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

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 parameterising 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 laboratory fluids (Pm 10 −4 ). Because it is a critical-layer 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 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, 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 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 counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability driven by energy transfered 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.

A Liquid Metal Flume for Free Surface Magnetohydrodynamic Experiments

“…Experiments performed using this design will be supplemented by a linear stability analysis 42 and three-dimensional simulations 43 to aid in understanding the role of a magnetic field in turbulent free surface channel flow.…”

Electromagnetically driven flow in unsupported electrolyte layers: lubrication theory and linear stability of annular flow

“…The theoretical description and modelling of electromagnetically driven flows in supported films with a free interface are now well developed (Morley & Abdou 1995;Morley & Roberts 1996;Morley & Abdou 1997;Gao, Morley & Dhir 2002;Morley, Smolentsev & Gao 2002;Miloshevsky & Hassanein 2010;Giannakis, Fischer & Rosner 2009a;Giannakis, Rosner & Fischer 2009b;Lunz & Howell 2019) and continue to attract attention mainly due to applications in plasma flows and tokamaks. In the absence of the Lorentz force, the pure hydrodynamic description of the flow in unsupported liquid films was initiated in Prévost & Gallez (1986), Sharma & Ruckenstein (1988) and later received a huge boost because of its relevance to nonlinear film rupture and two-dimensional turbulence problems (Couder, Chomaz & Rabaud 1989;Gharib & Derango 1989;Chomaz & Cathalau 1990;Erneux & Davis 1993;Sharma et al 1995;Van De Fliert, Howell & Ockenden 1995;Wu et al 1995) as reviewed in Kellay & Goldburg (2002) and Oron, Davis & Bankoff (1997).…”

Flow of a thin liquid-metal film in a toroidal magnetic field

“…Free-surface MHD experiments with liquid metal have been performed since the beginnings of MHD research (Nornberg et al 2008;Alpher et al 1960;Platacis et al 2014) but precise measurements are difficult to obtain. Free-surface MHD flows have also been modelled (see Morley & Abdou 1995;Morley & Abdou 1997;Morley & Roberts 1996;Giannakis et al 2009b, and references within) and simulated numerically (Morley et al 2004;Miloshevsky & Hassanein 2010;Gao et al , 2003Giannakis et al 2009a); however there is more ground to cover.…”