A novel semi-analytical solution to Jeffery-Hamel equation

A new approach based on the Adomian decomposition and the Fourier transform is introduced. The method suggests a solution for the well-known magneto-hydrodynamic (MHD) Jeffery-Hamel equation. Results of Adomian decomposition method combined with Fourier transform are compared with exact and numerical methods. The FTADM as an exclusive and new method satisfies all boundary and initial conditions over the entire spatial and temporal domains. Moreover, using the FTADM leads to rapid approach of approximate results toward the exact solutions is demonstrated. The second derivative of Jeffery-Hamel solution related to the similar number of items of recursive terms under a vast spatial domain shows the maximum error in the order of 10 − 5 comparing to exact and numerical solutions. The results also imply that the FTADM can be considered as a precise approximation for solving the third-order nonlinear Jeffery-Hamel equations.


Introduction
The Jeffery-Hamel mathematical foundation is the most suitable way of modeling the incompressible viscous fluid flow between two inclined plates under a magnetic field. Generally, engineering applications such as fluid mechanics and environmental science utilize Jeffery-Hamel flows. Because of its significant applications in liquid metals-based cooling systems' designation, magnetohydrodynamic generators, accelerators, pumps, and flow meters are of major interest of researchers [1][2][3][4]. For MHD Jeffery-Hamel flow, Jeffery [5] and Hamel [6] formulated a mathematical foundation for the problem solution in detail. To be or not to be an external magnetic field applied on the fluid flow leads to quite different behaviors of a conducting fluid flowing [7,8]. Controlling of the fluid flow by variations of the external magnetic field intensity is the main reason of the significance of the Jeffery-Hamel problems' study. Moreover, the approach of the problem solution must include the non-dimensional factors (the magnetic Reynolds and the Hartmann numbers) as well as the angle of the plates. Therefore, a wide range of the solutions exists comparing to classical problem. Heretofore, to obtain an approximate solution for classical Jeffery-Hamel flow equation, a few of approaches are suggested [9,10]. A variety of approaches such as the homotopy perturbation [11,12], the differential transformation [13] and homotopy analysis  are introduced to obtain analytical solution for the nonlinear problems, because of its strong nonlinearity so far.
The aim of the present work is to introduce a precise analytical solution for the third-order nonlinear Jeffery-Hamel equation by using FTADM [39]. The exclusivity of the FTADM, as a new method, is the satisfaction of all boundary and initial conditions over the entire spatial and temporal domains. Moreover, the well-known strongly nonlinear third-order Jeffery-Hamel type equations are solved using the FTADM where the trend of rapid approach of the approximate results toward the exact solutions is demonstrated. We solve the problem with the Reynolds number equal to 10 and the plates with the angles of a =   5 and a set of the Hartmann numbers includes: 0, 200, 400, 800, 1000, 2000. Furthermore, we investigate the Jeffery-Hamel problem's solutions by changing the values of the effective physical factors.

The Adomian decomposition method
The MHD Jeffery-Hamel problem, includes a confined fluid flow between two inclined sheets. The configuration of the flow depicts in figure 1.
ADM is introduced by Adomian [40][41][42]. The Solution of a differential equations (ordinary or partial) with the ADM is given as follows. Suppose a differential equation as below: where G is general operator. G could be separated into two individual linear and the nonlinear operators: The unknown function ( ) u x t , of the linear operator may be replaced by a series solution [43]: The recursive equations give the solution components (u , n  n 0). By equation (3), we can rewrite the linear part in the below form: Adomian polynomials can convert the nonlinear part into the infinite series as below [44][45][46][47][48][49][50]: With some algebra, equation (5) may be rewritten as:

Fundamentals of FTADM
Fourier transform of both sides of equation (7) is: Adomian polynomials, A , n have the below forms: The recursive relations, in light of equation (8), are: The last relations may be expand as: Here, we describe the solution step by step. Maple software package gives the value of  u 0 for the first part of equation (11). Then, we obtain the value of u 0 by inverse Fourier transform applying. In light of the first part of equation (9), it defines the Adomian polynomial, A . 0 The second part of equation (11) and Adomian polynomial A 0 give the value of  u . 1 the value of u , 1 arises by the inverse Fourier transform to  u . 1 It leads to define the Adomian polynomial A 1 based on the second part of equation (9) and so on. This in turn results the complete evaluation of the solution components u k upon using different corresponding parts of equations (5) and (11).

FTADM application on Jeffery-Hamel problem
Demonstration of the merits and validation of FTADM method are examined by solving the one-dimension, third-order and nonlinear Jeffery-Hamel problem in throughout of the domain. An accurate agreement exists between the solution of the Navier-Stokes equations, in the example of two-dimensional flow inside a channel with tilted sheets having a vertex and existence a source or sink, and the Jeffery Hamel problem. The form of the one-dimensional Jeffery-Hamel equation is [4,11]: where a is the angle between the two sheets, Re and H are the Reynolds and the Hartmann numbers, respectively. Solving the equation (12) is done with considering the Dirichlet and Neumann boundary conditions as below:

A precise analytical approximation approach (FTADM)
The Fourier transform of equation (12) gives: where F denotes the FT. The integration by parts in the equation (14) leads to: Replacing the equation (15) into the equation (14) gives: For solving the equation (16), we need to have ( )  f 0 . The left side of equation (12) undergoes the triple definite integration with the boundary condition ( ) = f 0 1,we have [11,12]: By taking the integration by parts, we rewrite equation (17) as: By applying the boundary conditions (equation (13)) and reordering the terms, the formulation for ( )  f 0 is: Replacing the equation (19) into the equation (16) and apply the boundary conditions defined by equation (13), we have: Utilizing of equations (12) and (18), (20) may be rewritten as follows: Using equation (21), we establish the recursive equations as below:  Then applying the FT to second part of equation (9), A , 1 gives the value of  A . 1 Having the values of f , 1 A 1 and  A 1 will define the fourth part of equation (22). Having the values of f , 1 A , 1  A , 1 the fourth part of equation (22) will provide the value of  f . 2 Then, applying the inverse FT provides the value of f 2 and so on. This in turn leads to the complete evaluation of the components of f , k  k 0, upon using different corresponding parts of equations (9) and (22). Solving the recursive equation, equation (22), gives:    a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a  a   +  -+  +  --+  +  -+  +  --+  +  -+  +  +  ----+  +  + -----+  -----+  + -   and so on.

Results and comparisons
By comparing the results of our method and well-known and valuable results of [8], we show the validation and accuracy of our method. Tables 1 and 2 depict our results for ( )  f 0 and the true value and the numerical data [8]   for six various Hartmann numbers and Re=10 while a =   5 , respectively. Obviously, there is a good similarity between the (FTADM) data, the true value and the numerical data [8]. Tables 3-12

Conclusions
In this paper, a new modification of the ADM, the Fourier transform Adomian decomposition method (FTADM), is proposed to solve the nonlinear MHD Jeffery-Hamel equation. The comparison of our results for ( )  f 0 at various Hartmann numbers obtained via the FTADM with the exact and numerical data [8] show  excellent agreement. The comparison of velocity obtained using the FTADM for Re=10 and a =   5 at different Hartmann numbers with those obtained by the numerical Runge-Kutta method show excellent agreements and the maximum error is on the order of -10 . 5 We conclude that the new FTADM is a capable and precise approximate semi-analytical approach for solving the nonlinear MHD Jeffery-Hamel equation.