 Research Article
 Open Access
A Note on the Solution of the Von Kármán Equations Using Series and Chebyshev Spectral Methods
 ZodwaG Makukula^{1},
 Precious Sibanda^{1}Email author and
 SandileSydney Motsa^{2}
https://doi.org/10.1155/2010/471793
© Zodwa G. Makukula et al. 2010
 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.
Keywords
 Chebyshev Polynomial
 Collocation Point
 Homotopy Analysis Method
 Injection Velocity
 Uniform Suction
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 NavierStokes 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, GalerkinBSpline, and Chebyshevcollocation methods have been used previously to find solutions of the von Kármán equations [3–6]. 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
Boundary Value Problems
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
The superscript denotes the transpose, is a diagonal matrix, and is an identity matrix of size . We implement boundary conditions (3.14) in rows 1, , and of in columns 1 through to by setting all entries in the remaining columns to be zero. The second set (3.15) is implemented in rows and , respectively, by setting , and setting all other columns to be zero. We further set entries of in rows , , , , and to zero.
which provides the initial approximation for the solution of (3.5).
The matrix is the matrix on the righthand side of (3.25) but with the boundary conditions incorporated by setting the first, , , , and , rows and columns to zero. Thus, starting from the initial approximation, which is obtained from (3.17), higherorder approximations for can be obtained through recursive formula (3.28).
4. Successive Linearisation Method
5. MHD Swirling Boundary Layer Flow
where is the magnetic interaction parameter due to the externally applied magnetic field and denotes uniform suction ( ) or blowing ( ) 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.
6. Results and Discussion
In this section we present the results for the velocity distributions and . 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 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 collocation points and .
Comparison of at different orders for the SLM approximations when , against the results of [13] for different values when .
 1st order  2nd order  3rd order  4th order  Numerical  Reference [13] 

2.0  0.28399669  0.29148466  0.29148082  0.29148082  0.29148082  0.29148086 
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 
Comparison of at different orders for the SLM approximations when against the results of [13] for different values when .
 1st order  2nd order  3rd order  4th order  Numerical  Reference [13] 

2.0  0.46621214  0.46571639  0.46571471  0.46571471  0.46571471  0.46571471 
1.0  0.69404148  0.69065793  0.69066292  0.69066292  0.69066292  0.69066292 
0.0  1.06924152  1.06907700  1.06905336  1.06905336  1.06905336  1.06905336 
1.0  1.61663439  1.65615591  1.65707514  1.65707580  1.65707580  1.65707588 
2.0  2.31476548  2.42896548  2.43136137  2.43136154  2.43136154  2.43136154 
Flow parameters and at different orders for the SLM approximations when , for different values when .


 

2nd order  4th order  Numerical  2nd order  4th order  Numerical  
5  0.17788071  0.17788125  0.17788125  0.20387855  0.20387920  0.20387920 
4  0.20924002  0.20924073  0.20924073  0.25452255  0.25452370  0.25452370 
3  0.24839904  0.24839882  0.24839882  0.33393576  0.33393640  0.33393640 
3  0.14238972  0.14422157  0.14422157  3.30816863  3.31056638  3.31056638 
4  0.11266351  0.11466456  0.11466456  4.23823915  4.24002059  4.24002059 
5  0.09266580  0.09447344  0.09447344  5.19357411  5.19480492  5.19480492 
Flow parameters and at different orders for the SLM approximations when , for different values when .


 

2nd order  4th order  Numerical  2nd order  4th order  Numerical  
0  0.39183500  0.38956624  0.38956624  1.17700614  1.17522084  1.17522083 
2  0.19726747  0.19756823  0.19756823  2.01809456  2.01847353  2.01847353 
4  0.14885275  0.14901611  0.14901611  2.56931412  2.56932504  2.56932504 
6  0.12469326  0.12476317  0.12476317  3.00455809  3.00452397  3.00452397 
8  0.10953285  0.10956389  0.10956389  3.37536371  3.37533046  3.37533046 
10  0.09887642  0.09889037  0.09889037  3.703823547  3.70379689  3.70379689 
Table 5 gives a comparison between the numerical and the SLM results for larger values of , up to when . 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, 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.
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 builtin Matlab bvp4c solver and against the homotopy analysis method and homotopyPadé 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 HomotopyPadé 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 . 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
References
 von Kármán T: Uberlaminare und turbulence reibung. Zeitschrift für Angewandte Mathematik und Mechanik 1921,52(1):233252.View ArticleMATHGoogle Scholar
 Cochran WG: The flow due to a rotating disc. Proceedings of the Cambridge Philisophical Society 1934, 30: 365375. 10.1017/S0305004100012561View ArticleMATHGoogle Scholar
 Chien CS, Shih YT: A cubic Hermite finite elementcontinuation method for numerical solutions of the von Kármán equations. Applied Mathematics and Computation 2009,209(2):356368. 10.1016/j.amc.2008.12.054MathSciNetView ArticleMATHGoogle Scholar
 Kirby RM, Yosibash Z: Solution of vonKármán dynamic nonlinear plate equations using a pseudospectral method. Computer Methods in Applied Mechanics and Engineering 2004,193(6–8):575599.MathSciNetView ArticleMATHGoogle Scholar
 Yosibash Z, Kirby RM, Gottlieb D: Collocation methods for the solution of vonKármán dynamic nonlinear plate systems. Journal of Computational Physics 2004,200(2):432461. 10.1016/j.jcp.2004.03.018MathSciNetView ArticleMATHGoogle Scholar
 Zerarka A, Nine B: Solutions of the von Kàrmàn equations via the nonvariational Galerkin–spline approach. Communications in Nonlinear Science and Numerical Simulation 2008,13(10):23202327. 10.1016/j.cnsns.2007.06.009MathSciNetView ArticleMATHGoogle Scholar
 Adomian G: A review of the decomposition method and some recent results for nonlinear equations. Computers & Mathematics with Applications 1991,21(5):101127. 10.1016/08981221(91)90220XMathSciNetView ArticleMATHGoogle Scholar
 Lyapunov AM: The General Problem of the Stability of Motion. Taylor & Francis, London, UK; 1992:x+270.MATHGoogle Scholar
 He JH: Homotopy perturbation method for solving boundary value problems. Physics Letters A 2006,350(12):8788. 10.1016/j.physleta.2005.10.005MathSciNetView ArticleMATHGoogle Scholar
 He JH: A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. International Journal of NonLinear Mechanics 2000,35(1):3743. 10.1016/S00207462(98)000857MathSciNetView ArticleMATHGoogle Scholar
 Liao S: Beyond Perturbation. Introduction to the Homotopy Analysis Method, CRC Series: Modern Mechanics and Mathematics. Volume 2. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2004:xii+322.MATHGoogle Scholar
 Yang C, Liao S: On the explicit, purely analytic solution of von Kármán swirling viscous flow. Communications in Nonlinear Science and Numerical Simulation 2006,11(1):8393. 10.1016/j.cnsns.2004.05.006MathSciNetView ArticleMATHGoogle Scholar
 Turkyilmazoglu M: Purely analytic solutions of magnetohydrodynamic swirling boundary layer flow over a porous rotating disk. Computers and Fluids 2010,39(5):793799. 10.1016/j.compfluid.2009.12.007View ArticleMATHGoogle Scholar
 Ackroyd JAD: On the steady flow produced by a rotating disc with either surface suction or injection. Journal of Engineering Physics 1978, 12: 207220.MATHGoogle Scholar
 Motsa SS, Sibanda P, Shateyi S: A new spectralhomotopy analysis method for solving a nonlinear second order BVP. Communications in Nonlinear Science and Numerical Simulation 2010,15(9):22932302. 10.1016/j.cnsns.2009.09.019MathSciNetView ArticleMATHGoogle Scholar
 Motsa SS, Sibanda P, Awad FG, Shateyi S: A new spectralhomotopy analysis method for the MHD JefferyHamel problem. Computers and Fluids 2010,39(7):12191225. 10.1016/j.compfluid.2010.03.004MathSciNetView ArticleMATHGoogle Scholar
 Makukula Z, Motsa SS, Sibanda P: On a new solution for the viscoelastic squeezing flow between two parallel plates. Journal of Advanced Research in Applied Mathematics 2010,2(4):3138. 10.5373/jaram.455.060310MathSciNetView ArticleGoogle Scholar
 Motsa SS, Sibanda P: A new algorithm for solving singular IVPs of LaneEmden type. Proceedings of the 4th International Conference on Applied Mathematics, Simulation, Modelling, July 2010, Corfu Island, Greece, NAUN International Conferences 176180.Google Scholar
 Boyd JP: Chebyshev and Fourier Spectral Methods. 2nd edition. Dover, Mineola, NY, USA; 2001:xvi+668.MATHGoogle Scholar
 Canuto C, Hussaini MY, Quarteroni A, Zang TA: Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics. Springer, New York, NY, USA; 1988:xiv+557.View ArticleMATHGoogle Scholar
 Don WS, Solomonoff A: Accuracy and speed in computing the Chebyshev Collocation Derivative. SIAM Journal on Scientific Computing 1995,16(6):12531268. 10.1137/0916073MathSciNetView ArticleMATHGoogle Scholar
 Kierzenka J, Shampine LF: A BVP solver based on residual control and the MATLAB PSE. ACM Transactions on Mathematical Software 2001,27(3):299316. 10.1145/502800.502801MathSciNetView ArticleMATHGoogle Scholar
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