Nonlinear energy stability of magnetohydrodynamics Couette and Hartmann shear flows: A contradiction and a conjecture

A nonlinear (energy) stability analysis is performed for a magnetized ferrofluid layer, heated and soluted from below, with stress-free boundaries. A rigorous nonlinear stability result is derived by introducing a suitable generalized energy functional. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. For ferrofluids we find that the existence of subcritical instabilities is possible, however, it is noted that, in case of a nonferrofluid, the global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of the magnetic parameters, this coincidence is immediately lost. The effects of the magnetic parameter, M3, and the solute gradient, S1, on the subcritical instability region have also been analyzed. It is shown that with the increase of the magnetic parameter the subcritical instability region between the two theories decreases quickly, while with the increase of the solute gradient the subcritical region expands. We also demonstrate the coupling between the buoyancy and magnetic forces in the nonlinear energy stability analysis.

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Nonlinear energy stability and convection near the density maximum

Acta Mechanica ◽

10.1007/bf01170801 ◽

1992 ◽

Vol 95 (1-4) ◽

pp. 9-28 ◽

Cited By ~ 6

Author(s):

G. McKay ◽

B. Straughan

Keyword(s):

Energy Stability ◽

Density Maximum ◽

Nonlinear Energy

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Bounds on the energy stability limit of plane parallel shear flows

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/pl00001562 ◽

2001 ◽

Vol 52 (4) ◽

pp. 573-596 ◽

Cited By ~ 3

Author(s):

R. Kaiser ◽

B. J. Schmitt

Keyword(s):

Shear Flows ◽

Stability Limit ◽

Energy Stability ◽

Plane Parallel ◽

Parallel Shear Flows

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A nonlinear stability analysis for magnetized ferrofluid heated from below

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.1906 ◽

2007 ◽

Vol 464 (2089) ◽

pp. 83-98 ◽

Cited By ~ 45

Author(s):

Sunil ◽

Amit Mahajan

Keyword(s):

Stability Analysis ◽

Rayleigh Number ◽

Nonlinear Stability ◽

Magnetic Parameter ◽

Linear Instability ◽

Instability Region ◽

Energy Stability ◽

Subcritical Instability ◽

Magnetized Ferrofluid ◽

Nonlinear Energy

A nonlinear (energy) stability analysis is performed for a magnetized ferrofluid layer heated from below, in the stress-free boundary case. By introducing a suitable generalized energy functional, a rigorous nonlinear stability result is derived for a thermoconvective magnetized ferrofluid. The mathematical emphasis is on how to control the nonlinear terms caused by the magnetic body and inertia forces. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of the linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in the case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M 3 , on subcritical instability region has also been analysed. It is shown that with the increase of magnetic parameter, M 3 , the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and the magnetic forces in the nonlinear energy stability analysis.

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