Magnetically driven flow in a liquid-metal battery

We investigate the flow within a liquid-metal battery induced by an externally imposed magnetic field, $B_0$. An analytical model for laminar flow is proposed and this is found to be in excellent agreement with numerical simulations, not only for weakly forced steady flow, but also for the time-averaged velocity in more strongly forced flows where the motion is either unsteady (but laminar) or else weakly turbulent. Our primary conclusion is that surprisingly weak magnetic fields are capable of destabilizing the flow and inducing turbulence, with turbulence first appearing at $B_0\approx 1.0\mathrm{G}$. By comparison, the earth's magnetic field is ∼0.5 G. This extreme sensitivity of liquid-metal pools carrying current to an external magnetic field has long been known in the context of other industrial processes.

Figure 1

Schematic of a liquid-metal battery. (a) A simple design. (b) An alternative design.

Figure 2

Our model of a liquid-metal battery.

Figure 3

The current distribution for our benchmark case. Note that very little of the current passes through the base of the container.

Figure 4

A comparison of the depth-average force distribution of Eq. (10) (yellow) and power-law approximation (14) (blue), with $p=6$, for the benchmark case.

Figure 5

The overall structure of the secondary flow.

Figure 6

The instantaneous azimuthal velocity for the case of $B_0=0.5\mathrm{G}$ at $t=2500\mathrm{s}$: (a) $z$/$d=0.23$; (b) midplane ($z$/$d=0.5$). The flow is computed for the applied forces: full force distribution [Eq. (9), curve 1, blue]; depth averaged [Eq. (10), curve 2, red]; power-law approximation [Eq. (14)] with $p=6$ (curve 3, yellow) and $p=3$ (curve 4, purple). The theoretical prediction of Eq. (23) with $p=6$ is plotted in curve 5 (green, dashed).

Figure 7

Variation of the time-averaged $u_{\theta }$ with vertical position for ${\mathrm{B}}_0=0.5\mathrm{G}$, at $r$/$R=0.78$, 0.89, 0.972. The symbol $\langle \rangle$ denotes a time average.

Figure 8

(a) The instantaneous velocity magnitude, $\left|\mathbf{u}\right|$, and streamlines in a vertical plane, for $B_0=0.1\mathrm{G}$ at $t=1000\mathrm{s}$. (b) The time-averaged velocity magnitude, and streamlines showing the time-averaged poloidal flow, for $B_0=0.5\mathrm{G}$.

Figure 9

The instantaneous velocity magnitude, $\left|\mathbf{u}\right|$, and streamlines in a vertical plane, for $B_0=0.5\mathrm{G}$ at times (a) $t=940\mathrm{s}$, (b) $t=1010\mathrm{s}$, (c) $t=1090\mathrm{s}$, and (d) $t=1160\mathrm{s}$.

Figure 10

The vertical velocity, for $B_0=0.5\mathrm{G}$, at $r$/$R=0.972$ (left) and on the horizontal plane $z$/$d=0.23$ (right) at (a) $t=940\mathrm{s}$, (b) $t=1010\mathrm{s}$, (c) $t=1090\mathrm{s}$, and (d) $t=1160\mathrm{s}$.

Figure 11

The maximum time-averaged azimuthal velocity as a function of $B_0$ (in gauss) for the field strengths $B_0=0.1–1.5\mathrm{G}$. The dotted line is $B_0^{2/3}$.

Figure 12

The instantaneous and time-averaged azimuthal velocity on the planes $z$/$d=0.23$ (left) and midplane (right) for $B_0=1\mathrm{G}$. The computed flows correspond to the full force distribution [Eq. (9), blue and red] and the power-law approximation [Eq. (14)] with $p=6$ (yellow and purple). The theoretical prediction of Eq. (23) is also shown (green, dashed). The symbol $\langle \rangle$ denotes a time average.

Figure 13

The instantaneous velocity magnitude, $\left|\mathbf{u}\right|$, and streamlines in a vertical plane, for $B_0=1\mathrm{G}$. The times are (a) $t=940\mathrm{s}$, (b) $t=1010\mathrm{s}$, (c) $t=1090\mathrm{s}$, and (d) $t=1160\mathrm{s}$.

Figure 14

The vertical velocity at $r/R=0.972$ (left) and on the horizontal plane $z$/$d=0.23$ (right) for $B_0=1\mathrm{G}$ and at times (a) $t=940\mathrm{s}$, (b) $t=1010\mathrm{s}$, (c) $t=1090\mathrm{s}$, and (d) $t=1160\mathrm{s}$.

Figure 15

Time series of $\left|\mathbf{u}\right|$ at (a) $z$/$d=0.23$ and (b) $z$/$d=0.5$ for $B_0=0.5$, 0.8, 1.0, and 1.5 G at the radius $r$/$R=0.972$. Dashed lines show the times in Figs. 9, 10, 13, and 14.

Figure 16

Time series of $\left|\mathbf{u}\right|$ at the same height, $z$/$d=0.23$, but at different radii, $r$/$R=0.78$, 0.89, 0.972. (a) $B_0=0.5\mathrm{G}$, (b) $B_0=0.8\mathrm{G}$, and (c) $B_0=1.0\mathrm{G}$. The dashed lines show the times for Figs. 9, 10, 13, and 14.