Last updated 06/06/2020

### 1. What is the EIVF

The electrovortex flow, named also “Electrically induced vortical flows” (or EIVF), is inducing in a conductive liquid media, when electrical current passes through it . Spatial distributed electrical current of density interacts with self magnetic field and generates an electromagnetic force field When a force field is non-potential then fluid elements get elementary rotation around local vector of force vorticity While fluid velocity is small, the vorticity of electromagnetic force is compensated by fluid friction tensions in each point of a flow.  So, a EIVF appear in a Stokes (linear) regime.

When the current strength increases, an electromagnetic vorticity and flow velocities are increasing too, and the flow patterns become depending strong on effects of nonlinear transfer of a fluid vorticity With this, nonstationary nonlinear effects manifest themselves (through bifurcations) in forms of self-oscillation or specific core structures and axisymmetric patterns are rotating around an axis usually.

As an example, the left part of fig 1 shows a scheme of axisymmetric toroidal vortex in a meridional plane of a cylindrical container (R2 – radius, L = R2 - depth): 1 – liquid metal, 2 – top electrode, 3 – nonconductive side wall of container, 4 – bottom electrode, .

On a scheme:
EA is an axis symmetry of container
AB is a boundary between top electrode (of radius R1) and liquid surface
BC is a free surface of liquid
CD is nonconductive side wall of container with
DE is a bottom electrode (of radius R2)

The origin of cylindrical coordinates is placed in a point O. Dash lines show approximated picture of electrical current lines (electrical stream function). A solid lines shows approximated hydrodynamic stream function (in a flow without an azimuthal rotation).

The right part of fig. 1 shows a photography a meridional cross section of toroidal vortex, placed on a free surface. Here the vertical axis symmetry is on the middle of picture, the radius of top electrode is R1 = 6 mm; the radius of bottom electrode R2 = 30 mm is equal to a bath radius and to a bath depth L = R2. Fig 1. The EIVF in a form of toroidal vortex in a cylindrical container of depth L and radius R2 = L. On the left - a scheme of meridional plane (AE – axis symmetry of a container): on the right - a photography of whole meridional cross section. Here: 1 - liquid metal, 2 - top electrode (radius R1), 3 - nonconductive side wall of container, 4 - bottom electrode (radius R2), .

Some other examples of axisymmetric EIVF in form of axisymmetric toroidal vortexes are shown on Fig.2 and Fig .3.

Fig:2. Top line: The same flow, like in fig. 1. b. Middle line, at the left: A meridional flow near partly submerged electrodes; at the right * – a EIVF near conical electrode of large diameter. Bottom line, at the left - a flow near fully submerged electrode, - the current discharge between side wall of electrode and bottom of bath; at the right – the same flow, like in the first photo above. (* Here and below: photos were gotten together with A. Chaikovsky and S. Andrienko) Fig. 3: On a top line and the last photo in a bottom line -  flow near conical electrode*. The EIVF has stable axisymmetric structure in a second photo only. In three others EIVF have unstable structure with broken axis symmetries. Bottom line at the left* – a EIVF toroidal vortex in a model of Electroslag Welding. The current discharges between partly submerged electrode and curvilinear bath bottom; in the middle – a EIVF in half-spherical container, the upper electrode is like in a fig 1, b. ### 2. EIVF in the industry

There are a lot of effective industrial technologies where electrical current is passed through a bath with liquid melts. Among them, there are

• Electrical Arc Furnaces (EAF): 3-phases, DC (European), DC UNG (Universal Next Generation, Russia)
• Electroslag remelting (ESR) with 1 to 6 remelting electrodes, DC / AC / 3-phases, Fig. 4 shows electrodes after ESR with bifilar power supply scheme
• Electroslag welding (ESW) and multi-electrode refacing (surface improvement) with axis symmetrical, bifilar, doubled and 3-phases power supply current lines. Fig. 5 shows a scheme of axis symmetrical ESW with two bathes with EIVF (slag and metal)
• Ore thermal furnaces, flux- and salt-melting furnaces. Fig.6 shows a common view of ore thermal furnace with two upper electrodes.
• Electric batteries Fig. 4. Photo of two large (height 3 m) electrodes after electro-slag remelting of large sized electrodes.   The melting bottom boundary was formed by strong local EIVF

Fig. 5. A scheme of Electroslaf Welding. The EIVF exist both in molten slag and in a weld pool [www.suntech.com]

Fig. 6. A photo of Ore-thermal furnace [http://www.optim-toledo.com/]

Note, that from one hand all constructive components of electro-aggregates, - form and relative position of electrodes or arcs, built-in and external current leads, bath geometry, ferromagnetic masses and others, - define 3D fields of current density and self magnetic field, distributed in a melt and, as a result, an EIVF there.

From another hand, the EIVF defines heat-mass-transfer in a melt, metal quality, energy efficiency, ecology and safety. So, the understanding of EIVF is a key factor for building an effective electrical metallurgical technology.

### 3. EIVF in physical tasks

Various effects of EIVF can be checked and studied in a cold physical and calculation models for:

• Physical and calculation models of electrical aggregates
• New physical experiments with cold liquid metals
• Exact and numerical solutions of EIVF equations system

Need to say some words about ANSYS multi-physics programs for complex physical task calculations. This is an excellent tool for science-research investigation of EIVF objects, like EIVF. But strong understanding of physics is required for getting effective results: from task formulation till calculated results explanation.

### 4. Governing MHD equations

Basic MGD (magnetohydrodynamics) equations includes  equations of

Momentum transport Heat transfer Concentration transport Continuity Maxwell’s equations Ohm’s law for a moving media Charge conservation and media statement conditions Here: are vectors fields of velocity, force of gravity, current density, magnetic and electrical correspondently; – pressure, density, temperature and thermo-conductivity coefficient; kinematic viscosity, specific heat capacity, electrical conductivity and concentration of impurity; – magnetic permittivity of vacuum, - a density of free charges, - dielectric constant, – diffusion coefficient; some differential operators: – gradient, – divergence, – vorticity (rotor), – Laplasian.

### 5. The system of EIVF equations

Various technological aggregates require the investigation of behaviour of such liquid media, like melted liquid metal, aluminium, flux, slag and so on. In a physical model – cold liquid metal is used, for example, GALINSTAN (alloy of indium, gallium and stanum). So, for physical modelling. Some rules are required for recalculation of results, gotten from physical modelling to a real conditions of electro-aggregates. For this we do some assumptions about properties of media.

Our assumptions include, [1-2]:

• Constant values of density, kinematic viscosity and electric conductivity: • Incompressible media • No mechanical impurities, chemical reactions and free charges
• Thermo-convection can be neglected due to strong electromagnetic stirring
• No strong deformation of free surface, (no back reaction from surface deformation to electromagnetic force field)
• Electrodynamics approximation • Electromagnetic fields do not depend on time. (Direct current – DC – is used)
• The hydrodynamic flows can be stationary or nonstationary

With mentioned above assumptions and after applying of operation rot to momentum transport equation, the system of MHD will consist [1-2] of:
Vorticity transfer equation  Incompressibility equation Maxwell’s equations (for DC) Ohm’s law in electrodynamics approximation Note that here:

• Electrodynamic and hydrodynamic fields have no influences each to another. So, fields of can be defined preliminary from Maxwell’s equations only. Then the result will be used in the right side of vorticity transfer equation.
• There is no a pressure in a task formulation. – The EIVF in a container appears due to vorticity of electromagnetic force and can’t be organized by pressure distribution
• Every mentioned assumption can be ignored. Then the system of equations will be adjusted by correspondent equation
• The EIVF can be organized by using an alternative current (AC) of industrial frequency. Then electromagnetic field of have to be averaged by time. If skin depth is more than bath depth, then EIVF under of AC will be the same like under equal value of in DC, .

### 6. The physical modelling of EIVF

The shows that there are five independent physical characteristics – – that consist of four independent dimensions - So, there is a single dimensionless parameter in the EIVF (analog to Reinolds number): Using characteristic velocity in form and defining other characteristic values as: - equations from point 5 can be written in a form, where all scalar and vector fields are dimensionless: With this, for real physical modelling experiment there are two conditions only, , :

1. Geometrical similarity of real bath and physical model
2. Similarity of electrical boundary conditions

In this case a characterises velocity is proportional to a value of common current

It means, that physical modelling can be provided by a single parameter S only.