Feasibility of Metal Pad Roll Instability Experiments at Room Temperature

Combining theoretical arguments and numerical simulations, we demonstrate that the metal pad roll instability can occur in a centimeter-scale setup with reasonable values of the magnetic field and electrical current and with metal pairs that are liquid at room temperature. We investigate two fluid pairs: gallium with mercury (immiscible pair) or gallium with GaInSn eutectic alloy (miscible pair).

Figure 1

Sketch of the cylindrical setup and instability mechanism. (a) A layer of gallium floats on a layer of mercury or eutectic GaInSn alloy. A current $\mathit{J}$ passes through the cell which is subjected to an external vertical magnetic field ${\mathit{B}}^e$. (b) Because of the conductivity difference (gallium is a better conductor), a small excess current $\mathit{j}$ appears as soon as the interface is inclined. This excess current interacts with the external magnetic field and generates a Lorentz force $\mathit{j}\times {\mathit{B}}^e$ (green arrows). (c) This force transfers power [$(\mathit{j}\times {\mathit{B}}^e)·\mathit{u}>0$] to a wave that rotates anticlockwise when seen from above.

Figure 2

Growth rate $\lambda$ in ${\mathrm{s}}^{-1}$ of the fundamental wave ($m=1$, ${\kappa }_{11}=1.841$) in the $J\text{−}{B}_z$ plane in a $R=H_1=H_2=2\mathrm{cm}$ setup with layers of Ga and Hg or Ga and GaInSn eutectic.

Figure 3

Growth rate $\lambda$ in ${\mathrm{s}}^{-1}$ of the fundamental wave ($m=1$, ${\kappa }_{11}=1.841$) as a function of $J$ or $I$ in a $R=H_1=H_2=2\mathrm{cm}$ setup with magnetic field $B_z=15\mathrm{mT}$.

Figure 4

Simulations in the immiscible Ga and Hg setup. (a) Time series of the velocity at $r_a=0$, ${\theta }_a=0$, $z_a=-5\mathrm{mm}$ gives $\omega$. (b) Time series of $e_{c,1}$ gives $\lambda$ ($e_{c,1}$ is the kinetic energy of the azimuthal Fourier mode $m=1$). (c) Isosurface $\rho =({\rho }_1+{\rho }_2)/2$ and turbulent structures in the vertical plane $x=0$ at $t=37\mathrm{s}$.

Figure 5

Emergence of a rotating gravity wave in Ga and GaInSn eutectic (miscible) setup, $J=1.5\times {10}^5\mathrm{A}{\mathrm{m}}^{-2}$. We show the isosurface $\rho =({\rho }_1+{\rho }_2)/2$, the density $\rho$ in color code in the vertical plane $x=0$, and lines of excess current $\mathit{j}$.

Figure 6

Simulation with Ga and GaInSn eutectic (miscible setup) using exaggerated diffusion $D_{\mathrm{fake}}=100D=1.7\times {10}^{-7}{\mathrm{m}}^2{\mathrm{s}}^{-1}$; $J=4\times {10}^5\mathrm{A}{\mathrm{m}}^{-2}$; $t=31.35\mathrm{s}$. The interface shown here is the isosurface $\rho =({\rho }_1+{\rho }_2)/2$.