%0 Journal Article
%T A Note on Improved Homotopy Analysis Method for Solving the Jeffery-Hamel Flow
%A Sandile Sydney Motsa
%A Precious Sibanda
%A Gerald T. Marewo
%A Stanford Shateyi
%J Mathematical Problems in Engineering
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/359297
%X This paper presents the solution of the nonlinear equation that governs the flow of a viscous, incompressible fluid between two converging-diverging rigid walls using an improved homotopy analysis method. The results obtained by this new technique show that the improved homotopy analysis method converges much faster than both the homotopy analysis method and the optimal homotopy asymptotic method. This improved technique is observed to be much more accurate than these traditional homotopy methods. 1. Introduction The mathematical study of the flow of a viscous incompressible two-dimensional fluid in a wedge-shaped channel with a sink or source at the vertex was pioneered by Jeffery [1] and Hamel [2]. The problem has since been studied extensively by, among others, Axford [3] who included the effects of an externally applied magnetic field and Rosenhead [4] who obtained a general solution containing elliptic functions. Instability and bifurcation are other aspects of the Jeffery-Hamel problem that have attracted widespread interest; see, for example, Akulenko and Kumakshev [5, 6]. Three-dimensional extensions to and bifurcations of the Jeffery-Hamel flow have been made by Stow et al. [7] while McAlpine and Drazin [8] presented a normal mode analysis of two-dimensional perturbations of a viscous incompressible fluid driven between inclined plane walls by a line source at the intersection of the walls. Banks et al. [9] investigated various perturbations and the linear temporal stability of such flows and found evidence of a strong interaction between the disturbances up- and down-stream if the angle between the planes exceeds a certain Reynolds-number-dependent critical value. Makinde and Mhone [10] investigated the temporal stability of MHD Jeffery-Hamel flows. They showed that an increase in the magnetic field intensity has a strong stabilizing effect on both diverging and converging channel geometries. A review of the theory of instabilities and bifurcations in channels is given by Drazin [11]. As with most problems in science and engineering, the equations governing the Jeffery-Hamel problem are highly nonlinear and so generally do not have closed form analytical solutions. Nonlinear equations can, in principle, be solved by any one of a wide variety of numerical methods. However, numerical solutions rarely give intuitive insights into the effects of various parameters associated with a problem. Consequently, most recent studies of flows in diverging and converging channels have centred on the use of the Jeffery-Hamel flow equations as a testing and
%U http://www.hindawi.com/journals/mpe/2010/359297/