×

zbMATH — the first resource for mathematics

Temporal stability of small disturbances in MHD Jeffery-Hamel flows. (English) Zbl 1175.76061
Summary: The temporal development of small disturbances in magnetohydrodynamic (MHD) Jeffery-Hamel flows is investigated, in order to understand the stability of hydromagnetic steady flows in convergent/divergent channels at very small magnetic Reynolds number \(R_m\). A modified form of normal modes that satisfy the linearized governing equations for small disturbance development asymptotically far downstream is employed [A. McAlpine and P.G. Drazin, Fluid Dyn. Res. 22, No. 3, 123–138 (1998; Zbl 1051.76554)]. The resulting fourth-order eigenvalue problem which reduces to the well known Orr-Sommerfeld equation in some limiting cases is solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The results indicate that a small divergence of the walls is destabilizing for plane Poiseuille flow while a small convergence has a stabilizing effect. However, an increase in the magnetic field intensity has a strong stabilizing effect on both diverging and converging channel geometry.

MSC:
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76W05 Magnetohydrodynamics and electrohydrodynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hartmann, J.; Lazarus, F., Kgl. danske videnskab. selskab. mat.-fys. medd., 15, 6,7, (1937)
[2] Makinde, O.D.; Alagoa, K.D., Effect of magnetic field on steady flow through an indented channel, AMSE, modelling, measurement & control, 68, 1, 25-32, (1999)
[3] Moreau, R., Magnetohydrodynamics, (1990), Kluwer Academic Publishers Dordrecht · Zbl 0714.76003
[4] Lock, R.C., The stability of the flow of an electrically conducting fluid between parallel plates under a transverse magnetic field, Proc. R. soc. lond. A, 233, 105, (1955)
[5] Makinde, O.D.; Motsa, S.S., Hydromagnetic stability of plane Poiseuille flow using Chebyshev spectral collocation method, J. ins. math. comput. sci., 12, 2, 175-183, (2001)
[6] Makinde, O.D., Magneto-hydrodynamic stability of plane-Poiseuille flow using multi-deck asymptotic technique, Math. comput. modelling, 37, 3-4, 251-259, (2003) · Zbl 1027.76020
[7] Kakutani, T., The hydromagnetic stability of the modified plane Couette flow in the presence of transverse magnetic field, J. phys. soc. Japan, 19, 1041, (1964)
[8] Makinde, O.D.; Motsa, S.S., Hydromagnetic stability of generalized plane-Couette flow, Far east J. appl. math., 6, 1, 77-88, (2002) · Zbl 1003.76032
[9] Takashima, M., The stability of the modified plane Couette flow in the presence of transverse magnetic field, Fluid dyn. res., 22, 105-121, (1998) · Zbl 1051.76563
[10] Takashima, M., The stability of the modified plane Poiseuille flow in the presence of transverse magnetic field, Fluid dyn. res., 17, 293-310, (1996) · Zbl 1051.76562
[11] Hamel, G., Spiralformige bewegung zahre flussikeiten, Jahresber. d. dt. mathematiker-vereinigung., 34, (1916)
[12] Jeffery, G.B., Steady motions of a viscous fluid, Philos. mag., 29, 455, (1996)
[13] Batchelor, G.K., An introduction to fluid mechanics, (1967), Cambridge University Press · Zbl 0152.44402
[14] Fraenkel, L.E., Laminar wall in symmetric channels with slightly curved walls. I. on the Jeffery Hamel solutions for flow between plane walls, Proc. R. soc. lond. A, 267, 119-138, (1962) · Zbl 0104.42403
[15] Banks, W.H.H.; Drazin, P.G.; Zaturska, M.B., On perturbations of jeffery – hamel flow, J. fluid mech., 186, 559-581, (1988) · Zbl 0648.76019
[16] Drazin, P.G., Flow through a diverging channel: instability and bifurcation, Fluid dyn. res., 24, 321-327, (1999) · Zbl 1051.76569
[17] McAlpine, A.; Drazin, P.G., On the spatio-development of small perturbations of jeffery – hamel flows, Fluid dyn. res., 22, 123-138, (1998) · Zbl 1051.76554
[18] Tam, K.K., Linear stability of the non-parallel bickley jet, Can. appl. math. Q., 3, 99-110, (1996) · Zbl 0866.76032
[19] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag New York · Zbl 0658.76001
[20] Wiedemann, J.A.C.; Reddy, S.C., A MATLAB differentiation matrix suite, ACM toms, 26, 4, 465-519, (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.