Thermocapillary convection in a cylindrical liquid-metal floating zone with a strong axial magnetic field and with a non-axisymmetric heat flux

This paper treats the steady three-dimensional thermocapillary convection in a cylindrical liquid-metal zone between the isothermal ends of two coaxial solid cylinders and surrounded by an atmosphere. There is a uniform steady magnetic field which is parallel to the common centrelines of the liquid zone and solid cylinders, and there is a non-axisymmetric heat flux into the liquid's free surface. The magnetic field is sufficiently strong that inertial effects and convective heat transfer are negligible, and that viscous effects are confined to thin boundary layers adjacent to the free surface and to the liquid–solid interfaces. With an axisymmetric heat flux, the axisymmetric thermocapillary convection is confined to the thin layer adjacent to the free surface, but with a non-axisymmetric heat flux, there is an azimuthal flow inside the free-surface layer from the hot spot to the cold spot with the circulation completed by flow across the inviscid central core region. This problem is related to the magnetic damping of thermocapillary convection for the floating-zone growth of semiconductor crystals in Space.

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The effect of uniform magnetic field on spatial-temporal evolution of thermocapillary convection with the silicon oil based ferrofluid

Thermal Science ◽

10.2298/tsci200223156y ◽

2020 ◽

Vol 24 (6 Part B) ◽

pp. 4159-4171

Author(s):

Shuo Yang ◽

Rui Ma ◽

Qiaosheng Deng ◽

Guofeng Wang ◽

Yu Gao ◽

...

Keyword(s):

Magnetic Field ◽

Free Surface ◽

Transverse Magnetic Field ◽

Liquid Bridge ◽

Thermocapillary Convection ◽

Surface Deformation ◽

Axial Magnetic Field ◽

Transverse Magnetic ◽

Surface Flow ◽

Silicon Oil

A uniform axial or transverse magnetic field is applied on the silicon oil based ferrofluid of high Prandtl number fluid (Pr ? 111.67), and the effect of magnetic field on the thermocapillary convection is investigated. It is shown that the location of vortex core of thermocapillary convection is mainly near the free surface of liquid bridge due to the inhibition of the axial magnetic field. A velocity stagnation region is formed inside the liquid bridge under the axial magnetic field (B = 0.3-0.5 T). The disturbance of bulk reflux and surface flow is suppressed by the increasing axial magnetic field. There is a dynamic response of free surface deformation to the axial magnetic field, and then the contact angle variation of the free surface at the hot corner is as following, ?hot, B = 0.5 T = 83.34? > ?hot, B = 0.3 T = 72.16? > > ?hot,B = 0.1 T = 54.21? > ?hot, B = 0 T = 43.33?. The results show that temperature distribution near the free surface is less and less affected by thermocapillary convection with the increasing magnetic field, and it presents a characteristic of heat-conduction. In addition, the transverse magnetic field does not realize the fundamental inhibition for thermocapillary convection, but it transfers the influence of thermocapillary convection to the free surface.

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Spin-Up from Rest of a Liquid Metal with Deformable Free Surface in a Cylinder under the Influence of a Uniform Axial Magnetic Field

Fluids ◽

10.3390/fluids6120438 ◽

2021 ◽

Vol 6 (12) ◽

pp. 438

Author(s):

Toshio Tagawa ◽

Kewei Song

Keyword(s):

Magnetic Field ◽

Free Surface ◽

Liquid Metal ◽

Hartmann Number ◽

Viscosity Ratio ◽

Axial Magnetic Field ◽

Density Ratio ◽

Striking Difference ◽

Gas Phases ◽

Component Velocity

Spin-up from rest of a liquid metal having deformable free surface in the presence of a uniform axial magnetic field is numerically studied. Both liquid and gas phases in a vertically mounted cylinder are assumed to be an incompressible, immiscible, Newtonian fluid. Since the viscous dissipation and the Joule heating are neglected, thermal convection due to buoyancy and thermocapillary effects is not taken into account. The effects of Ekman number and Hartmann number were computed with fixing the Froude number of 1.5, the density ratio of 800, and the viscosity ratio of 50. The evolutions of the free surface, three-component velocity field, and electric current density are portrayed using the level-set method and HSMAC method. When a uniform axial magnetic field is imposed, the azimuthal momentum is transferred from the rotating bottom wall to the core region directly through the Hartmann layer. This is the most striking difference from spin-up of the nonmagnetic case.

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Liquid-Metal MHD Open-Channel Flows

Journal of Applied Mechanics ◽

10.1115/1.3167557 ◽

1984 ◽

Vol 51 (1) ◽

pp. 13-18 ◽

Cited By ~ 5

Author(s):

P. R. Hays ◽

J. S. Walker

Keyword(s):

Magnetic Field ◽

Free Surface ◽

Magnetic Fields ◽

Liquid Metal ◽

Boundary Layers ◽

Open Channel ◽

Three Dimensional ◽

Viscous Effects ◽

Channel Bottom ◽

Dimensional Variable

Many metallurgical applications of magnetohydrodynamics (MHD) involve open-channel liquid-metal flows with magnetic fields. This paper treats the three-dimensional, variable-depth flow in a rectangular open channel having an electrically insulating bottom and perfectly conducting sides. A steady, uniform magnetic field is applied perpendicular to the channel bottom. Induced magnetic fields and surface tension effects are neglected, while the applied magnetic field is sufficiently strong that inertial effects are negligible everywhere. Viscous effects are confined to boundary layers adjacent to the bottom, sides, and free surface. Solutions are presented for the inviscid core and the boundary layers. The locations of the free surface above the core and above the boundary layers adjacent to the sides are obtained. The side-layer variables are rescaled into universal profile functions which depend on the coordinates in the channel’s cross section and on a parameter related to the local slopes of the bottom and the free surface. The solutions for the side layers in open channels are compared to the side-layer solutions for certain rectangular closed ducts in order to reveal the effects of the free surface. This comparison leads to a qualitative correspondence principle between open-channel and closed-duct side-layer solutions. The similarities and differences between corresponding open-channel and closed-duct side layers are discussed.

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

Journal of Fluid Mechanics ◽

10.1017/s0022112099005698 ◽

1999 ◽

Vol 394 ◽

pp. 281-302 ◽

Cited By ~ 26

Author(s):

M. PRANGE ◽

M. WANSCHURA ◽

H. C. KUHLMANN ◽

H. J. RATH

Keyword(s):

Magnetic Field ◽

Free Surface ◽

Magnetic Fields ◽

Prandtl Number ◽

Linear Stability ◽

Thermocapillary Convection ◽

Axial Magnetic Field ◽

Unstable Mode ◽

Linear Stability Theory ◽

The Stability

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.

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