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Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

Published online by Cambridge University Press:  25 September 2009

D. GIANNAKIS ,
R. ROSNER  and
P. F. FISCHER
D. GIANNAKIS*
Affiliation:
Department of Physics, University of Chicago, Chicago, IL 60637, USA
R. ROSNER
Affiliation:
Department of Physics, University of Chicago, Chicago, IL 60637, USA Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA Argonne National Laboratory, Argonne, IL 60439, USA
P. F. FISCHER
Affiliation:
Argonne National Laboratory, Argonne, IL 60439, USA
*
†Email address for correspondence: [email protected]
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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, assuming that the plate is either a perfect electrical insulator or a perfect conductor. We first confirm that the Squire transformation for magnetohydrodynamics 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, respectively parameterizing gravity and surface tension, 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 conducting laboratory fluids (Pm ≤ 10−4). Because it is a critical-layer instability (moderately modified by the presence of the free surface), the hard mode exhibits similar behaviour as 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 Rec on the Hartmann number Ha. In contrast, the structure of the soft mode's growth-rate contours in the (Re, α) plane, where α is the wavenumber, differs markedly between problems with small, but non-zero, Pm and their counterparts in the inductionless limit, Pm ↘ 0. 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 non-zero the Lorentz force originating from the steady-state current leads to a modification of Rec(Ha) to either a sub-linearly increasing or a decreasing function of Ha, respectively for problems with insulating or perfectly conducting walls. In insulating-wall problems we also observe pairs of counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability driven by energy transferred from the steady-state shear to both of the velocity and magnetic degrees of freedom. Movies are available with the online version of the paper.

JFM classification

Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 217 - 277
Copyright
Copyright © Cambridge University Press 2009

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References

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Giannakis et al. supplementary movie

Movie 1. Real and imaginary parts of the modal phase velocities c for inductionless free-surface Hartmann flow at Re=7×105, α=2×10-3, Ga=8.3×107, and Ca=0.07. The animation, which corresponds to the eigenvalue plots in figure 7(a, b), covers Hartmann numbers logarithmically spaced in the interval 0.1≤Ha≤50. Stable (Im(c)<0) and unstable (Im(c)>0) modes are respectively coloured blue and red. The A, P, and S branches of the spectrum, as well as mode F, which participates in the soft instability, are also indicated. As described in the paper, the main features of the evolution of this spectrum are (i) stabilisation of mode F, occurring when Ha≈5.9, (ii) collapse of the A eigenvalue branch, and (iii) and alignment of the P and S branches.

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Giannakis et al. supplementary movie

Movie 2. Real and imaginary parts of the modal phase velocities c for free-surface Hartmann flow with insulating wall at Re=7×105, α=2×10-3, Ga=8.3×107, Ca=0.07, and Ha=0.1. The animation, which corresponds to the eigenvalue plots in figure 14(a, b), covers magnetic Prandtl numbers logarithmically spaced in the interval 10-6≤Pm≤10-4. Stable (Im(c)<0) and unstable (Im(c)>0) modes are respectively coloured blue and red. Besides the A, F, P, and S modes present in the inductionless case in movie 1, the spectrum contains a magnetic mode, which is labelled M and plotted with a + marker. Mode M is singular in the inductionless limit Pm→0. As Pm is increased, its complex phase velocity moves along the S branch from large and negative values of Im(c) towards the upper part of the spectrum. At the same time, given the weakness of the applied field, the A, F, P, and S modes exhibit comparatively negligible variation.

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Giannakis et al. supplementary movie

Movie 3. Real and imaginary parts of the modal phase velocities c for free-surface Hartmann flow with insulating wall at Re=7×105, α=2×10-3, Ga=8.3×107, Ca=0.07, and Pm=10-5. The animation, which corresponds to the eigenvalue plots in figure 15(a–c), covers Hartmann numbers logarithmically spaced in the interval 0.1≤Ha≤50. Stable (Im(c)<0) and unstable (Im(c)>0) modes are respectively coloured blue and red. As Ha is increased, the least stable P mode and the second least stable A mode, plotted with * markers, become converted to travelling Alfvén waves (labelled L+ and L-), while the magnetic mode, plotted (as in movie 2) with a + marker, joins the P branch. Mode F becomes stabilised when Ha≈37, i.e. for significantly higher magnetic-field strengths than in the inductionless case in movie 1.

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Giannakis et al. supplementary movie

Movie 4. Real and imaginary parts of the modal phase velocities c for free-surface Hartmann flow with insulating wall at Re=6.3×106, α=1×10-4, Ga=8.3×107, Ca=0.07, and Pm=10-4. The animation, which corresponds to the eigenvalue and energy plots in figure 16, covers Hartmann numbers logarithmically spaced in the interval 0.1≤Ha≤600. Stable (Im(c)<0) and unstable (Im(c)>0) modes are respectively coloured blue and red. As in movie 3, when Ha is increased a pair of counter-propagating Alfvén modes develops in the spectrum. In this case, however, the downstream-propagating wave L+ originates from mode F (here plotted with a * marker), while the upstream-propagating wave L-, which is unstable for 9.5≤Ha≤21, originates from mode M. Moreover, the least stable P mode separates from the P branch, and takes over the role of mode F. The latter exchange of modal identities results to a short Hartmann-number

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