decomposed by an infinite series of polynomials given by where and .
The solution, is approximated by the truncated series of order , that is,
In our case and .
Now using NIM on (2.15) and (2.16), for .
We obtain the following: Considering the NIM 5thorder solution, we have the following: Using the boundary conditions at , we get the following value of and The approximate solution is as follows:
3.3. HPM
To illustrate the basic idea of homotopy perturbation method [17–21], we consider the following nonlinear differential equation: is a general differential operator, the operator can usually be divided into two parts and , where is linear, and is nonlinear: sois a known analytic function.
With the boundary condition .
is a boundary operator, and is the boundary of the domain .
Now we construct the following homotopy: where is an embedding parameter and is the first approximation that satisfied the boundary condition. To get an approximate solution, we expand in Taylor’s series about in the following manner: Plugging (3.25) into (3.24) and then equating the coefficient of like powers of, we get the following problems which are directly integrable.
Zerothorder problem:
Firstorder problem:
Secondorder problem: We consider the following 5thorder solution,
3.4. OHAM
According to [22–24], we consider the following differential equation: with the boundary conditions where is a differential operator, is a boundary operator, and is a known function of : . The operator can be written as , where is linear and is a nonlinear operator. In OHAM we first construct a homotopy equation, where is an embedding parameter, is a nonzero auxiliary function for and The solution of traces the solution curve continuously as approaches to 1, where is the solution of the zerothorder problem that will come in the next few lines. We next choose the auxiliary function in the following form: where are the convergence controlling constants which are to be determined. To get an approximate solution, we expand in Taylor’s series about in the following manner: Now after substituting the auxiliary function and in homotopy equation we compare the coefficient of like powers of, to obtain the following linear equations.
Zerothorder problem:
Firstorder problem:
Secondorder problem: and so on.
If the series is convergent at for suitable auxiliary constants , then
The result of the mthorder approximations are given by
Residual of the solution is If, will be the exact solution, but it does not happen specially in nonlinear problems. To find the optimal values of, many methods can be applied. We follow the method of least squares. According to the method of least squares, we first construct the functional and then minimizing it, we have where and are in the domain of the problem. With these constants known, the approximate solution (of order) is well determined.
Now applying OHAM to (2.15) and (2.16), we obtain the following problems which are directly integrable.
Zerothorder problem:
Firstorder problem:
Secondorder problem: Considering the OHAM 5thorder solution and using the method of least squares, we obtain . Hence the solution is
3.5. NDSolve
NDSolve is a mathematica code, utilized for solution of ordinary and partial differential equations. This code is also used for differentialalgebraic equations and system of ordinary differential equation. NDSolve gives solution on discrete points rather than for the function itself. List interpolation is used for the construction of approximating polynomial.
Apply NDSolve to (2.15) and (2.16), the following approximate solution is obtained:
3.6. DTM
According to [33, 34], the basic idea of differential transforms method (DTM) starts from the following definition.
If is a given function, its differential transform is defined as follows: The inverse transform of is defined by In actual application, the function is expressed by a finite series Equation (3.51) implies that . is negligibly small.
The fundamental operations of the DTM are given in Table 1.


3.6.1. Analysis of the Method
Consider a fourthorder boundary value problem with the boundary conditions: where are given values.
The differential transform of (3.52) is as follows: where is the differential transform of.
The transformed boundary conditions (3.53) are given by Using (3.54) and (3.55) values of are obtained which give the following series solution up to , applying DTM on (2.15) and (2.16), the transformed boundary conditions and differential are as follows: Using (3.57), we obtain the following values of .
For these values, the unknowns and are determined by the following system: We find, .
Now using the inverse differential transform, the following approximate solution of is obtained:
4. Conclusion
In this paper we have used Adomian’s decomposition method, new iterative method, homotopy perturbation method, optimal homotopy asymptotic method, and differential transform method to an axisymmetric squeezing flow problem. Though all the methods are based on Taylor’s series expansion, they produce different results because each one has its own environment. Homotopy methods combine homotopy from topology and perturbation method. It has been observed that homotopy perturbation method may suffer convergence in many problems while OHAM controls the convergence region by employing the auxiliary function. Differential transform method, Adomian decomposition method, and new iterative method are the straightforward application of Taylor’s series, and they suffer from divergence in general and particularly in initial value problems. Besides all these facts, ADM, HPM, and DTM can lead easily to closedform solutions of many problems.
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Copyright © 2012 Hamid Khan 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.