A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem

doi:10.1016/j.compfluid.2010.03.004

Computers & Fluids

Volume 39, Issue 7, August 2010, Pages 1219-1225

https://doi.org/10.1016/j.compfluid.2010.03.004 Get rights and content

Abstract

In this paper a novel hybrid spectral-homotopy analysis technique developed by Motsa et al. (2009) and the homotopy analysis method (HAM) are compared through the solution of the nonlinear equation for the MHD Jeffery–Hamel problem. An analytical solution is obtained using the homotopy analysis method (HAM) and compared with the numerical results and those obtained using the new hybrid method. The results show that the spectral-homotopy analysis technique converges at least twice as fast as the standard homotopy analysis method.

Introduction

The problem of an incompressible, viscous fluid between non-parallel walls, commonly known as the Jeffery–Hamel flow, is an example of one of the most applicable type of flows in fluid mechanics [3]. Consequently, this problem has been well studied in literature, see for example, [6], [23] for a survey of the early literature on the Jeffery–Hamel problem. The classical Jeffery–Hamel problem was extended in [1] to include the effects of an external magnetic field on an electrically conducting fluid. The magnetic field acts as a control parameter and, beside the flow Reynolds number and the angle of the walls, in MHD Jeffery–Hamel problems there are two additional non-dimensional parameters that determine the solutions, namely the magnetic Reynolds number and the Hartmann number. Potentially therefore, a much larger variety of solutions than in the classical problem could be expected.

The majority of the convergent–divergent channel problems studied so far do not yield precise analytical solutions [3]. In addition to the various numerical methods currently available, it has always been an interesting problem and a challenge to devise new and more efficient algorithms, particularly analytical or semi-analytical techniques for finding solutions of nonlinear equations. The Jeffery–Hamel problem is well suited for testing such new solution techniques. Apart from using numerical techniques such as in [18], [22], recent approaches to solving the Jeffery–Hamel flow equations include perturbation techniques [21], the Adomian decomposition method (ADM) [3], [17], [18], the variational iteration method (VIM), the homotopy perturbation method (HPM) [5] and the homotopy analysis method (HAM) [11], [12], [4]. Approximate solutions of the Jeffery–Hamel problem were found using the variational iteration method and using the homotopy perturbation method [5], [7], [8], [9], [10]. A comparison was made with the numerical solution and the results showed that the homotopy perturbation method gives better accuracy compared to the variational iteration method. We note here that previous studies such as [13] have shown that the HPM is equivalent to the HAM when the auxiliary parameter $ℏ = - 1$ . In [18] the influence of an arbitrary magnetic Reynolds number on Jeffery–Hamel flow was studied using a perturbation series summation and improvement technique. However, as shown in [15], [16] and elsewhere, the HPM, VIM and other non-perturbation techniques are prone to give divergent approximations and so cannot be trusted completely.

The homotopy analysis method has been used successfully to solve a variety of nonlinear BVPs, see [12], [14] for a detailed exposition, and was used recently to study Jeffery–Hamel flow in the absence of an applied external magnetic field [4]. The HAM however suffers from a number of restrictive measures, such as the requirement that the solution sought ought to conform to the so-called rule of solution expression and the rule of coefficient ergodicity. In a recent study, Motsa et al. [20] proposed a spectral modification of the homotopy analysis method, the spectral-homotopy analysis method (SHAM) that seeks to remove some restrictive assumptions associated with the implementation of the standard homotopy analysis method. In this paper we extend the work reported in [4] by determining the exact analytical solution of the MHD Jeffery–Hamel problem when an external magnetic field is present using the homotopy analysis method. The nonlinear equation is then solved using the hybrid spectral-homotopy analysis method and a comparison of the HAM, SHAM and numerical results proves the applicability, accuracy and efficiency of the spectral-homotopy analysis method.

Section snippets

Mathematical formulation

Consider the steady two-dimensional flow of an incompressible conducting viscous fluid between two rigid plane walls that meet at an angle $2 α$ as shown in Fig. 1. The rigid walls are considered to be divergent if $α > 0$ and convergent if $α < 0$ . We assume that the velocity is purely radial and depends on r and $θ$ so that $v = (u (r, θ), 0)$ only. The continuity equation and the Navier–Stokes equations in polar coordinates are $\frac{ρ}{r} \frac{\partial}{\partial r} (ru (r, θ)) = 0,$ $u (r θ) \frac{\partial u (r, θ)}{\partial r} = - \frac{1}{ρ} \frac{\partial P}{\partial r} + ν [\nabla^{2} u (r, θ) - \frac{u (r, θ)}{r^{2}}] - \frac{σ B_{0}^{2}}{ρ r^{2}} u (r, θ),$ $0 = - \frac{1}{ρ r} \frac{\partial P}{\partial θ} + 2 ν$

Solution by the homotopy analysis method

To solve the nonlinear ordinary differential Eq. (5) using the Homotopy analysis method (HAM) we choose, see [4], the initial approximation $F_{0} (y) = 1 - y^{2},$ which satisfies the boundary conditions (6). Using the method of highest order differential matching we consider an auxiliary linear operator of the form $L [ϕ (y; q)] = \frac{\partial^{3} ϕ (y; q)}{\partial y^{3}},$ with the property $L [c_{1} + c_{2} y + c_{3} y^{2}] = 0,$ where $c_{1} – c_{3}$ are arbitrary integration constants.

Furthermore, the governing Eq. (5) suggest that we define the following nonlinear operator $N ϕ ($

Spectral-homotopy analysis solution

The spectral-homotopy analysis method developed by Motsa et al. [20] blends Chebyshev pseudospectral collocation methods with some aspects of the HAM described in the previous section. The advantage of this modification is that we get a technique that is more efficient and does not depend on the rule of solution expression and the rule of ergodicity unlike the standard HAM. In addition, the range of admissible $ℏ$ values is much wider in the SHAM. In applying the SHAM, we begin by transforming

Results and discussion

As pointed out by Liao [12], the accuracy and convergence of the HAM series solution depends on the careful selection of the auxiliary parameter $ℏ$ . In this study, the admissible values of $ℏ$ were chosen from the so-called $ℏ$ -curve in which $F^{″} (0)$ and $F^{'} (1)$ were considered to be independent variables and plotted against $ℏ$ . The valid region for $ℏ$ , where the series converges is the horizontal segment of each $ℏ$ -curve. The $ℏ$ -curves for $F^{″} (0)$ and $F^{'} (1)$ are shown in Fig. 2 which illustrates the effect of

Conclusion

In this paper we have used a novel spectral-homotopy analysis method and the standard homotopy analysis method to solve the 3rd order nonlinear differential equation that governs the hydromagnetic Jeffery–Hamel problem. An exact analytical solution of the nonlinear differential equation has been found using the homotopy analysis method. A comparison of the convergence rates of the SHAM and HAM shows that the SHAM converges more rapidly – up to three times faster than the HAM. Furthermore, in

Acknowledgements

The authors wish to acknowledge financial support from the National Research Foundation (NRF) and the University of Venda.

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A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem

Abstract

Introduction

Section snippets

Mathematical formulation

Solution by the homotopy analysis method

Spectral-homotopy analysis solution

Results and discussion

Conclusion

Acknowledgements

Phys Lett A

Int J Non-Linear Mech

Comput Methods Appl Mech Eng

Int J Non-linear Mech

Appl Math Comput

Commun Nonlinear Sci Numer Simul

Commun Nonlinear Sci Numer Simul

Comput Phys Commun

Appl Math Comput

Comput Math Appl

Commun Nonlinear Sci Numer Simulat

Fluid Dyn Res