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A note on the solution of the von Kármán equations using series and Chebyshev spectral methods. (English) Zbl 1207.35248
Summary: 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M22 Spectral methods applied to problems in fluid mechanics
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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References:
[1] von Kármán, T, Uberlaminare und turbulence reibung, Zeitschrift für Angewandte Mathematik und Mechanik, 52, 233-252, (1921) · JFM 48.0968.01
[2] Cochran, WG, The flow due to a rotating disc, Proceedings of the Cambridge Philisophical Society, 30, 365-375, (1934) · JFM 60.0729.08
[3] Chien, C-S; Shih, Y-T, A cubic Hermite finite element-continuation method for numerical solutions of the von Kármán equations, Applied Mathematics and Computation, 209, 356-368, (2009) · Zbl 1158.74044
[4] Kirby, RM; Yosibash, Z, Solution of von-Kármán dynamic non-linear plate equations using a pseudo-spectral method, Computer Methods in Applied Mechanics and Engineering, 193, 575-599, (2004) · Zbl 1060.74660
[5] Yosibash, Z; Kirby, RM; Gottlieb, D, Collocation methods for the solution of von-Kármán dynamic non-linear plate systems, Journal of Computational Physics, 200, 432-461, (2004) · Zbl 1115.74344
[6] Zerarka, A; Nine, B, Solutions of the von Kàrmàn equations via the non-variational Galerkin-spline approach, Communications in Nonlinear Science and Numerical Simulation, 13, 2320-2327, (2008) · Zbl 1221.74086
[7] Adomian, G, A review of the decomposition method and some recent results for nonlinear equations, Computers & Mathematics with Applications, 21, 101-127, (1991) · Zbl 0732.35003
[8] Lyapunov AM: The General Problem of the Stability of Motion. Taylor & Francis, London, UK; 1992:x+270. · Zbl 0786.70001
[9] He, J-H, Homotopy perturbation method for solving boundary value problems, Physics Letters A, 350, 87-88, (2006) · Zbl 1195.65207
[10] He, J-H, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics, 35, 37-43, (2000) · Zbl 1068.74618
[11] 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. · Zbl 1051.76001
[12] Yang, C; Liao, S, On the explicit, purely analytic solution of von Kármán swirling viscous flow, Communications in Nonlinear Science and Numerical Simulation, 11, 83-93, (2006) · Zbl 1075.35059
[13] Turkyilmazoglu, M, Purely analytic solutions of magnetohydrodynamic swirling boundary layer flow over a porous rotating disk, Computers and Fluids, 39, 793-799, (2010) · Zbl 1242.76368
[14] Ackroyd, JAD, On the steady flow produced by a rotating disc with either surface suction or injection, Journal of Engineering Physics, 12, 207-220, (1978) · Zbl 0412.76022
[15] Motsa, SS; Sibanda, P; Shateyi, S, A new spectral-homotopy analysis method for solving a nonlinear second order BVP, Communications in Nonlinear Science and Numerical Simulation, 15, 2293-2302, (2010) · Zbl 1222.65090
[16] Motsa, SS; Sibanda, P; Awad, FG; Shateyi, S, A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem, Computers and Fluids, 39, 1219-1225, (2010) · Zbl 1242.76363
[17] 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, 2, 31-38, (2010)
[18] Motsa, SS; Sibanda, P, A new algorithm for solving singular IVPs of Lane-Emden type, 176-180, (2010) · Zbl 1343.65087
[19] Boyd JP: Chebyshev and Fourier Spectral Methods. 2nd edition. Dover, Mineola, NY, USA; 2001:xvi+668. · Zbl 0994.65128
[20] 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. · Zbl 0658.76001
[21] Don, WS; Solomonoff, A, Accuracy and speed in computing the Chebyshev collocation derivative, SIAM Journal on Scientific Computing, 16, 1253-1268, (1995) · Zbl 0840.65010
[22] Kierzenka, J; Shampine, LF, A BVP solver based on residual control and the MATLAB PSE, ACM Transactions on Mathematical Software, 27, 299-316, (2001) · Zbl 1070.65555
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