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 twodimensional. We then investigate numerically how a flownormal magnetic field, and the associated Hartmann steady state, affect the soft and hard instability modes of freesurface flow, working in
the lowmagneticPrandtlnumber regime of laboratory fluids (Pm <= 10[sup −4]). Because it is a criticallayer instability (moderately modified by the presence of the free surface), 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 neutralstability curve, and the dependence of its critical Reynolds number Re[sub c] on the Hartmann number Ha. In contrast, the structure of the soft mode�s growthrate contours in the (Re, α) plane, where α is the wavenumber, differs markedly between problems with small, but nonzero, Pm, and their counterparts in the inductionless limit. As derived from largewavelength 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 steadystate current leads to a modification of Re[sub 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 counterpropagating Alfven waves, the upstreampropagating wave undergoing an instability driven by energy transfered from the steadystate shear to both of the velocity and magnetic degrees of freedom. Movies are available with the online version of the paper.
