Electric interface condition for sliding and viscous contacts

The first principles of electromagnetism impose that the tangential electric field must be continuous at the interface between two media. The definition of the electric field depends on the frame of reference, leading to an ambiguity in the mathematical expression of the continuity condition when the two sides of the interface do not share the same rest frame. We briefly review the arguments supporting each choice of interface condition and illustrate how the most theoretically consistent choice leads to a paradox in induction experiments. We then present a model of sliding contact between two solids and between a fluid and a solid and show how this paradox can be lifted by taking into account the shear induced by the differential motion in a thin intermediate viscous layer at the interface, thereby also lifting the ambiguity in the electric interface condition. We present some guidelines regarding the appropriate interface condition to employ in magnetohydrodynamics applications, in particular for numerical simulations in which sliding contact is used as an approximation of the viscous interface between a conducting solid and a fluid of very low viscosity such as in planetary interior simulations.

Figure 1

(a) A rotating cylinder is immersed within a transverse uniform background magnetic field. (b) Solution for $\mathrm{Rm}=10$, based on condition CI and assuming continuity of the tangent magnetic field at the boundary. Streamlines show the magnetic field; colors show the normalized value of the axial current density, with blue and red pointing, respectively, inside and outside the page. (c) Same as in (b), but based on condition CII and allowing for surface currents. The internal magnetic field is zero in that case for all values of $\mathrm{Rm}$, and the current density is restricted to a vanishingly thin sheet at the cylinder's surface.

Figure 2

Local model of the interface between two semi-infinite conducting solids sliding on top of each other. The presence of a thin viscous layer between the two solids (in blue) radically affects the solution (see the text).

Figure 3

Local model of the solid-fluid interface. The finite viscosity of the fluid causes the appearance of a thin boundary layer in the fluid region represented by the dotted area (see main text).

Figure 4

Thickness of the boundary layer at the fluid-solid interface for parameters typical of Earth's upper fluid core as a function of the frequency (in cycles per day). The dashed line shows the steady limit corresponding to the Hartmann layer thickness ${\delta }_H=\sqrt{\nu \eta }/v_A$.

Figure 5

Top: velocity and magnetic field perturbations as a function of depth for three values of the frequency (in cycles per day). Bottom: relative jump in velocity and magnetic field. The black dashed lines indicate the depth of the Hartmann layer in the steady limit (see Fig. 4).

Figure 6

Top: electric current density divided by electric conductivity and electric field near the interface. Bottom: relative jump in current density and electric field. The black dashed lines indicate the depth of the Hartmann layer in the steady limit (see Fig. 4).

Figure 7

Velocity and electric current density near a solid-fluid interface using the boundary conditions (28a) and (28b) (solid curves) compared to the exact solution in Sec. 4 (dashed curves).

Figure 8

(a) The circuit contour $\mathcal{C}$ enclosing the surface $\mathcal{S}$ intersects the surface of discontinuity between two regions of space perpendicularly along the straight line $\sigma$. (b) It is always possible to decompose each of $\mathcal{C}$ and $\mathcal{S}$ into two parts if we specify appropriate junction conditions at $\sigma$ (see the text).

Figure 9

Similar to Fig. 8, but with an intermediate transition layer of thickness $\delta$ with cross section ${\mathcal{S}}^{\delta }$ enclosed by the con- tour ${\mathcal{C}}^{\delta }$.