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Large-wavelength instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers | | D. Giannakis
; R. Rosner
; P. Fischer
; | | Date: |
8 Aug 2008 | | Abstract: | We study the linear stability of the flow of a viscous electrically
conducting capillary fluid on a planar fixed plate in the presence of gravity
and a uniform magnetic field. We first confirm that the Squire transformation
for MHD is compatible with the stress and insulating boundary conditions at the
free surface, but argue that unless the flow is driven at fixed Galilei and
capillary numbers, the critical mode is not necessarily two-dimensional. We
then investigate numerically how a flow-normal magnetic field, and the
associated Hartmann steady state, affect the soft and hard instability modes of
free surface flow, working in the low magnetic Prandtl number regime of
laboratory fluids. Because it is a critical layer instability, the hard mode is
found to exhibit similar behaviour to the even unstable mode in channel
Hartmann flow, in terms of both the weak influence of Pm on its neutral
stability curve, and the dependence of its critical Reynolds number Re_c on the
Hartmann number Ha. In contrast, the structure of the soft mode’s growth rate
contours in the (Re, alpha) plane, where alpha is the wavenumber, differs
markedly between problems with small, but nonzero, Pm, and their counterparts
in the inductionless limit. As derived from large wavelength approximations,
and confirmed numerically, the soft mode’s critical Reynolds number grows
exponentially with Ha in inductionless problems. However, when Pm is nonzero
the Lorentz force originating from the steady state current leads to a
modification of Re_c(Ha) to either a sublinearly increasing, or decreasing
function of Ha, respectively for problems with insulating and conducting walls.
In the former, we also observe pairs of Alfven waves, the upstream propagating
wave undergoing an instability at large Alfven numbers. | | Source: | arXiv, 0808.1130 | | Services: | Forum | Review | PDF | Favorites | |
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