Analytical Approximate Solution of Nonlinear Differential Equation Governing Jeffery-Hamel Flow with High Magnetic Field by Adomian Decomposition Method

The magnetohydrodynamic Jeffery-Hamel flow is studied analytically. The traditional NavierStokes equation of fluid mechanics and Maxwell’s electromagnetism governing equations reduce to nonlinear ordinary differential equations to model this problem. The analytical tool of Adomian decomposition method is used to solve this nonlinear problem. The velocity profile of the conductive fluid inside the divergent channel is studied for various values of Hartmann number. Results agree well with the numerical Runge-Kutta method results, tabulated in a table. The plots confirm that the method used is of high accuracy for different α, Ha, and Re numbers.


Introduction
The flow of fluid through a divergent channel is called Jeffery-Hamel flow since introducing this problem by Jeffery 1 andHamel 2 in 1915 and1916, respectively.On the other hand, the term of magnetohydrodynamic MHD was first introduced by Bansal 3 in 1994.The theory of magnetohydrodynamics is inducing current in a moving conductive fluid in presence of magnetic field; such induced current results in force on ions of the conductive fluid.The theoretical study of magnetohydrodynamic MHD channel has been a subject of great interest due to its extensive applications in designing cooling systems with liquid metals, MHD generators, accelerators, pumps, and flow meters 4-7 .
In fluid mechanics most of the problems are nonlinear.It is very important to develop efficient methods to solve them.Up to now, it is very difficult to obtain analytical approximations of nonlinear partial differential equations even though there are highperformance computers and computation software.The small disturbance stability of Magnetohydrodynamic stability of plane Poiseuille flow has been investigated by Makinde and Motsa 8 and Makinde 9 for generalized plane Couette flow.Their results show that magnetic field has stabilizing effects on the flow.Considerable efforts have been done to study the MHD theory for technological application of fluid pumping system in which electrical energy forces the working conductive fluid.Damping and controlling of electrically conducting fluid can be achieved by means of an electromagnetic body force Lorentz force produced by the interaction of an applied magnetic field and an electric current that usually is externally supplied.Harada  The Adomian decomposition method ADM is used to solve a wide range of physical problems.One of the semiexact methods which does not need linearization or discretization is Adomian decomposition method, and several modifications have improved its ability 14-16 .An advantage of this method is that it can provide analytical approximation or an approximated solution to a rather wide class of nonlinear and stochastic equations without linearization, perturbation, closure approximation, or discretization methods.Unlike the common methods, that is, weak nonlinearity and small perturbation which change the physics of the problem due to simplification, ADM gives the approximated solution of the problem without any simplification.Thus, its results are more realistic 17 .ADM abilities have attracted many authors to use this method for solving fluid dynamic problems.
Jafari and Daftardar-Gejji 18 presented a modified ADM to solve a system of nonlinear equations, which yielded a series of solutions with faster accelerated convergence than the series obtained by the standard ADM.Bulut  Chang 26 presented a decomposition solution for fins with temperature-dependent surface heat flux.Arslanturk 27 inspected on the fins efficiency of convective straight fins with temperature-dependent thermal conductivity using the decomposition method.ADM also has been used by several researchers to solve a wide range of physical problems in various engineering fields such as fluid flow and porous media simulation 28-31 and other nonlinear systems 32-38 .In recent years some researchers used new methods to solve these kinds of problems 39-41 .
In this paper, we have applied ADM to find the approximate solutions of nonlinear differential equations governing the MHD Jeffery-Hamel flow, and a comparison between the results and the numerical solution has been provided.The numerical results of this problem are done using Maple 12.

Governing Equations
Consider a system of cylindrical polar coordinates r, θ, z which steady two-dimensional flow of an incompressible conducting viscous fluid from a source or sink at channel walls, lie in planes, and intersect in the axis of z, assuming purely radial motion which means that there is no change in the flow parameter along the z direction.The flow depends on r, θ, and further assume that there is no magnetic field in the z-direction.In the reduced form of continuity, Navier-Stokes and Maxwell's equations are 40 1 r ∂u r, θ ∂r where B 0 is the electromagnetic induction, σ the conductivity of the fluid, u r the velocity along radial direction, P the fluid pressure, υ the coefficient of kinematic viscosity, and ρ the fluid density.Considering u θ 0 for purely radial flow, one can define the velocity parameter as f θ ru r .

2.4
Introducing the η θ/α as the dimensionless degree, the dimensionless form of the velocity parameter can be obtained by dividing that to its maximum value as ISRN Mathematical Analysis Substituting 2.5 into 2.2 and 2.3 and eliminating P , one can obtain the ordinary differential equation for the normalized function profile as 10 with the following reduced form of boundary condition We introduce the Reynolds number and the Hartmann number based on the electromagnetic parameter as follows, respectively: Ha σB 2 0 ρυ .2.9

Fundamentals of Adomian Decomposition Method (ADM)
Consider equation Fu t g t , where F represents a general nonlinear ordinary or partial differential operator including both linear and nonlinear terms.The linear terms are decomposed into L R, where L is easily invertible usually the highest order derivative and R is the remainder of the linear operator.Thus, the equation can be written as 14 : where Nu indicates the nonlinear terms.By solving this equation for Lu, since L is invertible, we can write If L is a second-order operator, L −1 is twofold in definite integral.By solving 3.2 , we have 14 where A and B are constants of integration and can be found from the boundary or initial conditions.Adomian method assumes that the solution u can be expanded into infinite series as Also, the nonlinear term Nu will be written as where A n are the special Adomian polynomials.By specified A n , the next component of u can be determined: Finally, after some iteration and getting sufficient accuracy, the solution of the equation can be expressed by 3.5 .In 3.6 , the Adomian polynomials can be generated by several means.Here we used the following recursive formulation 41 : , n 0, 1, 2, 3, . . . .

3.7
Since the method does not resort to linearization or assumption of weak nonlinearity, the solution generated is in general more realistic than those achieved by simplifying the model of the physical problem.

Application
According to 3.1 , 2.6 must be written as follows: where the differential operator L is given by L d 3 /dη 3 .Assume that the inverse of the operator L exists and it can be integrated from 0 to η, that is, L −1 η 0 • dη dη dη.Operating with L −1 on 4.1 and after exerting boundary condition on it, we have where

ISRN Mathematical Analysis
ADM introduced the following expression: To determine the components of f m η the f 0 η is defined by applying the boundary condition of 2.7 , and by assuming f 0 β, Using f η

4.8
According to 4.8 , the accuracy of ADM solution increases by increasing the number of solution terms m .For the complete solution of 4.8 , β should be determined, with boundary condition of f 0 β.

Results and Discussion
In this study the objective was to apply Adomian decomposition method to obtain an explicit analytic solution of the MHD Jeffery-Hamel problem.The magnetic field acts as a control parameter such as the flow Reynolds number and the angle of the walls, in MHD Jeffery-Hamel problems.There is an additional nondimensional parameter that determines the solutions, namely, the Hartmann number.Table 1 shows the value of constant β for different α, Ha, and Re numbers at the divergent channel.
For comparison, a few limited cases of the ADM solutions are compared with the numerical results.The comparison between the numerical results and ADM solution for velocity when Re 25 and α 5 • is shown in Table 2.The error bar shows an acceptable agreement between the results observed, which confirms the validity of the ADM.In these tables the error is introduced as follows: %Error of ADM for f η at different steps when α 5, Re 50, and Ha 100 can be seen in Figure 2; it shows that at first we have large error then it decreases and after 12 steps the error becomes minimized.Step %Error Figures 3, 4, and 5 show the magnetic field effect on the velocity profiles for divergent channels.There are good agreements between the numerical solution obtained by the fourthorder Runge-Kutta method and the differential transformation method.
Under magnetic field the Lorentz force effect is opposite of the momentum's direction that stabilizes the velocity profile.
The results show moderate increases in the velocity with increasing Hartmann numbers at small angle α 2.5 • and differences between velocity profiles are more noticeable at greater angles.Backflow is excluded in converging channels 42 but it may occur for large Reynolds numbers in diverging channels.For specified opening angle, after a critical Reynolds number, we observe that separation and backflow are started.
Figures 6-8 show the magnetic field effects at constant α and different Reynolds numbers.At α 5 • , Re 75 with increasing Hartmann number the velocity profile becomes flat and thickness of boundary layer decreases, but at this Reynolds number no backflow is observed as shown in Figure 6.It can be seen in Figure 9 that without magnetic field at α 5 • , Re 150 the backflow starts and that with increasing Hartmann number this phenomenon eliminates.By increasing Reynolds number the backflow expands and so greater magnetic field is needed in order to eliminate it.As shown in Figures 7 and 8

Conclusion
In this paper, magnetohydrodynamic Jeffery-Hamel flow has been solved via a sort of analytical method, Adomian decomposition method ADM .Also this problem is solved by a numerical method the Runge-Kutta method of order 4 , and some conclusions are summarized as follows.
a Adomian decomposition method is a powerful approach for solving MHD Jeffery-Hamel flow in high magnetic field, and it can be observed that there is a good agreement between the present and numerical results.
et al. 10 studied the fundamental characteristics of linear Faraday MHD theoretically and numerically.In 2005, Anwari et al. 11 continued the Harada et al. 10 work numerically and theoretically, for various loading configurations.Jang and Lee 12 emphasized on the idea that, in such problems, the moving ions drag the bulk fluid with themselves and such MHD system induces continues pumping of conductive fluid without any moving part.Homsy et al. 13 worked and developed the same idea mentioned above.The purpose of the current work is to study the mechanics of the fluid through a divergent channel in presence of electromagnetic field Figure 1 .

Table 2 :
Comparison between the numerical results and ADM solution for velocity when Re 25, α 5 • .

Figure 9 :
Figure 9: The ADM solution for velocity in divergent channel for α 5 • , Re 150.

at α 5 •
, Re 225 the back flow eliminates at Ha 1000 while at α 5 • , Re 300 this occurs at Ha 2000.

b
Increasing Reynolds numbers leads to adverse pressure gradient which causes velocity reduction near the walls.c Increasing Hartmann number will lead to backflow reduction.In greater angles or Reynolds numbers high Hartmann number is needed for the reduction of backflow.Nomenclature B 0 : Magnetic field wb • m 2 β: Constant Lu: Linear term Nu: Nonlinear term Ru: The remainder of linear operator F: number r, θ: Cylindrical coordinates U max : Maximum value of velocity u, v: Velocity components along x-, y-axes, respectively.Greek Symbols υ: Kinematic viscosity τ: Viscous stresses α: Angle of the channel θ: Any angle η: Dimensionless angle.

et al. 19 studied viscous incompressible flow through orifice. Allen and Syam 20 and Wang 21 investigated nonhomogeneous and classical Blasius equation by ADM. Soh 22 applied ADM to solve thin film equation. Hashim 23 presented the Adomian decomposition method for solving BVPs for fourth
order integrodifferential equations and the Blasius equation 24 .Kechil and Hashim 25 presented a nonperturbative solution of free-convective boundary-layer equation by ADM.

Table 1 :
Value of f 0 β at various Re, Ha, and α.