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The magnetohydrodynamic Jeffery-Hamel flow is studied analytically. The traditional Navier-Stokes 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

The flow of fluid through a divergent channel is called Jeffery-Hamel flow since introducing this problem by Jeffery [

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 high-performance computers and computation software. The small disturbance stability of Magnetohydrodynamic stability of plane Poiseuille flow has been investigated by Makinde and Motsa [

Geometry of the MHD Jeffery-Hamel flow.

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 [

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 [

Jafari and Daftardar-Gejji [

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.

Consider a system of cylindrical polar coordinates

Consider equation

In (

According to (

Operating with

and

Using

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

Value of

Re | Ha | Ha | Ha | |

25 | 2.5 | −2.42483 | −2.54507 | −2.66789 |

5 | −3.09259 | −3.57685 | −4.09816 | |

7.5 | −3.98607 | −5.14061 | −6.44257 | |

50 | 2.5 | −2.77059 | −2.87965 | −2.99167 |

5 | −3.85619 | −4.25077 | −4.69303 | |

7.5 | −5.16973 | −6.04706 | −7.13483 |

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

Comparison between the numerical results and ADM solution for velocity when

Ha | Ha | Ha | Ha | |||||||||

Numerical | ADM | %Error | Numerical | ADM | Error | Numerical | ADM | Error | Numerical | ADM | %Error | |

0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |

0.1 | 0.986671 | 0.9866365 | 0.000035 | 0.987488 | 0.9867671 | 0.00073 | 0.988606 | 0.9870331 | 0.001591 | 0.99022 | 0.992695 | 0.002475 |

0.2 | 0.947258 | 0.9471292 | 0.000136 | 0.950396 | 0.9477282 | 0.002807 | 0.9547 | 0.9488372 | 0.006141 | 0.960933 | 0.970544 | 0.009611 |

0.3 | 0.883419 | 0.8831778 | 0.000273 | 0.890009 | 0.884749 | 0.00591 | 0.899076 | 0.8873853 | 0.013003 | 0.912273 | 0.912273 | 0.006304 |

0.4 | 0.797697 | 0.7973476 | 0.000438 | 0.808288 | 0.8005834 | 0.009532 | 0.822925 | 0.8132968 | 0.0117 | 0.844383 | 0.832683 | 0.009146 |

0.5 | 0.693233 | 0.6928205 | 0.000595 | 0.70764 | 0.6984237 | 0.013024 | 0.727664 | 0.7175749 | 0.013865 | 0.757286 | 0.743421 | 0.010234 |

0.6 | 0.573424 | 0.5730174 | 0.000709 | 0.590631 | 0.5813941 | 0.015639 | 0.614709 | 0.6086584 | 0.009843 | 0.650719 | 0.643816 | 0.006902 |

0.7 | 0.441593 | 0.4412649 | 0.000743 | 0.459695 | 0.4520834 | 0.016558 | 0.485232 | 0.4810561 | 0.008606 | 0.523909 | 0.515303 | 0.018548 |

0.8 | 0.300674 | 0.300475 | 0.000662 | 0.316855 | 0.3121316 | 0.014907 | 0.33989 | 0.3351125 | 0.014056 | 0.37529 | 0.361234 | 0.013709 |

0.9 | 0.152979 | 0.1529137 | 0.000427 | 0.163467 | 0.1618745 | 0.009742 | 0.178555 | 0.1772346 | 0.007395 | 0.202125 | 0.19473 | 0.009368 |

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

%Error of ADM for

Figures

The ADM solution for velocity in divergent channel for

The ADM solution for velocity in divergent channel for

The ADM solution for velocity in divergent channel for

Under magnetic field the Lorentz force effect is in 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 (

Figures

The ADM solution for velocity in divergent channel for

The ADM solution for velocity in divergent channel for

The ADM solution for velocity in divergent channel for

The ADM solution for velocity in divergent channel for

By increasing Reynolds number the backflow expands and so greater magnetic field is needed in order to eliminate it. As shown in Figures

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.

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.

Increasing Reynolds numbers leads to adverse pressure gradient which causes velocity reduction near the walls.

Increasing Hartmann number will lead to backflow reduction. In greater angles or Reynolds numbers high Hartmann number is needed for the reduction of backflow.

Magnetic field (

Constant

Linear term

Nonlinear term

The remainder of linear operator

General nonlinear operator

Adomian polynominal

Dimensionless velocity

Hartmann number

Density

Pressure term

Reynolds number

Cylindrical coordinates

Maximum value of velocity

Velocity components along

Kinematic viscosity

Viscous stresses

Angle of the channel

Any angle

Dimensionless angle.