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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.

76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76W05 Magnetohydrodynamics and electrohydrodynamics
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