Fluid-rigid body interaction in an incompressible electrically conducting fluid

TL;DR

This paper establishes the existence of weak solutions for a coupled fluid-structure and magnetohydrodynamics model involving an insulating rigid body in an incompressible conducting fluid, using advanced mathematical techniques.

Contribution

It introduces a novel mathematical framework and proof for the existence of solutions to a complex fluid-structure-electromagnetic interaction system.

Findings

Existence of weak solutions proven for the coupled system.

Application of Rothe method for decoupling the PDE system.

Use of Brinkman penalization to handle mechanical aspects.

Abstract

We analyze a mathematical model that describes the interaction between an insulating rigid body and an incompressible electrically conducting fluid surrounding it. The model as well as the mathematical analysis involve the fields of fluid-structure interaction and of magnetohydrodynamics. Our main result shows the existence of a weak solution to the corresponding system of partial differential equations. The proof relies on a use of a time discretization via the Rothe method to decouple the system. This allows us to deal with test functions, depending on the position of the moving body and therefore on the solution of the system, in the weak formulation of the induction equation. The proof moreover makes use of the Brinkman penalization in order to cope with the mechanical part of the problem.