Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

Abstract

The stability of axisymmetric steady thermocapillary convection of electrically conducting fluids in half-zones under the influence of a static axial magnetic field is investigated numerically by linear stability theory. In addition, the energy transfer between the basic state and a disturbance is considered in order to elucidate the mechanics of the most unstable mode. Axial magnetic fields cause a concentration of the thermocapillary flow near the free surface of the liquid bridge. For the low Prandtl number fluids considered, the most dangerous disturbance is a non-axisymmetric steady mode. It is found that axial magnetic fields act to stabilize the basic state. The stabilizing effect increases with the Prandtl number and decreases with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic field also influences the azimuthal symmetry of the most unstable mode.

Rayleigh–Bénard instability in a vertical cylinder with a vertical magnetic field

“…They presented accurate numerical results for the steady axisymmetric base flow for Ha 6 200. The instability of the steady, axisymmetric thermocapillary convection treated by Prange et al (1999) is fundamentally different from the Rayleigh-Bénard instability treated here, but we mention their work in order to illustrate that different types of linear stability analyses are limited to small values of Ha because of numerical resolution problems. Gelfgat & Bar-Yoseph (2001) considered small, planar perturbations to the steady, planar buoyant convection in a rectangle with thermally insulated top and bottom walls and with isothermal endwalls at different temperatures.…”

Effect of an axial magnetic field on the flow pattern in a cylindrical floating zone

“…The Lorentz force per unit volume generated by the magnetic field isF L ¼j ÂB,j being the current density vector, defined byj ¼ rðÀr/ þṼ ÂBÞ, where / is the induced electric potential andṼ the velocity vector. For axial magnetic fields and axisymmetric flows, no electric potential is induced (Prange et al, 1999), hence the Lorentz force reduces toF L ¼ rðṼ ÂBÞ ÂB, which for the axial field becomes…”

“…Lan (1996) and Lan and Yeh (2004) consider a global formulation in a mirror furnace, taking into account heat losses by radiation, but the redistribution of this energy by the mirror (that reflects it back to the sample) is not considered, so that the thermal boundary condition is simplified as well. The influence of axial magnetic fields on the stability of steady axisymmetric flows in liquid bridges is studied in Prange et al (1999), although in a half-zone model (another simplified model where the two supporting disks are kept at different temperatures).…”

Numerical linear stability analysis of a thermocapillary-driven liquid bridge with magnetic stabilization

“…Lan and Yeh [2004; performed quite complete full-zone modeling involving three-dimensional radiation, a deformable free surface and melting interfaces, dopant distribution, and axial and transverse magnetic damping. Prange et al [1999] studied the half-zone instability with axial magnetic field stabilization up to Ha = 25.…”