This section starts with a presentation of the equations that are implemented in SFEMaNS.

Equations considered by SFEMaNS

The following equations are implemented in SFEMaNS.

The incompressible Navier-Stokes equations. In a domain $\Omega$, these equations are written as follows:
\begin{align*} \partial_t\bu+\left(\ROT\bu\right)\CROSS\bu - \frac{1}{\Re}\LAP \bu +\GRAD p &=\bef, \\ \DIV \bu &= 0, \end{align*}

with $\bu$ the velocity field, $p$ the pressure, $\Re$ the kinetic Reynolds number and $\bef$ a source term.

Remark: One can also consider the abovee equations with variable density and viscosity as described in Extension to multiphase flow problem
The heat equation. In a domain $\Omega$, these equations are written as follows:
\begin{align*} C \partial_t T+ \DIV(T \bu) - \DIV (\lambda \GRAD T) &= f_T, \end{align*}

with $T$ the temperature, $\bu$ the velocity field, $C$ the volumetric heat capacity, $\lambda$ the thermal conducitivty and $f_T$ a source term.
The Maxwell equations. In a conducting domain $\Omega_c$, these equations are written as follows:
\begin{align*} \partial_t (\mu^c \bH^c) + \nabla \times \left(\frac{1}{\Rm \sigma} \nabla \times \bH^c \right) = \nabla\times (\bu \times \mu^c \bH^c) + \nabla \times \left(\frac{1}{\Rm \sigma}\mathbf{j}^s \right), \\ \text{div} (\mu^c \bH^c) = 0 , \end{align*}

with $\bH^c$ the magnetic field, $\bu$ the velocity field, $\textbf{j}^s$ a source term, $\mu^c$ the magnetic permeability, $\sigma$ the electrical conductivity and $\Rm$ the magnetic Reynolds number. If the magnetic permeability is discontinuous across a surface denoted $\Sigma_\mu$, the following equations have to be satisfied on $\Sigma_\mu$:
\begin{align*} \bH^c_1 \times \bn_1 + \bH^c_2 \times \bn_2 = 0,\\ \mu^c_1\bH^c_1 \cdot \bn_1 + \mu^c_2 \bH^c_2 \cdot \bn_2 = 0 ,\\ \end{align*}

where $\bn_1, \bn_2$ are outward normal to the surface $\Sigma_\mu$. $\bn_1$ points from $ $\Omega_1$ to $ $\Omega_2$ and $\bn_1=-\bn_2$.
If the conductivity in a subdomain of the magnetic domain is zero, we adopt a potential representation for the magnetic field. Let $\Omega_v$ be the insulating domain in question (assumed to be simply connected). $\Omega_v$ is henceforth referred to as vacuum. The Maxwell equations are then written as follows in $\Omega_v$:
\begin{align*} -\mu^v \partial_t \LAP \phi = 0 , \end{align*}
where $\phi$ the scalar potential such that $\bH=\GRAD \phi$ in the vacuum. The following continuity conditions across the interface $\Sigma=\Omega_c \cap \Omega_v$ have to be satisfied:
\begin{align*} \bH^c \times \bn^c + \nabla \phi \times \bn^v = 0 , \\ \mu^c \bH^c \cdot \bn^c + \mu ^v \nabla \phi \cdot \bn^v = 0 , \end{align*}
with $\bn^c$ and $\bn^v$ the outward normals to the surface $\Sigma$.

Remark: the above equations are supplemented by initial and boundaries conditions.

Key structural assumptions

Domain geometry and axisymmetric hypothesis

The code SFEMaNS uses cylindrical coordinates $(r,\theta,z)$ and assumes a priori that the computational domain is axi-symmetric. The approximation method consists of using a Fourier decomposition in the azimuthal direction and Lagrange finite elements in the meridian section.

Domain decomposition and simply connected insulating sub-domain hypothesis

The computational domain $\Omega$ is divided into the following three sub-domains:

A conducting fluid domain, denoted by $\Omega_{c,f}$, where the conductivity, permeability, viscosity and density of the fluid are constant and positive.
A conducting solid domain, denoted by $\Omega_{c,s}$, where the velocity of the solid is imposed. This sub-domain is assumed to be a finite union of disjoint solid axisymmetric domains $\Omega_{c,s}^i$ with positive constant conductivity $\sigma_i$ and permeability $\mu_i$. We denote by $I$ the set that contains the integers $i$.
An insulating domain, called vacuum and denoted by $\Omega_v$, where the electrical conductivity $\sigma$ is zero and the relative magnetic permeability $\mu^v$ is 1 by default (but can be set to an other value by the user).

The insulating sub-domain $\Omega_v$ is assumed to be simply connected so the magnetic field are written $\bH=\GRAD\phi$. The scalar potential $\phi$ can be proved to be the solution of the following equation in $\Omega_v$:

\begin{align*} -\mu^v \partial_t \LAP \phi = 0. \end{align*}

Remarks:

The Navier-stokes equations are approximated in the fluid domain $\Omega_{c,f}$.
The heat equation is approximated in a domain $\Omega_T=\Omega_{c,f} \cup (\underset{j\in J}{\cup}\Omega_{c,s}^j)$ with $J$ a set included in $I$.
The magnetic field $\bH^c$ is approximated in $\Omega_c=\Omega_{c,f} \cup (\underset{i\in I}{\cup}\Omega_{c,s}^i)$.
The scalar potential $\phi$ is approximated in $\Omega_v$.

SFEMaNS's possibilities

The following set ups can be considered by the code SFEMaNS:

Hydrodynamics (NST). The Navier-Stokes equations are approximated. Thermal effect can also be considered.
Magnetism (MXW). The Maxwell equations are approximated with a given velocity field.
Magnetohydrodynamics (MHD). The Navier-Stokes and the Maxwell equations are approximated. The Lorentz force $\textbf{f}_\text{L}= (\ROT \bH) \times (\mu\bH)$ is added to the source term $\textbf{f}$ of the Navier-Stokes equations. Thermal effect can also be considered when solving the temperature equation.
Ferrohydrodynamics (FHD). All of the above equations are approximated. The Kelvin force $\textbf{f}_\text{fhd}=g(T) \GRAD(\frac{\bH^2}{2})$, with $g(T)$ a user-defined scalar function, is added to the source term $\textbf{f}$ of the Navier-Stokes equations. The action of the magnetic field on the temperature can be taken into account. The change of temperature can also be taken into account in the Maxwell equations.

The following extensions are also available in SFEMaNS but require a good knowledge of the code to be used properly:

Stabilization method, called entropy viscosity, for problems with large kinetic Reynolds numbers.
Non axisymmetric geometry.
Multiphase flow problem with variable density, dynamical viscosity and electrical conductivity.
Magnetic permeability depending of the time and the azimuthal direction. We note the variation in $\theta$ is smooth while jumps in the $(r,z)$ direction can be considered.
Quasi-static approximation of the MHD equations.

The approximation methods of the above setting are described in the section Numerical approximation. The use is referred to this section for more details on the quasi-static approximation of the MHD equations.