Electro-Vortex Flow Dynamics

• Electro-vortex flows are forced MHD circulations arising from nonuniform current distributions that generate unbalanced Lorentz forces in conducting liquids.
• Studies show that EVF strength and topology depend on current collector geometry, electrical boundary conditions, and even weak geomagnetic fields, affecting battery mixing and interface stability.
• Experimental and numerical analyses reveal scaling laws (e.g., S parameter and Re ~ S^0.5) that quantify EVF behavior, highlighting its dual role in enhancing mixing or threatening layer integrity.

Electro-vortex flow (EVF) is a current-driven magnetohydrodynamic circulation in an electrically conducting liquid that arises when the imposed current is not spatially uniform. In cylindrical conductors and liquid metal battery analogues, the nonuniformity typically originates from a mismatch between the size of a current collector or electrode and the size of the fluid domain, so that current lines converge or diverge and acquire radial components. The resulting Lorentz force, $\mathbf{J}\times\mathbf{B}$, is not everywhere pressure-balanced and drives a poloidal circulation that commonly includes a central jet and compensating return flow; under weak axial background fields, that circulation can be transformed into a swirling, time-dependent helical state. EVF is therefore both a canonical laboratory MHD flow and a practically important transport mechanism in liquid metal batteries, where it may either enhance mixing or threaten interface integrity (Cheng et al., 2021, Liu et al., 2022, Soni et al., 4 Aug 2025).

1. Definition and physical origin

EVF appears when the electric current does not remain purely axial or laterally uniform as it passes through a conducting liquid. This occurs near changes in cross-section, feeding lines, or solid–liquid current transfer regions. In the formulation used for liquid metal batteries, current entering from a thin wire or a small current collector spreads in a wider current collector and then in the liquid metal; as a result, the current acquires radial components and the Lorentz force develops a non-irrotational part that cannot be balanced by pressure alone. In cylindrical cells with a smaller top collector than the vessel cross-section, the flow is drawn radially toward the smaller current collector, and the forcing intensifies near that collector, often forming a downward jet on the centerline with return flow along the sidewall (Weber et al., 2014, Cheng et al., 2021, Weber et al., 2017).

A recurring point across the literature is that EVF is not an instability in the threshold sense. It is a forced flow caused by current redistribution. One study states explicitly that there is no threshold current for the onset of electro-vortex flows and that EVF starts as soon as the current is applied; another emphasizes that EVF cannot arise if the current is homogeneous, no matter how large it is, because then the Lorentz force is irrotational and can be balanced by pressure (Weber et al., 2014, Liu et al., 2019, Soni et al., 4 Aug 2025).

This distinction is central in battery applications. The flow is generated by realistic current-collector geometry and electrical boundary conditions rather than by a separate bifurcation criterion. A plausible implication is that EVF is best regarded as an intrinsic consequence of current delivery architecture in conducting-liquid devices.

2. Governing formulations and force decomposition

The standard EVF description is based on incompressible MHD with Lorentz forcing. A representative formulation writes

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$

with

$\vec f = \vec f_L = \vec J \times \vec B.$

In the low-magnetic-Reynolds-number regime, the magnetic field and current density are commonly decomposed as

$$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$

and the quasistatic approximation is used,

$$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$

Charge conservation then yields a Poisson equation for the perturbed potential in the liquid metal,

$$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$

while in solid current collectors one solves

$\Delta \varphi_{cc} = 0.$

The induced magnetic field is then obtained from the current density through the Biot–Savart law,

$$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$

These formulations are common to EVF solvers for coupled solid–liquid conductors and to cylindrical battery models (Weber et al., 2014, Weber et al., 2017, Liu et al., 2019).

When thermal effects are retained, the momentum equation is augmented by buoyancy and the temperature field obeys an advection–diffusion equation with Joule heating,

$$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$

or, in the notation of a coupled Rayleigh–Bénard-convection model,

$$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$

The buoyancy force is introduced by the Boussinesq or Boussinesq-Overbeck approximation. These extensions are used when EVF is studied together with thermal convection or Joule-heating-induced density variations (Liu et al., 2019, Keogh et al., 2020).

3. Jet formation, transient structure, and geomagnetic-field-induced swirl

A detailed cylindrical realization was studied in a GaInSn vessel supplied with a constant electrical current of $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$0, entering through the bottom cable and leaving through the top cable. The geometry contains top and bottom cables, top and bottom electrodes, and the GaInSn pool, with a tapered top electrode and a mismatch between electrode and container diameters. This geometry forces the current to converge strongly near the upper center, sharply increasing $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$1 there and causing the EVF jet to originate close to the top center (Liu et al., 2022).

In that notation, the dominant EVF-driving force is the self-interaction term

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$2

called Lf. Its geometry determines the classical jet. The radially inward part of $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$3 interacting with the toroidal $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$4 produces a downward force, while the axial part of $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$5 with the same toroidal field produces an inward radial force. Together these components drive a strong downward-flowing jet near the top center. This axisymmetric picture is adequate only when the external axial field is absent or negligible (Liu et al., 2022).

The transient behavior becomes qualitatively different when a vertical background magnetic field component $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$6 is present. A second Lorentz-force contribution appears,

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$7

called lf. Although this force is much weaker than Lf by roughly two to three orders of magnitude, it acts in the toroidal direction and twists the jet. In the representative case $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$8, streamlines become spiral, the lower part of the jet exhibits bifurcation, and the high-velocity region swings irregularly around the centerline rather than remaining a fixed central plume. As $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$9 increases, the jet region becomes shorter, more strongly twisted, and more irregularly displaced from the centerline (Liu et al., 2022).

The combined experimental and numerical study preceding that transient analysis reached a closely related conclusion. In a cylindrical GaInSn vessel with a $\vec f = \vec f_L = \vec J \times \vec B.$0 top electrode and a $\vec f = \vec f_L = \vec J \times \vec B.$1 bottom electrode, the geomagnetic field inside the vessel was measured as approximately $\vec f = \vec f_L = \vec J \times \vec B.$2, $\vec f = \vec f_L = \vec J \times \vec B.$3, and $\vec f = \vec f_L = \vec J \times \vec B.$4. The simulations showed that $\vec f = \vec f_L = \vec J \times \vec B.$5 strongly modifies the EVF structure, whereas $\vec f = \vec f_L = \vec J \times \vec B.$6 and $\vec f = \vec f_L = \vec J \times \vec B.$7 are much less important for the velocity magnitude and overall flow shape. As $\vec f = \vec f_L = \vec J \times \vec B.$8 becomes more negative from $\vec f = \vec f_L = \vec J \times \vec B.$9 to $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$0, the maximum velocity decreases, the location of the velocity maximum shifts in the UDV projection, and the jet shortens from about $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$1 to about $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$2 of the total height. Numerical profiles agree well with the measured velocity distributions when using $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$3, $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$4, and $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$5 (Liu et al., 2019).

These results establish a specific misconception to avoid: the external field need not be large to matter. Geomagnetic-strength $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$6 can significantly alter the EVF topology in laboratory-scale liquid-metal columns.

4. Characteristic parameters, scalings, and geometry dependence

A standard nondimensional measure of EVF forcing is

$$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$7

where $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$8 is vacuum permeability, $$\vec B = \vec B_0 + \vec b, \qquad \vec J = \vec J_0 + \vec j,$$9 is the applied current, $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$0 the density, and $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$1 the kinematic viscosity. In this interpretation, $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$2 measures how strongly magnetic forcing competes with viscous dissipation. A liquid-gallium laboratory model for battery layers supplied currents from $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$3 to $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$4 and reached $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$5 up to $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$6; in EVF-only runs it defined a characteristic velocity by

$$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$7

and obtained

$$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$8

which was reported as very close to the theoretical exponent $$\vec E = -\nabla \Phi, \qquad \vec j = \sigma(-\nabla \Phi + \vec v \times \vec B).$$9 and consistent with previous EVF studies. That work also cites earlier predictions of $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$0 at lower $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$1 and $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$2 at higher $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$3, with a swirl transition between regimes (Cheng et al., 2021).

Transient swirl and three-dimensionality have been quantified by helicity,

$$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$4

which measures the linkage or knottedness of the velocity field. In the cylindrical GaInSn study with external axial field, helicity increases with $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$5, but not uniformly: from $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$6 to about $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$7 the increase is slow; from about $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$8 to $$\Delta \varphi = \nabla \cdot (\vec v \times \vec B),$$9 it rises steeply; it then levels off around $\Delta \varphi_{cc} = 0.$0; another steep rise occurs from $\Delta \varphi_{cc} = 0.$1 to $\Delta \varphi_{cc} = 0.$2; beyond that it largely plateaus, with only mild growth up to $\Delta \varphi_{cc} = 0.$3. In the plane $\Delta \varphi_{cc} = 0.$4, the angular velocity was defined as

$\Delta \varphi_{cc} = 0.$5

and the maximum angular velocity was found to depend approximately linearly on $\Delta \varphi_{cc} = 0.$6 with the piecewise fit

$\Delta \varphi_{cc} = 0.$7

where $\Delta \varphi_{cc} = 0.$8 is nondimensionalized by $\Delta \varphi_{cc} = 0.$9 and $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$0 by $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$1 (Liu et al., 2022).

A later theoretical analysis for cylindrical electrodes refined the classical $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$2-based description by introducing the current-collector radius ratio

$$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$3

In a confined cylindrical domain with a co-axially placed current collector, the strength of EVF depends not only on $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$4, $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$5, and $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$6, but also on $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$7. The limiting behavior is explicit: as $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$8, current lines become nearly vertical and EVF should vanish; as $$\vec b(\vec r) = \frac{\mu_0}{4\pi}\int dV' \, \frac{\vec j(\vec r') \times (\vec r - \vec r')} {\left|\vec r - \vec r'\right|^3}.$$9, current diverges strongly and EVF is strongest. Using the vorticity transport equation in the high-$$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$0 regime, that study derived

$$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$1

and proposed a modified EVF parameter

$$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$2

with the revised scaling

$$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$3

For $$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$4 and $$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$5, the agreement between the theoretical estimate and numerical simulations was reported as $$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$6 for varying $$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$7, $$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$8 for varying $$\dot{T} + (u \cdot \nabla)T = \nabla \cdot \left(\frac{\lambda}{\rho C_p}\nabla T\right) + \frac{1}{\rho C_p}\frac{J^2}{\sigma},$$9, and $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$0 for the combined dataset. The same study notes that for $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$1, finite-domain effects become dominant and the canonical EVF structure changes qualitatively (Soni et al., 4 Aug 2025).

5. Experimental platforms and numerical methods

The experimental literature uses room-temperature liquid metals as analogues for battery electrodes. One platform consists of a cylindrical layer of liquid gallium with main cell diameter $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$2 and height $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$3 varied from $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$4 to $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$5, giving $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$6 from $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$7 to $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$8. Current is driven from a top negative electrode of selectable diameter to a bottom copper current collector, so that current lines spread strongly and create the radial current divergence needed for EVF. Thermal control is achieved with copper plates soldered to copper coils acting as heat exchangers; the coils are wound in a non-inductive double-spiral pattern to minimize unwanted horizontal temperature gradients. Since copper is attacked by gallium, the contact surfaces are coated with CuO, and the sidewall is Delrin to provide thermal resistance and electrical insulation. A further design choice routes the power leads vertically for a long distance so that the stray lateral magnetic field is below $$\frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T = \alpha \nabla^2 T + \frac{Q}{\rho_0 c_p}, \qquad Q = \frac{|\mathbf{J}|^2}{\sigma}.$$9, smaller than Earth’s field (Cheng et al., 2021).

Diagnostic access is constrained by the opacity of liquid metals, so Ultrasonic Doppler Velocimetry is central. The liquid-gallium battery analogue uses nine UDV transducers arranged to capture the distinct morphologies of convection and EVF, together with K-type thermocouples embedded in the plates. In the cylindrical GaInSn experiment, a DOP 3010 system with four sensors mounted at holes 1, 3, 5, and 7 measured transient velocity along the beam lines non-invasively. These studies also emphasize post-processing procedures that map sparse probe data back onto the physical geometry and interpolate them into movie-like flow visualizations (Cheng et al., 2021, Liu et al., 2019).

Numerically, EVF modeling has converged on coupled solid–liquid conductor formulations. A prominent OpenFOAM strategy employs a global parent mesh for electric potential across all conductive regions and a child mesh for fluid dynamics in the liquid. Electric potential is solved on the parent mesh with spatially varying conductivity, current density is mapped to the fluid mesh, and the magnetic field and Lorentz force are then evaluated where needed. At sharp conductivity jumps, the interface is placed on cell faces and harmonic interpolation is used,

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$00

For Poisson or Laplace problems with Neumann conditions, a deflated PCG regularization,

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$01

was introduced as a robust alternative to reference-cell regularization. The magnetic field is computed by a hybrid strategy that combines Biot–Savart for the static field with induction or vector-potential equations for the induced part; this was reported to be about $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$02 faster than volume Biot–Savart in one test, while full volume Biot–Savart was about $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$03 slower than the vector-potential approach on a $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$04-cell mesh (Weber et al., 2017).

Other numerical implementations extend these ideas to thermal coupling and realistic boundary conditions. The GaInSn model with Joule heating solved electric potential, current density, and temperature on a parent mesh spanning the conductive geometry, used the PIMPLE algorithm and Euler time discretization, and adopted residual criteria of $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$05 for the energy equation and $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$06 for momentum and electric potential. The current-collector study used multi-domain support with Dirichlet–Neumann partitioning and an interface potential

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$07

with iterative convergence to $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$08. Validation includes comparison against Opera for current distribution, the mercury-bath experiment of Zhilin et al. for EVF, and the Pb-Bi experiment of Ashour et al. for coupled convection and EVF (Liu et al., 2019, Weber et al., 2014, Keogh et al., 2020).

6. Interactions, related phenomena, and liquid-metal-battery relevance

In liquid metal batteries, EVF is one of the major flow drivers during charging and discharging. The broader hydrodynamic setting includes thermal-gradient-driven convection, because the electrolyte is much more resistive than the metal layers and Joule heating concentrates there. In the anode, unstable thermal stratification can drive convection; in the cathode, the thermal gradient may be stabilizing and suppress motion. One criterion discussed for the EVF–convection competition is Davidson’s estimate that convection dominates when

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$09

where

$$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$10

The alternative viewpoint is that the transition occurs when characteristic velocities are comparable (Cheng et al., 2021).

Direct coupled simulations show that the interaction is not trivial. In a Pb-Bi electrode model, Rayleigh–Bénard convection without current was found to be unsteady and transient. The introduction of a $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$11 current stabilized the convection cells, whereas a $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$12 current led to the dominance of EVF in the central region of the electrode. In that mixed regime, EVF dominated centrally while convection remained present near the edges, and the central jet was swept sideways by neighboring convection rolls and oscillated from side to side. When the imposed vertical background magnetic field was increased from $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$13 to $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$14, the model produced a swirl flow rather than a purely axial EVF structure (Keogh et al., 2020).

EVF must also be distinguished from the Tayler instability. The Tayler instability is a kink-type current-driven instability with a threshold; for the canonical cylindrical liquid-metal case, onset occurs beyond a critical Hartmann number $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$15, and the associated Reynolds number scales like $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$16. EVF, by contrast, is a forced flow with no critical current and a Reynolds number that was found to be approximately linear in Hartmann number in one studied regime. Because EVF appears immediately and can be strong before the Tayler mode grows, it can suppress Tayler-instability development. Short or well-coupled current collectors favor this outcome, whereas taller collectors weaken EVF and allow Tayler-instability growth to emerge later (Weber et al., 2014).

For design, current collectors and feeding lines are decisive. Tall current collectors allow current to spread over a larger vertical distance and reduce radial current gradients in the liquid; flat or shallow collectors force stronger current spreading and stronger EVF. The conductivity ratio $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$17 also matters: high-conductivity collectors weaken EVF by redistributing current inside the collector, while poorly conducting collectors shift more current closure into the liquid and strengthen EVF. The strongest change occurs near $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$18. One study states that flattening the current collectors below an aspect ratio of $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$19 increases the Lorentz forces further and would definitely lead to an instationary electro-vortex flow, whereas collectors with $$\partial_t \vec v + (\vec v\cdot\nabla)\vec v = -\frac{1}{\rho}\nabla p + \nu \Delta \vec v + \frac{1}{\rho}\vec f, \qquad \nabla\cdot \vec v = 0,$$20 or larger are much more favorable (Weber et al., 2014).

These findings explain the ambivalent role of EVF in batteries. Moderate motion can enhance mixing and transport, helping homogenize the electrode and reduce concentration buildup, but sufficiently strong EVF can deform the fluid interfaces between battery layers and trigger failure. The literature therefore treats EVF not merely as an unavoidable side effect of current supply, but as a design-sensitive transport mechanism whose strength, topology, and coupling to other flows depend on current magnitude, current-collector geometry, electrical boundary conditions, and even geomagnetic-strength background fields (Cheng et al., 2021, Liu et al., 2022, Weber et al., 2014).