MODELING OF FLUID-RIGID BODY INTERACTION IN AN ELECTRICALLY CONDUCTING FLUID
JAN SCHERZ1,2,3, ANJA SCHLÖMERKEMPER1

  * 1

Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074
Würzburg, Germany

  * 2

Department of Mathematical Analysis, Faculty of Mathematics and Physics,
Charles University in Prague, Sokolovská 83, Prague 8, 18675, Czech Republic

  * 3

Mathematical Institute, Czech Academy of Sciences, Žitná 25, Prague 1, 11567,
Czech Republic

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.

##  1 Introduction

The study of electrically conducting fluids interacting with solid materials
is motivated by biomechanical applications such as capsule endoscopy, cf.
[14]. This medical procedure constitutes a minimally invasive method for the
detection of diseases by propelling small capsule shaped camera devices
through parts of the human body such as veins or arteries. In the electrically
conducting blood it is possible to generate the drive and control the
navigation of these devices remotely by applying electromagnetic forces. A
similar application is given by a procedure known as remote drug delivery, cf.
[13, Section 4.4]: Robots of a microscopic scale can be used to transport
medication through the blood stream in the human body directly to the region
of the body in which it is needed while damage through the medication to
healthy tissue is avoided.

In this article we study the mathematical modeling of the motion of several
insulating rigid bodies through an electrically conducting diamagnetic
dielectric viscous non-homogeneous fluid surrounded by a perfect conductor. In
the mathematical examination of this and similar phenomena, the considered
models are typically simplified subject to certain physical restrictions in
order to make their mathematical analysis feasible. The main contribution of
the present article is the rigorous derivation of specific models used in the
literature (see below) from a universal model describing the situation in
generality. Further, the physical assumptions under which this derivation is
justified are elaborated on.

The interaction between electrically conducting fluids and insulating rigid
bodies constitutes a combination of two types of physical phenomena; fluid-
structure interaction (FSI) and magnetohydrodynamics (MHD). The research field
of FSI deals with interactions between rigid or deformable solids and fluids
contained in, adjacent to or surrounding the solids, cf. [3, 4]. In the case
of rigid bodies, FSI is commonly modeled via a coupling of the (incompressible
or compressible) Navier-Stokes equations for the description of the fluid
motion and the balances of linear and angular momentum of the bodies. The
mathematical literature is rich in analytical results on this model: We
specifically mention the articles [9, 10], wherein the global-in-time
existence of weak solutions to the problem was proved in the compressible and
the incompressible setting, respectively. We further mention the article [26],
in which the same result was shown in the two-dimensional incompressible
setting with an additional examination of the question of the possibility of
collisions between the bodies.

MHD instead describes the study of the interaction between electrically
conducting fluids and electromagnetic fields. The behaviour of electromagnetic
fields is described by the Maxwell equations, cf. for example [8]. In MHD,
these equations are then coupled with the Navier-Stokes equations for the
description of the fluid, cf. [5, 7, 21]. The equations in this coupling are
further simplified in comparison to the original systems under several
physical assumptions. The latter procedure is commonly referred to as the
magnetohydrodynamic approximation. A justification of this approximation is
for example given in [19, 20]. Mathematically, the MHD problem is also well
investigated and we refer to [12, 27] for the proofs of the global-in-time
existence of weak solutions in the case of an incompressible and a
compressible fluid, respectively.

Despite the extensively worked out theory in both FSI and MHD, not many
results on the combination between these two research areas appear to be
available. A first mathematical exploration of this uncharted territory was
made in [17, 18]. In those articles the flow of an electrically conducting
incompressible fluid around an insulating rigid body is studied in two and
three spatial dimensions, respectively, and the existence of weak solutions to
the corresponding models is proved. The rigid body in those articles, however,
is assumed to be immovable; an extension of the result to the case of a freely
moving body was achieved by the authors of the present article, in joint work
with Barbora Benešová and Šárka Nečasová, in [1]. A further extension to the
setting of a compressible fluid was obtained later in [28], see also [29].

In the present article we derive the models used in the latter two papers (and
thus of [17, 18] when assuming that the rigid bodies are not moveable
anymore). More precisely, we provide the mathematical derivation of a model
for the motion of several insulating rigid bodies through an electrically
conducting diamagnetic dielectric viscous non-homogeneous fluid surrounded by
a perfect conductor. The individual partial differential equations in this
model, e.g. the Navier-Stokes equations and the Maxwell’s equations, are well-
known. In particular, as mentioned above, there exists plenty of literature on
magnetohydrodynamics (see again [5, 7, 21]), which lies in the center of
interest of our investigations. We thus remark that the findings of this
article do not constitute a new result. Instead, we provide deeper
mathematical insight in the derivation than we were able to find in the
literature.

We further point out that the present article is a self-contained variant of a
chapter from the PhD thesis [29] of the first author, in which the derivation
of the desired model is given in the case of an incompressible fluid. Here, we
focus on the compressible case, which is achieved by the same procedure as in
the incompressible case.

The outline of the article is as follows: In Section 2, we present the desired
model analyzed mathematically in [28], the mathematical derivation of which is
the main contribution of this article. Our starting point in the derivation of
this model is a more general system composed of the Maxwell equations, the
compressible Navier-Stokes equations and the balances of linear and angular
momentum of the rigid bodies in their most universal forms, which we present
in Section 3. This system models the interaction between the fluid, the rigid
bodies and the electromagnetic fields inside of both materials in full
generality. After a first adjustment of the system to the properties of the
materials we consider in our specific setting - in particular the non-
conductivity of the solid domain - our procedure consists of two main steps:
Firstly, in Section 4, we derive boundary and interface conditions for the
electromagnetic fields from the Maxwell equations. Secondly, in Section 5, we
simplify the system by identifying several insignificant terms and dropping
them from the equations. This step, which is achieved via a
nondimensionalization, in particular constitutes the classical
magnetohydrodynamic approximation in the fluid part of the domain. In Section
6, we summarize the final system we achieve through this approach. This system
constitutes a slightly more general version of the model from Section 2. The
latter system is then obtained after a few more (mathematical) adjustments
discussed at the end of Section 6. Finally, in Section 7, we give a short
recap of the result on the existence of weak solutions to the derived model,
which was originally stated in [28].

##  2 Model used in the mathematical analysis

We study several insulating rigid bodies moving through an electrically
conducting viscous non-homogeneous compressible fluid. Additionally, we assume
the fluid to be a linearly magnetic material and a linearly dielectric, cf.
the relations (3.36) below. The latter conditions are in particular satisfied
for diamagnetic linearly dielectrics such as for example blood, cf. [11, 24].
We also assume the fluid to be surrounded by a (rigid) perfect conductor. In
the present section we showcase a mathematical model of this situation which
has been analyzed mathematically in [28]. This model, however, is subject to
certain further physical and mathematical assumptions. The main contribution
of this article is the derivation of this model from a universal model
describing the above situation in full generality, cf. Section 3 below.

We consider a domain $\Omega\subset\mathbb{R}^{3}$, a time interval $[0,T]$
with $T>0$ and we denote by $Q$ the time space domain $Q:=(0,T)\times\Omega$.
The domain $\Omega$ is filled, at each time $t\in[0,T]$, with $N\in\mathbb{N}$
rigid bodies $S^{i}(t)\subset\Omega$, $i=1,...,N$ as well as the fluid
$F(t):=\Omega\setminus\overline{S(t)}$, where

$\displaystyle S(t):=\bigcup_{i=1}^{N}S^{i}(t)$

denotes the solid domain at time $t$. Similarly, we divide the time space
domain $Q$ into its solid part $Q^{s}$ and its fluid part $Q^{f}$,

$\displaystyle Q^{s}:=\left\\{(t,x)\in Q:\ x\in S(t)\right\\},\quad\quad
Q^{f}:=Q\setminus\overline{Q^{s}}.$

For any function defined on $Q$ we indicate, whenever it is necessary to
stress the difference, its restrictions to $Q^{s}$ and $Q^{f}$ by the
superscripts $s$ and $f$, respectively. The quantities which model the
interaction between the electromagnetic fields, the fluid and the rigid bodies
can be distinguished between electromagnetic and mechanical quantities. On the
electromagnetic side we have the magnetic induction
$B:Q\rightarrow\mathbb{R}^{3}$, the magnetic field
$H:Q\rightarrow\mathbb{R}^{3}$, the electric field
$E:Q\rightarrow\mathbb{R}^{3}$ and the electric current density
$j:Q\rightarrow\mathbb{R}^{3}$. The mechanical quantities are the density
$\rho:Q\rightarrow\mathbb{R}$, the velocity field
$u:Q\rightarrow\mathbb{R}^{3}$ and the pressure
$p=p^{f}:Q^{f}\rightarrow\mathbb{R}$. The evolution of these quantities is
determined by a system of partial differential equations consisting of the
Maxwell equations

$\displaystyle\text{curl}H$
$\displaystyle=\left\\{\begin{array}[]{ll}j+J&\text{in }Q^{f},\\\ 0&\text{in
}Q^{s},\end{array}\right.$ (2.3) $\displaystyle\partial_{t}B+\text{curl}E$
$\displaystyle=0\quad\quad\quad\quad\quad\text{in }Q^{f}\ \text{and }Q^{s},$
(2.4) $\displaystyle\ \text{div}E$
$\displaystyle=0\quad\quad\quad\quad\quad\text{in }Q^{s},$ (2.5)
$\displaystyle\text{div}B$ $\displaystyle=0\quad\quad\quad\quad\quad\text{in
}Q^{f}\ \text{and }Q^{s},$ (2.6)

the compressible Navier-Stokes equations

$\displaystyle\partial_{t}\rho+\operatorname{div}(\rho u)=$ $\displaystyle
0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ \text{in
}Q^{f},$ (2.7) $\displaystyle\partial_{t}(\rho u)+\text{div}(\rho u\otimes
u)+\nabla p=$ $\displaystyle\text{div}\mathbb{T}+\rho
g+\frac{1}{\mu}\text{curl}B\times B\quad\quad\text{in }Q^{f}$ (2.8)

and the balances of linear and angular momentum

$\displaystyle m^{i}\frac{d}{dt}V^{i}(t)=\frac{d}{dt}\int_{S^{i}(t)}\rho u\
dx=\int_{\partial S^{i}(t)}\left[\mathbb{T}-p\ \text{id}\right]\text{n}\
dA+\int_{S^{i}(t)}\rho g\ dx,$ $\displaystyle\quad t\in[0,T],$ (2.9)
$\displaystyle\frac{d}{dt}\left(\mathbb{J}^{i}(t)w^{i}(t)\right)=\frac{d}{dt}\int_{S^{i}(t)}\rho\left(x-X^{i}\right)\times
u\ dx\ $ $\displaystyle=\int_{\partial
S^{i}(t)}(x-X^{i})\times\left[\mathbb{T}-p\ \text{id}\right]\text{n}\
dA+\int_{S^{i}(t)}\rho\left(x-X^{i}\right)\times g\ dx,$ $\displaystyle\quad
t\in[0,T]$ (2.10)

for $i=1,...,N$ in combination with the relations

$\displaystyle j=\sigma(E+u\times B)\quad\text{in }Q^{f}\ \text{and
}Q^{s},\quad\quad\sigma=\left\\{\begin{array}[]{ll}\sigma^{f}>0&\text{in
}Q^{f},\\\ \sigma^{s}=0&\text{in }Q^{s},\end{array}\right.$ (2.13)
$\displaystyle\quad\quad\quad\quad\quad\quad\ \ B=\mu
H,\quad\quad\mu>0\quad\text{in }Q$ (2.14)

as well as the boundary and interface conditions

$\displaystyle B(t)\cdot\text{n}$ $\displaystyle=0\quad\text{on
}\partial\Omega,\quad\quad$ $\displaystyle\ \ \ \ \ \ \ \ \
B^{f}(t)-B^{s}(t)=0\quad\text{on }\partial S(t),$ (2.15) $\displaystyle
E(t)\times\text{n}$ $\displaystyle=0\quad\text{on }\partial\Omega,\quad\quad$
$\displaystyle\left(E^{f}(t)-E^{s}(t)\right)\times\text{n}=0\quad\text{on
}\partial S(t),$ (2.16) $\displaystyle u(t)$ $\displaystyle=0\quad\text{on
}\partial\Omega,\quad\quad$ $\displaystyle\ \ \ \ \ \ \ \ \ \
u^{f}(t)-u^{s}(t)=0\quad\text{on }\partial S(t).$ (2.17)

The Maxwell system (2.3)–(2.6) consists of Ampère’s law (2.3) - in which
$J:Q\rightarrow\mathbb{R}^{3}$ constitutes an external forcing term -, the
Maxwell-Faraday equation (2.4), Gauss’s law (2.5) and Gauss’s law for
magnetism (2.6). The Navier-Stokes system (2.7), (2.8) is composed of the
continuity equation (2.7) and the momentum equation (2.8). The latter equation
contains two forcing terms: On the one hand, the quantity $\rho g$, in which
the $g:Q\rightarrow\mathbb{R}^{3}$ denotes a given function, represents an
external force, e.g. the gravitational force. On the other hand, the influence
of the electromagnetic fields on the motion of the fluid shows itself in the
Lorentz force $\frac{1}{\mu}\operatorname{curl}B\times B$. The stress tensor
$\mathbb{T}$ in the momentum equation is defined as

$\displaystyle\mathbb{T}=\mathbb{T}(u):=2\nu\mathbb{D}(u)+\lambda\text{id}\
\operatorname{div}u,\quad\quad\mathbb{D}(u):=\frac{1}{2}\nabla
u+\frac{1}{2}(\nabla u)^{T},$

where the constants $\nu>0$ and $\lambda\geq-\nu$ are referred to as the
viscosity coefficients. Moreover, the pressure $p$ in the momentum equation is
assumed to depend only on the density $\rho$. More specifically, it is assumed
to satisfy the isentropic constitutive relation

$\displaystyle p=p\left(\rho^{f}\right)=a\left(\rho^{f}\right)^{\gamma},\quad
a>0,\ \gamma>\frac{3}{2}.$ (2.18)

In the balances (2.9), (2.10) of linear and angular momentum, the quantities

$\displaystyle m^{i}:=$ $\displaystyle\int_{S^{i}(t)}\rho(t,x)\ dx,\quad
X^{i}(t):=\frac{1}{m^{i}}\int_{S^{i}(t)}\rho(t,x)x\ dx,$
$\displaystyle\mathbb{J}^{i}(t)a\cdot b:=$
$\displaystyle\int_{S^{i}(t)}\rho(t,x)\left[a\times\left(x-X^{i}(t)\right)\right]\cdot\left[b\times\left(x-X^{i}(t)\right)\right]\
dx,\quad a,b\in\mathbb{R}^{3},$

denote the mass $m^{i}$, the center of mass $X^{i}$ and the inertia tensor
$\mathbb{J}^{i}$ of the $i$-th rigid body, respectively. These equations
determine the translational velocity $V^{i}$ as well as the rotational
velocity $w^{i}$ of the $i$-th rigid body, the overall velocity of which is
then given as

$\displaystyle
u(t,x)=u^{s^{i}}(t,x):=V^{i}(t)+w^{i}(t)\times\left(x-X^{i}(t)\right)\quad\quad\text{for
all }t\in[0,T]\ \text{and }x\in S^{i}(t).$

In particular, the balances of linear and angular momentum show how the motion
of the insulating bodies is influenced by the fluid, however, not - directly -
by the electromagnetic fields. The influence of the fluid on the
electromagnetic fields is seen in Ohm’s law (2.13). In this relation, $\sigma$
stands for the electric conductivity. Since $\sigma=\sigma^{s}=0$ in the
insulating solids, the equation further shows that the bodies have no (direct)
impact on the behavior of the electromagnetic fields. The constant $\mu$ in
the linear relation (2.14) represents the magnetic permeability and, finally,
we point out that in the interface and boundary conditions (2.15)–(2.17) the
no slip condition (2.17) for the velocity field displays the influence of the
rigid bodies on the fluid motion.

As explained above, the system (2.3)–(2.17) does not model the interaction
between insulating rigid bodies and an electrically conducting fluid in full
generality but rather under certain additional assumptions. In the following
section we will present a universal model and subsequently we will precisely
work out the assumptions under which this model can be modified such that it
turns into the model (2.3)–(2.17).

##  3 General model

In the present section we showcase a universal model for the motion of
insulating rigid bodies through an electrically conducting linearly dielectric
and linearly magnetic viscous non-homogeneous compressible fluid surrounded by
a perfect conductor. The notation for this model essentially coincides with
the one for the model (2.3)–(2.17), however, it requires a few additions which
we summarize in the following. The domain of the perfect conductor is
$\mathbb{R}^{3}\setminus\overline{\Omega}$, we denote the associated time
space domain by

$\displaystyle
Q^{\operatorname{ext}}:=(0,T)\times\left(\mathbb{R}^{3}\setminus\overline{\Omega}\right).$

The restriction of functions defined on $(0,T)\times\mathbb{R}^{3}$ to
$Q^{\operatorname{ext}}$ is indicated by the superscript $\operatorname{ext}$.
Further, for the desription of the electromagnetic effects in the general
setting, we require several additional physical quantities: We denote by
$D:(0,T)\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ the electric induction,
by $\rho_{c}:(0,T)\times\mathbb{R}^{3}\rightarrow\mathbb{R}$ the density of
electric charges, by $M:(0,T)\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$
the magnetization and by
$P:(0,T)\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ the polarization. The
system of partial differential equations modelling the considered fluid-rigid
body interaction in full generality consists of the Maxwell equations

$\displaystyle\operatorname{curl}H=$
$\displaystyle\partial_{t}D+j+J\quad\quad\text{in }Q^{f},\ Q^{s}\text{ and
}Q^{\text{ext}},$ (3.1) $\displaystyle\partial_{t}B+\operatorname{curl}E=$
$\displaystyle 0\quad\quad\quad\quad\quad\quad\ \ \text{in }Q^{f},\
Q^{s}\text{ and }Q^{\text{ext}},$ (3.2) $\displaystyle\operatorname{div}D=$
$\displaystyle\rho_{c}\quad\quad\quad\quad\quad\quad\ \text{in }Q^{f},\
Q^{s}\text{ and }Q^{\text{ext}},$ (3.3) $\displaystyle\operatorname{div}B=$
$\displaystyle 0\quad\quad\quad\quad\quad\quad\ \ \text{in }Q^{f},\
Q^{s}\text{ and }Q^{\text{ext}},$ (3.4)

the compressible Navier-Stokes equations

$\displaystyle\partial_{t}\rho+\operatorname{div}\left(\rho u\right)=$
$\displaystyle
0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
\text{in }Q^{f},$ (3.5) $\displaystyle\partial_{t}\left(\rho
u\right)+\operatorname{div}\left(\rho u\otimes u\right)+\nabla p=$
$\displaystyle\operatorname{div}\mathbb{T}+\rho
g+\rho_{c}E+\left(j+J\right)\times B\quad\quad\text{in }Q^{f},$ (3.6)

and the balances of linear and angular momentum

$\displaystyle m^{i}\frac{d}{dt}V^{i}(t)=\frac{d}{dt}\int_{S^{i}(t)}\rho u\
dx=\int_{\partial S^{i}(t)}\left[\mathbb{T}-p\ \text{id}\right]\text{n}\
dA+\int_{S^{i}(t)}\rho g\ dx,$ $\displaystyle\quad t\in[0,T],$ (3.7)
$\displaystyle\frac{d}{dt}\left(\mathbb{J}^{i}(t)w^{i}(t)\right)=\frac{d}{dt}\int_{S^{i}(t)}\rho\left(x-X^{i}\right)\times
u\ dx\ $ $\displaystyle=\int_{\partial
S^{i}(t)}(x-X^{i})\times\left[\mathbb{T}-p\ \text{id}\right]\text{n}\
dA+\int_{S^{i}(t)}\rho\left(x-X^{i}\right)\times g\ dx,$ $\displaystyle\quad
t\in[0,T]$ (3.8)

for $i=1,...,N$ in combination with the relations

$\displaystyle j=\left\\{\begin{array}[]{ll}\sigma(E+u\times B)&\text{in
}Q^{f}\ \text{and }Q^{s},\\\ \sigma E&\text{in
}Q^{\text{ext}},\end{array}\right.$
$\displaystyle\quad\quad\sigma=\left\\{\begin{array}[]{ll}\ \
\sigma^{f}>0&\text{in }Q^{f},\\\ \ \ \sigma^{s}:=0&\text{in }Q^{s},\\\
\sigma^{\text{ext}}:=+\infty&\text{in }Q^{\text{ext}},\end{array}\right.$
(3.14) $\displaystyle H=\frac{1}{\mu_{0}}B-M\quad\quad\text{in }Q^{f},\
Q^{s}\text{ and }Q^{\text{ext}},\quad\quad\quad$ $\displaystyle\quad
D=\epsilon_{0}E+P\quad\quad\text{in }Q^{f},\ Q^{s}\text{ and }Q^{\text{ext}}$
(3.15)

as well as the boundary and interface conditions

$\displaystyle u(t)=0\quad\quad\text{on }\partial\Omega,\quad\quad\quad\quad
u^{f}(t)-u^{s}(t)=0\quad\quad$ $\displaystyle\text{on }\partial S(t).$ (3.16)

The most prominent differences in this system as opposed to the system
(2.3)–(2.17) lie in the more general form of the Maxwell equations (3.1)–(3.4)
and the momentum equation (3.6). All of the Maxwell equations are now
formulated in all three time space domains $Q^{s}$, $Q^{f}$ and
$Q^{\operatorname{ext}}$ and they contain the additional quantities
$\partial_{t}D$ and $\rho_{c}$ on the right-hand sides of Ampère’s law (3.1)
and Gauss’s law (3.3), respectively. In the momentum equation, the Lorentz
force is now to be found in its general form $\rho_{c}E+(j+J)\times B$.
Additionally, on the external force $J$ we here impose the assumptions

$\displaystyle\operatorname{div}J=0\quad\quad\text{in
}Q^{f},\quad\quad\quad\quad J=0\quad\quad\text{in }Q^{s}\text{ and
}Q^{\text{ext}}.$ (3.17)

In Ohm’s law (3.14) in the exterior domain, the choice
$\sigma=\sigma^{\operatorname{ext}}=+\infty$ is in accordance with the
exterior domain being occupied by a perfect conductor. The absence of the
velocity field $u$ in this relation is explained by the fact that no motion
occurs in this part of the domain. In the relations (3.15), the first identity
constitutes the general form of the linear relation (2.14) between $B$ and
$H$, in which $\mu_{0}$ denotes the value of the magnetic permeability $\mu$
in vaccum. The second identity in (3.15), in which $\epsilon_{0}$ stands for
the value of the dielectric permittivity in vacuum, constitutes a
corresponding relation between $D$ and $E$. Finally, we point out that the
boundary and interface conditions (2.15), (2.16) for the electromagnetic
fields are neglected in the general system. This is because these conditions,
as opposed to the no-slip condition (3.16) for the velocity field, which
constitutes a standard assumption in fluid-structure interaction models (see
for example [10, 26]), are inherent to the Maxwell equations. This can be seen
in Section 4 below, where we derive them explicitly.

###  3.1 Mathematical assumptions

For the mathematical calculations in the following sections we impose certain
regularity assumptions on the involved functions and domains. These
assumptions can be summarized as

The d $\displaystyle\text{omains}\ \Omega\ \text{and}\ S(t)\ \text{are of
class}\ C^{2,1}\ \text{at each time }t\in[0,T],$ (3.18)
$\displaystyle\rho,u,B,H,D,E$ $\displaystyle\ \text{and}\ J\ \text{are twice
continuously differentiable in}\ Q^{f},Q^{s}\ \text{and}\ Q^{\text{ext}}\
\text{and}$ (3.19) $\displaystyle\rho$ $\displaystyle,u,B,H,D,E,J\ \text{and
all their derivatives are bounded.}$ (3.20)

We mainly require these assumptions for the derivation of the interface
conditions of the electromagnetic fields in Section 4 below. These conditions
are derived from integrated versions of the Maxwell equations. The assumptions
(3.18)–(3.20) are used for passing to the limit in the equations when the
domain of integration shrinks to a point.

###  3.2 Adjustments to the material assumptions

The physical properties assumed for the materials in consideration lead to
some immediate simplifications of certain aspects of the model (3.1)–(3.16).
In the perfect conductor in the exterior domain we may assume, in accordance
with the physical literature (cf. [6, Chapter 1, Part A, §4.2.4.3]), that

$\displaystyle B=E=j=0\quad\text{in }Q^{\text{ext}}.$ (3.21)

Indeed, in the (immovable) exterior domain we can set $u=0$. Moreover, by
definition of the electrical conductivity $\sigma$ in (3.15), we know that
$\sigma=\sigma^{\text{ext}}=+\infty$ in $Q^{\text{ext}}$. Thus, due to the
boundedness of $E$ assumed in (3.20), Ohm’s law (3.14) implies that $E=j=0$.
The Maxwell-Faraday equation (3.2) then further shows that $\partial_{t}B=0$.
Assuming, without loss of generality, that $B=B(0)=0$ at the initial time
$t=0$, we thus infer that $B=B(t)=0$ at all times $t\in[0,T]$. As a
consequence of the trivial relations (3.21) the equations for $B$, $E$ and $j$
in the exterior domain become superfluous in our system. In particular, under
exploitation of the assumption (3.17) on $J$, the Maxwell system (3.1)–(3.4)
in the exterior domain reduces to the equations

$\displaystyle\operatorname{curl}H=\partial_{t}D\quad\quad$
$\displaystyle\text{in }Q^{\text{ext}},$ (3.22)
$\displaystyle\operatorname{div}D=\rho_{c}\ \ \ \quad\quad$
$\displaystyle\text{in }Q^{\text{ext}}.$ (3.23)

In the insulating solid domain the electrical conductivity $\sigma=\sigma^{s}$
vanishes, cf. (3.15). Consequently, by Ohm’s law (3.14), it holds that $j=0$
in $Q^{s}$. The non-conductivity of the solid also allows us to regard this
region as a vacuum from the electromagnetic point of view, within which there
exist no electric charges. In particular this means that the electric charge
density vanishes in the solid region, i.e. it holds that

$\displaystyle\rho_{c}=0\quad\quad\text{in }Q^{s}.$ (3.24)

Further, by the assumptions (3.17), we know that $J=0$ in $Q^{s}$. Under
consideration of these facts, the Maxwell system (3.1)–(3.4) in the solid
domain reduces to

$\displaystyle\operatorname{curl}H=\partial_{t}D\quad\quad$
$\displaystyle\text{in }Q^{s},$ (3.25)
$\displaystyle\partial_{t}B+\operatorname{curl}E=0\quad\quad\quad$
$\displaystyle\text{in }Q^{s},$ (3.26)
$\displaystyle\operatorname{div}D=0\quad\quad\quad$ $\displaystyle\text{in
}Q^{s},$ (3.27) $\displaystyle\operatorname{div}B=0\quad\quad\quad$
$\displaystyle\text{in }Q^{s}.$ (3.28)

The fact that the solid domain is considered as a vacuum from the
electromagnetic point of view moreover means that the magnetization and the
polarization vanish in this region,

$\displaystyle M=P=0\quad\quad\text{in }Q^{s}.$ (3.29)

In the fluid domain, the assumption of the fluid being a linear (magnetic)
material and a linear dielectric imply a linear relation between the
magnetization and the magnetic field as well as between the polarization and
the electric field, respectively. Namely, it holds that

$\displaystyle M=\chi_{m}^{f}H\quad\quad\text{in }Q^{f},\quad\quad\quad\quad
P=\epsilon_{0}\chi_{e}^{f}E\quad\quad\text{in }Q^{f},$ (3.30)

where $-1\ll\chi_{m}^{f}<0$ and $0<\chi_{e}^{f}$ denote the magnetic and the
electric susceptibility of the fluid, respectively, cf. [16, Section 4.4.1,
Section 6.4.1]). The latter quantities are related to the magnetic
permeability $\mu^{f}$ and the dielectric permittivity $\epsilon^{f}$ of the
fluid via the relations

$\displaystyle\mu^{f}=\mu_{0}\mu_{r}^{f}>0,\quad\quad\epsilon^{f}=\epsilon_{0}\epsilon_{r}^{r}>0,$

where $\mu_{r}^{f}:=1+\chi_{m}^{f}$ denotes the relative permeability and
$\epsilon_{r}^{f}:=1+\chi_{e}^{f}$ the relative permittivity of the fluid.
Since the magnetic permeability and the dielectric permittivity in the
insulating solid region coincide with the magnetic permeability $\mu_{0}$ and
the dielectric permittivity $\epsilon_{0}$ in vacuum, respectively, we may
thus write

$\displaystyle\mu=\left\\{\begin{array}[]{ll}\mu^{f}=\mu_{0}\mu_{r}^{f}&\text{in
}Q^{f},\\\ \mu^{s}=\mu_{0}&\text{in
}Q^{s},\end{array}\right.\quad\quad\epsilon=\left\\{\begin{array}[]{ll}\epsilon^{f}=\epsilon_{0}\epsilon_{r}^{f}&\text{in
}Q^{f},\\\ \epsilon^{s}=\epsilon_{0}&\text{in }Q^{s}.\end{array}\right.$
(3.35)

This, together with the relations (3.29) and (3.30), reduces the nonlinear
identities (3.15) in the fluid and the solid region to the linear equations

$B=\mu H\quad\quad\text{in }Q^{f}\ \text{and}\ Q^{s},\quad\quad\quad\quad
D=\epsilon E\quad\quad\text{in }Q^{f}\ \text{and}\ Q^{s}.$ (3.36)

##  4 Interface conditions for the electromagnetic fields

In this section we derive the interface conditions for the electromagnetic
fields on $\partial\Omega$ and $\partial S(t)$, $t\in[0,T]$. We emphasize,
that here we do not refer to the conditions on $\partial\Omega$ as boundary
conditions. This is because we study the electromagnetic fields not only
inside of $\Omega$ but also in the perfect conductor
$\mathbb{R}^{3}\setminus\overline{\Omega}$. The full set of interface
conditions for the electromagnetic fields reads

$\displaystyle E^{f}(t)\times\text{n}$ $\displaystyle=0\quad\quad\ \text{on
}\partial\Omega,\quad\quad$
$\displaystyle\left(E^{f}(t)-E^{s}(t)\right)\times\text{n}=0\quad\quad\ \
\text{on }\partial S(t),$ (4.1)
$\displaystyle\left(H^{\text{ext}}(t)-H^{f}(t)\right)\times\text{n}$
$\displaystyle=k(t)\quad\text{on }\partial\Omega,\quad\quad$
$\displaystyle\left(H^{f}(t)-H^{s}(t)\right)\times\text{n}=0\quad\quad\
\text{on }\partial S(t),$ (4.2) $\displaystyle B^{f}(t)\cdot\text{n}$
$\displaystyle=0\quad\quad\ \text{on }\partial\Omega,\quad\quad$
$\displaystyle\ \ \left(B^{f}(t)-B^{s}(t)\right)\cdot\text{n}=0\quad\
\quad\text{on }\partial S(t),$ (4.3)
$\displaystyle\left(D^{\text{ext}}(t)-D^{f}(t)\right)\cdot\text{n}$
$\displaystyle=\omega(t)\quad\text{on }\partial\Omega,\quad\quad$
$\displaystyle\ \
\left(D^{f}(t)-D^{s}(t)\right)\cdot\text{n}=\omega(t)\quad\text{on }\partial
S(t),$ (4.4)

for the surface current density $k$ on $\partial\Omega$ and the surface charge
density $\omega$ on $\partial\Omega$ and $\partial S(t)$. We remark that here
the surface charge density $\omega$ stands for the charge per unit area,
whereas the surface current density $k$ is defined as the current per unit
length. The latter definition becomes clear in the derivation of the condition
(4.2) below, cf. the formula (4.16). We further remark that the fields
$E^{\text{ext}}$ and $B^{\text{ext}}$ do not appear in the conditions on
$\partial\Omega$ in (4.1) and (4.3), which is due to the fact that these
fields vanish in the exterior domain, cf. (3.21). In the following we deduce
the above conditions from the Maxwell equations. In this procedure we restrict
ourselves to the conditions (4.2) and (4.4), since the remaining conditions
(4.1) and (4.3) can be derived via the exact same techniques. We first derive
the conditions (4.2) for the tangential component of $H$. At some arbitrary
but fixed time $t\in[0,T]$ we pick an arbitrary point $x\in\partial S(t)$ and
consider it the origin $x=(0,0,0)^{T}$ of a local coordinate system with axes
$\theta$, $\zeta$ and $\eta$, where $\zeta$ is orthogonal to $\partial S(t)$
at $x$ while $\theta$ and $\eta$ are tangential to $\partial S(t)$ at $x$. We
point out that in this construction the axis $\zeta$ is determined uniquely
(except for its orientation). The axis $\eta$ may be chosen as an arbitrary
axis intersecting $\zeta$ orthogonally in $x$ and subsequently the axis
$\theta$ is obtained as the unique axis intersecting both $\zeta$ and $\eta$
orthogonally in $x$. Further, we introduce the outer unit normal vector
$\text{n}_{x}=(0,1,0)^{T}$ on $\partial S(t)$ at $x$ and the unit vector
$\text{n}_{x}^{\prime}=(0,0,1)^{T}$, normal to the $\theta\zeta$-plane. Due to
the smoothness assumptions (3.18) on $\partial S(t)$ there exists some small
$\Delta s>0$ and a twice continuously differentiable function

$\displaystyle\phi:(-\Delta s,\Delta
s)\rightarrow\mathbb{R},\quad\quad\phi(0)=\phi^{\prime}(0)=0,$

such that the intersection of a small open neighborhood of $x=(0,0,0)^{T}$ in
$\partial S(t)$ with the $\theta\zeta$-plane can be expressed as the set

$\displaystyle\left\\{\gamma(\theta):=\left(\begin{array}[]{c}\theta\\\
\phi(\theta)\\\ 0\\\ \end{array}\right):\ \theta\in(-\Delta s,\Delta
s)\right\\}.$ (4.8)

Without loss of generality we assume points below the curve $\gamma$ defined
by (4.8) (i.e. points $(\theta,\zeta,0)^{T}\in(-\Delta s,\Delta
s)\times\mathbb{R}\times\\{0\\}$ with $\zeta<\phi(\theta)$) to lie inside of
$S(t)$, while the points above $\gamma$ (i.e. points
$(\theta,\zeta,0)^{T}\in(-\Delta s,\Delta s)\times\mathbb{R}\times\\{0\\}$
with $\zeta\geq\phi(\theta)$) are assumed to lie outside of $S(t)$. Shifting
the curve $\gamma$ upwards as well as downwards along the axis $\zeta$ over a
small distance $\Delta l>0$, respectively, we introduce a curved rectangle
$\Delta F$, enclosed by the four edges

$\displaystyle S_{1}$ $\displaystyle=\left\\{\gamma_{1}(\zeta):=(\Delta
s,\phi(\Delta s)+\zeta,0)^{T}:\ \zeta\in(-\Delta l,\Delta l)\right\\},$ (4.9)
$\displaystyle S_{2}$
$\displaystyle=\left\\{\gamma_{2}(\theta):=(-\theta,\phi(-\theta)+\Delta
l,0)^{T}:\ \theta\in(-\Delta s,\Delta s)\right\\},$ (4.10) $\displaystyle
S_{3}$ $\displaystyle=\left\\{\gamma_{3}(\zeta):=(-\Delta s,\phi(-\Delta
s)-\zeta,0)^{T}:\ \zeta\in(-\Delta l,\Delta l)\right\\},$ (4.11)
$\displaystyle S_{4}$
$\displaystyle=\left\\{\gamma_{4}(\theta):=(\theta,\phi(\theta)-\Delta
l,0)^{T}:\ \theta\in(-\Delta s,\Delta s)\right\\},$ (4.12)

in the $\theta\zeta$-plane, cf. Figure 1. We multiply Ampère’s law in the
fluid domain (see (3.1)) and in the solid domain (see (3.25)) by the vector
$\text{n}_{x}^{\prime}$ \- which is normal to $\Delta F$ \- and integrate the
result over $\Delta F$. This yields the identity

Figure 1: The curved rectangle $\Delta F$.

$\displaystyle\int_{S_{1}}H\cdot ds+\int_{S_{2}}H\cdot ds+\int_{S_{3}}H\cdot
ds+\int_{S_{4}}H\cdot ds$ $\displaystyle=$ $\displaystyle\int_{\partial\Delta
F}H\cdot ds=\int_{\Delta F}(\operatorname{curl}H)\cdot\text{n}_{x}^{\prime}\
dA=\int_{\Delta F}\left(\partial_{t}D+j+J\right)\cdot\text{n}_{x}^{\prime}\
dA,$ (4.13)

where, for the sake of readability, we neglect the argument $t$ in the
notation of the involved functions. Using the parametrizations (4.9)–(4.12) we
can rearrange this equation by rewriting the integrals on the left-hand side.
Using further the boundedness assumptions (3.20) on $H$, $D$, $J$ as well as
the $C^{2}$-regularity of $\phi$, we then deduce the inequality

$\displaystyle\left|\tau_{x}\cdot H\left(\begin{pmatrix}0\\\ \Delta l\\\
0\end{pmatrix}\right)-\tau_{x}\cdot H\left(\begin{pmatrix}0\\\ -\Delta l\\\
0\end{pmatrix}\right)-\frac{1}{2\Delta s}K(\Delta F)\right|\leq c(\Delta
l+\Delta s),$ (4.14)

where $\tau_{x}:=\text{n}_{x}^{\prime}\times\text{n}_{x}=(-1,0,0)^{T}$, $c>0$
is a constant independent of $\Delta l$ and $\Delta s$ and

$\displaystyle K(\Delta F):=\int_{\Delta F}j\cdot\text{n}_{x}^{\prime}\ dA$
(4.15)

denotes the current in the curved rectangle $\Delta F$. For the details of the
derivation of this estimate we refer to Section 8.1 in the appendix. We let
first $\Delta l$ and subsequently $\Delta s$ tend to zero. Then
$\frac{1}{2\Delta s}K(\Delta F)$ converges to the surface current density
(i.e. the current per unit length)

$\displaystyle k=k_{x}:=\lim_{\Delta s\rightarrow 0}\lim_{\Delta l\rightarrow
0}\frac{1}{2\Delta s}K(\Delta F)$ (4.16)

in the origin $x=(0,0,0)^{T}$ of the local coordinate system and we infer that

$\displaystyle\left(H^{f}\left(\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix}\right)-H^{s}\left(\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix}\right)\right)\cdot\left(\text{n}_{x}^{\prime}\times\text{n}_{x}\right)=\left(H^{f}\left(\begin{pmatrix}0\\\
0\\\ 0\end{pmatrix}\right)-H^{s}\left(\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix}\right)\right)\cdot\tau_{x}=k_{x}.$

Due to the arbitrary choice of the axis $\eta$ and hence the vector
$\text{n}_{x}^{\prime}$ as well as the arbitrary choice of the origin
$x=(0,0,0)^{T}\in\partial S(t)$ of the local coordinate system we may drop the
index $x$ and infer the interface condition

$\left(H^{f}-H^{S}\right)\times\text{n}=k\quad\text{on }\partial S(t).$ (4.17)

Here, we assume that $k=0$ on $\partial S(t)$, i.e. the right-hand side of the
identity (4.17) is zero and consequently we infer the second equation for $H$
in (4.2). This is in accordance with the literature, according to which the
surface current density $k$ vanishes on the surfaces of most materials. More
specifically, for example in [16, Section 9.4.2], it is stated that the
equality (4.17) with zero right-hand side holds true on the interfaces between
Ohmic conductors, i.e. conductors (with finite conductivity $\sigma<\infty$)
in which Ohm’s law holds true. Indeed, provided that Ohm’s law is also
satisfied on the interface itself, the identity $k=0$ follows directly from
the definition of $k$ in (4.16) and the boundedness assumptions (3.20) on $E$,
$u$ and $B$. However, we point out that the applicability of Ohm’s law on the
interface poses an additional assumption, the validity of which does not seem
to be generally accepted. It remains an open problem to mathematically justify
$k=0$ in the case that Ohm’s law is not satisfied at the interface.

In general, $k$ does not necessarily need to vanish. From the mathematical
point of view, $k$ may take values different from zero if $j$ becomes infinite
on the considered surface, which can be expressed mathematically via the use
of a Dirac delta distribution. Also according to the physical literature there
exist materials, such as superconductors, on the surfaces of which $k$ takes
non-zero values, cf. [15, Chapter 11]. In the condition for $H$ on
$\partial\Omega$ in (4.2), which is derived by the same arguments as the
identity (4.17), we thus refrain from the restrictive assumption $k=0$.
Indeed, while on $\partial S(t)$ we make this assumption in order for the
interface condition to match the one used in the model 2.3–(2.17) for the
analytical work in the article [28], on $\partial\Omega$ we regard it more
appropriate to allow $k$ to take non-zero values.

Next we deduce the conditions (4.4) for the normal component of $D$. Again we
choose an arbitrary point $x\in\partial S(t)$ as the origin $x=(0,0,0)^{T}$ of
a local coordinate system with axes $\theta$, $\zeta$ and $\eta$, where
$\zeta$ is orthogonal to $\partial S(t)$ at $x$ while $\theta$ and $\eta$ are
tangential to $\partial S(t)$ at $x$. We denote by $B_{r}(0)$ the
$2$-dimensional open ball with radius $r>0$ centered at $x=(0,0,0)^{T}$ in the
$\theta\eta$-plane of this local coordinate system. Due to the smoothness
assumptions (3.18) on $S(t)$ we can choose $r$ sufficiently small and find
some twice continuously differentiable function

$\displaystyle\Phi:B_{r}(0)\rightarrow\mathbb{R},\quad\quad\Phi(0,0)=\partial_{i}\Phi(0,0)=0\quad\text{for
}i=1,2,$

where $\partial_{i}\Phi$ denotes the derivative of $\Phi$ with respect to the
$i$-th variable, such that a small open neighborhood of $x$ in $\partial S(t)$
can be written as the set

$\displaystyle\left\\{\Gamma(\theta,\eta):=\begin{pmatrix}\theta\\\
\Phi(\theta,\eta)\\\ \eta\end{pmatrix}:\ (\theta,\eta)\in B_{r}(0)\right\\}.$
(4.18)

Without loss of generality we assume points below the surface $\Gamma$ defined
by (4.18) (i.e. points $(\theta,\zeta,\eta)^{T}$ with $(\theta,\eta)\in
B_{r}(0)$ and $\zeta<\Phi(\theta,\eta)$) to lie inside of $S(t)$ while the
points above $\Gamma$ (i.e. points $(\theta,\zeta,\eta)^{T}$ with
$(\theta,\eta)\in B_{r}(0)$ and $\zeta>\Phi(\theta,\eta)$) are assumed to lie
outside of $S(t)$. By shifting the surface $\Gamma$ upwards and downwards
along the axis $\zeta$ over a small distance $\Delta l>0$ we define a cylinder
$C$ with the curved bases

$\displaystyle\Delta_{\text{top}}$
$\displaystyle:=\left\\{\Gamma_{\text{top}}(s,\alpha):=\begin{pmatrix}s\cos(\alpha)\\\
\Phi(s\cos(\alpha),s\sin(\alpha))+\Delta l\\\ s\sin(\alpha)\end{pmatrix}:\
0\leq s\leq r,\ 0\leq\alpha\leq 2\pi\right\\},$ (4.19)
$\displaystyle\Delta_{\text{bottom}}$
$\displaystyle:=\left\\{\Gamma_{\text{bottom}}(s,\alpha):=\begin{pmatrix}s\cos(-\alpha)\\\
\Phi(s\cos(-\alpha),s\sin(-\alpha))-\Delta l\\\ s\sin(-\alpha)\end{pmatrix}:\
0\leq s\leq r,\ 0\leq\alpha\leq 2\pi\right\\}$ (4.20)

and the lateral area

$\displaystyle\Delta_{\text{side}}$
$\displaystyle:=\left\\{\Gamma_{\text{side}}(\alpha,h):=\begin{pmatrix}r\cos(\alpha)\\\
\Phi(r\cos(\alpha),r\sin(\alpha))+h\\\ r\sin(\alpha)\end{pmatrix}:\
0\leq\alpha<2\pi,\ -\Delta l\leq h\leq\Delta l\right\\},$ (4.21)

cf. Figure 2. Moreover, we denote by $\text{n}_{C}$ the outer unit normal
vector on $\partial C$, which we split into the outer unit normal vector
$\text{n}_{\text{top}}$ on $\Delta_{\text{top}}$, the outer unit normal vector
$\text{n}_{\text{bottom}}$ on $\Delta_{\text{bottom}}$ and the outer unit
normal vector $\text{n}_{\text{side}}$ on $\Delta_{\text{side}}$.

Figure 2: The cylinder $C$ with curved bases.

We integrate Gauss’s law in the fluid domain (cf. (3.3)) and in the solid
domain (cf. (3.27)) over the cylinder $C$. This leads to the identity

$\int_{\Delta_{\text{top}}}D\cdot\text{n}_{\text{top}}\
dA+\int_{\Delta_{\text{bottom}}}D\cdot\text{n}_{\text{bottom}}\
dA+\int_{\Delta_{\text{side}}}D\cdot\text{n}_{\text{side}}\ dA=\int_{\partial
C}D\cdot\text{n}_{C}\ dA=\int_{C}\text{div}D\ dx=\int_{C}\rho_{c}\ dx.$ (4.22)

This equation can be rearranged by rewriting the integrals on the left-hand
side under usage of the parametrizations (4.19)–(4.21). Then, under
exploitation of the assumed boundedness of $B$ (see (3.20)) as well as the
$C^{2}$-regularity of $\Phi$, we deduce the estimate

$\displaystyle\left|\text{n}_{x}\cdot\left[D\left(\begin{pmatrix}0\\\ -\Delta
l\\\ 0\end{pmatrix}\right)-D\left(\begin{pmatrix}0\\\ \Delta l\\\
0\end{pmatrix}\right)\right]-\frac{1}{\pi r^{2}}W(C)\right|\leq c(r+\Delta l)$
(4.23)

for the outer unit normal vector $\text{n}_{x}=(0,1,0)^{T}$ on $\partial S(t)$
in the origin $x=(0,0,0)^{T}$ of the local coordinate system, a constant $c>0$
independent of $r$ and $\Delta l$ and the electric charge

$\displaystyle W(\Delta F):=\int_{C}\rho_{c}\ dX$ (4.24)

of the cylinder $C$. More details on the derivation of this estimate are given
in Section 8.2 in the appendix. We let $r$ and $\Delta l$ tend to zero in the
inequality (4.23). This limit passage turns the quantity $\frac{1}{\pi
r^{2}}W(C)$ into the surface charge density

$\displaystyle\omega=\omega_{x}:=\lim_{r\rightarrow 0}\lim_{\Delta
l\rightarrow 0}\frac{1}{\pi r^{2}}W(C)$

in the origin $x=(0,0,0)^{T}$ of the local coordinate system. Since
$x\in\partial S(t)$ was chosen arbitrarily we thus infer the second condition
for $D$ in (4.4). The first condition follows by the same arguments.

##  5 Magnetohydrodynamic approximation via nondimensionalization

In this section we carry out a nondimensionalization of various equations from
the system introduced in Section 3 in order to single out several negligibly
small terms. We will then neglect these terms from the system in order to
obtain a simplified model. In particular, in the fluid domain this
simplification constitutes the classical magnetohydrodynamic approximation, an
alternative derivation of which can be found in [19, 20]. For any physical
quantity $a$ we denote by $\overline{a}>0$ an associated characteristic scale.
Introducing the dimensionless variables

$\displaystyle t^{\prime}$ $\displaystyle:=\frac{t}{\overline{t}},\quad\quad
T^{\prime}:=\frac{T}{\overline{t}},\quad\quad
x^{\prime}:=\frac{x}{\overline{x}}$ (5.1)

we may then define nondimensionalized versions of our several domains,

$\Omega^{\prime}:=\frac{1}{\overline{x}}\Omega,\quad\quad
S^{\prime}(t^{\prime}):=\frac{1}{\overline{x}}S(t^{\prime}\overline{t})=\frac{1}{\overline{x}}S(t),\quad\quad
F^{\prime}(t^{\prime}):=\Omega^{\prime}\setminus\overline{S^{\prime}(t^{\prime})}=\frac{1}{\overline{x}}F(t)$

and

$Q^{\prime}:=(0,T^{\prime})\times\Omega^{\prime},\quad\quad{Q^{s}}^{\prime}:=\left\\{(t^{\prime},x^{\prime})\in
Q^{\prime}:\ x^{\prime}\in
S^{\prime}(t^{\prime})\right\\},\quad\quad{Q^{f}}^{\prime}:=\left\\{(t^{\prime},x^{\prime})\in
Q^{\prime}:\ x^{\prime}\in F^{\prime}(t^{\prime})\right\\}.$

We remark that

$(t,x)\in Q\Leftrightarrow(t^{\prime},x^{\prime})\in
Q^{\prime},\quad\quad(t,x)\in
Q^{S}\Leftrightarrow(t^{\prime},x^{\prime})\in{Q^{S}}^{\prime},\quad\quad(t,x)\in
Q^{f}\Leftrightarrow(t^{\prime},x^{\prime})\in{Q^{f}}^{\prime}.$

Moreover, for any $(t^{\prime},x^{\prime})\in Q^{\prime}$ we introduce the
dimensionless mechanical quantities

$\displaystyle\rho^{\prime}$
$\displaystyle(t^{\prime},x^{\prime}):=\frac{\rho(t,x)}{\overline{\rho}},\ \ \
u^{\prime}(t^{\prime},x^{\prime}):=\frac{u(t,x)}{\overline{u}},\ \ \
p^{\prime}(t^{\prime},x^{\prime}):=\frac{p(t,x)}{\overline{p}},\ \ \
g^{\prime}(t^{\prime},x^{\prime}):=\frac{g(t,x)}{\overline{g}},$ (5.2)

and for all $(t^{\prime},x^{\prime})\in(0,T^{\prime})\times\mathbb{R}^{3}$ we
introduce the dimensionless electromagnetic quantities

$\displaystyle D^{\prime}(t^{\prime}$
$\displaystyle,x^{\prime}):=\frac{D(t,x)}{\overline{D}},\ \ \
E^{\prime}(t^{\prime},x^{\prime}):=\frac{E(t,x)}{\overline{E}},\ \ \
B^{\prime}(t^{\prime},x^{\prime}):=\frac{B(t,x)}{\overline{B}},\ \ \
H^{\prime}(t^{\prime},x^{\prime}):=\frac{H(t,x)}{\overline{H}},$ (5.3)
$\displaystyle\quad\quad\quad\ \
\rho_{c}^{\prime}(t^{\prime},x^{\prime}):=\frac{\rho_{c}(t,x)}{\overline{\rho}_{c}},\
\ \ j^{\prime}(t^{\prime},x^{\prime}):=\frac{j(t,x)}{\overline{j}},\ \ \
J^{\prime}(t^{\prime},x^{\prime}):=\frac{J(t,x)}{\overline{j}}.$ (5.4)

###  5.1 The Maxwell system

In the classical magnetohydrodynamic approximation, the Maxwell system in the
fluid domain is simplified by dropping the quantity $\partial_{t}D$ in
Ampère’s law. The purpose of this section is to justify this simplification.
Furthermore, proceeding similarly in the solid region, we carry out the same
reduction in Ampère’s law in the solid domain. We start by nondimensionalizing
the Maxwell-Faraday equation (3.2) in the fluid and the solid domain.
Expressing the magnetic induction $B$ and the electric field $E$ through the
dimensionless variables introduced in (5.1) and (5.3) and making use of the
chain rule, the equation (3.2) becomes

$\frac{\overline{B}}{\overline{t}}\partial_{t^{\prime}}B^{\prime}(t^{\prime},x^{\prime})+\frac{\overline{E}}{\overline{x}}\nabla_{x^{\prime}}\times
E^{\prime}(t^{\prime},x^{\prime})=0\quad\quad\text{in }{Q^{f}}^{\prime}\
\text{and}\ {Q^{s}}^{\prime}.$

In this relation we assume the two quantities
$\partial_{t^{\prime}}B^{\prime}$ and $\nabla_{x^{\prime}}\times E^{\prime}$
to be equally significant, meaning that the coefficients in front of these
terms have to coincide,

$\frac{\overline{B}}{\overline{t}}=\frac{\overline{E}}{\overline{x}}.$ (5.5)

This relation, together with the linear relations (3.36) and the decomposition
(3.35) of $\mu$ and $\epsilon$, allows us to express Ampère’s law (3.1) in the
fluid domain and (3.25) in the solid domain as

$\displaystyle\nabla_{x^{\prime}}\times
B^{\prime}\left(t^{\prime},x^{\prime}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{\mu_{0}\mu_{f}^{r}\overline{x}\epsilon_{0}\epsilon_{f}^{r}\overline{E}}{\overline{t}\
\overline{B}}\partial_{t^{\prime}}E^{\prime}\left(t^{\prime},x^{\prime}\right)+\frac{\mu^{f}\overline{x}\
\overline{j}}{\overline{B}}j^{\prime}\left(t^{\prime},x^{\prime}\right)+\frac{\mu^{f}\overline{x}\
\overline{j}}{\overline{B}}J^{\prime}\left(t^{\prime},x^{\prime}\right)&\text{in
}Q^{f},\\\ \frac{\mu_{0}\overline{x}\epsilon_{0}\overline{E}}{\overline{t}\
\overline{B}}\partial_{t^{\prime}}E^{\prime}\left(t^{\prime},x^{\prime}\right)&\text{in
}Q^{s}.\end{array}\right.$ (5.8)

In the fluid domain, we assume the quantities $j^{\prime}$ and
$\nabla_{x^{\prime}}\times B^{\prime}$ to be equally significant, meaning that

$\overline{B}=\mu^{f}\overline{x}\overline{j},$ (5.9)

Denoting by $c>0$ the speed of light, which is known to satisfy

$\displaystyle c=(\mu_{0}\epsilon_{0})^{-\frac{1}{2}},$ (5.10)

we further assume that

$\overline{u}=\frac{\overline{x}}{\overline{t}},\quad\quad\overline{u}\ll
c,\quad\quad\mu_{r}^{f}\epsilon_{r}^{f}\approx 1.$ (5.11)

We point out that the latter of the assumptions in (5.11) is reasonable for
isotropic liquid electrical conductors (i.e. for example electrolytes, molten
salts and liquid metals), cf. [22, Section II.1]. Combining the relation (5.5)
with the first assumption in (5.11), we infer that

$\frac{\overline{E}}{\overline{B}}=\overline{u}.$ (5.12)

This, together with the relations (5.9)–(5.11), allows us to simplify the
nondimensionalized version (5.8) of Ampère’s law to

$\displaystyle\nabla_{x^{\prime}}\times
B^{\prime}\left(t^{\prime},x^{\prime}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{\overline{u}^{2}}{c^{2}}\partial_{t^{\prime}}E^{\prime}\left(t^{\prime},x^{\prime}\right)+j^{\prime}\left(t^{\prime},x^{\prime}\right)+J^{\prime}\left(t^{\prime},x^{\prime}\right)&\text{in
}Q^{f},\\\
\frac{\overline{u}^{2}}{c^{2}}\partial_{t^{\prime}}E^{\prime}\left(t^{\prime},x^{\prime}\right)&\text{in
}Q^{s}.\end{array}\right.$ (5.15)

Here, the term $\frac{\overline{u}^{2}}{c^{2}}\partial_{t^{\prime}}E^{\prime}$
is negligibly small according to the second assumption in (5.11). Formally, we
may hence neglect it, which reduces the identity (5.15) to

$\displaystyle\nabla_{x^{\prime}}\times
B^{\prime}\left(t^{\prime},x^{\prime}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}j^{\prime}\left(t^{\prime},x^{\prime}\right)+J^{\prime}\left(t^{\prime},x^{\prime}\right)&\text{in
}Q^{f},\\\ 0&\text{in }Q^{s}.\end{array}\right.$ (5.18)

The desired reduced version of Ampère’s law is then obtained by converting
this identity back into a dimensional form,

$\displaystyle\nabla\times H\left(t,x\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}j\left(t,x\right)+J\left(t,x\right)&\text{in
}Q^{f},\\\ 0&\text{in }Q^{s}.\end{array}\right.$ (5.21)

###  5.2 The Navier-Stokes system

In the Navier-Stokes system we nondimensionalize the momentum equation (3.6)
with the aim of simplifying the Lorentz force $\rho_{c}E+(j+J)\times B$ to the
expression $\frac{1}{\mu}\operatorname{curl}B\times B$ as it is common in the
magnetohydrodynamic approximation. A straight forward calculation under
exploitation of the relations (5.1), (5.2), (5.3) and (5.4) allows us to
express the momentum equation (3.6) in the form

$\displaystyle\frac{\overline{\rho}\
\overline{u}}{\overline{t}}\partial_{t^{\prime}}\left(\rho^{\prime}(t^{\prime},x^{\prime})u^{\prime}(t^{\prime},x^{\prime})\right)+\frac{\overline{\rho}\
\overline{u}^{2}}{\overline{x}}\nabla_{x^{\prime}}\cdot\left(\rho^{\prime}(t^{\prime},x^{\prime})u^{\prime}(t^{\prime},x^{\prime})\otimes
u^{\prime}(t^{\prime},x^{\prime})\right)+\frac{\overline{p}}{\overline{x}}\nabla_{x^{\prime}}p^{\prime}(t^{\prime},x^{\prime})$
$\displaystyle=$
$\displaystyle\frac{2\nu\overline{u}}{\overline{x}^{2}}\nabla_{x^{\prime}}\cdot\mathbb{D}_{x^{\prime}}\left(u^{\prime}(t^{\prime},x^{\prime})\right)+\frac{\lambda\overline{u}}{\overline{x}^{2}}\nabla_{x^{\prime}}\cdot\left(\operatorname{id}\left(\nabla_{x^{\prime}}\cdot
u^{\prime}\left(t^{\prime},x^{\prime}\right)\right)\right)+\overline{\rho}\
\overline{g}\rho^{\prime}(t^{\prime},x^{\prime})g(t,x)+\overline{\rho}_{c}\overline{E}\rho_{c}^{\prime}(t^{\prime},x^{\prime})E^{\prime}(t^{\prime},x^{\prime})$
$\displaystyle+\overline{j}\
\overline{B}\left(j^{\prime}(t^{\prime},x^{\prime})+J^{\prime}(t^{\prime},x^{\prime})\right)\times
B^{\prime}(t^{\prime},x^{\prime})\quad\quad\text{in }Q^{f^{\prime}}.$ (5.22)

We proceed by taking the divergence of Ohm’s law (3.14) in the fluid domain,
which, under exploitation of Gauss’s law (3.3) and the linear relations
(3.36), yields the identity

$\displaystyle\frac{1}{\sigma}\nabla\cdot j(t,x)-\nabla\cdot\left(u(t,x)\times
B(t,x)\right)=\frac{1}{\epsilon^{f}}\nabla\cdot
D(t,x)=\frac{\rho_{c}(t,x)}{\epsilon^{f}}\quad\quad\text{in }Q^{f}.$

Nondimensionalizing this relation we obtain

$\displaystyle\frac{\overline{j}}{\sigma\overline{x}}\nabla_{x^{\prime}}\cdot
j^{\prime}(t^{\prime},x^{\prime})-\frac{\overline{u}\overline{B}}{\overline{x}}\nabla_{x^{\prime}}\cdot\left(u^{\prime}(t^{\prime},x^{\prime})\times
B^{\prime}(t^{\prime},x^{\prime})\right)=\frac{\overline{\rho}_{c}}{\epsilon^{f}}\rho_{c}^{\prime}(t^{\prime},x^{\prime})\quad\quad\text{in
}{Q^{f}}^{\prime}.$

Here the first term on the left-hand side is equal to zero since
$\nabla_{x^{\prime}}\cdot j^{\prime}=0$ due to Ampère’s law (5.18) and the
fact that (by the assumptions on $J$ in (3.17)) $J^{\prime}$ is divergence-
free. We infer that

$\frac{\overline{\rho}_{c}\overline{x}}{\epsilon^{f}\overline{u}\overline{B}}\rho^{\prime}_{c}(t^{\prime},x^{\prime})=-\nabla_{x^{\prime}}\cdot\left(u^{\prime}(t^{\prime},x^{\prime})\times
B^{\prime}(t^{\prime},x^{\prime})\right)\quad\text{in }Q^{f^{\prime}}.$

In this identity we assume the quantities $\rho^{\prime}_{c}$ and
$-\nabla_{x^{\prime}}\cdot\left(u^{\prime}\times B^{\prime}\right)$ to be
equally significant, meaning that

$\overline{\rho}_{c}=\frac{\epsilon^{f}\overline{u}\overline{B}}{\overline{x}}.$
(5.23)

This yields the equation

$\displaystyle\overline{\rho}_{c}\overline{E}=\frac{\epsilon^{f}\overline{u}\overline{B}}{\overline{x}}\overline{E}=\frac{\epsilon^{f}\overline{B}^{2}\overline{u}^{2}}{\overline{x}}=\frac{\epsilon_{0}\epsilon_{r}^{f}\mu_{0}\mu_{r}^{f}\overline{x}\overline{j}\
\overline{B}\overline{u}^{2}}{\overline{x}}\approx\frac{\overline{u}^{2}}{c^{2}}\overline{j}\
\overline{B},$

where we used the relation (5.12) for the second identity, the relations
(3.35) and (5.9) for the third identity and the relation (5.10) as well as the
third assumption in (5.11) for the last identity. For the (nondimensionalized)
Lorentz force this means that

$\displaystyle\overline{\rho}_{c}\
\overline{E}\rho_{c}^{\prime}(t^{\prime},x^{\prime})E^{\prime}(t^{\prime},x^{\prime})+\overline{j}\
\overline{B}\left(j^{\prime}(t^{\prime},x^{\prime})+J^{\prime}(t^{\prime},x^{\prime})\right)\times
B^{\prime}(t^{\prime},x^{\prime})$ $\displaystyle\approx$
$\displaystyle\frac{\overline{u}^{2}}{c^{2}}\overline{j}\
\overline{B}\rho_{c}^{\prime}(t^{\prime},x^{\prime})E^{\prime}(t^{\prime},x^{\prime})+\overline{j}\
\overline{B}\left(j^{\prime}(t^{\prime},x^{\prime})+J^{\prime}(t^{\prime},x^{\prime})\right)\times
B^{\prime}(t^{\prime},x^{\prime})\quad\quad\text{in }Q^{f^{\prime}}.$

Due to the second assumption in (5.11) we see that the first term on the
right-hand side of this relation is negligibly small compared to the second
one. Hence, formally, we may neglect it and the nondimensionalized momentum
equation (5.22) reduces to

$\displaystyle\frac{\overline{\rho}\
\overline{u}}{\overline{t}}\partial_{t^{\prime}}\left(\rho^{\prime}(t^{\prime},x^{\prime})u^{\prime}(t^{\prime},x^{\prime})\right)+\frac{\overline{\rho}\
\overline{u}^{2}}{\overline{x}}\nabla_{x^{\prime}}\cdot\left(\rho^{\prime}(t^{\prime},x^{\prime})u^{\prime}(t^{\prime},x^{\prime})\otimes
u^{\prime}(t^{\prime},x^{\prime})\right)+\frac{\overline{p}}{\overline{x}}\nabla_{x^{\prime}}p^{\prime}(t^{\prime},x^{\prime})$
$\displaystyle=$
$\displaystyle\frac{2\nu\overline{u}}{\overline{x}^{2}}\nabla_{x^{\prime}}\cdot\mathbb{D}_{x^{\prime}}\left(u^{\prime}(t^{\prime},x^{\prime})\right)+\frac{\lambda\overline{u}}{\overline{x}^{2}}\nabla_{x^{\prime}}\cdot\left(\operatorname{id}\left(\nabla_{x^{\prime}}\cdot
u^{\prime}\left(t^{\prime},x^{\prime}\right)\right)\right)+\overline{\rho}\
\overline{g}\rho^{\prime}(t^{\prime},x^{\prime})g(t,x)$
$\displaystyle+\overline{j}\
\overline{B}\left(j^{\prime}(t^{\prime},x^{\prime})+J^{\prime}(t^{\prime},x^{\prime})\right)\times
B^{\prime}(t^{\prime},x^{\prime})\quad\quad\text{in }Q^{f^{\prime}}.$

Transforming this equation back into a dimensional form and exploiting
Ampère’s law (5.21) and the linear relation (3.36) to rewrite the remaining
part of the Lorentz force, we obtain the desired simplified momentum equation

$\displaystyle\partial_{t}\left(\rho(t,x)u(t,x)\right)+\nabla\cdot\left(\rho(t,x)u(t,x)\otimes
u(t,x)\right)+\nabla p(t,x)$ $\displaystyle=$ $\displaystyle
2\nu\nabla\cdot\mathbb{D}\left(u(t,x)\right)+\lambda\nabla\cdot\left(\operatorname{id}\left(\nabla\cdot
u\left(t,x\right)\right)\right)+\rho(t,x)g(t,x)+\frac{1}{\mu}\left(\nabla\times
B(t,x)\right)\times B(t,x)\quad\quad\text{in }Q^{f}.$ (5.24)

##  6 Summary of the derived system

In the following we present a summary of the system derived in the previous
sections. The mechanical part of this system coincides with the mechanical
part (3.5)–(3.8), (3.16) of the original system except for the momentum
equation, which has been simplified in the course of the magnetohydrodynamic
approximation, cf. (5.24). In the electromagnetic part of the derived system,
the Maxwell-Faraday equation, Gauss’s law and Gauss’s law for magnetism in the
fluid domain are adopted directly from the original model, cf. (3.2)–(3.4). In
the solid domain, these equations have been adjusted to the assumption of the
solid being insulating in (3.26)–(3.28). Ampère’s law has been adjusted to the
same assumption and has undergone, in both the fluid and the solid domain, the
magnetohydrodynamic approximation; the resulting equation is stated in (5.21).
The Maxwell system in the exterior domain in our derived model consists, in
accordance with the assumption of the exterior domain being a perfect
conductor, only of Ampère’s law (3.22) and Gauss’s law (3.23), since the
remaining equations become superfluous due to the trivial relations (3.21).
Ohm’s law in the derived system keeps its form (3.14) from the original model,
however it can be omitted in the exterior domain because of the trivial
relations (3.21). The constitutive relations (3.15) from the original model
reduce according to the trivial relations (3.21) in the exterior domain while
they take the form (3.36) in the fluid and the solid domain. Finally, our
derived model also includes the interface conditions (4.1)–(4.4) for the
electromagnetic fields.

Before we present the derived system in its complete form we recall the
conditions which we obtained through the scaling assumptions we used for the
magnetohydrodynamic approximation in Section 5. These conditions consist of
the relations

$\displaystyle\frac{\overline{B}}{\overline{t}}=\frac{\overline{E}}{\overline{x}},\quad\quad\overline{B}=\mu^{f}\overline{x}\overline{j},\quad\quad\overline{u}=\frac{\overline{x}}{\overline{t}},\quad\quad\overline{u}\ll
c,\quad\quad\mu_{r}^{f}\epsilon_{r}^{f}\approx
1,\quad\quad\overline{\rho}_{c}=\frac{\epsilon^{f}\overline{B}\overline{u}}{\overline{x}}$

for the characteristic scales $\overline{a}>0$ of the physical quantities $a$
in our model, cf. (5.5), (5.9), (5.11) and (5.23).

The full system we derived under these conditions for the modeling of the
motion of several insulating rigid bodies through an electrically conducting
diamagnetic dielectric viscous non-homogeneous and compressible fluid
surrounded by a perfect conductor is composed of the Maxwell equations

$\displaystyle\text{curl}H$
$\displaystyle=\left\\{\begin{array}[]{ll}j+J&\text{in }Q^{f},\\\ 0&\text{in
}Q^{s},\\\ \partial_{t}D&\text{in }Q^{\operatorname{ext}},\end{array}\right.$
(6.4) $\displaystyle\partial_{t}B+\operatorname{curl}E$
$\displaystyle=0\quad\quad\quad\quad\quad\text{in }Q^{f}\text{ and }Q^{s},$
(6.5) $\displaystyle\operatorname{div}D$
$\displaystyle=\left\\{\begin{array}[]{ll}\rho_{c}&\text{in }Q^{f}\text{ and
}Q^{\operatorname{ext}},\\\ 0&\text{in }Q^{s},\end{array}\right.$ (6.8)
$\displaystyle\operatorname{div}B$
$\displaystyle=0\quad\quad\quad\quad\quad\text{in }Q^{f}\text{ and }Q^{s},$
(6.9)

the compressible Navier-Stokes equations

$\displaystyle\partial_{t}\rho+\operatorname{div}(\rho u)=$ $\displaystyle
0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ \text{in
}Q^{f},$ (6.10) $\displaystyle\partial_{t}(\rho u)+\text{div}(\rho u\otimes
u)+\nabla p=$ $\displaystyle\text{div}\mathbb{T}+\rho
g+\frac{1}{\mu}\text{curl}B\times B\quad\quad\text{in }Q^{f},$ (6.11)

the balances of linear and angular momentum

$\displaystyle m^{i}\frac{d}{dt}V^{i}(t)=$
$\displaystyle\frac{d}{dt}\int_{S^{i}(t)}\rho u\ dx=\int_{\partial
S^{i}(t)}[\mathbb{T}-p\text{ id}]\text{n}\ dA+\int_{S^{i}(t)}\rho g\ dx,\quad
t\in[0,T],$ (6.12)
$\displaystyle\frac{d}{dt}\left(\mathbb{J}^{i}(t)w^{i}(t)\right)=$
$\displaystyle\frac{d}{dt}\int_{S^{i}(t)}\rho\left(x-X^{i}\right)\times u\ dx$
$\displaystyle=$ $\displaystyle\int_{\partial
S^{i}(t)}(x-X^{i})\times[\mathbb{T}-p\ \text{id}]\text{n}\
dA+\int_{S^{i}(t)}\rho\left(x-X^{i}\right)\times g\ dx,\quad t\in[0,T],$
(6.13)

for $i=1,...,N$, in combination with the trivial relations

$\displaystyle B=E=j=0\quad\quad\text{in }Q^{\text{ext}},$ (6.14)

the relations

$\displaystyle j=$ $\displaystyle\sigma(E+u\times B)\quad\text{in }Q^{f}\text{
and }Q^{s},\quad\quad\sigma=\left\\{\begin{array}[]{ll}\sigma^{f}>0&\text{in
}Q^{f},\\\ \sigma^{s}=0&\text{in }Q^{s},\end{array}\right.$ (6.17)
$\displaystyle B=$ $\displaystyle\mu H\quad\quad\text{in }Q^{f}\ \text{and
}Q^{s},\quad\quad\quad\quad\ D=\epsilon E\quad\quad\text{in }Q^{f}\ \text{and
}Q^{s},$ (6.18) $\displaystyle H=$ $\displaystyle-M\quad\ \text{in
}Q^{\text{ext}},\quad\quad\quad\quad\quad\quad\quad D=P\quad\quad\ \ \text{in
}Q^{\text{ext}}$ (6.19)

and the boundary and interface conditions

$\displaystyle E^{f}(t)\times\text{n}$ $\displaystyle=0\quad\quad\ \text{on
}\partial\Omega,\quad\quad$
$\displaystyle\left(E^{f}(t)-E^{s}(t)\right)\times\text{n}=0\quad\quad\ \
\text{on }\partial S(t),$ (6.20)
$\displaystyle\left(H^{\text{ext}}(t)-H^{f}(t)\right)\times\text{n}$
$\displaystyle=k(t)\quad\text{on }\partial\Omega,\quad\quad$
$\displaystyle\left(H^{f}(t)-H^{s}(t)\right)\times\text{n}=0\quad\quad\
\text{on }\partial S(t),$ (6.21) $\displaystyle B^{f}(t)\cdot\text{n}$
$\displaystyle=0\quad\quad\ \text{on }\partial\Omega,\quad\quad$
$\displaystyle\ \ \left(B^{f}(t)-B^{s}(t)\right)\cdot\text{n}=0\quad\
\quad\text{on }\partial S(t),$ (6.22)
$\displaystyle\left(D^{\text{ext}}(t)-D^{f}(t)\right)\cdot\text{n}$
$\displaystyle=\omega(t)\quad\text{on }\partial\Omega,\quad\quad$
$\displaystyle\ \
\left(D^{f}(t)-D^{s}(t)\right)\cdot\text{n}=\omega(t)\quad\text{on }\partial
S(t),$ (6.23) $\displaystyle u^{f}(t)$ $\displaystyle=0\quad\ \quad\text{on
}\partial\Omega,\quad\quad$ $\displaystyle\quad\quad\ \quad\ \
u^{f}(t)-u^{s}(t)=0\quad\ \quad\text{on }\partial S(t).$ (6.24)

In this system, the external force
$J:(0,T)\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ in Ampère’s law (6.4)
in the fluid domain is assumed to satisfy

$\displaystyle\nabla\cdot J=0\quad\quad\text{in }Q^{f},\quad\quad\quad\quad
J=0\quad\quad\text{in }Q^{s}\text{ and }Q^{\text{ext}}.$ (6.25)

The magnetic permeability $\mu$ and the dielectric permittivity $\epsilon$ in
the relations (6.18) are given by

$\displaystyle\quad\mu:=\left\\{\begin{array}[]{ll}\mu^{f}:=\mu_{0}\mu_{r}^{f}>0&\text{in
}Q^{f},\\\ \mu^{s}:=\mu_{0}>0&\text{in
}Q^{s},\end{array}\right.\quad\epsilon=\left\\{\begin{array}[]{ll}\epsilon^{f}:=\epsilon_{0}\epsilon_{r}^{f}>0&\text{in
}Q^{f},\\\ \epsilon^{s}:=\epsilon_{0}>0&\text{in }Q^{s},\end{array}\right.$
(6.30)

wherein $\mu_{r}^{f}$ and $\epsilon_{r}^{f}$ denote the relative permeability
and the relative permittivity of the fluid. Finally, in the interface
conditions (6.21), (6.23) the quantities $k$ and $\omega$ denote the surface
current density and the surface charge density, respectively, on
$\partial\Omega$ and $\partial S(t)$.

The derived system (6.4)–(6.24) finds itself in an intermediate state between
the general system (3.1)–(3.16) and the system (2.3)–(2.17) from the
mathematical analysis, the derivation of which is the ultimate goal of the
present article. The system (3.1)–(3.16) models the interaction between an
electrically conducting fluid and an insulating rigid body in full generality.
The equations (2.3)–(2.17), as a reduced version of this system, are less
general, however, they bear the advantage of admitting weak solutions as the
authors were able to show in [28]. The disadvantage of the latter system lies
in the fact that some of its modifications in comparison to the system
(3.1)–(3.16) are made for purely mathematical reasons. This, in turn, shows
why the system (6.4)–(6.24) is interesting: It constitutes an intermediate
result in the derivation of the system (2.3)–(2.17) from the system
(3.1)–(3.16) containing only those modifications for which we have been able
to provide - under certain assumptions - physical arguments. However, we point
out that existence of weak solutions to the system (6.4)–(6.24) remains a
challenging problem due to the jump of the magnetic permeability $\mu$ across
the interface between the fluid and the solid in this system, which would
require a new variational formulation and, probably, a new methodology.

We end this section by discussing the final adjustments which need to be made
in order to turn the system (6.4)–(6.24) into the system (2.3)–(2.17). In the
latter system, as it is common practice in many mathematical works in
magnetohydrodynamics (cf. for example [2]), the Maxwell equations are only
considered inside of the domain $\Omega$. This is due to the additional
assumption of the linear relations (6.18) holding true also in the exterior
domain,

$\displaystyle B=\mu H\quad\quad\text{in }Q^{\text{ext}},\quad\quad\quad\quad\
D=\epsilon E\quad\quad\text{in }Q^{\text{ext}}$ (6.31)

for the magnetic permeability $\mu>0$ and the dielectric permittivity
$\epsilon>0$ in the perfect conductor
$\mathbb{R}^{3}\setminus\overline{\Omega}$. Indeed, under this assumption the
trivial relations (6.14) and the equation (6.8) immediately imply that also
the quantities $H$, $D$ and $\rho_{c}$ vanish in $Q^{\text{ext}}$, cf. for
example [6, Chapter 1: Part A: §4.2.4.3]. In this case the whole
electromagnetic subsystem in the exterior domain becomes trivial and it is
sufficient to examine the problem in the interior domain. Thus, in the system
(2.3)–(2.17), in which the condition (6.31) is assumed implicitly, all
electromagnetic equations in the exterior domain are neglected. However, we
point out that there does not seem to be a physical argument for the validity
of the linear relations (6.31) in a perfect conductor. For this reason we here
decided to formulate the system (6.4)–(6.24) in a more general form, without
the condition (6.31) and with the electromagnetic subsystem in the exterior
domain.

Furthermore, Gauss’s law (6.8) in the fluid domain is not included in the
system (2.3)–(2.17). This is explained as follows: In mathematical works in
magnetohydrodynamics the (reduced) Maxwell system is commonly compressed into
two equations for the magnetic induction $B$, Gauss’s law for magnetism (6.9)
and the so-called induction equation,

$\displaystyle\operatorname{div}B=0\quad\text{in
}Q^{f},\quad\quad\partial_{t}B+\nabla\times(B\times
u)+\frac{1}{\sigma}\nabla\times\left(\frac{1}{\mu}\nabla\times\left(B\right)-J\right)=0\quad\text{in
}Q^{f}.$ (6.32)

Indeed, the latter equation results directly from a combination of Ohm’s law
(6.17), Ampère’s law (6.4) and the Maxwell-Faraday equation (6.5). The idea
behind this further reduction is that the unknown $B$ may be determined
independently of all the other unknowns from the Maxwell system in the fluid
domain. After determining $B$, we have the magnetic field $H$ given explicitly
from the relation (6.18) and consequently also the electric current density
$j$ from Ampère’s law (6.4). Subsequently, the electric field $E$ and the
electric induction $D$ can be computed directly from Ohm’s law (6.17) and the
relation (6.18) and we can use Gauss’s law (6.8) to immediately obtain the
density of electric charges $\rho_{c}$. Therefore, the Maxwell system in the
fluid domain can be solved by solving only the system (6.32) and in particular
Gauss’s law (6.8) becomes superfluous for the mathematical analysis. For this
reason Gauss’s law in the fluid domain is neglected in the system
(2.3)–(2.17).

As opposed to in the fluid domain, Gauss’s law (6.8) cannot be dropped in the
solid domain. Indeed, in the solid domain $\rho_{c}$ is equal to $0$ (cf.
(3.24)), while $E$ cannot be determined via Ohm’s law, so Gauss’s law rather
constitutes a condition required for determining $D$ and $E$ than for
determining $\rho_{c}$. We remark that in the system (2.3)–(2.17) this
equation is expressed in terms of $E$ instead of $D$, cf. (2.5). Thus the only
remaining equation involving $D$ is the linear relation between $D$ and $E$ in
(6.18), which therefore becomes redundant for the analysis and consequently
does not appear in the system (2.3)–(2.17).

The linear relation between $B$ and $H$ in (6.18) instead constitutes a
crucial component of the system (2.3)–(2.17). In the latter system, however,
it is modified in the sense that $\mu$ is assumed to take the same value in
both $Q^{f}$ and $Q^{s}$, i.e. $\mu$ is a constant in the whole domain $Q$,
cf. (2.14). This modification, which is only justified if the magnetic
permeability in $Q^{f}$ and $Q^{s}$ is (almost) the same, is made for purely
mathematical reasons. Namely, in combination with the interface conditions for
$H$ and $B$ on $\partial S(t)$ in (6.21) and (6.22), it ensures continuity of
$B$ across the fluid-solid interface. This is necessary for the weak
formulation of the system in [28, Definition 2.1] below, in which the magnetic
induction is assumed to be a Sobolev function over the whole domain $Q$.

Moreover, we remark that the interface conditions for $H$ on $\partial\Omega$
in (6.21) as well as the interface conditions (6.23) on $D$ do not appear in
the system (2.3)–(2.17) as they do not enter its weak formulation in [28,
Definition 2.1]. Finally, we point out that also the conditions (6.25) on $J$
are left out of the system (2.3)–(2.17) as the analysis in [28] can also be
carried out without them. This concludes the derivation of the system
(2.3)–(2.17).

##  7 Existence of weak solutions

The remainder of the article is dedicated to a brief summary of the existence
result on weak solutions to the system (2.3)–(2.17) which was published by the
first author in [28], see also [29]. We begin by introducing some additional
notation required for the definition of weak solutions. The rigidity of the
solid bodies implies the existence of orientation preserving isometries
$\eta^{i}(t):\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ such that the position
$S^{i}(t)$ of the $i$-th body at the time $t\in[0,T]$ may be expressed as

$\displaystyle S^{i}(t):=\eta^{i}\left(t,S_{0}^{i}\right),$

where $S_{0}^{i}$ denotes the corresponding initial position. The solid region
$S(t)$ at time $t\in[0,T]$ can then be expressed with the help of the set-
valued function

$\displaystyle S:\ [0,T]\rightarrow 2^{\mathbb{R}^{3}},\quad\quad
S(t):=\eta\left(t,S_{0}\right),$

where $S_{0}:=\bigcup_{i=1}^{N}S_{0}^{i}$ and

$\displaystyle\eta(t,\cdot):\
S_{0}\rightarrow\mathbb{R}^{3},\quad\quad\eta(t,\cdot)|_{S^{i}_{0}}:=\eta^{i}(t,\cdot).$

In order to establish a suitable connection between the isometries
$\eta^{i}(t)$ and the velocity field $u$ in the problem, we demand
compatibility of $u$ with the system $\\{S_{0}^{i},\eta^{i}\\}_{i=1}^{N}$,
meaning that each $\eta^{i}$, $i=1,...,N$, constitutes the Carathéodory
solution to an initial value problem

$\displaystyle\frac{d\eta^{i}(t,x)}{dt}=u^{s^{i}}\left(t,\eta^{i}(t,x)\right),\quad\quad\eta^{i}(0,x)=x$
(7.1)

for some rigid velocity field $u^{s^{i}}(t,\cdot)$ satisfying

$\displaystyle u(t,\cdot)=u^{s^{i}}(t,\cdot)\quad\quad\text{a.e. in }S^{i}(t)\
\text{for a.a. }t\in[0,T].$ (7.2)

The compatibility between $u$ and $\\{S_{0}^{i},\eta^{i}\\}_{i=1}^{N}$
guarantees that the rigid bodies travel along the flow curves of $u$ and
cannot penetrate each other, cf. [9, Lemma 3.1, Lemma 3.2]. Finally, by

$\displaystyle H_{\text{div}}^{1}\left(\Omega\right):=\left\\{b\in
H^{1}(\Omega):\ \operatorname{div}b=0\ \text{in }\Omega\right\\},\quad\quad
L_{\text{div}}^{2}\left(\Omega\right):=\left\\{b\in L^{2}(\Omega):\
\operatorname{div}b=0\ \text{in
}\mathcal{D}^{\prime}\left(\Omega\right)\right\\}$

we denote the space of $H^{1}(\Omega)$-functions and
$L^{2}(\Omega)$-functions, respectively, which are additionally divergence-
free and, for the weak formulation of the momentum equation and the induction
equation, we introduce the test function spaces

$\displaystyle\mathcal{Z}(S):=$
$\displaystyle\left\\{\phi\in\mathcal{D}\left([0,T)\times\Omega\right):\
\mathbb{D}(\phi)=0\quad\text{in an open neighborhood of
}\overline{Q^{s}(S)}\right\\},$ (7.3) $\displaystyle\mathcal{Y}(S):=$
$\displaystyle\left\\{b\in\mathcal{D}\left([0,T)\times\Omega\right):\
\operatorname{curl}b=0\quad\text{in an open neighborhood of
}\overline{Q^{s}(S)}\right\\}.$ (7.4)

Our definition of weak solutions to the system (2.3)–(2.17) reads as follows.

######  Definition 7.1.

[28, Definition 2.1] Let $T>0$, let $\Omega\subset\mathbb{R}^{3}$ be a bounded
domain and let $S_{0}=\bigcup_{i=1}^{N}S_{0}^{i}$, where
$S_{0}^{i}\subset\Omega$ for $i=1,...,N\in\mathbb{N}$ are bounded domains such
that

$\displaystyle\emptyset\neq S_{0}^{i}\ \text{is open and connected, }|\partial
S_{0}^{i}|=0\text{ and
}\overline{S_{0}^{i}}\bigcap\overline{S_{0}^{j}}=\emptyset\quad\quad\forall
i,j=1,...,N,\ i\neq j.$ (7.5)

Assume $\nu,\lambda,a,\gamma,\sigma,\mu\in\mathbb{R}$ to satisfy

$\displaystyle\nu,a,\sigma,\mu>0,\quad\nu+\lambda\geq
0,\quad\gamma>\frac{3}{2},$ (7.6)

consider some external data $g,J\in L^{\infty}(Q)$ and consider some initial
data $0\leq\rho_{0}\in L^{\gamma}(\Omega)$, $(\rho u)_{0}\in L^{1}(\Omega)$
and $B_{0}\in L^{2}_{\operatorname{div}}(\Omega)$ satisfying

$\displaystyle\frac{\left|(\rho u)_{0}\right|^{2}}{\rho_{0}}\in
L^{1}(\Omega),\quad(\rho u)_{0}=0\ \ \text{a.e. in }\left\\{x\in\Omega:\
\rho_{0}(x)=0\right\\},\quad B_{0}\cdot\operatorname{n}=0\ \text{on
}\partial\Omega.$ (7.7)

Then the system (2.3)–(2.17) is said to admit a weak solution on $[0,T]$ if
there exists a function

$\displaystyle\eta:\ [0,T]\times
S_{0}\rightarrow\mathbb{R}^{3},\quad\left.\eta(t,\cdot)\right|_{S_{0}^{i}}=\eta^{i}(t,\cdot)\quad\quad\forall
i=1,...,N,\ t\in[0,T],$ (7.8)

where each $\eta^{i}(t,\cdot):\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ denotes
an orientation preserving isometry, and if there exist functions

$\displaystyle 0\leq\rho$ $\displaystyle\in
L^{\infty}\left(0,T;L^{\gamma}(\Omega;\mathbb{R})\right)\bigcap
C\left([0,T];L^{1}(\Omega;\mathbb{R})\right),$ (7.9) $\displaystyle u$
$\displaystyle\in\left\\{\phi\in
L^{2}\left(0,T;H_{0}^{1}\left(\Omega;\mathbb{R}^{3}\right)\right):\
\mathbb{D}(\phi)=0\ \text{in }Q^{s}(S)\right\\},$ (7.10) $\displaystyle B$
$\displaystyle\in\left\\{b\in
L^{\infty}\left(0,T;L^{2}\left(\Omega;\mathbb{R}^{3}\right)\right)\bigcap
L^{2}\left(0,T;H_{\operatorname{div}}^{1}\left(\Omega;\mathbb{R}^{3}\right)\right):\
\operatorname{curl}b=0\ \text{in }Q^{s}(S),\ b\cdot\operatorname{n}=0\
\text{on }\partial\Omega\right\\},$ (7.11)

where $S=S(\cdot)=\eta(\cdot,S_{0})$, such that $\rho$ and $u$, extended by
$0$ in $\mathbb{R}^{3}\setminus\Omega$, satisfy the continuity equation

$\displaystyle-\int_{0}^{T}\int_{\Omega}\rho\partial_{t}\psi
dxdt-\int_{\Omega}\rho_{0}\psi(0,x)\ dx=$
$\displaystyle\int_{0}^{T}\int_{\Omega}(\rho u)\cdot\nabla\psi\ dxdt$ (7.12)

for all $\psi\in\mathcal{D}([0,T)\times\Omega)$ as well as the renormalized
continuity equation

$\displaystyle\partial_{t}\zeta(\rho)+\operatorname{div}\left(\zeta\left(\rho\right)u\right)+\left[\zeta^{\prime}\left(\rho\right)\rho-\zeta\left(\rho\right)\right]\operatorname{div}u=0\quad$
$\displaystyle\text{in
}\mathcal{D}^{\prime}\left((0,T)\times\mathbb{R}^{3}\right),$ (7.13)

for any

$\displaystyle\zeta\in
C^{1}\left([0,\infty)\right):\quad\left|\zeta^{\prime}(r)\right|\leq
cr^{\lambda_{1}}\quad\forall r\geq 1,\quad\quad\text{where }c>0,\
\lambda_{1}>-1,$ (7.14)

such that the momentum equation and the induction equation,

$\displaystyle-\int_{0}^{T}\int_{\Omega}\rho u\cdot\partial_{t}\phi\
dxdt-\int_{\Omega}(\rho u)_{0}\cdot\phi(0,x)\ dx=$
$\displaystyle\int_{0}^{T}\int_{\Omega}\left(\rho u\otimes
u\right):\mathbb{D}(\phi)+a\rho^{\gamma}\operatorname{div}\phi-2\nu\mathbb{D}(u):\mathbb{D}(\phi)$
$\displaystyle-\lambda\operatorname{div}u\operatorname{div}\phi+\rho
g\cdot\phi+\frac{1}{\mu}\left(\operatorname{curl}B\times B\right)\cdot\phi\
dxdt,$ (7.15) $\displaystyle-\int_{0}^{T}\int_{\Omega}B\cdot\partial_{t}b\
dxdt-\int_{\Omega}B_{0}\cdot b(0,x)\ dx=$
$\displaystyle\int_{0}^{T}\int_{\Omega}\left[-\frac{1}{\sigma\mu}\operatorname{curl}B+u\times
B+\frac{1}{\sigma}J\right]\cdot\operatorname{curl}b\ dxdt,$ (7.16)

are satisfied for any $\phi\in\mathcal{Z}(S)$ and any $b\in\mathcal{Y}(S)$
and, finally, such that the system $\\{S_{0}^{i},\eta^{i}\\}_{i=1}^{N}$ is
compatible with the velocity field $u$.

The existence of weak solutions to the system (2.3)–(2.17) in the sense of
Definition 7.1 is guaranteed by the following result.

######  Theorem 7.1.

[28, Theorem 2.3] 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 (7.5). Assume moreover the
coefficients $\sigma,\mu,\nu,\lambda,a,\gamma\in\mathbb{R}$ to satisfy the
conditions (7.6) and the data $g,J\in L^{\infty}(Q)$, $\rho_{0}\in
L^{\gamma}(\Omega)$, $(\rho u)_{0}\in L^{1}(\Omega)$ and $B_{0}\in
L^{2}_{\operatorname{div}}(\Omega)$ to satisfy the conditions (7.7). Then the
system (2.3)–(2.17) admits a weak solution $(\eta,\rho,u,B)$ on $[0,T]$ in the
sense of Definition 7.1 which in addition satisfies the energy inequality

$\displaystyle\int_{\Omega}\frac{1}{2}\rho(\tau)\left|u(\tau)\right|^{2}+\frac{a}{\gamma-1}\rho^{\gamma}(t)+\frac{1}{2\mu}\left|B(\tau)\right|^{2}\
dx+\int_{0}^{\tau}\int_{\Omega}2\nu|\mathbb{D}\left(u\right)|^{2}+\lambda\left|\operatorname{div}u\right|^{2}+\frac{1}{\sigma\mu^{2}}\left|\operatorname{curl}B\right|^{2}\
dxdt$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}\frac{1}{2}\frac{\left|\left(\rho
u\right)_{0}\right|^{2}}{\rho_{0}}+\frac{a}{\gamma-1}\rho_{0}^{\gamma}+\frac{1}{2\mu}\left|B_{0}\right|^{2}\
dx+\int_{0}^{\tau}\int_{\Omega}\rho g\cdot
u+\frac{1}{\sigma\mu}J\cdot\operatorname{curl}B\ dxdt$ (7.17)

for almost all $\tau\in[0,T]$.

Proof

As a courtesy to the readers we sketch the main ideas of the (quite technical)
proof of Theorem 7.1; the whole proof is given in [28] and - with some
additional details - in [29]. The proof is accomplished via an approximation
method: First, an easily solvable approximation to the original system is
constructed. After solving the approximate system, a solution to the original
problem is then recovered by passing to the limit in the approximation. Our
approximation consists of five different levels, characterized by five
parameters $\Delta t>0$, $n,m\in\mathbb{N}$, $\epsilon,\alpha>0$:

  * •

On the $\Delta t$-level, the induction equation is discretized with respect to
the time variable via the Rothe method (see [25, Section 8.2]), while the
mechanical part of the problem is split up into a series of time-dependent
problems on the small intervals between the discrete times. To this end we fix
$\Delta t>0$ with $\frac{T}{\Delta t}\in\mathbb{N}$ and split up the interval
$[0,T]$ into the discrete times $k\Delta t$, $k=1,...,\frac{T}{\Delta t}$.

  * •

On the $n$-level, a Galerkin method is carried out in order to solve the
approximate momentum equation.

  * •

The approximation levels associated to $m,\epsilon$ and $\alpha$ correspond to
the approximation used in [9] for the purely mechanical problem: The $m$-level
describes a penalization method which allows us to pass from a fluid-only
system to a system containing both a fluid and rigid bodies. On the
$\epsilon$\- and $\alpha$-levels, the system is regularized through the
addition of multiple regularization terms as well as an artificial pressure.

The demand for a time discretization on the $\Delta t$-level can be explained
in the following way: The fact that the test functions in both the momentum
equation and the induction equation depend on the solution of the system (cf.
(7.3), (7.4)) prevents us from solving all of the equations in the original
system simultaneously. In the case of the momentum equation this problem is
circumvented via the application of a penalization method (see the $m$-level
below). In the case of the induction equation, however, no similar method
appears to be available. Instead we decouple the system via a time
discretization, which allows us to solve the equations one after another by
using time-lagging functions in the coupling terms. In this way, at each
discrete time $k\Delta t$, $k=1,...,\frac{T}{\Delta t}$, in our discretization
we are able to first determine the position $S_{\Delta t,k}(k\Delta t)$ of the
solid and subsequently solve the discrete induction equation at this specific
time with test functions chosen accordingly. More specifically, as discrete
approximations of the test functions (7.4) we consider test functions $b\in
H^{1}(\Omega)$ satisfying

$\displaystyle\text{curl}\ b=0\ \text{in }S_{\Delta t,k}(k\Delta
t)\bigcap\Omega,\quad b\cdot\text{n}|_{\partial\Omega}=0.$

Then, a discretized version of the induction equation of the form

$\displaystyle-\int_{\Omega}\frac{B_{\Delta t}^{k}-B_{\Delta t}^{k-1}}{\Delta
t}\cdot b\ dx=$
$\displaystyle\int_{\Omega}\left[\frac{1}{\sigma\mu}\operatorname{curl}B_{\Delta
t}^{k}-u_{\Delta t}^{k}\times B_{\Delta t}^{k-1}-\frac{1}{\sigma}J_{\Delta
t}^{k}\right]\cdot\operatorname{curl}b\ dx,$ (7.18)

wherein $B_{\Delta t}^{k}$, $u_{\Delta t}^{k}$ and $J_{\Delta t}^{k}$ denote
discrete approximations of the unknown $B$ and $u$ and the given force $J$,
can be solved under the application of classical methods. The deployment of a
time discretization, however, leads to further problems. More precisely, the
well-known techniques for showing non-negativity of the density in the
construction of weak solutions to the compressible Navier-Stokes equations
(see [23, Sections 7.6.5, 7.6.6, 7.7.4.2]) do not seem to be compatible with
discrete equations. In particular, it does not seem possible to discretize the
mechanical part of our system in such a way that a meaningful energy
inequality can be derived, which in turn is required for obtaining uniform
bounds of the approximate solution and passing to the limit in the
discretization.

This issue motivates the use of a hybrid approximation scheme, in which the
induction equation is discretized as in (7.18) while the mechanical equations
are treated as continuous equations on the small intervals $[(k-1)\Delta
t,k\Delta t]$ between the consecutive discrete time points. Consequently, the
problem of the solution-dependence of the test functions in the induction
equation can be handled as described above while the non-negativity of the
density can be achieved under exploitation of the classical existence theory
for the compressible Navier-Stokes equations. A suitable overall energy
inequality for the approximate problem can then be derived under the
consideration of piecewise linear interpolants of the discrete functions,
which allows us to deduce a continuous energy estimate from the discrete
induction equation. This can subsequently be added together with the
corresponding mechanical estimate to obtain the full energy inequality.

The Galerkin method on the $n$-level constitutes a classical procedure in the
solving of the continuous compressible Navier-Stokes equations. In addition,
the higher regularity of the velocity field on the Galerkin level is of
benefit during the limit passage in the time discretization.

The penalization method on the $m$-level is what allows us to handle the
solution dependent test functions (7.3) in the momentum equation. For the same
purpose it has been used previously in the purely mechanical problem in, for
example, [9] in the compressible case and [26] in the incompressible setting.
In this method, a fluid is considered in the whole domain and the rigid bodies
are approximated by letting the viscosity of this fluid rise to infinity in
the later solid region. The approximate all-fluid problem can then be solved
with classical test functions independent of the solution and it suffices to
switch to the solution-dependent test functions (7.3) in the limit passage
with respect to $m\rightarrow\infty$. For the mathematical implementation of
this penalization method, we define the approximate solid region $S(t)$ at any
time $t\in[0,T]$ through the flow curves of a suitable regularization
$R_{\delta}[u]$, $\delta>0$, of the (fluid) velocity $u$: Namely, we set

$\displaystyle S(t):=\bigcup_{i=1}^{N}S^{i}(t),$

where

$\displaystyle
S^{i}(t):=\left(O^{i}(t)\right)^{\delta}=\left\\{x\in\mathbb{R}^{3}:\
\operatorname{dist}\left(x,O^{i}(t)\right)<\delta\right\\},\quad
O^{i}(t):=\eta\left(t,O^{i}\right),$

denotes the position of the approximate $i$-th solid body, $\eta$ is
determined via the initial value problem

$\displaystyle\frac{d}{dt}\eta(t,x)=$ $\displaystyle
R_{\delta}\left[u\right]\left(t,\eta(t,x)\right),\quad\quad\eta(0,x)=x$ (7.19)

and

$\displaystyle O^{i}:=\left(S_{0}^{i}\right)_{\delta}:=\left\\{x\in
S_{0}^{i}:\ \operatorname{dist}\left(x,\partial
S_{0}^{i}\right)>\delta\right\\}$

denotes the $\delta$-kernel of the initial position $S_{0}^{i}$ of the $i$-th
body. We point out that the regularized velocity field $R_{\delta}[u]$ can be
chosen such that the implication

$\displaystyle\mathbb{D}(u(t,\cdot))=0\quad\text{in
}U\quad\quad\Rightarrow\quad\quad
R_{\delta}[u](t,\cdot)=u(t,\cdot)\quad\text{in }U_{\delta}=\left\\{x\in U:\
\operatorname{dist}\left(x,\partial U\right)>\delta\right\\}$ (7.20)

holds true for any domain $U\subset\mathbb{R}^{3}$. We further denote by

$\displaystyle\chi(t,x):=\operatorname{dist}\left(x,\overline{\mathbb{R}^{3}\setminus
S(t)}\right)-\operatorname{dist}\left(x,\overline{S(t)}\right)$

the signed distance function of $S(t)$ and fix some convex function $H\in
C^{\infty}(\mathbb{R})$ such that $H(z)=0$ for $z\in(-\infty,0]$ and $H(z)>0$
otherwise. Then, we replace the viscosity coefficients $\nu$ and $\lambda$ in
the momentum equation (7.15) by the quantities

$\displaystyle\nu\left(\chi\right):=\nu+mH\left(\chi\right),\quad\quad\lambda\left(\chi\right):=\lambda+mH\left(\chi\right),$

which are referred to as variable viscosity coefficients. In the limit passage
with respect to $m\rightarrow\infty$, the coefficients $\nu(\chi)$ and
$\lambda(\chi)$ rise to infinity in the (approximate) solid region. The
associated energy inequality then shows that, in each solid body $S^{i}(t)$,
the limit velocity $u$ has a vanishing symmetric gradient $\mathbb{D}(u)=0$
and is thus equal to a rigid velocity field $u^{S_{i}}$. Since, due to the
relation (7.19), the bodies $S^{i}(t)$ in the limit are again determined via
the flow curves of $R_{\delta}[u]$, the implication (7.20) shows that
$R_{\delta}[u]=u$ in the sets $O^{i}(t)\subset S^{i}(t)$. It follows that

$\displaystyle u^{S_{i}}=u=R_{\delta}[u]\quad\quad\text{in }O^{i}(t).$

This implies rigidity of the bodies $S^{i}(t)$, the motion of which is
determined by $R_{\delta}[u]$, as well as compatibility of $u$ with the system
$\\{S_{0}^{i},\eta^{i}\\}_{i=1}^{N}$ for certain orientation preserving
isometries $\eta^{i}(t,\cdot):\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$.

On the $\epsilon$-level, several regularization methods with different
purposes are combined into one step. The continuity equation is regularized by
the addition of the Laplacian $\epsilon\Delta\rho$ of the density to its
right-hand side. This parabolic regularization constitutes the classical
approach from the theory of the compressible Navier-Stokes equations for the
construction of a non-negative density, see [23, Sections 7.6.5, 7.6.6,
7.7.4.2]. In order to compensate for this additional term in the derivation of
the energy inequality, the momentum equation is moreover supplemented by the
quantity $\epsilon\nabla u\nabla\rho$. Furthermore, the quantity

$\displaystyle\epsilon|u|^{2}u$ (7.21)

is added to the momentum equation. This is due to the fact that, while in the
continuous system the mixed terms from the momentum equation and the induction
equation cancel each other in the energy inequality, the same is not true in
our hybrid approximation. Therefore, in order to obtain uniform bounds for the
approximate solution, these terms need to be controlled by the non-negative
left-hand side of the energy inequality. This is possible via an estimate of
the form

$\displaystyle\int_{\Omega}\frac{1}{\mu}\left(u_{\Delta t}^{k-1}\times
B_{\Delta t}^{k-1}\right)\cdot\operatorname{curl}B_{\Delta t}^{k}\
dx\leq\frac{1}{\sqrt{\mu}\epsilon}\left\|B_{\Delta
t}^{k-1}\right\|_{L^{2}(\Omega)}^{2}+\frac{\epsilon}{8}\left\|u_{\Delta
t}^{k-1}\right\|_{L^{4}(\Omega)}^{4}+\frac{\epsilon}{8\mu^{3}}\left\|\operatorname{curl}B_{\Delta
t}^{k}\right\|_{L^{4}(\Omega)}^{4}$ (7.22)

for the mixed term in the discrete induction equation (7.18) and a similar
estimate for the mixed term in the momentum equation. Here, the term
$\frac{\epsilon}{8}\|u_{\Delta t}^{k-1}\|_{L^{4}(\Omega)}^{4}$ can be absorbed
by the left-hand side of the energy inequality on the $\Delta t$-level thanks
to the regularization term (7.21). In order to control the term
$\frac{1}{\epsilon 8\mu^{3}}\|\operatorname{curl}B_{\Delta
t}^{k}\|^{4}_{L^{4}(\Omega)}$ in the estimate (7.22) in the same way, the
induction equation is regularized similarly through the addition of the
$4$-double-curl

$\displaystyle\frac{\epsilon}{\mu^{2}}\text{curl}\left(|\text{curl}B|^{2}\text{curl}B\right)$

of the magnetic induction. The term $\frac{1}{\mu\epsilon}\|B_{\Delta
t}^{k-1}\|^{2}_{L^{2}(\Omega)}$ in the estimate (7.22) can then simply be
controlled via the Gronwall lemma, which leads to the desired bounds uniform
with respect to $\Delta t$. The induction equation is further regularized
through the addition of the term

$\displaystyle\epsilon\operatorname{curl}(\Delta(\operatorname{curl}B)).$

With the help of this term, the induction equation can be expressed via a
weakly continuous and coercive operator, which guarantees its solvability.

Finally, the addition of an artificial pressure to the momentum equation on
the $\alpha$-level is also well-known from the classical theory for the
compressible Navier-Stokes system, see for example [23, Section 7.3.8]. This
artificial pressure, which is chosen of the form $\alpha\rho^{\beta}$ for a
sufficiently large $\beta>\max\\{4,\gamma\\}$, helps us to pass to the limit
in the quantity $\epsilon(\nabla u\nabla\rho)$ from the $\epsilon$-level by
guaranteeing a higher integrability of the density, cf. [23, Section 7.8.2].
The higher integrability moreover enables the use of the regularization
technique by DiPerna and Lions (cf. [23, Lemma 6.8, Lemma 6.9]) and thus
facilitates the limit passage with respect to $\epsilon\rightarrow 0$.

##  8 Appendix

Here we present some calculations needed for the derivation of the boundary
and interface conditions in Section 4. These are mainly elementary and follow
the ideas of the standard pill box argument in the physics literature. Here we
present the calculations for the case of a curved instead of a flat interface.

###  8.1 Proof of the estimate (4.14)

In this section we prove the estimate (4.14), used in the derivation of the
boundary and interface conditions (4.1) for the tangential component of $H$ in
Section 4. To this end we first exploit the fact that $\phi$ is twice
continuously differentiable with $\phi(0)=\phi^{\prime}(0)=0$ and $H$ is
continuously differentiable in both the fluid and the solid domain, cf.
(3.19). This allows us to use the fundamental theorem of calculus and
calculate the integrals on the left-hand side of the integrated version (4.13)
of Ampère’s law as

$\displaystyle\int_{S_{1}}H\cdot ds=$ $\displaystyle\int_{-\Delta l}^{\Delta
l}H\left(\begin{pmatrix}\Delta s\\\ \phi(\Delta s)+\zeta\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\end{array}\right)\ d\zeta$ (8.4) $\displaystyle=$ $\displaystyle\int_{-\Delta
l}^{\Delta l}H\left(\begin{pmatrix}0\\\ \zeta\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\end{array}\right)+\int_{0}^{\Delta
s}\frac{d}{d\xi}H\left(\begin{pmatrix}\xi\\\ \phi(\xi)+\zeta\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\end{array}\right)\ d\xi d\zeta,$ (8.11) $\displaystyle\int_{S_{2}}H\cdot ds=$
$\displaystyle\int_{-\Delta s}^{\Delta s}H\left(\begin{pmatrix}-\theta\\\
\phi(-\theta)+\Delta l\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}-1\\\
-\phi^{\prime}(-\theta)\\\ 0\\\ \end{array}\right)\ d\theta$ (8.15)
$\displaystyle=$ $\displaystyle\int_{-\Delta s}^{\Delta
s}H\left(\begin{pmatrix}0\\\ \Delta l\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}-1\\\ 0\\\ 0\\\
\end{array}\right)+\int_{0}^{\theta}\frac{d}{d\xi}\left[H\left(\begin{pmatrix}-\xi\\\
\phi(-\xi)+\Delta l\\\ 0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}-1\\\
-\phi^{\prime}(-\xi)\\\ 0\\\ \end{array}\right)\right]\ d\xi d\theta,$ (8.22)
$\displaystyle\int_{S_{3}}H\cdot ds=$ $\displaystyle\int_{-\Delta l}^{\Delta
l}H\left(\begin{pmatrix}0\\\ -\zeta\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}0\\\ -1\\\ 0\\\
\end{array}\right)+\int_{0}^{\Delta
s}\frac{d}{d\xi}H\left(\begin{pmatrix}-\xi\\\ \phi(-\xi)-\zeta\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}0\\\ -1\\\ 0\\\
\end{array}\right)\ d\xi d\zeta,$ (8.29) $\displaystyle\int_{S_{4}}H\cdot ds=$
$\displaystyle\int_{-\Delta s}^{\Delta s}H\left(\begin{pmatrix}0\\\ -\Delta
l\\\ 0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
\end{array}\right)+\int_{0}^{\theta}\frac{d}{d\xi}\left[H\left(\begin{pmatrix}\xi\\\
\phi(\xi)-\Delta l\\\ 0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}1\\\
\phi^{\prime}(\xi)\\\ 0\\\ \end{array}\right)\right]\ d\xi d\theta.$ (8.36)

We remark that for the first terms on the right-hand sides of the equations
(8.11) and (8.29), respectively, it holds that

$\displaystyle\int_{-\Delta l}^{\Delta l}H\left(\begin{pmatrix}0\\\ \zeta\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\end{array}\right)\ d\zeta+\int_{-\Delta l}^{\Delta
l}H\left(\begin{pmatrix}0\\\ -\zeta\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}0\\\ -1\\\ 0\\\
\end{array}\right)\ d\zeta=0.$ (8.43)

With the identities (8.11)–(8.43) at hand we rearrange the equation (4.13).
From the boundedness of $H$, $D$, $J$ and $\phi$ as well as their derivatives
(cf. (3.20)) we then infer that

$\displaystyle\left|\int_{-\Delta s}^{\Delta s}H\left(\begin{pmatrix}0\\\
\Delta l\\\ 0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}-1\\\ 0\\\ 0\\\
\end{array}\right)\ d\theta+\int_{-\Delta s}^{\Delta
s}H\left(\begin{pmatrix}0\\\ -\Delta l\\\
0\end{pmatrix}\right)\cdot\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
\end{array}\right)\ d\theta-\int_{\Delta F}j\cdot\text{n}_{x}^{\prime}\
dA\right|$ (8.50) $\displaystyle\leq$ $\displaystyle c\left(\Delta s\Delta
l+(\Delta s)^{2}\right)$

for some constant $c>0$ independent of $\Delta s$ and $\Delta l$. Dividing
this inequality by $2\Delta s$ we conclude that

$\displaystyle\left|\tau_{x}\cdot H\left(\begin{pmatrix}0\\\ \Delta l\\\
0\end{pmatrix}\right)-\tau_{x}\cdot H\left(\begin{pmatrix}0\\\ -\Delta l\\\
0\end{pmatrix}\right)-\frac{1}{2\Delta s}K(\Delta F)\right|\leq c(\Delta
l+\Delta s),$

where $\tau_{x}=\text{n}_{x}^{\prime}\times\text{n}_{x}=(-1,0,0)^{T}$ and
$K(\Delta F)$ is defined by (4.15), i.e. the desired inequality (4.14).

###  8.2 Proof of the estimate (4.23)

In this section we show the estimate (4.23), needed in the deduction of the
boundary and interface conditions (4.4) for the normal component of $D$ in
Section 4. To this end we rewrite the integrals in the integrated version
(4.22) of Gauss’s law under usage of the parametrizations (4.19)–(4.21). We
start by calculating, from the parametrization (4.19),

$\displaystyle\left|\frac{\partial\Gamma_{\text{top}}(s,\alpha)}{\partial
s}\times\frac{\partial\Gamma_{\text{top}}(s,\alpha)}{\partial\alpha}\right|\text{n}_{\text{top}}(s,\alpha)=s\begin{pmatrix}\partial_{1}\Phi(s\cos(\alpha),s\sin(\alpha))\\\
-1\\\ \partial_{2}\Phi(s\cos(\alpha),s\sin(\alpha))\end{pmatrix}.$

This in combination with the parametrization (4.19), the fundamental theorem
of calculus - which may be applied since $D$ is continuously differentiable in
both the fluid and the solid domain and $\Phi$ is twice continuously
differentiable - and the identities
$\Phi(0,0)=\partial_{1}\Phi(0,0)=\partial_{2}\Phi(0,0)=0$ allows us to
calculate the first integral on the left-hand side of the identity (4.22) as

$\displaystyle\int_{\Delta_{\text{top}}}D\cdot\text{n}_{\text{top}}\ dA=$
$\displaystyle\int_{0}^{2\pi}\int_{0}^{r}sD\left(\begin{pmatrix}s\cos(\alpha)\\\
\Phi(s\cos(\alpha),s\sin(\alpha))+\Delta l\\\
s\sin(\alpha)\end{pmatrix}\right)\cdot\begin{pmatrix}\partial_{1}\Phi(s\cos(\alpha),s\sin(\alpha))\\\
-1\\\ \partial_{2}\Phi(s\cos(\alpha),s\sin(\alpha))\end{pmatrix}\ dsd\alpha$
$\displaystyle=$ $\displaystyle\pi r^{2}D\left(\begin{pmatrix}0\\\ \Delta l\\\
0\end{pmatrix}\right)\cdot\begin{pmatrix}0\\\ -1\\\ 0\end{pmatrix}$
$\displaystyle+\int_{0}^{2\pi}\int_{0}^{r}$ $\displaystyle
s\int_{0}^{s}\frac{d}{d\xi}\left[D\left(\begin{pmatrix}\xi\cos(\alpha)\\\
\Phi(\xi\cos(\alpha),\xi\sin(\alpha))+\Delta l\\\
\xi\sin(\alpha)\end{pmatrix}\right)\cdot\begin{pmatrix}\partial_{1}\Phi(\xi\cos(\alpha),\xi\sin(\alpha))\\\
-1\\\ \partial_{2}\Phi(\xi\cos(\alpha),\xi\sin(\alpha))\end{pmatrix}\right]\
d\xi dsd\alpha.$ (8.51)

Analogously, for the second integral on the left-hand side of the identity
(4.22), we calculate, from the parametrization (4.20),

$\displaystyle\int_{\Delta_{\text{bottom}}}D\cdot\text{n}_{\text{bottom}}\
dA=\pi r^{2}D\left(\begin{pmatrix}0\\\ -\Delta l\\\
0\end{pmatrix}\right)\cdot\begin{pmatrix}0\\\ 1\\\ 0\end{pmatrix}$
$\displaystyle+\int_{0}^{2\pi}\int_{0}^{r}s\int_{0}^{s}\frac{d}{d\xi}\left[D\left(\begin{pmatrix}\xi\cos(-\alpha)\\\
\Phi(\xi\cos(-\alpha),\xi\sin(-\alpha))-\Delta l\\\
\xi\sin(-\alpha)\end{pmatrix}\right)\cdot\begin{pmatrix}-\partial_{1}\Phi(\xi\cos(-\alpha),\xi\sin(-\alpha))\\\
1\\\ -\partial_{2}\Phi(\xi\cos(-\alpha),\xi\sin(-\alpha))\end{pmatrix}\right]\
d\xi dsd\alpha.$ (8.52)

Finally, a similar calculation for the third integral on the left-hand side of
(4.22) under exploitation of the parametrization (4.21) leads to

$\displaystyle\int_{\Delta_{\text{side}}}D\cdot\text{n}_{\text{side}}\ dA$
$\displaystyle=$ $\displaystyle\int_{-\Delta l}^{\Delta
l}\int_{0}^{2\pi}r\left[D\left(\begin{pmatrix}0\\\ h\\\
0\end{pmatrix}\right)\cdot\begin{pmatrix}-\cos(\alpha)\\\ 0\\\
-\sin(\alpha)\end{pmatrix}\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.+\int_{0}^{r}\frac{d}{d\xi}D\left(\begin{pmatrix}\xi\cos(\alpha)\\\
\Phi(\xi\cos(\alpha),\xi\sin(\alpha))+h\\\
\xi\sin(\alpha)\end{pmatrix}\right)\cdot\begin{pmatrix}-\cos(\alpha)\\\ 0\\\
-\sin(\alpha)\end{pmatrix}d\xi\right]\ d\alpha dh$ $\displaystyle=$
$\displaystyle\int_{-\Delta l}^{\Delta
l}\int_{0}^{2\pi}r\int_{0}^{r}\frac{d}{d\xi}D\left(\begin{pmatrix}\xi\cos(\alpha)\\\
\Phi(\xi\cos(\alpha),\xi\sin(\alpha))+h\\\
\xi\sin(\alpha)\end{pmatrix}\right)\cdot\begin{pmatrix}-\cos(\alpha)\\\ 0\\\
-\sin(\alpha)\end{pmatrix}\ d\xi d\alpha dh.$ (8.53)

We use the identities (8.51), (8.52) and (8.53) to reformulate the terms on
the left-hand side of the equation (4.22). Rearranging the resulting equation
and using the fact that $D$ and $\phi$ as well as their derivatives are
bounded we estimate

$\displaystyle\left|\pi r^{2}D\left(\begin{pmatrix}0\\\ \Delta l\\\
0\end{pmatrix}\right)\cdot\begin{pmatrix}0\\\ 1\\\ 0\end{pmatrix}-\pi
r^{2}D\left(\begin{pmatrix}0\\\ -\Delta l\\\
0\end{pmatrix}\right)\cdot\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix}-\int_{C}\rho_{c}\ dx\right|\leq c\left(r^{3}+r^{2}\Delta
l\right)$

for a constant $c>0$ independent of $r$ and $\Delta l$. Dividing this
inequality by $\pi r^{2}$ we arrive at the estimate

$\displaystyle\left|\text{n}_{x}\cdot\left[D\left(\begin{pmatrix}0\\\ -\Delta
l\\\ 0\end{pmatrix}\right)-D\left(\begin{pmatrix}0\\\ \Delta l\\\
0\end{pmatrix}\right)\right]-\frac{1}{\pi r^{2}}W(C)\right|\leq c(r+\Delta
l),$

where $\text{n}_{x}=(0,1,0)^{T}$ on $\partial S(t)$ constitutes the outer unit
normal vector in the origin $x=(0,0,0)^{T}$ of the local coordinate system and
$W(C)$ is defined by (4.24). This proves the desired estimate (4.23).

Acknowledgment The main essence of this work was obtained at the University of
Würzburg and forms parts of the first author’s PhD thesis [29]. Some of the
writing process was conducted while J.S. was working at the University of
Prague and the Czech Academy of Sciences, respectively. J.S. acknowledges the
support by Praemium Academiæ of Šárka Nečasová. The Institute of Mathematics,
CAS is supported by RVO:67985840.

## References

  * [1] B. Benešová, Š. Nečasová, J. Scherz and A. Schlömerkemper, Fluid-rigid body interaction in an incompressible electrically conducting fluid, SIAM Journal on Mathematical Analysis 55 no. 2: 929–965 (2023)
  * [2] X. Blanc and B. Ducomet, Weak and strong solutions of equations of compressible magnetohydrodynamics, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids: 2869–2925, Springer, Cham (2016)
  * [3] T. Bodnár, G.P. Galdi, Š. Nečasová, Fluid-Structure Interaction and Biomedical Applications, Birkhäuser, Basel (2014)
  * [4] H.-J. Bungartz, M. Schäfer: Fluid-Structure Interaction: Modelling, Simulation, Optimisation, Springer-Verlag, Berlin Heidelberg (2006)
  * [5] H. Cabannes: Theoretical Magnetohydrodynamics, Academic Press, New York (1970)
  * [6] R. Dautray and P.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 1: Physical Origins and Classical Methods, Springer-Verlag, Berlin (2000)
  * [7] P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, United Kingdom (2001)
  * [8] A.C. Eringen and G.A. Maugin, Electrodynamics of Continua II, Springer-Verlag, New York (1990)
  * [9] E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid, Arch. Rational Mech. Anal. 167: 281–308 (2003)
  * [10] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ. 3: 419–441 (2003)
  * [11] E.P. Furlani, Magnetophoretic separation of blood cells at the microscale, Journal of Physics D: Applied Physics 40 no. 5: 1313–1319 (2007)
  * [12] J.F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford (2006)
  * [13] T.S. Gregory, R. Cheng, G. Tang, L. Mao and Z.T.H. Tse, The magnetohydrodynamic effect and its associated material designs for biomedical applications: a state-of-the-art review, Adv. Funct. Mater. 26: 3942–3952 (2016)
  * [14] T.S. Gregory, K.J. Wu, J. Yu, J.B. Box, R. Cheng, L. Mao, G. Tang and Z.T.H. Tse, Magnetohydrodynamic-Driven Design of Microscopic Endocapsules in MRI, IEEE/ASME Transactions on Mechatronics 20 no. 6: 2691–2698 (2015)
  * [15] W. Greiner, Classical Electrodynamics, Springer-Verlag, New York (1998)
  * [16] D.J. Griffiths, Introduction to Electrodynamics, 3rd ed., Prentice Hall, Upper Saddle River (1999)
  * [17] J.L. Guermond and P.D. Minev, Mixed Finite Element Approximation of an MHD Problem Involving Conducting and Insulating Regions: The 2D Case, ESAIM: Mathematical Modelling and Numerical Analysis 136 no. 33: 517–536 (2002)
  * [18] J.L. Guermond and P.D. Minev, Mixed Finite Element Approximation of an MHD Problem Involving Conducting and Insulating Regions: The 3D Case, Numer. Meth. PDE 19: 709–731 (2003)
  * [19] S. Jiang and F. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asymptotic Analysis 95: 161–185 (2015)
  * [20] S. Jiang and F. Li, Rigorous derivation of the compressible magnetohydrodynamic equations from the electromagnetic fluid system, Nonlinearity 25: 1735–1752 (2012)
  * [21] A.G. Kulikovskiy and G.A. Lyubimov, Magnetohydrodynamics, Addison Wesley, Reading, Massachussets (1965)
  * [22] R. Moreau, Magnetohydrodynamics, Springer, Dordrecht (1990)
  * [23] A. Novotný and I. Straškraba, Introduction to the mathematical theory of compressible flow, Oxford University Press, Oxford (2004)
  * [24] H. Pauly and H.P. Schwan, Dielectric properties and ion mobility in erythrocytes, Biophys J. 6 no. 5: 621–639 (1966)
  * [25] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin (2005)
  * [26] J.A. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Rational Mech. Anal. 161: 93–112 (2002)
  * [27] R. Sart, Existence of finite energy weak solutions for the equations MHD of compressible fluids, Applicable Analysis 19 no. 3: 357–379 (2009)
  * [28] J. Scherz, Fluid-rigid body interaction in a compressible electrically conducting fluid, Math. Nachr. (2023)
  * [29] J. Scherz, Weak Solutions to Mathematical Models of the Interaction between Fluids, Solids and Electromagnetic Fields, PhD thesis, University of Würzburg and Charles University in Prague (2024)