The role of boundary conditions on the free surface of a liquid metal in an electrovortex flow

An electro-vortex flow between two hemispherical electrodes is considered. The influence of the type of boundary condition on the surface of a conducting liquid medium on the velocity field in the volume is studied numerically. The dependences of the velocity on the axis of the vessel on the radius of the small electrode and the parameter of the electric vortex flow are obtained for various types of boundary conditions on the surface.


Introduction
For carrying out magnetohydrodynamic experiments, metals that remain liquid at room temperature are most convenient -mercury and various alloys based on indium and gallium. At the same time, on the one hand, gallium alloys have an advantage over mercury due to their non-toxicity, on the other hand, the open liquid metal surface is oxidized in air.
This oxide film behaves like a flexible film, i. e. the adhesion condition is realized on it, and the surface deformation under certain conditions can be neglected. Observations of an open surface (using appropriate experimental setups) can provide information on the flow structure if oxides on the surface are chemically eliminated, and the type of boundary condition changes. We considered the effect of the type of boundary condition on the surface on the electrovortex flow (EVF) in a hemispherical container.
EVF is formed as a result of interaction of the non-uniform electric current passing through the liquid metal with own magnetic field of this current [1]. Such flows significantly affect many processes in mechanical engineering (electro-welding) and electrometallurgy (electroslag remelting, various electric melting furnaces) [2].
We consider consider the system where electric current I0 propagates from the small hemispherical electrode with the radius R1 through the conductive media (liquid metal) to the big hemispherical electrode with the radius R2 ( Figure 1). Electromagnetic force F=J×B (where J is the current density and B is the total magnetic field consisting of the self-magnetic field and the external magnetic fields) causes the liquid to move. In such an axisymmetric system (without the influence of external magnetic fields), EVF has the form of a one toroidal vortex. Various aspects of this problem were considered, for example in [3][4][5].

Numerical model
We considered steady, two-dimensional, axisymmetric (∂.../∂φ = 0) system of equations. To numerically solve the Navier -Stokes equation, we used the finite volume method. Area calculation area (figure 1) was a quarter circle with internal radius R1 (small electrode) and external radius R2=1 (large electrode) and consisted of 2500-20000 quadrangular cells for different grids. The wall boundary conditions were set on both electrode and we considered two types of the boundary condition on the upper surface. Wall boundary condition when U|θ=π/2=0 and slip boundary condition when Uθ|θ=π/2=0, Figure 1. Scheme of the electrovortex flow in the hemisphere Calculations were carried out with the electrodynamical approximation, when the influence of the magnetic field on the velocity field was considered only by adding the source of the electromagnetic force to the motion equation F = J × B, excluding electric currents induced by the moving fluid. Possibility of using this approach is due to the small value of magnetic Reynolds number: Rem=μ0σUR2<<1. For typical value of velocityU ~0.1 m/s, Rem~0.04. Here μ0 -magnetic constant, σelectrical conductivity.
Applying these scales to the Navier-Stokes equation written in spherical coordinates (3) as in [2] we can get specific dimensionless parameters: so-called the parameter of the electrovortex flow S. Here e -unit vector.

Results
In Figure 2, the dependence of the axial velocity Uz on z at several numbers S at radius R1=0.001 are presented. These data are quite correlated with the results obtained in [6]. We can see that the maximum speed is located approximately in the same place on the z-axis for different types of boundary condition on the surface.  Figure 2. The dependence of the axial velocity on z coordinate. 1 -S=1; 2 -1e3, 3 -1e6. Index "a" -wall boundary condition, no index -slip boundary condition. Figure 3 shows the dependences of the maximum axial velocity on the number S for different values of the small electrode radius R1 for different types of boundary conditions. We limited ourselves to calculations for R1 <0.1, since at large R1, the flow significantly changes the geometry in the nonlinear case, and a flow in a large volume turns into a flow in a slot, which is already noticeable for R1 = 0.1. Up to S ~ 1000, the flow is in a linear mode. In this mode, the shape of the velocity profile Uz (z) does not depend on the parameter S, which in this case is only a scale factor. It can be seen from the graphs that with increasing current, the influence of the type of boundary conditions on the velocity field decreases. The degree of influence can be characterized by the value of γBC.
(2) Figure 4. shows the dependence of the value of γBC on the parameter S for different small electrode radii R1. Then γBC = 0 means the absence of the influence of the type of the boundary condition, and the maximum value, which means the maximum effect of the type of the boundary condition, is achieved at S between 0 and 1000. The radius of the small electrode R1 affects the maximum speed, with decreasing R1, the maximum speed increases, thus the dependence of the limiting values of γBC (influence parameter of the type of boundary condition).
In Figure 5. shows the dependence of the maximum value of γBC on the size of the small electrode is presented. An increase in the velocity due to a decrease in the radius of the small electrode leads to a decrease in the influence of the boundary condition at the same parameter S.

Conclusions
Calculations of the axial velocity of an electric vortex flow in a hemisphere were carried out for different currents at different values of the radius of the small electrode for different types of boundary conditions on the surface. It is shown that the degree of influence of the type of boundary condition decreases with increasing current and decreasing the radius of the small electrode. The maximum limiting values of the parameter of the influence of the boundary conditions on the axial velocity are obtained. Futher, these results will be verified by comparison with the analytical solutions presented in [1], [7], [8].