On the expedient solution of the magneto-hydrodynamic Jeffery-Hamel flow of Casson fluid

The equation of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel is derived and solved using a new modified Adomian decomposition method (ADM). So far in all problems where semi-analytical methods are used the boundary conditions are not satisfied completely. In the present research, a hybrid of the Fourier transform and the Adomian decomposition method (FTADM), is presented in order to incorporate all boundary conditions into our solution of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel flow. The effects of various emerging parameters such as channel angle, stretching/shrinking parameter, Casson fluid parameter, Reynolds number and Hartmann number on velocity profile are considered. The results using the FTADM are compared with the results of ADM and numerical Range-Kutta fourth-order method. The comparison reveals that, for the same number of components of the recursive sequences over a wide range of spatial domain, the relative errors associated with the new method, FTADM, are much less than the ADM. The results of the new method show that the method is an accurate and expedient approximate analytic method in solving the third-order nonlinear equation of Jeffery-Hamel flow of non-Newtonian Casson fluid.


Mathematical Formulation
We consider the two dimensional MHD flow of an incompressible non-Newtonian Casson fluid from a source or sink between two stretching or shrinking walls. Angle between the walls is 2α. Where α < 0, α > 0 represent the convergent walls and divergent walls, respectively. Figure (1) shows the geometry of the MHD Jeffery-Hamel flow. The walls stretch/shrink at a rate s with radial distance of r and the velocity at the walls is: The constitutive equation describing an incompressible non-Newtonian Casson fluid is written as follow 22 : In Eq. (2), where μ B denotes the plastic dynamic viscosity of the non-Newtonian fluid, P y is the yield stress of the fluid, π is the product of the component of deformation rate with itself, namely, π = e ij e ij where e ij is the (i, j) th component of the deformation rate and π c is a critical value of this product based on the non-Newtonian model. We assume that the flow is purely radial and there is no magnetic field in the z-direction. The conservation laws of mass and momentum for this problem are: Consider the previously mentioned assumptions, Eqs (2)(3)(4) reduce to:  Table 1. Comparison between the present results of f ″(0) with the exact and numerical results of Abbasbandy 5 for different values of Hartmann numbers of α = +5°, β = ∞, Re = 10 and C = 0.
where, u is the velocity component in radial direction, p is the pressure, υ is kinematic viscosity, β =   where, u c is the centerline rate of movement, u w is the velocity at the channel walls and s represent the stretching/ shrinking rate. From Eq. (5): Using dimensionless parameters, c By eliminating the pressure terms from Eqs (6) and (7) and then using Eqs (10) and (11), we obtain the following third-order nonlinear differential equation: Subject to the boundary conditions, where α is the angle between the two planes, β is the Casson fluid parameter and = c Values for skin friction coefficient, defined as:

The Adomian Decomposition Method
The Adomian decomposition method (ADM) was created by Adomian [29][30][31][32] . The basic idea of the ADM for solving ordinary and partial differential equations is as follows. Assume a differential equation in the following form: where G is an arbitrary operator. The operator G may generally be partitioned into two separate operators, the linear and the nonlinear operators as, n n 0 However, the nonlinear term, in Eq. (17), may be expressed as an infinite series using the Adomian polynomials as [34][35][36][37][38][39][40] :   The components of u can be easily obtained by solving the recursive equations obtained from Eq. (22).

Basic idea of the FTADM
Taking the Fourier transform of both sides of Eq. (22), we obtain 28 : where the Adomian polynomials, A n are:

Case Study of the Jeffery-Hamel problem
We solve the one-dimensional third-order nonlinear differential equation of the Jeffery-Hamel flow of non-Newtonian Casson fluid to demonstrate the effectiveness and the validity of the FTADM method, over the entire domain. The aforesaid equation in the one-dimensional case is written as: Equation (27) is solved subject to the mixed set of Dirichlet and Neumann boundary conditions as: Where, C is the stretching or shrinking parameter.
The FTADM as an accurate approximate analytic method. Taking the Fourier transform of Eq. (27), we obtain the following: Where F stands for the Fourier transform. Using the method of integration by parts, the Fourier transform of each individual part in Eq. (29) takes the following form:   Using integration by parts, Eq. (32) may be rewritten as: Upon using the boundary conditions given in Eq. (28) and rearranging terms, we obtain the relation for f ″(0) as:      Where the hat symbol denotes the Fourier transform. Using Eq. (36), we construct the following recursive equations: where   Tables  (1) and (2) it is clear that there is an excellent agreement between the present results (FTADM), the exact solutions and numerical results of Abbasbandy 5 . Table 3 shows comparison of the present results of velocity obtained from the FTADM with the numerical Runge-Kutta fourth-order results for convergent and divergent channels.
In addition, Fig. 2 displays comparison of the FTADM results and the numerical Runge-Kutta fourth-order results for Re = 10, α = 5°, β = 0.5, H = 800, C = 0 graphically. Our comparison shows excellent agreement with the numerical Runge-Kutta results. Table 4 gives a comparison of the convergence rate of the FTADM at different number of truncated terms (N) against the numerical approximations. The relative errors given in Table 4, is evaluated as follow: A reasonable trend of approach and excellent agreement of our solution with the numerical solution are evident. Table 5 shows a comparison between relative errors of analytical solutions obtained by the ADM and the FTADM methods. It is evident from Table 5 that for the same number of truncated terms (N = 6), the maximum relative error associated with the ADM is in the order of 10 −4 , while the maximum relative error associated with the FTADM is in the order of 10 −8 . Figure 3 depicts the comparison of the errors of ADM with FTADM results of velocity for different recursive components. According to this figure, the errors associated with the FTADM are much less than the ordinary ADM.  4, 5, 6 and 7 demonstrate the effect of stretching/shrinking parameter on the fluid velocity profile in the convergent and divergent channel, respectively. It is clear that for stretching channel, velocity profile increases with the increasing of stretching parameter. In the shrinking channel, the velocity profile decreases due to an increase in the absolute value of shrinking parameter. In the case of stretching divergent channel, with increasing stretching parameter or increasing Reynolds number the velocity increases. This nonlinear increment in velocity is probably due to more drag force acting on the plate at large values of stretching parameter. Figures 8 and 9 depict the variations of the fluid velocity with an opening angle for stretching/shrinking divergent/convergent channels. According to these figures, the velocity profiles act as a decreasing function of opening angle for stretching/shrinking divergent channel. In the stretching/shrinking convergent channel, velocity profile increases with the increase of absolute value of opening angle. Figures 10 and 11 illustrate the effect of Casson fluid parameter on the fluid velocity profiles in the divergent and convergent channels, respectively. In the divergent channel, as the Casson fluid parameter increases the velocity profiles decrease significantly. On the other word, Fig. 10 shows that higher values of Casson fluid parameter have the tendency to decelerate the velocity of fluid flow. It is expected that an increase in Casson fluid parameter is used to decrease the stress that increases the value of dynamic viscosity, thereby produce a resistance in the fluid flow. It is interesting to mention that when the Casson fluid parameter increases indefinitely, the problem reduces to a Newtonian fluid case. In the convergent channel, an opposite behavior is observed. The effect of Reynolds number on the fluid velocity is shown in Figs 12 and 13. The graphs show that in stretching/ shrinking divergent channels, as the Reynolds number increases the fluid velocity decreases. An opposite trend is seen for stretching/shrinking convergent channels, where the velocity is an increasing function of Reynolds number. nents of the recursive sequence are compared with those obtained by the ADM. The comparison shows that the relative errors associated with the FTADM are much less than the ADM. • We conclude that FTADM is more accurate than ADM and therefore the FTADM is an effective and expedient approximate semi-analytical method for solving the nonlinear equation of MHD Jeffery-Hamel flow of non-Newtonian Casson fluid. • Our results show that in the case of stretching divergent channel, with increasing stretching parameter or increasing Reynolds number the velocity increases. This nonlinear increment in velocity is probably due to more drag force acting on the plate at large values of stretching parameter.