# Variational Iteration Method for the Magnetohydrodynamic Flow over a Nonlinear Stretching Sheet

Authors: Lan Xu; Eric W. M. Lee
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The variational iteration method (VIM) is applied to solve the boundary layer problem of magnetohydrodynamic flow over a nonlinear stretching sheet. The combination of the VIM and the Padé approximants is shown to be a powerful method for solving two-point boundary value problems consisting of systems of nonlinear differential equations. And the comparison of the obtained results with other available results shows that the method is very effective and convenient for solving boundary layer problems.


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## Body

## 1. Introduction

It is well known that most of the phenomena that arise in mathematical physics and engineering fields can be described by partial differential equations. Recent advances of partial differential equations are stimulated by new examples of applications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and physics. There are many traditional and recently developed methods to give numerical and analytical approximate solutions of nonlinear differential equations such as Euler method, Runge-Kutta method, Taylor series method, Adomian decomposition method [1], Variational iteration method [2, 3], Hankel-Padé method [4], DTM-Padé method [5], homotopy perturbation method [6], and Hamiltonian method [7].In this paper, we consider the model proposed by authors in [1] describing the problem of the boundary layer flow of an incompressible viscous fluid over a nonlinear stretching sheet. The boundary layer flow is often encountered in many engineering and industrial processes. Such processes include the aerodynamic extrusion of plastic sheets, hot rolling, glass fiber production, and so on [1, 4, 5]. And various aspects of the stretching flow problem were discussed by various investigators. Chiam [8] analyzed the MHD flow of a viscous fluid bounded by a stretching surface with power law velocity. He presented the numerical solution of the boundary value problem by utilizing the Runge-Kutta shooting algorithm with Newton iteration. Here, we aim to solve the MHD flow caused by a sheet with nonlinear stretching. The approximate solution of the nonlinear problem is obtained by the variational iteration method.The variational iteration method [2] is a type of Lagrange multiplier method to find analytical solutions. The method gives the possibility to solve many kinds of non linear equations. In this method, general Lagrange multipliers are introduced to construct correction functional for the problems. The multipliers can be identified optimally via variational theory. It has been used to solve effectively, easily, and accurately a large class of nonlinear problems with approximation [9].


## 2. Basic Idea of the VIM

The basic idea was systematically illustrated and discussed in [9, 10]. To illustrate the basic idea of the VIM, we consider the following general nonlinear system:
(1)L[u(t)]+N[u(t)]=g(t),
where L, N, and g(t) are the linear operator, the nonlinear operator, and a given continuous function, respectively. The basic character of the method is to construct a correction functional for the system, which reads
(2)un+1(t)=un(t)+∫0tλ(s)[Lun(s)+Nu~n(s)-g(s)]ds,
where λ is a Lagrange multiplier which can be identified optimally via the variational theory. The subscript n indicates the nth approximation, and u~n denotes a restricted variation, that is, δu~n=0.


## 3. Problem Statement and Governing Equations

We consider the magnetohydrodynamic (MHD) flow of an incompressible viscous fluid over a stretching sheet aty=0. The fluid is electrically conducting under the influence of an applied magnetic field B(x) normal to the stretching sheet. The induced magnetic field is neglected. The resulting boundary layer equations are as follows [1]:
(3)∂u∂x+∂v∂y=0,(4)u∂u∂x+v∂u∂y=ν∂2u∂y2-σB2(x)ρu,
where u and v are the velocity components in the x and y directions, respectively, ν is the kinematic viscosity, ρ is the fluid density, and σ is the electrical conductivity of the fluid. In (4), the external electric field and the polarization effects are negligible, and in [8]
(5)B(x)=B0x(n-1)/2.
The boundary conditions corresponding to the nonlinear stretching of a sheet are
(6)u(x,0)=cxn,v(x,0)=0,u(x,y)⟶0asy⟶∞.
Upon making use of the following substitutions:
(7)η=c(n+1)2νx(n-1)/2y,u=cxnf′(η),(8)v=-cν(n+1)2x(n-1)/2[f(η)+n-1n+1ηf′(η)],
Substituting (8) into (3)–(6), the resulting nonlinear differential system can be written in the following form:
(9)f′′′+ff′′-βf′2-Mf′=0,(10)f(0)=0,f′(0)=1,f′(∞)=0,
where
(11)β=2n1+n,M=2σB02ρc(1+n).
The parameter β is a measure of the pressure gradient, and M is the magnetic parameter. Positive β denotes the favorable negative pressure gradient, and negative β denotes the unfavorable positive pressure gradient; naturally, β=0 denotes the flat plate. For the special case of β=1, the exact analytical solution of (9) is [11]
(12)f(η)=1-exp(-1+Mη)1+M.


## 4. Approximate Solution by the VIM

In order to obtain VIM solution of (9), we construct a correction functional which reads
(13)fn+1(η)=fn(η)+∫0ηλ(τ)[∂3fn(τ)∂τ3+f~n(τ)∂2f~n(τ)∂τ2-β=fn(η)+λ(τ)×××(∂f~n(τ)∂τ)2-M∂f~n(τ)∂τ]dτ,
where λ(τ) is the general Lagrangian multiplier which can be identified optimally via the variational theory. And f~n(τ) is considered as a restricted variation, that is, δf~n(τ)=0. We omit asterisks for simplicity. Its stationary conditions can be obtained as follows:
(14)1+λ′′(τ)|τ=η=0,λ′(τ)|τ=η=0,λ′′′(τ)=0.
The Lagrange multipliers can be readily identified as the following form:
(15)λ(τ)=-12(τ-η)2.
As a result, we obtain the following variational iteration formula
(16)fn+1(η)=fn(η)-12∫0η(τ-η)2[∂3fn(τ)∂τ3+f~n(τ)∂2f~n(τ)∂τ2-β=fn(η)-121212(τ-η)2×(∂f~n(τ)∂τ)2-M∂f~n(τ)∂τ]dτ.
Now, we assume that an initial approximation
(17)f0(η)=a+bη+cη2.
where a,b, and c are unknown constants to be further determined.By the iteration formula (16) and the initial approximation (17), we can obtain directly the first-order approximate solution as follows:
(18)f1(η)=f0(η)-12∫0η(τ-η)2=f0(η)-12∫0η×[∂3f0(τ)∂τ3+f0(τ)∂2f0(τ)∂τ2=f0(η)-12∫0η×-β(∂f0(τ)∂τ)2-M∂f0(τ)∂τ]dτ=a+bη+cη2-c230η5+bM6η3+cM12η4η5+βb26η3+βc215-ac3η3-bc12η4+bcβ6η4=a+bη+cη2+bM+βb2-2ac6η3+cM+bc(2β-1)12η4+(2β-1)c230η5.
Making use of the initial conditions f(0)=0,f′(0)=1, we can readily obtain the results as follows:
(19)a=0,b=1,c=12f′′(0),
where f′′(0)=α will be examined in this work, according the initial condition f′(∞)=0.Then,(20)f1(η)=η+12αη2+M+β6η3+α(M+2β-1)24η4+(2β-1)α2120η5.
And the following second-order approximate solution can be obtained
(21)f2(η)=f1(η)-12∫0η(τ-η)2=f1(η)-12∫0η×[∂3f1(τ)∂τ3+f1(τ)∂2f1(τ)∂τ2=f1(η)-12∫0η×-β(∂f1(τ)∂τ)2-M∂f1(τ)∂τ]dτ=η+12αη2+M+β6η3+α(M+2β-1)24η4+[(2β-1)α2120+β260+βM40-160+M2120-M60]η5+(β2α72+βMα72-βα60+M2α720-Mα90+α240)η6+(β3840+β2M420+β2α2252-β21260+βM2840+βMα2504-βM630-2βα2315-M21260-Mα2630+11α25040)η7+(β3α1008+β2Mα672+β2α32016-5β2α4032+βM2α2016-13βMα8064-7βα31260+βα2688-M2α2688+Mα2688+11α340320)η8+(β3α22592+β2Mα22592-37β2α260480+βM2α218144-13βMα225920+53βα2181440-M2α224192+Mα26480-α224192)η9+(β3α312960+β2Mα325920-β2α37200-13βMα3259200+7βα386400+Mα364800-α364800)η10+(β3α4142560-β2α479200+7βα4950400-α4712800)η11.
Therefore, according to (13), we can easily obtain higher-order approximate solution as follows:
(22)f(η)=r0+r1η+r2η2+r3η3+r4η4+r5η5+⋯,
by using mathematical software such as MATLAB.It is evident that the main problem for solving (21) is to obtain the value of f′′(0), then we can resort to any numerical integration routine to obtain the solution of the problem. For this purpose, we will employ the Padé method to determine this unknown value with high accuracy.


## 5. Padé Approximation

It is well known that Padé approximations [12] have the advantage of manipulating the polynomial approximation into a rational function of polynomials. This manipulation provides us with more information about the mathematical behavior of the solution. Besides that, power series are not useful for large values of η, say η=∞. This can be attributed to the possibility that the radius of convergence may not be sufficiently large to contain the boundaries of the domain. Therefore, the combination of the series solution through the decomposition method or any other series solution method with the Padé approximation provides an effective tool for handling boundary value problems on infinite or semi-infinite domains. Furthermore, it is noted that Padé approximants can be easily evaluated by using Matlab.Therefore, we suppose that the solutionf(η) can be expanded as a Taylor series about η=0(23)f(η)=∑j=0∞fjηj.
Padé approximant, symbolized by [S/N], is a rational function defined by
(24)[SN](η)=∑j=0Spjηj∑j=0Nqjηj.
If we selected S=N, then the approximants [N/N] are called diagonal approximants. More importantly, the diagonal approximants are the most accurate approximants; therefore, we have to construct only diagonal approximants.Then,(25)p0+p1η+p2η2+p3η3+⋯+pNηNq0+q1η+q2η1+q3η3+⋯+qNηN=r0+r1η+r2η2+r3η3+r4η4+⋯.
By using cross multiplication in (25), we find
(26)p0+p1η+p2η2+p3η3+⋯+pNηN=r0q0+(r1q0+q1r0)η+(r2q0+q1r1+q2r0)η2+(r3q0+q1r2+q2r1+q3r0)η3+⋯.
Using the boundary condition f′(∞)=0, the diagonal approximant [N/N] vanishes if the coefficient of η with the highest power in the numerator vanishes. By putting the coefficients of the highest power of η equal to zero, we can easily obtain the values of f′′(0) listed in Tables 1 and 2 and Figure 1, using Matlab. The order of Padé approximation [12/12] has sufficient accuracy; on the other hand, if the order of Padé approximation increases, the accuracy of the solution increases.Table 1
Comparison of the values off′′(0) obtained by the variational iteration method and other methods [1] for various values of Mwhen β=1.
M
VIM
ADM [1]
Exact [1]
1.0
−1.41421
−1.41421
−1.41421
5.0
−2.44948
−2.44948
−2.44948
10.0
−3.31662
−3.31662
−3.31662
50.0
−7.14142
−7.14142
−7.14142
100.0
−10.04987
−10.04987
−10.04987
500.0
−22.38302
−22.38302
−22.38302Table 2
Comparison of the values off′′(0) obtained by the variational iteration method and the modified Adomian decomposition method [1] for various values of β andM.
M
β
=
-
1
.
5
β
=
1
.
5
β
=
5
VIM
ADM [1]
VIM
ADM [1]
VIM
ADM [1]
1.0
−0.6530
−0.6532
-
1.5253
−1.5252
-
2.152
9
−2.1528
5.0
−2.0852
−2.0852
-
2.5162
−2.5161
−2.9414
−2.9414
10
−3.0562
−3.0562
-
3.3663
−3.3663
-
3.695
7
−3.6956
50
−7.0239
−7.0239
-
7.1647
−7.1647
−7.3256
−7.3256
100
-
9.966
7
−9.9666
-
10.0776
−10.0776
−10.1816
−10.1816
500
−22.3458
−22.3457
-
22.3905
−22.3904
-
22.442
6
−22.4425Figure 1
Comparison between the approximate solution by the VIM and exact solution forβ=1 and M=10.Substituting (21) and the value of f′′(0) into (8), we can easily obtain the second-order approximate solution of (3)-(4).


## 6. Conclusion

In this paper, the variational iteration method is used to obtain approximate solutions of magnetohydrodynamics boundary layer equations. The analytical solutions of the governing nonlinear boundary layer problem are obtained. Without using the Padé approximation, the analytical solution that were obtained by the VIM cannot satisfy the boundary condition at infinityf′(∞)=0. The combination of the VIM and the Padé approximants is shown to be a powerful method for solving two-point boundary value problems consisting of systems of nonlinear differential equations. And the obtained solutions are in good agreement with exact values.



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