Electric interface condition for sliding and viscous contacts

TL;DR

This paper examines the electric interface conditions at sliding contacts, highlighting ambiguities due to reference frames, and proposes a viscous layer model to resolve paradoxes and improve magnetohydrodynamics simulations.

Contribution

It introduces a viscous layer model at interfaces to clarify electric boundary conditions in sliding contacts, resolving theoretical paradoxes.

Findings

Theoretical ambiguities in electric interface conditions are addressed.

A viscous layer model resolves paradoxes in induction experiments.

Guidelines for applying interface conditions in magnetohydrodynamics are provided.

Abstract

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 consistant 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 where sliding contact is used as an approximation to the viscous interface between a conducting solid and a fluid of very low viscosity such as in planetary interior simulations.