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

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 (𝑟,𝜃,𝑧) 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 2𝛼, 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 𝑟 and 𝜃, 𝑉(𝑢(𝑟,𝜃),0) [3–5], using the continuity Navier-Stokes equations in polar coordinates, the governing equations are𝜌𝑟𝜕((𝜕𝑟𝑟𝑢𝑟,𝜃))=0, (2.1)𝑢()()𝑟,𝜃𝜕𝑢𝑟,𝜃1𝜕𝑟=−𝜌𝜕𝑝𝜕𝜕𝑟+𝑣²𝑢(𝑟,𝜃)𝜕𝑟²+1𝑟𝜕𝑢(𝑟,𝜃)+1𝜕𝑟𝑟²𝜕²𝑢(𝑟,𝜃)𝜕𝜃²−𝑢(𝑟,𝜃)𝑟², (2.2)−1𝜌𝑟𝜕𝑝+𝜕𝜌2𝑣𝑟²𝜕𝑢(𝑟,𝜃)𝜕𝜃=0, (2.3) where 𝜌 is the fluid density, 𝑝 is the pressure, and 𝑣 is the kinematic viscosity. From (2.1) and using dimensionless parameters we get𝑓(𝜃)=𝑟𝑢(𝑟,𝜃), (2.4)𝐹(𝑥)=𝑓(𝜃)𝑓_max𝜃,𝑥=𝛼. (2.5)

Substituting (2.5) into (2.2) and (2.3) and eliminating the pressure, we obtain an ordinary differential equation for the normalized function profile 𝐹(𝑥):𝐹^′′′(𝑥)+2𝛼Re𝐹(𝑥)𝐹^′(𝑥)+4𝛼²𝐹^′(𝑥)=0, (2.6) where prime denotes derivative with respect to 𝑥 and the Reynolds number isRe=𝛼𝑓_max𝑣=𝑢_max𝑣divergentchannel∶𝛼>0,𝑢_max>0convergentchannel∶𝛼<0,u_max<0 (2.7) and 𝑢_max is the maximum velocity at the centre of the channel.

The boundary conditions for (2.6) are𝐹(0)=1,𝐹^′(0)=0,𝐹(1)=0. (2.8)

3. Fundamentals of the OHAM

We consider the following nonlinear differential equation [10, 11]:𝐿(𝐹(𝑥))+𝑓(𝑥)+𝑁(𝐹(𝑥))=0 (3.1) subject to a boundary condition𝐵(𝐹)=0, (3.2) where 𝐿 is a linear operator, 𝑓(𝑥) is a known analytical function, 𝑁 is a nonlinear operator, and 𝐵 is a boundary operator. By means of the OHAM one constructs a family of equations:1−𝑝𝐿𝜙𝑥,𝑝+𝑓(𝑥)=ℎ𝑥,𝑝𝐿𝜙𝑥,𝑝+𝑓(𝑥)+𝑁𝜙𝑥,𝑝, (3.3) and the boundary condition is𝐵𝜙𝑥,𝑝=0. (3.4)

In (3.3), 𝜙(𝑥,𝑝) is an unknown function, 𝑝∈[0,1] is an embedding parameter, and ℎ(𝑥,𝑝) is an auxiliary function such that ℎ(𝑥,0) = 0 and ℎ(𝑥,𝑝)≠0 for 𝑝≠0. When 𝑝 increases from 0 to 1, the solution 𝜙(𝑥,𝑝), changes from the initial approximation 𝐹₀(𝑥) to the solution 𝐹(𝑥). Obviously, when 𝑝=0 and 𝑝=1 it holds that𝜙(𝑥,0)=𝐹₀(𝑥),𝜙(𝑥,1)=𝐹(𝑥). (3.5)

Expanding 𝜙(𝑥,𝑝) in series with respect to the parameter 𝑝, one has𝜙𝑥,𝑝=𝐹₀(𝑥)+𝑝𝐹₁(𝑥)+𝑝²𝐹₂(𝑥)+…. (3.6)

If the initial approximation 𝐹₀(𝑥) and the auxiliary function ℎ(𝑥,𝑝) are properly chosen so that the series (3.6) converges at 𝑝=1, one has𝐹(𝑥)=𝐹₀(𝑥)+𝐹₁(𝑥)+𝐹₂(𝑥)+…. (3.7)

Notice that the series (3.6) contains the auxiliary function ℎ(𝑥,𝑝) which determines their convergence regions. The results of the 𝑚th-order approximations are given by𝐹(𝑥)≈𝐹₀(𝑥)+𝐹₁(𝑥)+𝐹₂(𝑥)+⋯+𝐹_𝑚(𝑥). (3.8)

We propose an auxiliary function ℎ(𝑥,𝑝) of the formℎ𝑥,𝑝=𝑝𝐾₁(𝑥)+𝑝²𝐾₂(𝑥)+⋯+𝑝^𝑚𝐾_𝑚(𝑥), (3.9) where 𝐾_𝑖(𝑥), 𝑖=1,2,…,𝑚 can be functions on the variable 𝑥.

Substituting (3.6) into (3.1) we obtain𝐿(𝐹(𝑥))+𝑓(𝑥)+𝑁(𝐹(𝑥))=𝑁₀𝐹₀(𝑥)+𝑝𝑁₁𝐹₀(𝑥),𝐹₁(𝑥)+𝑝²𝑁₂𝐹₀(𝑥),𝐹₁(𝑥),𝐹₂(𝑥)+…. (3.10)

If we substitute (3.9) and (3.10) into (3.3) and we equate to zero the coefficients of various powers of 𝑝, we obtain the following linear equations:𝐿𝐹₀(𝑥)+𝑓(𝑥)=0,𝐵𝐹₀(𝑥)=0, (3.11)𝐿𝐹_𝑖−𝐿𝐹_𝑖−1−^𝑖_𝑗=1𝐾_𝑗𝑁_𝑖−𝑗𝐹₀,𝐹₁,…,𝐹_𝑖−𝑗=0,𝐵𝐹_𝑖=0,𝑖=1,2,…,𝑚−1 (3.12)𝐿𝐹_𝑚−𝐿𝐹_𝑚−1−^𝑚−1_𝑗=1𝐾_𝑗𝑁_{𝑚−1−𝑗}−𝐾_𝑚𝑁₀=0,𝐵𝐹_𝑚=0. (3.13) At this moment, the 𝑚th-order approximate solution given by (3.8) depends on the functions 𝐾₁,𝐾₂,…,𝐾_𝑚. The constants 𝐶₁, 𝐶₂, …, 𝐶_𝑞 which appear in the expression of 𝐾_𝑖(𝑥) can be identified via various methodologies such as the least square method, the Galerkin method, and the collocation method.

The constants 𝐶₁, 𝐶₂, …, 𝐶_𝑞 could be determined, for example, if we substitute (3.8) into (3.1) resulting in the following residual:𝑅𝑥,𝐶_𝑖=𝐿𝐹𝑥,𝐶_𝑖+𝑓(𝑥)+𝑁𝐹𝑥,𝐶_𝑖,𝑖=1,2,…. (3.14)

For 𝑥_𝑖∈ (𝑎,𝑏) where 𝑎 and 𝑏 are two values depending on the given problem and we substitute 𝑥_𝑖 into (3.14), we obtain the system of equations𝑅𝑥₁,𝐶_𝑖=𝑅𝑥₂,𝐶_𝑖=⋯=𝑅𝑥_𝑞,𝐶_𝑖,𝑖=1,2,…,𝑞, (3.15) where 𝑞 is the number of constants 𝐶_𝑖 which appear in the expression of the functions 𝐾₁(𝑥), 𝐾₂(𝑥),…,𝐾_𝑚(𝑥).

One can observe that our procedure contains the auxiliary function ℎ(𝑥,𝑝) 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 𝐾₁,…,𝐾_𝑚(𝑥) 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 𝑓(𝑥)=0 and the linear operator𝐿𝜙=𝜕𝑥,𝑝³𝜙𝑥,𝑝𝜕𝑥³. (4.1)

The nonlinear operator is𝑁𝜙𝑥,𝑝=2𝛼Re𝜙𝑥,𝑝𝜕𝜙𝑥,𝑝𝜕𝑥+4𝛼²𝜕𝜙𝑥,𝑝𝜕𝑥, (4.2) and the boundary conditions are𝜙0,𝑝=1,𝜕𝜙0,𝑝𝜕𝑥=0,𝜙1,𝑝=0. (4.3) Equation (3.11) becomes𝐹₀^′′′(𝑥)𝐹=0,₀(0)=1,𝐹^′₀(0)=0,𝐹₀(1)=0. (4.4) It is obtained that𝐹₀(𝑥)=1−𝑥². (4.5) From (4.2) and (3.10), we obtain the following expression:𝑁₀(𝑥)=𝐹₀^′′′(𝑥)+2𝛼Re𝐹₀(𝑥)𝐹^′₀(𝑥)+4𝛼²𝐹^′₀(𝑥). (4.6) If we substitute (4.5) into (4.6), we obtain𝑁₀(𝑥)=4𝛼Re𝑥³−4𝛼Re+2𝛼²𝑥. (4.7)

There are many possibilities to choose the functions 𝐾_𝑖, 𝑖=1,2,…. The convergence of the solutions 𝐹_𝑖, 𝑖=1,2,…𝑚 and consequently the convergence of the approximate solution 𝐹(𝑥) given by (3.8) depend on the auxiliary functions 𝐾_𝑖. Basically, the shape of 𝐾_𝑖(𝑥) should follow the terms appearing in (4.7), (3.12), and (3.13) which are polynomial functions. We consider the following cases (𝑚=2).

Case 1. If 𝐾₁(𝑥) is of the form 𝐾₁(𝑥)=𝐶₁, (4.8) where 𝐶₁ is an unknown constant at this moment, then (3.12) for 𝑖=1 becomes 𝐹₁^′′′(𝑥)−𝐹₀^′′′(𝑥)−𝐾₁𝑁₀𝐹₀=0. (4.9)
Substituting (4.5), (4.7), and (4.8) into (4.9), we obtain the equation in 𝐹₁: 𝐹₁^′′′(𝑥)−4𝐶₁𝛼Re𝑥³−4𝐶₁𝛼Re+2𝛼²𝑥=0,𝐹₁(0)=𝐹^′₁(0)=𝐹₁(1)=0. (4.10)
The solution of (4.10) is given by 𝐹₁(𝑥)=𝐶₁𝛼Re𝑥30⁶−𝐶₁𝛼Re+2𝛼²6𝑥⁴+2𝛼Re+5𝛼²𝐶15¹𝑥². (4.11) Equation (3.13) for 𝑚=2 can be written in the form 𝐹₂^′′′(𝑥)−𝐹₁^′′′(𝑥)−𝐶₁𝑁₁𝐹₀,𝐹₁−𝐾₂(𝑥)𝑁₀𝐹₀=0, (4.12) where 𝑁₁ is obtained from (3.10): 𝑁₁𝐹₀,𝐹₁=𝐹₁^′′′+2𝛼Re𝐹₀𝐹^′₁+𝐹^′₀𝐹₁+4𝛼²𝐹^′₁. (4.13) If we consider 𝐾₂(𝑥)=𝐶₂, (4.14) where 𝐶₂ is an unknown constant, then from (4.5), (4.11), (4.12), (4.13), and (4.14) we obtain the following equation in 𝐹₂: 𝐹₂^′′′+8𝛼²Re²𝐶²₁𝑥15⁷−12𝛼²Re²+24𝛼²Re5𝐶²₁𝑥⁵−4𝛼Re2𝐶₁+𝐶₂+60𝛼Re+36𝛼²Re²+40𝛼³Re−16𝛼⁴𝐶15²¹𝑥³+4𝛼Re+2𝛼²𝐶₁+𝐶₂+60𝛼Re+120𝛼²−8𝛼²Re²−36𝛼³Re−40𝛼⁴𝐶15²¹𝑥=0. (4.15)
So, the solution of (4.15) is given by 𝐹₂(𝑥)=−𝛼²Re²𝐶²₁𝑥1350¹⁰+𝛼²Re²+2𝛼²Re𝐶140²¹𝑥⁸+𝛼Re2𝐶₁+𝐶₂+3015𝛼Re+9𝛼²Re²+10𝛼³Re−4𝛼⁴𝐶450²¹𝑥⁶+−𝛼Re+2𝛼²𝐶₁+𝐶₂6+2𝛼²Re²+9𝛼³Re+10𝛼⁴−15𝛼Re−3𝛼²𝐶90²¹𝑥⁴+2𝛼Re+5𝛼²𝐶15¹+𝐶₂+2520𝛼Re−1932𝛼⁴+360𝛼²−919𝛼²Re²−2580𝛼³Re𝑥18900². (4.16)
The second-order approximate solution (𝑚=2) is obtained from (3.8) 𝐹(𝑥)=𝐹₀(𝑥)+𝐹₁(𝑥)+𝐹₂(𝑥), (4.17) where 𝐹₀, 𝐹₁, and 𝐹₂ are given by (4.5), (4.11), and (4.16), respectively.

Case 2. In this case we consider 𝐾₁(𝑥)=𝐶₁, (4.18)𝐾₂(𝑥)=𝐶₂𝑥+𝐶₃, (4.19) where 𝐶₁, 𝐶₂, and 𝐶₃ are unknown constants.
It is clear that the function 𝐹₁ is given by (4.11). Equations (3.13) or (4.12) becomes 𝐹₂^′′′+8𝛼²Re²𝐶²₁𝑥15⁷−12𝛼²Re²+24𝛼²Re5𝐶²₁𝑥⁵−4𝛼Re𝐶₃𝑥⁴−4𝛼Re𝐶₁+𝐶₂+60𝛼Re−36𝛼²Re²−120𝛼³Re−80𝛼⁴𝐶15²¹𝑥³+4𝛼Re+2𝛼²𝐶₃𝑥²+4𝛼Re+2𝛼²𝐶₁+𝐶₂+60𝛼Re+12𝛼²−8𝛼²Re²−36𝛼³Re−40𝛼⁴𝐶15²¹𝐹𝑥=0,₂(0)=𝐹^′₂(0)=𝐹₂(1)=0 (4.20) and has the solution 𝐹₂(𝑥)𝛼=−²Re²𝐶²₁𝑥1350¹⁰+𝛼²Re²+2𝛼²Re𝐶140²¹𝑥⁸+2𝛼Re𝐶105³𝑥⁷+𝛼Re2𝐶₁+𝐶₂+3015𝛼Re+9𝛼²Re²+10𝛼³Re−4𝛼⁴𝐶450²¹𝑥⁶+𝛼Re+2𝛼²𝐶15³𝑥⁵+−𝛼Re+2𝛼²𝐶₁+𝐶₂6+2𝛼²Re²+9𝛼³Re+10𝛼⁴−15𝛼Re−3𝛼²𝐶90²¹𝑥⁴+2𝛼Re+5𝛼²𝐶15¹+𝐶₂+5𝛼Re+14𝛼²𝐶105³+2520𝛼Re−1932𝛼⁴+360𝛼²−919𝛼²Re−2580𝛼³Re𝐶18900²¹𝑥². (4.21)
The second-order approximate solution becomes 𝐹(𝑥)=𝐹₀(𝑥)+𝐹₁(𝑥)+𝐹₂(𝑥), (4.22) where 𝐹₀, 𝐹₁, and 𝐹₂ are given by (4.5), (4.11), and (4.21), respectively.

Case 3. In the third case we consider 𝐾₁(𝑥)=𝐶₁+𝐶₂𝐾𝑥,₂(𝑥)=𝐶₃+𝐶₄𝑥+𝐶₅𝑥². (4.23) Equation (3.12) for 𝑖=1 or (4.9) can be written as 𝐹₁^′′′−4𝛼Re𝐶₂𝑥⁴−4𝛼Re𝐶₁𝑥³+4𝛼Re+2𝛼²𝐶₂𝑥²+4𝛼Re+2𝛼²𝐶₁𝐹𝑥=0,₁(0)=𝐹^′₁(0)=𝐹₁(1)=0. (4.24) From (4.24) we have 𝐹₁(𝑥)=2𝛼Re𝐶₂𝑥105⁷+𝛼Re𝐶₁𝑥30⁶−𝛼Re+2𝛼²𝐶₂𝑥15⁵−𝛼Re+2𝛼²𝐶₁6𝑥⁴+𝛼Re+2𝛼²5𝐶₁+2𝐶₂−30𝛼Re7𝐶₁+4𝐶₂𝑥210². (4.25) Equation (3.13) becomes 𝐹₂^′′′(𝑥)+12𝛼²Re²𝐶35²²𝑥⁹+92𝛼²Re²𝐶₁𝐶₂𝑥105⁸+8𝛼²Re²𝐶²₁−156𝛼²Re²+12𝛼³Re5𝐶²₂𝑥⁷−18𝛼²Re²+36𝛼³Re5𝐶₁𝐶₂𝑥⁶−4𝛼Re𝐶₅+12𝛼²Re²+24𝛼³Re5𝐶²₁+12𝛼Re−2𝛼²Re²−8𝛼³Re−8𝛼⁴3𝐶²₂𝑥⁵−4𝛼Re𝐶₂+𝐶₄+120𝛼Re−46𝛼²Re²−160𝛼³Re−120𝛼⁴𝐶15¹𝐶₂−40𝛼²Re²+112𝛼³Re𝐶105²²𝑥⁴−4𝛼Re𝐶₁+𝐶₃−4𝛼Re+2𝛼²𝐶₃+60𝛼Re−36𝛼²Re²−120𝛼³Re−80𝛼⁴𝐶15²¹−40𝛼²Re²+112𝛼³Re𝐶105¹𝐶₂−4𝛼Re+2𝛼²𝐶²₂𝑥³+4𝛼Re+2𝛼²𝐶₂+𝐶₄+120𝛼Re+240𝛼²−8𝛼²Re²−36𝛼³Re−40𝛼⁴𝐶15¹𝐶₂+20𝛼²Re²+96𝛼³Re+112𝛼⁴𝐶105²²𝑥²+4𝛼Re+2𝛼²𝐶₁+𝐶₃+60𝛼Re+120𝛼²−8𝛼²Re²−36𝛼³Re−40𝛼⁴𝐶15²¹−20𝛼²Re²+96𝛼³Re+112𝛼⁴𝐶105¹𝐶₂𝑥,𝐹₂(0)=𝐹^′₂(0)=𝐹₂(1)=0. (4.26)
The solution of (4.26) is 𝐹₂(𝑥)𝛼=−²Re²𝐶²₂𝑥3850¹²−46𝛼²Re²𝐶₁𝐶₂𝑥51975¹¹+−𝛼²Re²𝐶²₁+𝛼1350²Re²+2𝛼³Re𝐶600²²𝑥¹⁰+𝛼²Re²+2𝛼³Re𝐶420¹𝐶₂𝑥⁹+𝛼Re𝐶₅+𝛼84²Re²+2𝛼³Re𝐶140²¹+6𝛼Re−𝛼²Re²−4𝛼³Re−4𝛼⁴𝐶504²²𝑥⁸+2𝛼Re𝐶₂+𝐶₄+10560Re−23𝛼²Re²−80𝛼³Re−60𝛼⁴𝐶1575¹𝐶₂−20𝛼²Re²+56𝛼³Re𝐶11025²²𝑥⁷+𝛼Re𝐶₁+𝐶₃−30𝛼Re+2𝛼²𝐶₅+3015𝛼Re−9𝛼²Re²−30𝛼³Re−20𝛼⁴𝐶450²¹+5𝛼²Re²+14𝛼³Re𝐶1575¹𝐶₂−𝛼Re+2𝛼²𝐶30²²𝑥⁶−𝛼Re+2𝛼²𝐶₂+𝐶₄+1530𝛼Re+60𝛼²−2𝛼²Re²−9𝛼³Re−10𝛼⁴𝐶225¹𝐶₂−5𝛼²Re²+24𝛼³Re+28𝛼⁴𝐶1575²²𝑥⁵−𝛼Re+2𝛼²𝐶₁+𝐶₃6+15𝛼Re+30𝛼²−2𝛼²Re²−9𝛼³Re−10𝛼⁴𝐶90²¹−5𝛼²Re²+24𝛼³Re+28𝛼⁴𝐶630¹𝐶₂𝑥⁴+2𝛼Re+5𝛼²𝐶₁+𝐶₃+155𝛼Re+14𝛼²𝐶₂+𝐶₄+1059𝛼Re+28𝛼²𝐶420⁵+2520𝛼Re−1260𝛼⁴+6300𝛼²−163𝛼²Re²−900𝛼³Re𝐶18900²¹+19800𝛼Re+55440𝛼²−1433𝛼²Re²−8514𝛼³Re−10560𝛼⁴𝐶207900¹𝐶₂−4774𝛼⁴−32340𝛼²−10395𝛼Re+2695𝛼³Re+380𝛼²Re²𝐶485100²²𝑥². (4.27)
The second-order approximate solution is 𝐹(𝑥)=𝐹₀(𝑥)+𝐹₁(𝑥)+𝐹₂(𝑥), (4.28) where 𝐹₀, 𝐹₁, and 𝐹₂ are given by (4.5), (4.24), and (4.26), respectively.

Case 4. In the last case, we consider 𝐾₁(𝑥)=𝐶₁+𝐶₂𝑥, (4.29)𝐾₂(𝑥)=𝐶₃+𝐶₄𝑥+𝐶₅𝑥²+𝐶₆𝑥³+𝐶₇𝑥⁴+𝐶₈𝑥⁵+𝐶₉𝑥⁶. (4.30)

The solution of 𝐹₁(𝑥) is given by (4.24). On the other hand, (3.13) has the solution 𝐹₂(𝑥)=𝛼Re𝐶330⁹−𝛼²Re²𝐶²₂𝑥3850¹²+2𝛼Re𝐶495⁸−46𝛼²Re²𝐶51975¹𝐶₂𝑥¹¹+𝛼Re𝐶₇−Re+2𝛼²𝐶₉−𝛼180²Re²𝐶1350²¹+𝛼²Re²+2𝛼³Re𝐶600²²𝑥¹⁰+𝛼Re𝐶₆−𝛼Re+2𝛼²𝐶₈+𝛼126²Re²+2𝛼³Re𝐶420¹𝐶₂𝑥⁹+𝛼Re𝐶₅−𝛼Re+2𝛼²𝐶₇+𝛼84²Re²+2𝛼³Re𝐶140²¹+6𝛼Re−𝛼²Re²−4𝛼³Re−4𝛼⁴𝐶504²²𝑥⁸+2𝛼Re𝐶₄−2𝛼Re+2𝛼²𝐶₆+1052𝛼Re𝐶₂+10560𝛼Re−23𝛼²Re²−80𝛼³Re−60𝛼⁴𝐶1575¹𝐶₂−20𝛼²Re²+56𝛼³Re𝐶11025²²𝑥⁷+𝛼Re𝐶₁+𝐶₃−𝛼Re+2𝛼²𝐶₅+3015𝛼Re−9𝛼²Re²−30𝛼³Re−20𝛼⁴𝐶450²¹−5𝛼²Re²+14𝛼³Re𝐶1575¹𝐶₂−𝛼Re+2𝛼²𝐶²₂𝑥30⁶+2𝛼²Re²+9𝛼³Re+10𝛼⁴−30𝛼Re−60𝛼²𝐶225¹𝐶₂−5𝛼²Re²+24𝛼³Re+28𝛼⁴𝐶1575²²−𝛼Re+2𝛼²𝐶₂+𝐶₄𝑥15⁵+2𝛼²Re²+9𝛼³Re+10𝛼⁴−15𝛼Re−30𝛼²𝐶90²¹+5𝛼²Re²+24𝛼³Re+28𝛼⁴𝐶630¹𝐶₂−𝛼Re+2𝛼²𝐶₁+𝐶₃6𝑥⁴+5𝛼Re+14𝛼²𝐶105²+𝐶₄+2𝛼Re+5𝛼²𝐶15¹+2𝛼Re+15𝛼²𝐶15³+9𝛼Re+28𝛼²𝐶420⁵+7𝛼Re+24𝛼²𝐶630⁶+4𝛼Re+15𝛼²𝐶630⁷+27𝛼Re+110𝛼²𝐶6930⁸+2520𝛼Re−1260𝛼⁴+6300𝛼²−163𝛼²Re²−900𝛼³Re𝐶18900²¹+5𝛼Re+22𝛼²𝐶1980⁹+10395𝛼Re−380𝛼²Re²+32340𝛼²−2685𝛼³Re−4774𝛼⁴𝐶485100²²+19800𝛼Re−1443𝛼²Re²+55440𝛼²−8514𝛼³Re−10560𝛼⁴𝐶207900¹𝐶₂𝑥². (4.31)

The second-order approximate solution in this case is given by 𝐹(𝑥)=𝐹₀(𝑥)+𝐹₁(𝑥)+𝐹₂(𝑥), (4.32) where 𝐹₀, 𝐹₁, and 𝐹₂ 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 𝐶_𝑖 and we will show that the error of the solution decreases when the number of terms in the auxiliary function ℎ(𝑥,𝑝) 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 Re=50 and 𝛼=5 in Case 1 it is obtained two solutions for the constants 𝐶₁ and 𝐶₂:(a)𝐶₁=0.017506079𝐶₂=−0.047286881, (b)𝐶₁=−0.017506079𝐶₂=0.022737435but the second-order approximate solution (4.16) is the same in both cases: 𝐹(𝑥)≈1−1.767845893𝑥²+1.236877181𝑥⁴−0.619019693𝑥⁶+0.164176501𝑥⁸−0.014188096𝑥¹⁰ (5.1)

Example 5.2. For Re=50 and 𝛼=5 in the Case 2, we obtain(a)𝐶₁=0.007977212𝐶₂=−0.041469373𝐶₃=0.022761122, (b)𝐶₁=−0.007977212𝐶₂=−0.009560525𝐶₃=0.022761122.
The second-order approximate solution (4.18) becomes 𝐹(𝑥)≈1−1.769527092𝑥²+1.40514047𝑥⁴−0.45522244𝑥⁵−0.319921792𝑥⁶+0.108386295𝑥⁷+0.03409066𝑥⁸−0.002946101𝑥¹⁰. (5.2)

Example 5.3. For Re=50 and 𝛼=5 in Case 3, the second-order approximate solution (4.27) can be written in the form 𝐹(𝑥)≈1−1.769777647𝑥²+1.295738478𝑥⁴−0.010406134𝑥⁵−0.871743272𝑥⁶+0.178832164𝑥⁷+0.299266794𝑥⁸−0.090606944𝑥⁹−0.040442155𝑥¹⁰+0.00935599𝑥¹¹−0.0002178𝑥¹². (5.3)

Example 5.4. For Re=80 and 𝛼=−5 in Case 4, the second-order approximate solution (4.31) becomes 𝐹(𝑥)≈1−0.399291819𝑥²−0.461970063𝑥⁴−0.014703786𝑥⁵−0.12415397𝑥⁶−0.07325724𝑥⁷−0.08278982𝑥⁸+0.45101379𝑥⁹−0.648015234𝑥¹⁰+0.47466473𝑥¹¹−0.121496588𝑥¹² (5.4)
The profile of the 𝐹(𝑥) function is presented in Figure 2 for Re=50 and 𝛼=5.
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 𝐹(𝑥) for Re=50, 𝛼=5.

Table 2 

Comparison between the OHAM and numerical solutions for Re=50 and 𝛼=5(𝑒𝑟𝑟𝑜𝑟=|𝐹(𝑥)_num−𝐹(𝑥)_app|).

Table 3 

Comparison between Differential Transformation Method (DTM) [4], Homotopy Perturbation Method (HPM) [4], Homotopy Analysis Method [4], and OHAM-(5.4) for Re=80, 𝛼=−5.

Table 4 

Comparison between OHAM (5.4) and numerical solutions [4] for Re=80, 𝛼=−5.

Figure 2 

Sample profile of the 𝐹(𝑥) function for Re=50 and 𝛼=5, 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 ℎ(𝑥,𝑝) involving several constants 𝐶_𝑖 which are optimally determined.

From the results presented above, we can conclude that the following. (1)When 𝛼>0 and steep of the channel is divergent, stream in value of Reynolds number is caused by decreasing in velocity.(2)When 𝛼<0 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.