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Modeling of fluid-rigid body interaction in an electrically conducting fluid

Jan Scherz, Anja Schlömerkemper

Abstract

We derive a mathematical model for the motion of several insulating rigid bodies through an electrically conducting fluid. Starting from a universal model describing this phenomenon in generality, we elaborate (simplifying) physical assumptions under which a mathematical analysis of the model becomes feasible. Our main focus lies on the derivation of the boundary and interface conditions for the electromagnetic fields as well as the derivation of the magnetohydrodynamic approximation carried out via a nondimensionalization of the system.

Modeling of fluid-rigid body interaction in an electrically conducting fluid

Abstract

We derive a mathematical model for the motion of several insulating rigid bodies through an electrically conducting fluid. Starting from a universal model describing this phenomenon in generality, we elaborate (simplifying) physical assumptions under which a mathematical analysis of the model becomes feasible. Our main focus lies on the derivation of the boundary and interface conditions for the electromagnetic fields as well as the derivation of the magnetohydrodynamic approximation carried out via a nondimensionalization of the system.
Paper Structure (14 sections, 1 theorem, 123 equations, 2 figures)

This paper contains 14 sections, 1 theorem, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model used in the mathematical analysis
  3. General model
  4. Mathematical assumptions
  5. Adjustments to the material assumptions
  6. Interface conditions for the electromagnetic fields
  7. Magnetohydrodynamic approximation via nondimensionalization
  8. The Maxwell system
  9. The Navier-Stokes system
  10. Summary of the derived system
  11. Existence of weak solutions
  12. Appendix
  13. Proof of the estimate \ref{['wefujwe09uuj09']}
  14. Proof of the estimate \ref{['boundaryestimate']}

Key Result

Theorem 7.1

compressiblepaper Let $T > 0$, assume $\Omega \subset \mathbb{R}^3$ to be a simply connected domain of class $C^{2,\xi}$, $\xi \in (0,1)$, and assume $S_0^i \subset \Omega$, $i=1,...,N \in \mathbb{N}$ to be bounded domains of class $C^2$ which satisfy the conditions -326. Assume moreover the coeffic for almost all $\tau \in [0,T]$.

Figures (2)

  • Figure 1: The curved rectangle $\Delta F$.
  • Figure 2: The cylinder $C$ with curved bases.

Theorems & Definitions (2)

  • Definition 7.1
  • Theorem 7.1