Free-surface Hartmann flow is the parallel flow of a viscous, electrically conducting, capillary fluid on a planar surface, subject to gravity and a flow-normal magnetic field. This type of flow arises in a variety of industrial and astrophysical contexts, including liquid-metal walls in fusion devices, heavy-ion accelerator targets, and surface layers of white dwarfs and neutron stars. Typically, the Reynolds number, Re>104, is high, and the background magnetic field is strong (Ha>100, where the Hartmann number, Ha, measures the square root of the ratio of electromagnetic to viscous forces). On the other hand, the magnetic Prandtl number, Pm (the ratio of viscous to magnetic diffusivity), of laboratory fluids is small (e.g., Pm<10-4 for liquid metals), as is the case in a number of astrophysical models.
When the background magnetic field is zero, free-surface Hartmann flow exhibits the so-called soft and hard instability modes; the former being a surface wave destabilized by viscous stresses acting on the free surface, whereas the latter is a shear mode destabilized by positive Reynolds stress associated with an internal critical layer. We study in detail the influence of the external magnetic field on these two instabilities, working in the regime Pm<10-4. We also consider flows in the inductionless limit, Pr→0, where magnetic field perturbations diffuse infinitely fast, and the sole MHD effect is a Lorentz force arising from currents induced by the perturbed fluid motion within the background magnetic field.
We have developed a spectral Galerkin method to solve the coupled Orr-Sommerfeld and induction equations, which, in conjunction with suitable stress conditions at the free surface and continuity conditions for the magnetic field, govern the linear stability of free-surface Hartmann flow. Our scheme's discrete bases for the velocity and magnetic fields consist of linear combinations of Legendre polynomials, chosen according to the order of the Sobolev spaces of the continuous problem. The orthogonality properties of the bases solve the matrix-coefficient growth problem of the discrete stability operators, and eigenvalue-eigenfunction pairs can be computed stably at spectral orders at least as large as p=3000 with p-independent roundoff error.
We find that, because it is a critical-layer instability (moderately modified by the presence of the free surface), the hard mode exhibits similar behavior to the even unstable mode in the corresponding closed-channel flow, in terms of both the weak influence of Pm on its neutral-stability curve and the monotonic increase of its critical Reynolds number, Rec, with the Hartmann number. In contrast, the soft mode's stability properties exhibit the novel behavior of differing markedly between problems with small, but nonzero, Pm and their counterparts in the inductionless limit. Notably, the critical Reynolds number of the soft mode grows exponentially with Ha in inductionless problems, but when Pm is nonzero that growth is suppressed to either a sublinearly increasing, or a decreasing function of Ha (respectively when the lower wall is an electrical insulator or a perfect conductor). In the insulating-wall case, we also observe pairs of counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability at high Alfvén numbers.
We attribute the observed Pm-sensitivity of the soft instability to the strong-field behavior of the participating inductionless mode, which, even though stabilized by the magnetic field, approaches neutral stability as Ha grows. This near-equilibrium is consistent with a balance between Lorentz and gravitational forces, and renders the mode susceptible to effects associated with the dynamical response of the magnetic field to the flow (which vanishes in the inductionless limit), even when the magnetic diffusivity is large. The boundary conditions play a major role in the magnetic field response to the flow, since they determine (i) the properties of the steady-state induced current, which couples magnetic perturbations to the velocity field, and (ii) the presence or not of magnetic modes in the spectrum (these modes are not part of the spectrum of conducting-wall problems), which interact with the hydrodynamic ones, including the soft mode. In general, our analysis indicates that the inductionless approximation must be used with caution when dealing with free-surface MHD.