Electro-vortex flows (EVF) arise in conducting fluids due to diverging/converging current lines and the nonconservative Lorentz force, resulting from the interaction of this current with its own magnetic field. They are typically characterized by the EVF parameter, defined as $S={\mu }_0{I}^2/\left(4{\pi }^2\rho {\nu }^2\right)$ (${\mu }_0$ is the magnetic permeability), where it is known that $\mathit{\text{Re}}\propto \sqrt{S}$ for large $\mathit{\text{Re}}$. However, the strength of the EVF in a confined cylindrical cell with a coaxially placed current collector (CC) depends also on the ratio of the CC radius to the cylinder radius, $K=r_0/R$, in addition to the current magnitude, $I$, fluid density, $\rho$, and kinematic viscosity, $\nu$. Using the inertia-Lorentz balance in the vorticity transport equation, we derive a theoretical estimate of the r.m.s. EVF velocity and find that $u\propto I(1-K)/\sqrt{K}$, applicable for the aspect ratio of 2 (radius equals height). Our estimate leads to a modified EVF parameter, $S_M$, that incorporates $K$. We also explore the relationship between $K$ and the curvature in Lorentz force. Three-dimensional direct numerical simulations performed with our in-house OpenFOAM code confirm the validity of the estimate, with excellent agreement observed between theory and simulations across the range $S_M\in [63,9125]\times {10}^4$. We also discuss the key EVF characteristics using the vorticity dynamics: its driving mechanism, formation, and evolution. Intriguingly, our numerical simulation results reveal distinct flow features for $K\ge 0.75$, not reported earlier in the literature. Our results suggest that the scaling relationship should actually be $\mathit{\text{Re}}\propto \sqrt{S_M}$.

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• Magnetohydrodynamics
• Pattern formation
• Vortex dynamics