Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 169056, 16 pagesdoi:10.1155/2011/169056

Research Article

An Optimal Homotopy Asymptotic Approach Applied to Nonlinear MHD Jeffery-Hamel Flow

Vasile Marinca1,2 and Nicolae Herişanu1,2

¹Department of Mechanics and Vibration, Politehnica University of Timişoara, Bulevardul Mihai Viteazul, No. 1, 300222 Timişoara, Romania²Department of Electromechanics and Vibration, Center for Advanced and Fundamental Technical Research, Romanian Academy, Timisoara Branch, Bulevardul Mihai Viteazul, No. 24, 300223 Timişoara, Romania

Received 3 August 2011; Accepted 22 October 2011

Academic Editor: Paulo Batista Gonçalves

Copyright © 2011 Vasile Marinca and Nicolae Herişanu. 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.

A simple and effective procedure is employed to propose a new analytic approximate solution for nonlinear MHD Jeffery-Hamel flow. This technique called the Optimal Homotopy Asymptotic Method (OHAM) does not depend upon any small/large parameters and provides us with a convenient way to control the convergence of the solution. The examples given in this paper lead to the conclusion that the accuracy of the obtained results is growing along with increasing the number of constants in the auxiliary function, which are determined using a computer technique. The results obtained through the proposed method are in very good agreement with the numerical results.

1. Introduction

In various fields of science and engineering, nonlinear evolution equations, as well as their analytic and numerical solutions, are fundamentally important. The problem of an incompressible, viscous fluid between nonparallel walls with a sink or source at the vertex was pioneered by Jeffery [1] and Hamel [2]. Hamel mentioned an example of an exact nonsteady solution of the Navier-Stokes equations which describes the process of decay of a vortex through the action of the viscosity and considered the distribution of the tangential velocity component with respect to the radial distance and time and a particular case of the flow through a divergent channel was discussed and exactly solved. Jeffery-Hamel flows are exact similarity solution as the Navier-Stokes equations in the special case of two-dimensional flow through a channel with inclined plane walls meeting at a vertex with a source or sink at the vertex and have been studied by several authors and discussed in many text books and articles [3–5]. Sadri [6] denoted that Jeffery-Hamel flow used an asymptotic boundary condition to examine steady two-dimensional flow of a viscous fluid in a channel by means of certain symmetric solution of the flow although asymmetric solution are both possible and of physical interest [7].

The classical Jeffery-Hamel problem was extended in [8] to include the effects of external magnetic field in conducted fluid. The magnetic field acts as a control parameter, along with the flow, Reynolds number, and the angle of the walls.

Most scientific problems such as Jeffery-Hamel flows and other fluid mechanics problems are inherently nonlinear. Excepting a limited number of these problems, most do not have analytical solutions. Therefore, these nonlinear equations should be solved using other methods [9].

The aim of the present work is to propose an accurate approach to the Jeffery-Hamel flow problem using an analytical technique, namely, OHAM [10, 11].

The efficiency of our procedure, which does not require a small parameter in the equation, is based on the construction and determination of the auxiliary functions combined with a convenient way to optimally control the convergence of the solution.

2. Problem Statement and Governing Equation

We consider a system of cylindrical polar coordinates with a steady two-dimensional flow of an incompressible conducting viscous fluid from a source or sink at channel walls lying in planes, with angle , as shown in Figure 1.

Figure 1: Geometry of the Jeffery-Hamel flow problem.

Assuming that the velocity is only along the radial direction and depends on and , [3–5], using the continuity Navier-Stokes equations in polar coordinates, the governing equations are where is the fluid density, is the pressure, and is the kinematic viscosity. From (2.1) and using dimensionless parameters we get

Substituting (2.5) into (2.2) and (2.3) and eliminating the pressure, we obtain an ordinary differential equation for the normalized function profile : where prime denotes derivative with respect to and the Reynolds number is and is the maximum velocity at the centre of the channel.

The boundary conditions for (2.6) are

3. Fundamentals of the OHAM

We consider the following nonlinear differential equation [10, 11]: subject to a boundary condition where is a linear operator, is a known analytical function, is a nonlinear operator, and is a boundary operator. By means of the OHAM one constructs a family of equations: and the boundary condition is

In (3.3), is an unknown function, is an embedding parameter, and is an auxiliary function such that = 0 and for . When increases from 0 to 1, the solution , changes from the initial approximation to the solution . Obviously, when and it holds that

Expanding in series with respect to the parameter , one has

If the initial approximation and the auxiliary function are properly chosen so that the series (3.6) converges at , one has

Notice that the series (3.6) contains the auxiliary function which determines their convergence regions. The results of the th-order approximations are given by

We propose an auxiliary function of the form where , can be functions on the variable .

Substituting (3.6) into (3.1) we obtain

If we substitute (3.9) and (3.10) into (3.3) and we equate to zero the coefficients of various powers of , we obtain the following linear equations: At this moment, the th-order approximate solution given by (3.8) depends on the functions ,,. The constants , , , which appear in the expression of can be identified via various methodologies such as the least square method, the Galerkin method, and the collocation method.

The constants , , , could be determined, for example, if we substitute (3.8) into (3.1) resulting in the following residual:

For () where and are two values depending on the given problem and we substitute into (3.14), we obtain the system of equations where is the number of constants which appear in the expression of the functions , ,.

One can observe that our procedure contains the auxiliary function which provides us with a simple but rigorous way to adjust and control the convergence of the solution. It must be underlined that it is very important to properly choose the functions which appear in the approximation (3.8).

4. Application of the Jeffery-Hamel Flow Problem

We introduce the basic ideas of the proposed method by considering (2.6) and (2.8). We choose and the linear operator

The nonlinear operator is and the boundary conditions are Equation (3.11) becomes It is obtained that From (4.2) and (3.10), we obtain the following expression: If we substitute (4.5) into (4.6), we obtain

There are many possibilities to choose the functions , . The convergence of the solutions , and consequently the convergence of the approximate solution given by (3.8) depend on the auxiliary functions . Basically, the shape of should follow the terms appearing in (4.7), (3.12), and (3.13) which are polynomial functions. We consider the following cases ().

Case 1. If is of the form where is an unknown constant at this moment, then (3.12) for becomes Substituting (4.5), (4.7), and (4.8) into (4.9), we obtain the equation in : The solution of (4.10) is given by Equation (3.13) for can be written in the form where is obtained from (3.10): If we consider where is an unknown constant, then from (4.5), (4.11), (4.12), (4.13), and (4.14) we obtain the following equation in : So, the solution of (4.15) is given by The second-order approximate solution () is obtained from (3.8) where , , and are given by (4.5), (4.11), and (4.16), respectively.

Case 2. In this case we consider where , , and are unknown constants. It is clear that the function is given by (4.11). Equations (3.13) or (4.12) becomes and has the solution The second-order approximate solution becomes where , , and are given by (4.5), (4.11), and (4.21), respectively.

Case 3. In the third case we consider Equation (3.12) for or (4.9) can be written as From (4.24) we have Equation (3.13) becomes The solution of (4.26) is The second-order approximate solution is where , , and are given by (4.5), (4.24), and (4.26), respectively.

Case 4. In the last case, we consider

The solution of is given by (4.24). On the other hand, (3.13) has the solution

The second-order approximate solution in this case is given by where , , and are given by (4.5), (4.24), and (4.30), respectively.

5. Numerical Examples

In the following, using the algorithm described in Section 3, with the help of a computer program which implement the procedure presented above, we will obtain the convergence-control constants and we will show that the error of the solution decreases when the number of terms in the auxiliary function increases. Obviously, the computational effort increases along with increasing the number of convergence-control constants, but a significant improvement of the accuracy of results is observed.

Example 5.1. For and in Case 1 it is obtained two solutions for the constants and :(a), (b)but the second-order approximate solution (4.16) is the same in both cases:

Example 5.2. For and in the Case 2, we obtain(a), (b). The second-order approximate solution (4.18) becomes

Example 5.3. For and in Case 3, the second-order approximate solution (4.27) can be written in the form

Example 5.4. For and in Case 4, the second-order approximate solution (4.31) becomes The profile of the function is presented in Figure 2 for and .It is easy to verify the accuracy of the obtained solutions if we compare these analytical solutions with the numerical ones or with results obtained by other procedures.It can be seen from Tables 1, 2, 3, and 4 that the analytical solutions of Jeffery-Hamel flows obtained by OHAM are very accurate.

Table 1: The results of the second-order approximate solutions (5.1), (5.2), and (5.3) and numerical solution of for , .

, (5.1)
, (5.2)
, (5.3)
Numerical solution

0
1
1
1
1

0.1
0.98244611
0.982440382
0.982430842
0.98243124

0.2
0.931225969
0.931302469
0.931225959
0.93122597

0.3
0.850471997
0.850810709
0.850611445
0.85061063

0.4
0.746379315
0.747074996
0.746790784
0.74679081

0.5
0.626298626
0.627192084
0.626947253
0.62694818

0.6
0.497665923
0.498340984
0.498235028
0.49823446

0.7
0.366966345
0.366966353
0.366970088
0.36696635

0.8
0.238952034
0.238148782
0.238142322
0.23812375

0.9
0.116313019
0.115260361
0.115219025
0.11515193

1
0
0
0
0

Table 2: Comparison between the OHAM and numerical solutions for and .

Error of the solution (5.1)
Error of the solution (5.2)
Error of the solution (5.3)

0
0
0
0

0.1
0.00001487
0.000009142
0.000000398

0.2
0.000000001
0.000076499
0.000000011

0.3
0.000138633
0.0002
0.000000815

0.4
0.000411495
0.000284186
0.000000026

0.5
0.000649554
0.000243904
0.000000927

0.6
0.000568537
0.000106524
0.000000568

0.7
0.000000005
0.000000003
0.000003738

0.8
0.000828284
0.000024962
0.000018572

0.9
0.0001161089
0.000108431
0.000067095

1
0
0
0

Table 3: Comparison between Differential Transformation Method (DTM) [4], Homotopy Perturbation Method (HPM) [4], Homotopy Analysis Method [4], and OHAM-(5.4) for , .

(DTM)
(HPM)
(HAM)
(OHAM)
Numerical

0
1
1
1
1
1

0.1
0.9959603887
0.9960671874
0.9995960242
0.995960605
0.9959606278

0.2
0.9832745481
0.9836959424
0.9832755258
0.983275548
0.9832755383

0.3
0.9601775551
0.9610758773
0.9601798911
0.960179914
0.96017991139

0.4
0.9235170706
0.9249245156
0.9235215737
0.923521643
0.9235215894

0.5
0.8684511349
0.8701997697
0.8684588997
0.868458963
0.86845887772

0.6
0.7880785402
0.7898325937
0.7880910186
0.788090923
0.78809092032

0.7
0.673248448
0.6745334968
0.6731437690
0.673143633
0.6731436346

0.8
0.5119644061
0.5128373095
0.5119909939
0.511991107
0.5119910891

0.9
0.2915280122
0.2918936991
0.2915580178
0.291558742
0.29155874261

1
0
0
−0.000001149
0
0

Table 4: Comparison between OHAM (5.4) and numerical solutions [4] for , .

, (5.4)
Numerical
Error

0
1
1
0

0.1
0.995960605
0.9959606278
0.000000022

0.2
0.983275548
0.9832755383
0.000000009

0.3
0.960179914
0.96017991139
0.000000002

0.4
0.923521643
0.9235215894
0.000000053

0.5
0.868458963
0.86845887772
0.000000085

0.6
0.788090923
0.78809092032
0.000000002

0.7
0.673143633
0.6731436346
0.000000001

0.8
0.511991107
0.5119910891
0.000000017

0.9
0.291558742
0.29155874261
0.000000006

1
0
0
0

Figure 2: Sample profile of the function for and , given by (5.1).

6. Conclusions

In this paper the Optimal Homotopy Asymptotic Method (OHAM) is employed to propose a new analytic approximate solution for the nonlinear MHD Jeffery-Hamel flow problems. The proposed procedure is valid even if the nonlinear equation does not contain any small or large parameters.

OHAM provides us with a simple and rigorous way to control and adjust the convergence of the solution through the auxiliary functions involving several constants which are optimally determined.

From the results presented above, we can conclude that the following. (1)When and steep of the channel is divergent, stream in value of Reynolds number is caused by decreasing in velocity.(2)When and steep of the channel is convergent, the results are inverse. Increase in value of Reynolds number is caused by increasing in velocity.

The examples related to the Jeffery-Hamel flow problem presented in this paper lead to the very important conclusion that the accuracy of the obtained results is growing along with increasing the number of constants in the auxiliary function. This paper confirmed that DTM, HPM, or HAM gives a good accuracy, but OHAM is by far the best method delivering faster convergence and better accuracy. In the proposed procedure, iterations are performed in a very simple manner by identifying some coefficients, and therefore very good approximations are obtained in few terms. Actually the capital strength of the proposed procedure is its fast convergence, since after only two iterations it converges to the exact solution, which proves that this method is very effective in practice. This version of the method proves to be very rapid and effective, and this is proved by comparing the analytic solutions obtained through the proposed method with the solutions obtained via numerical simulations or other known procedures.

References

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