Hermite-Padé approximation approach to MHD Jeffery-Hamel flows. (English) Zbl 1102.76049

Summary: We study the nonlinear flow of incompressible conducting viscous fluid in convergent-divergent channels under the influence of externally applied homogeneous magnetic field. The governing equations are obtained and solved using a special type of Hermite-Padé approximation semi-numerical approach. This technique offers some advantages over solutions obtained by using traditional methods such as finite differences, spectral method, shooting method, etc. It reveals the analytical structure of the solution and important properties of overall flow structure, including velocity field, flow reversal control and bifurcations.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76W05 Magnetohydrodynamics and electrohydrodynamics
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
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[1] Baker, G. A.; Graves-Morris, P., Padé Approximants (1996), Cambridge University Press · Zbl 0923.41001
[2] Banks, W. H.H.; Drazin, P. G.; Zaturska, M. B., On perturbation of Jeffery-Hamel flow, J. Fluid Mech., 186, 559-581 (1988) · Zbl 0648.76019
[3] Batchelor, G. K., An Introduction to Fluid Dynamics (1967), Cambridge University Press · Zbl 0152.44402
[4] Common, A. K., Applications of Hermite-Padé approximants to water waves and the harmonic oscillator on a lattice, J. Phys. A, 15, 3665-3677 (1982) · Zbl 0519.41020
[5] Fraenkel, L. E., Laminar flow in symmetrical channels with slightly curved walls. I: On the Jeffery-Hamel solutions for flow between plane walls, Proc. R. Soc. Lond., A267, 119-138 (1962) · Zbl 0104.42403
[6] Guttamann, A. J., Asymptotic analysis of power – series expansions, (Domb, C.; Lebowitz, J. K., Phase Transitions and Critical Phenomena (1989), Academic Press: Academic Press New York), 1-234
[7] Hamadiche, M.; Scott, J.; Jeandel, D., Temporal stability of Jeffery-Hamel flow, J. Fluid Mech., 268, 71-88 (1994) · Zbl 0809.76039
[8] Hamel, G., Spiralförmige Bewgungen Zäher Flüssigkeiten, Jahresbericht der Deutschen Math. Vereinigung, 25, 34-60 (1916)
[9] Hunter, D. L.; Baker, G. A., Methods of series analysis III: Integral approximant methods, Phys. Rev. B, 19, 3808-3821 (1979)
[10] Jeffery, G. B., The two-dimensional steady motion of a viscous fluid, Philos. Mag., 6, 455-465 (1915)
[11] Khan, M. A.H.; Drazin, P. G.; Tourigny, Y., The summation of series in several variable and its applications in fluid dynamics, Fluid Dyn. Res., 33, 191-205 (2003) · Zbl 1032.76664
[12] Makinde, O. D., Laminar flow in a channel of varying width with permeable boundaries, Romanian J. Phys., 40, 403-417 (1995)
[13] Makinde, O. D., Steady flow in a linearly diverging asymmetrical channel, CAMES, 4, 157-165 (1997) · Zbl 0976.76513
[14] Makinde, O. D., Extending the utility of perturbation series in problems of laminar flow in a porous pipe and a diverging channel, J. Austral. Math. Soc. Ser. B, 41, 118-128 (1999) · Zbl 0956.76074
[15] Makinde, O. D., Strongly exothermic explosions in a cylindrical pipe: a case study of series summation technique, Mech. Res. Commun., 32, 191-195 (2005) · Zbl 1098.92077
[16] Moreau, R., Magnetohydrodynamics (1990), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0714.76003
[17] Sobey, I. J.; Drazin, P. G., Bifurcations of two-dimensional channel flows, J. Fluid Mech., 171, 263-287 (1986) · Zbl 0609.76050
[18] M.M. Vainberg, V.A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations, Noordoff, Leyden, 1974.; M.M. Vainberg, V.A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations, Noordoff, Leyden, 1974. · Zbl 0274.47033
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