A Note on the Solution of the Von Kármán Equations Using Series and Chebyshev Spectral Methods

  • ZodwaG Makukula1,

    Affiliated with

    • Precious Sibanda1Email author and

      Affiliated with

      • SandileSydney Motsa2

        Affiliated with

        Boundary Value Problems20102010:471793

        DOI: 10.1155/2010/471793

        Received: 23 March 2010

        Accepted: 2 October 2010

        Published: 4 October 2010

        Abstract

        The classical von Kármán equations governing the boundary layer flow induced by a rotating disk are solved using the spectral homotopy analysis method and a novel successive linearisation method. The methods combine nonperturbation techniques with the Chebyshev spectral collocation method, and this study seeks to show the accuracy and reliability of the two methods in finding solutions of nonlinear systems of equations. The rapid convergence of the methods is determined by comparing the current results with numerical results and previous results in the literature.

        1. Introduction

        Most natural phenomena can be described by nonlinear equations that, in general, are not easy to solve in closed form. The search for computationally efficient, robust, and easy to use numerical and analytical techniques for solving nonlinear equations is therefore of great interest to researchers in engineering and science. The study of the steady, laminar, and axially symmetric viscous flow induced by an infinite disk rotating steadily with constant angular velocity was pioneered by von Kármán [1]. He showed that the Navier-Stokes equations could be reduced to a set of ordinary differential equations and solved using an approximate integral method. His solution, however, contained errors that were later corrected by Cochran [2] by patching together two series expansions.

        Numerical and semianalytical methods including the cubic Hermite finite element, pseudospectral, Galerkin-B-Spline, and Chebyshev-collocation methods have been used previously to find solutions of the von Kármán equations [36]. These methods have their shortcomings, including instability, and hence the last few decades have seen the popularization of a number of new perturbation or nonperturbation techniques such as the Adomian decomposition method [7], the Lyapunov artificial small parameter method [8], the homotopy perturbation method [9, 10], and the homotopy analysis method [11].

        The homotopy analysis method (HAM) was used recently by Yang and Liao [12] to find explicit, purely analytic solutions of the swirling von Kármán equations. Turkyilmazoglu [13] used the homotopy analysis method to solve the equations governing the flow of a steady, laminar, incompressible, viscous, and electrically conducting fluid due to a rotating disk subjected to a uniform suction and injection through the walls in the presence of a uniform transverse magnetic field. For this extended form of the von Kármám problem, the homotopy analysis method, however, produced secular terms in the series solution. Turkyilmazoglu [13] overcame this weakness by using initial guesses based on Ackroyd's (see the work of Ackroyd [14]) exponentially decaying functions, and a new linear operator which resulted in a method capable of tracking the shape of the exact solution. An alternative approach that serves to address these and other limitations of the HAM is the spectral homotopy analysis method; see the work of Motsa et al. [15, 16]. It is an efficient hybrid method that blends the HAM algorithm with Chebyshev spectral methods. The method retains the proven qualities of the HAM while effectively using Chebyshev polynomials as basis functions to ensure rapid convergence of the solution series. A novel quasilinearisation method—the successive linearisation method (see the work of Makukula et al. [17] and Motsa and Sibanda [18])—promises further improvement in accuracy and convergence rates compared to both the HAM and the SHAM.

        In this study we apply the spectral homotopy analysis method (SHAM) and the successive linearisation method (SLM) to solve the von Kármán equations. The results are compared with those in the literature [11, 12] and against numerical approximations. Comparison of current results is further made with the recent results of Turkyilmazoglu [13] that include suction/injection and an applied magnetic field. We show, inter alia, that notwithstanding the fact that these two methods may involve more computations per step than the HAM, both the SHAM and SLM are efficient, robust, and converge much more rapidly compared to the standard homotopy analysis method.

        2. Governing Equations

        Our focus in this section is on the original von Kármán equation for the steady, laminar, axially symmetric viscous flow induced by an infinite disk rotating steadily with angular velocity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq1_HTML.gif about the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq2_HTML.gif -axis with the fluid confined to the half-space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq3_HTML.gif above the disk. In cylindrical coordinates http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq4_HTML.gif the equations of motion are given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ1_HTML.gif
        (2.1)

        Boundary Value Problems

        subject to the nonslip boundary conditions on the disk and boundary conditions at infinity
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ2_HTML.gif
        (2.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq5_HTML.gif is the fluid density, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq6_HTML.gif is the kinematic viscosity coefficient, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq7_HTML.gif is the pressure, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq9_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq10_HTML.gif are the velocity components in the radial, azimuthal, and axial directions, respectively, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq11_HTML.gif is the constant angular velocity. Using von Kármán's similarity transformations [1]
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ3_HTML.gif
        (2.3)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq12_HTML.gif is a nondimensional distance measured along the axis of rotation, the governing partial differential equations (2) reduce to a set of ordinary differential equations:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ4_HTML.gif
        (2.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ5_HTML.gif
        (2.5)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ6_HTML.gif
        (2.6)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ7_HTML.gif
        (2.7)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ8_HTML.gif
        (2.8)
        Substituting (2.7) into (2.4) and (2.5) yields
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ9_HTML.gif
        (2.9)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ10_HTML.gif
        (2.10)

        Equations (2.9) with the prescribed boundary conditions (2.10) are sufficient to give the three components of the flow velocity. The pressure distribution, if required, can be obtained from (2.6). This fully coupled and highly nonlinear system was solved using the spectral homotopy analysis method and the successive linearisation method. The results were validated using the Matlab bvp4c numerical routine and against results in the literature.

        3. The Spectral Homotopy Analysis Method

        Following Boyd [19], we begin by transforming the domain of the problem from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq13_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq14_HTML.gif using the domain truncation method. This approximates http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq15_HTML.gif by the computational domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq16_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq17_HTML.gif is a fixed length that is taken to be larger than the thickness of the boundary layer. The interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq18_HTML.gif is then transformed to the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq19_HTML.gif using the algebraic mapping
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ11_HTML.gif
        (3.1)
        For convenience we make the boundary conditions homogeneous by applying the transformations
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ12_HTML.gif
        (3.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq21_HTML.gif are chosen so as to satisfy boundary conditions (2.10). The chain rule gives
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ13_HTML.gif
        (3.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ14_HTML.gif
        (3.4)
        Substituting (3.2) and (3.3)-(3.4) in the governing equations gives
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ15_HTML.gif
        (3.5)
        where prime denotes derivative with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq22_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ16_HTML.gif
        (3.6)
        As initial guesses we employ the exponentially decaying functions used by Yang and Liao [12], namely,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ17_HTML.gif
        (3.7)
        The initial solution is obtained by solving the linear parts of (3.5), namely,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ18_HTML.gif
        (3.8)
        subject to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ19_HTML.gif
        (3.9)
        The system (3.8)-(3.9) is solved using the Chebyshev pseudospectral method where the unknown functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq24_HTML.gif are approximated as truncated series of Chebyshev polynomials of the form
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ20_HTML.gif
        (3.10)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq26_HTML.gif are the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq27_HTML.gif th Chebyshev polynomials with coefficients http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq29_HTML.gif , respectively, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq30_HTML.gif are Gauss-Lobatto collocation points defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ21_HTML.gif
        (3.11)
        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq31_HTML.gif is the number of collocation points. Derivatives of the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq32_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq33_HTML.gif at the collocation points are represented as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ22_HTML.gif
        (3.12)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq34_HTML.gif is the order of differentiation and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq35_HTML.gif is the Chebyshev spectral differentiation matrix (see, e.g., [20, 21]). Substituting (3.10)–(3.12) in (3.8)-(3.9) yields
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ23_HTML.gif
        (3.13)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ24_HTML.gif
        (3.14)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ25_HTML.gif
        (3.15)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ26_HTML.gif
        (3.16)

        The superscript http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq36_HTML.gif denotes the transpose, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq37_HTML.gif is a diagonal matrix, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq38_HTML.gif is an identity matrix of size http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq39_HTML.gif . We implement boundary conditions (3.14) in rows 1, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq40_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq41_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq42_HTML.gif in columns 1 through to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq43_HTML.gif by setting all entries in the remaining columns to be zero. The second set (3.15) is implemented in rows http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq45_HTML.gif , respectively, by setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq47_HTML.gif and setting all other columns to be zero. We further set entries of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq48_HTML.gif in rows http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq51_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq52_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq53_HTML.gif to zero.

        The values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq54_HTML.gif are determined from the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ27_HTML.gif
        (3.17)

        which provides the initial approximation for the solution of (3.5).

        We now seek the approximate solutions of (3.5) by first defining the following linear operators:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ28_HTML.gif
        (3.18)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq55_HTML.gif is the embedding parameter and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq57_HTML.gif are unknown functions. The zero th-order deformation equations are given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ29_HTML.gif
        (3.19)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq58_HTML.gif is the nonzero convergence controlling auxiliary parameter and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq60_HTML.gif are nonlinear operators given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ30_HTML.gif
        (3.20)
        The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq61_HTML.gif th-order deformation equations are given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ31_HTML.gif
        (3.21)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ32_HTML.gif
        (3.22)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ33_HTML.gif
        (3.23)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ34_HTML.gif
        (3.24)
        Applying the Chebyshev pseudospectral transformation to (3.21)–(3.23) gives
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ35_HTML.gif
        (3.25)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ36_HTML.gif
        (3.26)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq62_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq63_HTML.gif are as defined in (3.16) and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ37_HTML.gif
        (3.27)
        Boundary conditions (3.26) are implemented in matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq64_HTML.gif on the left-hand side of (3.25) in rows http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq65_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq66_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq68_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq69_HTML.gif , respectively, as before with the initial solution above. The corresponding rows, all columns, of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq70_HTML.gif on the right-hand side of (3.25), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq72_HTML.gif are all set to be zero. This results in the following recursive formula for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq73_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ38_HTML.gif
        (3.28)

        The matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq74_HTML.gif is the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq75_HTML.gif on the right-hand side of (3.25) but with the boundary conditions incorporated by setting the first, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq77_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq78_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq79_HTML.gif , rows and columns to zero. Thus, starting from the initial approximation, which is obtained from (3.17), higher-order approximations http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq80_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq81_HTML.gif can be obtained through recursive formula (3.28).

        4. Successive Linearisation Method

        The spectral homotopy analysis method, just like the original HAM, depends for its convergence rate on the careful selection of an embedded arbitrary parameter http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq82_HTML.gif . Turkyilmazoglu [13] showed that the solution of the von Kármán problem by the homotopy analysis method is prone to wild oscillations when suction/injection is present. In this section we apply the successive linearisation method that requires no artificial parameters to control convergence to solve the governing equations (2.9)-(2.10). The method assumes that the unknown functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq84_HTML.gif can be expanded as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ39_HTML.gif
        (4.1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq86_HTML.gif are unknown functions and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq88_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq89_HTML.gif ) are approximations that are obtained by recursively solving the linear part of the equation system that results from substituting (4.1) in the governing equations (2.9)-(2.10). Substituting (4.1) in the governing equations gives
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ40_HTML.gif
        (4.2)
        where the coefficient parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq91_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq92_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq93_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq94_HTML.gif are defined as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ41_HTML.gif
        (4.3)
        To facilitate direct comparison of the methods, we use the same initial approximations as in the case of the spectral homotopy analysis method of Yang and Liao [12]:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ42_HTML.gif
        (4.4)
        The solutions for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq95_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq96_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq97_HTML.gif , are obtained by successively solving the linearized form of (4.2), namely,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ43_HTML.gif
        (4.5)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ44_HTML.gif
        (4.6)
        Once each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq99_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq100_HTML.gif ) has been found, the approximate solutions for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq102_HTML.gif are obtained as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ45_HTML.gif
        (4.7)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq103_HTML.gif is the order of the SLM approximation. In coming up with (4.7), we have assumed that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ46_HTML.gif
        (4.8)
        Equations (4.5)-(4.6) can be solved using analytical techniques (whenever possible) or any numerical method. In this work the equations were solved using the Chebyshev spectral collocation method in the manner described in the previous section. This leads to the matrix equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ47_HTML.gif
        (4.9)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq104_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq105_HTML.gif square matrix and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq107_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq108_HTML.gif column vectors defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ48_HTML.gif
        (4.10)
        with
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ49_HTML.gif
        (4.11)
        In the above definitions, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq109_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq110_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq111_HTML.gif ) are diagonal matrices of size http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq113_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq114_HTML.gif being the Chebyshev spectral differentiation matrix. After modifying the matrix system (4.9) to incorporate boundary conditions, the solution is obtained as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ50_HTML.gif
        (4.12)

        5. MHD Swirling Boundary Layer Flow

        The study of the magnetohydrodynamic swirling boundary layer flow over a rotating disk with suction or injection through the porous surface of the disk has recently been undertaken by Turkyilmazoglu [13]. In this case the Navier-Stokes equations reduce to a set of ordinary differential equations
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ51_HTML.gif
        (5.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ52_HTML.gif
        (5.2)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ53_HTML.gif
        (5.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ54_HTML.gif
        (5.4)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ55_HTML.gif
        (5.5)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq115_HTML.gif is the magnetic interaction parameter due to the externally applied magnetic field and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq116_HTML.gif denotes uniform suction ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq117_HTML.gif ) or blowing ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq118_HTML.gif ) through the surface of the disk.

        Turkyilmazoglu [13] utilized a twin strategy, using Ackroyd's series expansion and the homotopy analysis method to find purely analytic solutions to (5.1)–(5.5). In this study we use the SLM to obtain solutions to this system of equations.

        Eliminating http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq119_HTML.gif in (5.1) and (5.2) using (5.4) gives the following system of equations:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ56_HTML.gif
        (5.6)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ57_HTML.gif
        (5.7)
        subject to the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ58_HTML.gif
        (5.8)
        The SLM is applied to (5.6) to (5.8) in the manner described in Section 4, and for brevity we omit the finer details. The intrinsic parameters of the SLM are essentially the same as those defined in Section 4 except for the following:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ59_HTML.gif
        (5.9)
        An appropriate initial approximation for finding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq120_HTML.gif in this case is
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ60_HTML.gif
        (5.10)

        6. Results and Discussion

        In this section we present the results for the velocity distributions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq122_HTML.gif . To check the accuracy of the successive linearisation method and the spectral homotopy analysis method, comparison is made with numerical solutions obtained using the Matlab http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq123_HTML.gif routine, which is an adaptive Lobatto quadrature scheme (see [22]). The current results are compared with previously published results by Liao [11], Yang and Liao [12], and Turkyilmazoglu [13]. The results presented in this work were generated using mostly http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq124_HTML.gif collocation points and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq125_HTML.gif .

        Table 1 gives a comparison of the values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq126_HTML.gif obtained at different orders of the SLM and the SHAM approximations against the homotopy analysis method results, the homotopy-Padé results, and the numerical results. Our finding is that the SLM results converge most rapidly to the numerical result of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq127_HTML.gif 0.884474. Full convergence is achieved at the very low third order. Comparatively, convergence (to 6 decimal places) was achieved at the twentieth order using the homotopy analysis method and at the fifteenth order in the case of the homotopy-Padé method. When the same http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq128_HTML.gif value is used, convergence of the spectral homotopy analysis method is achieved at the eighth order compared to the twentieth order for the homotopy analysis method approximations. This suggests that the SLM is a very useful computational tool that converges much more rapidly than the homotopy analysis method, the homotopy-Padé method, and the spectral homotopy analysis method, although, the SLM may, in fact, require more computations per step than the other methods.
        Table 1

        Comparison of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq129_HTML.gif at different orders of the HAM [12], Homotopy-Padé [11], SHAM, and the SLM approximations when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq131_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq132_HTML.gif .

        Order

        HAM [12]

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq133_HTML.gif

        Hom-Padé [11]

        Order

        SHAM

        Order

        SLM

        Numerical

        0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq134_HTML.gif 1

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq135_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq136_HTML.gif 0.885308

        2

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq137_HTML.gif 0.884944

        1

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq138_HTML.gif 0.871912

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq139_HTML.gif 0.884474

        5

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq140_HTML.gif 0.9173

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq141_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq142_HTML.gif 0.884475

        4

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq143_HTML.gif 0.884449

        2

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq144_HTML.gif 0.884521

         

        10

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq145_HTML.gif 0.8747

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq146_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq147_HTML.gif 0.884474

        6

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq148_HTML.gif 0.884476

        3

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq149_HTML.gif 0.884474

         

        15

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq150_HTML.gif 0.8833

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq151_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq152_HTML.gif 0.884474

        8

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq153_HTML.gif 0.884474

        4

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq154_HTML.gif 0.884474

         

        20

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq155_HTML.gif 0.8845

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq156_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq157_HTML.gif 0.884474

        10

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq158_HTML.gif 0.884474

        5

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq159_HTML.gif 0.884474

         
        Table 2 gives a comparison of the pressure difference http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq160_HTML.gif at different orders of the homotopy analysis method, SHAM, and SLM against the numerical results. A similar pattern as in Table 1 emerges where the SLM results converge rapidly to the numerical result of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq161_HTML.gif with full convergence achieved at the third order. In the case of the HAM, convergence up to four decimal places was achieved at the tenth order. For the same http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq162_HTML.gif values, the SHAM converges at the sixth order.
        Table 2

        Comparison of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq163_HTML.gif obtained at different orders for the HAM [12], SHAM, and SLM approximations when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq164_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq165_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq166_HTML.gif .

        HAM [12] order

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq167_HTML.gif order

        SHAM order

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq168_HTML.gif

        SLM

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq169_HTML.gif

        Numerical

        0

        0.3901

        2

        0.391563

        1

        0.380115

        0.391147

        5

        0.3910

        4

        0.391125

        2

        0.391189

         

        10

        0.3911

        6

        0.391149

        3

        0.391147

         

        15

        0.3911

        8

        0.391147

        4

        0.391147

         

        20

        0.3911

        10

        0.391147

        5

        0.391147

         
        Tables 36 give a comparison between the SLM and the results reported by Turkyilmazoglu [13] for several suction/injection velocities and magnetic parameter values. Comparison of the results of Turkyilmazoglu [13] with the SLM seems most appropriate since the former study also partly utilizes a linearizing technique, the Newton-Raphson method to compute elements of the solutions. Turkyilmazoglu [13] showed that for large injection velocities, the number of terms required to attain convergence of the series solution increases dramatically, for instance, for injection velocities http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq170_HTML.gif , up to 2000 terms are required to achieve convergence of the series solution method, and hence the study resorts to the Chebyshev collocation method to solve the governing equations. Nonetheless, our findings indicate that with only a few terms of the SLM series good levels of accuracy are achieved for all suction and injection velocities. For the suction and injection velocities in the range http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq172_HTML.gif in Tables 3-4 there is an excellent agreement between the fourth-order SLM, the numerical, and the results reported by Turkyilmazoglu [13].
        Table 3

        Comparison of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq173_HTML.gif at different orders for the SLM approximations when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq174_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq175_HTML.gif against the results of [13] for different http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq176_HTML.gif values when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq177_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq178_HTML.gif

        1st order

        2nd order

        3rd order

        4th order

        Numerical

        Reference [13]

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq179_HTML.gif 2.0

        0.28399669

        0.29148466

        0.29148082

        0.29148082

        0.29148082

        0.29148086

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq180_HTML.gif 1.0

        0.31835562

        0.32165707

        0.32166220

        0.32166220

        0.32166220

        0.32166220

        0.0

        0.31619804

        0.30929864

        0.30925799

        0.30925798

        0.30925798

        0.30925798

        1.0

        0.26848288

        0.25115842

        0.25104369

        0.25104397

        0.25104397

        0.25104397

        2.0

        0.19789006

        0.18779923

        0.18871806

        0.18871902

        0.18871902

        0.18871903

        Table 4

        Comparison of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq181_HTML.gif at different orders for the SLM approximations when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq182_HTML.gif against the results of [13] for different http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq183_HTML.gif values when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq184_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq185_HTML.gif

        1st order

        2nd order

        3rd order

        4th order

        Numerical

        Reference [13]

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq186_HTML.gif 2.0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq187_HTML.gif 0.46621214

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq188_HTML.gif 0.46571639

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq189_HTML.gif 0.46571471

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq190_HTML.gif 0.46571471

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq191_HTML.gif 0.46571471

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq192_HTML.gif 0.46571471

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq193_HTML.gif 1.0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq194_HTML.gif 0.69404148

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq195_HTML.gif 0.69065793

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq196_HTML.gif 0.69066292

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq197_HTML.gif 0.69066292

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq198_HTML.gif 0.69066292

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq199_HTML.gif 0.69066292

        0.0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq200_HTML.gif 1.06924152

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq201_HTML.gif 1.06907700

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq202_HTML.gif 1.06905336

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq203_HTML.gif 1.06905336

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq204_HTML.gif 1.06905336

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq205_HTML.gif 1.06905336

        1.0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq206_HTML.gif 1.61663439

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq207_HTML.gif 1.65615591

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq208_HTML.gif 1.65707514

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq209_HTML.gif 1.65707580

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq210_HTML.gif 1.65707580

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq211_HTML.gif 1.65707588

        2.0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq212_HTML.gif 2.31476548

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq213_HTML.gif 2.42896548

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq214_HTML.gif 2.43136137

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq215_HTML.gif 2.43136154

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq216_HTML.gif 2.43136154

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq217_HTML.gif 2.43136154

        Table 5

        Flow parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq218_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq219_HTML.gif at different orders for the SLM approximations when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq221_HTML.gif for different http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq222_HTML.gif values when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq223_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq224_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq225_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq226_HTML.gif

         

        2nd order

        4th order

        Numerical

        2nd order

        4th order

        Numerical

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq227_HTML.gif 5

        0.17788071

        0.17788125

        0.17788125

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq228_HTML.gif 0.20387855

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq229_HTML.gif 0.20387920

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq230_HTML.gif 0.20387920

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq231_HTML.gif 4

        0.20924002

        0.20924073

        0.20924073

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq232_HTML.gif 0.25452255

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq233_HTML.gif 0.25452370

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq234_HTML.gif 0.25452370

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq235_HTML.gif 3

        0.24839904

        0.24839882

        0.24839882

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq236_HTML.gif 0.33393576

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq237_HTML.gif 0.33393640

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq238_HTML.gif 0.33393640

        3

        0.14238972

        0.14422157

        0.14422157

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq239_HTML.gif 3.30816863

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq240_HTML.gif 3.31056638

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq241_HTML.gif 3.31056638

        4

        0.11266351

        0.11466456

        0.11466456

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq242_HTML.gif 4.23823915

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq243_HTML.gif 4.24002059

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq244_HTML.gif 4.24002059

        5

        0.09266580

        0.09447344

        0.09447344

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq245_HTML.gif 5.19357411

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq246_HTML.gif 5.19480492

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq247_HTML.gif 5.19480492

        Table 6

        Flow parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq248_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq249_HTML.gif at different orders for the SLM approximations when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq250_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq251_HTML.gif for different http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq252_HTML.gif values when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq253_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq254_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq255_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq256_HTML.gif

         

        2nd order

        4th order

        Numerical

        2nd order

        4th order

        Numerical

        0

        0.39183500

        0.38956624

        0.38956624

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq257_HTML.gif 1.17700614

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq258_HTML.gif 1.17522084

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq259_HTML.gif 1.17522083

        2

        0.19726747

        0.19756823

        0.19756823

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq260_HTML.gif 2.01809456

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq261_HTML.gif 2.01847353

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq262_HTML.gif 2.01847353

        4

        0.14885275

        0.14901611

        0.14901611

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq263_HTML.gif 2.56931412

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq264_HTML.gif 2.56932504

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq265_HTML.gif 2.56932504

        6

        0.12469326

        0.12476317

        0.12476317

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq266_HTML.gif 3.00455809

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq267_HTML.gif 3.00452397

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq268_HTML.gif 3.00452397

        8

        0.10953285

        0.10956389

        0.10956389

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq269_HTML.gif 3.37536371

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq270_HTML.gif 3.37533046

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq271_HTML.gif 3.37533046

        10

        0.09887642

        0.09889037

        0.09889037

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq272_HTML.gif 3.703823547

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq273_HTML.gif 3.70379689

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq274_HTML.gif 3.70379689

        Table 5 gives a comparison between the numerical and the SLM results for larger values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq275_HTML.gif , up to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq276_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq277_HTML.gif . Moderate increases in the suction/injection velocities appear to have no effect on the precision of the method with convergence again achieved at the fourth order of the SLM series. In Table 6, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq278_HTML.gif is fixed and the magnetic parameter varied. We compare the convergence rate of the SLM to the numerical computations and show that increasing this parameter has no effect either on the convergence rate of the successive linearisation method.

        Figure 1 gives a comparison between the fourth-order SHAM, second-order SLM, and numerical results for the dimensionless velocity distributions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq279_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq280_HTML.gif , respectively. There is an excellent agreement among the three sets of results. For purposes of comparison, it is worth noting that in case of the HAM in the work of Yang and Liao [12], agreement between the numerical and the HAM results was only observed at the 30th order of approximation for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq281_HTML.gif and at the 10th order for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq282_HTML.gif . As with most iterative methods, it is worth noting that the convergence rate may depend on the initial approximation used. However, since we have used the same initial approximations as Yang and Liao [12], the graphical results suggest that the SLM converges much more rapidly than both the HAM and SHAM. This may, however, be offset by the fact that the SLM may require more computations per step than the other two methods.
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Fig1_HTML.jpg
        Figure 1

        Comparison between the SHAM, SLM, and numerical solution of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq283_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq284_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq285_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq286_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq287_HTML.gif . The open circles represent the SHAM 4th-order solution, the filled circles represent the 2nd-order SLM solution, and the solid line represent the numerical solution.

        7. Conclusions

        In this work two relatively new methods, the spectral homotopy analysis method and the successive linearisation method, have been successfully used to solve the von Kármán nonlinear equations for swirling flow with and without suction/injection across the disk walls and an applied magnetic field. The velocity components were compared with numerical results generated using the built-in Matlab bvp4c solver and against the homotopy analysis method and homotopy-Padé results in the literature. The results indicate that both the spectral homotopy analysis method and the successive linearisation method may give accurate and convergent results using only few solution terms compared with the homotopy analysis method and the Homotopy-Padé methods. Comparison has also been made with the recent findings by Turkyilmazoglu [13]. The successive linearisation method gives better accuracy at lower orders than the spectral homotopy analysis method. The tradeoff, however, is that both the spectral homotopy analysis method and the successive linearisation method may involve more computations per step compared to the methods in the literature.

        Nonetheless, the successive linearisation method has been shown to be very efficient in that it rapidly converges to the numerical results. The study by Turkyilmazoglu [13] shows that whenever suction/blowing through the disk walls is present, the homotopy analysis method is prone to give wildly oscillating solutions. These oscillations have to be controlled by a careful choice of the embedded parameter http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq288_HTML.gif . The advantage of the successive linearisation method is that such a parameter does not exist and no such oscillations are observed in the solution of the von Kármán equations for swirling flow.

        Declarations

        Acknowledgment

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

        Authors’ Affiliations

        (1)
        School of Mathematical Sciences, University of KwaZulu-Natal
        (2)
        Department of Mathematics, University of Swaziland

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        © Zodwa G. Makukula et al. 2010

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