Open Access

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

  • ZodwaG Makukula1,
  • Precious Sibanda1Email author and
  • SandileSydney Motsa2
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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq1_HTML.gif about the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq3_HTML.gif above the disk. In cylindrical coordinates https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ2_HTML.gif
(2.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq5_HTML.gif is the fluid density, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq6_HTML.gif is the kinematic viscosity coefficient, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq7_HTML.gif is the pressure, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq9_HTML.gif , and https://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 https://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]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ3_HTML.gif
(2.3)
where https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ4_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ5_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ6_HTML.gif
(2.6)
https://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
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq13_HTML.gif to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq15_HTML.gif by the computational domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq16_HTML.gif where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq18_HTML.gif is then transformed to the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq19_HTML.gif using the algebraic mapping
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ12_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq20_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ13_HTML.gif
(3.3)
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq22_HTML.gif and
https://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,
https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ18_HTML.gif
(3.8)
subject to
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq23_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ20_HTML.gif
(3.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq26_HTML.gif are the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq27_HTML.gif th Chebyshev polynomials with coefficients https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq29_HTML.gif , respectively, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ21_HTML.gif
(3.11)
and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq32_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ22_HTML.gif
(3.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq34_HTML.gif is the order of differentiation and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ24_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ25_HTML.gif
(3.15)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ26_HTML.gif
(3.16)

The superscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq36_HTML.gif denotes the transpose, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq37_HTML.gif is a diagonal matrix, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq38_HTML.gif is an identity matrix of size https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq40_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq41_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq42_HTML.gif in columns 1 through to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq44_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq45_HTML.gif , respectively, by setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq46_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq48_HTML.gif in rows https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq49_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq51_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq52_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq53_HTML.gif to zero.

The values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq54_HTML.gif are determined from the equation
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ28_HTML.gif
(3.18)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq55_HTML.gif is the embedding parameter and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq56_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ29_HTML.gif
(3.19)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq60_HTML.gif are nonlinear operators given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ30_HTML.gif
(3.20)
The https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ32_HTML.gif
(3.22)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ33_HTML.gif
(3.23)
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ36_HTML.gif
(3.26)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq62_HTML.gif and https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq65_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq67_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq68_HTML.gif , and https://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 https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq71_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq73_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ38_HTML.gif
(3.28)

The matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq74_HTML.gif is the matrix https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq77_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq78_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq80_HTML.gif for https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq84_HTML.gif can be expanded as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ39_HTML.gif
(4.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq85_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq86_HTML.gif are unknown functions and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq88_HTML.gif ( https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ40_HTML.gif
(4.2)
where the coefficient parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq91_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq92_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq93_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq94_HTML.gif are defined as
https://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]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ42_HTML.gif
(4.4)
The solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq95_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq96_HTML.gif , https://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,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ44_HTML.gif
(4.6)
Once each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq99_HTML.gif ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq102_HTML.gif are obtained as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ45_HTML.gif
(4.7)
where https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ47_HTML.gif
(4.9)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq104_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq105_HTML.gif square matrix and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq107_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq108_HTML.gif column vectors defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ48_HTML.gif
(4.10)
with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ49_HTML.gif
(4.11)
In the above definitions, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq110_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq111_HTML.gif ) are diagonal matrices of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq113_HTML.gif with https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ51_HTML.gif
(5.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ52_HTML.gif
(5.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ53_HTML.gif
(5.3)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ55_HTML.gif
(5.5)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq116_HTML.gif denotes uniform suction ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq117_HTML.gif ) or blowing ( https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_Equ56_HTML.gif
(5.6)
https://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
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq120_HTML.gif in this case is
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq121_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq124_HTML.gif collocation points and https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq131_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq132_HTML.gif .

Order

HAM [12]

https://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

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

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

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

2

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

1

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

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

5

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

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

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

4

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

2

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

 

10

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

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

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

6

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

3

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

 

15

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

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

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

8

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

4

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

 

20

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

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

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

10

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

5

https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq165_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq166_HTML.gif .

HAM [12] order

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

SHAM order

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

SLM

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq171_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq174_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq176_HTML.gif values when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq177_HTML.gif .

https://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]

https://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

https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq183_HTML.gif values when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq184_HTML.gif .

https://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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

0.0

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

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

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

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

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

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

1.0

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

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

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

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

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

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

2.0

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

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

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

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

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

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

Table 5

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

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

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

https://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

https://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

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

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

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

https://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

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

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

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

https://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

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

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

https://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

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

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

https://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

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

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

https://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

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

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

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

Table 6

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

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

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

https://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

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

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

https://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

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

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

https://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

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

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

https://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

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

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

https://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

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

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

https://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

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq275_HTML.gif , up to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq276_HTML.gif when https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq279_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq281_HTML.gif and at the 10th order for https://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.
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq283_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq284_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq285_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471793/MediaObjects/13661_2010_Article_929_IEq286_HTML.gif , and https://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 https://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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