SFEMaNS  version 4.1 (work in progress) Reference documentation for SFEMaNS
SFEMaNS presentation

This section starts with a presentation of the equations that are implemented in SFEMaNS. The restrictions on the domain of computation and an overview of the features available is given.

# Equations considered by SFEMaNS

The following equations are implemented in SFEMaNS.

1. The Navier-Stokes equations. In a domain $$\Omega$$, these equations can be 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.
2. The heat equation. In a domain $$\Omega$$, these equations can be 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.
3. The Maxwell equations. In a conducting domain $$\Omega_c$$, these equations can be 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} \nabla\times \mathbf{j} \right), \\ \text{div} (\mu^c \bH^c) = 0 , \end{align*}

with $$\bH^c$$ the magnetic field, $$\bu$$ the velocity field, $$\textbf{j}$$ a source term, $$\mu^c$$ the magnetic permeability, $$\sigma$$ the electrical conductivity and $$\Rm$$ the magnetic Reynolds number. If the magnetic permeability presents jumps 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 we define outward normals $$\bn_1, \bn_2$$ to the surface $$\Sigma_\mu$$.

In a simply connected insulating domain $$\Omega_v$$, referred as vacuum, these equations can be written as follows:

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

with $$\phi$$ a scalar potential such that $$\GRAD \phi$$ is equal to the magnetic field $$\bH$$ 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 $$\Sigma$$.

Remarks:

1. The above equations are supplemented by initial and boundaries conditions.
2. The boundary conditions on the magnetic field involve the electric field $$\textbf{E}$$ that satisfies:

\begin{align*} - \nabla \times \textbf{E} = \partial_t (\mu \bH) \text{ in } \Omega,\\ \int_{\Gamma_{i,v}} \textbf{E} \cdot \textbf{n} = 0 \text{ for } 1 \leq i \leq J, \\ \end{align*}

where $$(\Gamma_{i,v})_{1\leq i \leq J}$$ are the connected components of $$\partial \Omega_v$$ with outward normal denoted $$\textbf{n}$$.

# Frame of work

## Domain geometry and axisymmetric hypothesis

The code SFEMaNS uses cylindrical coordinates $$(r,\theta,z)$$ so a spectral/finite element method can be applied. This method consists in using a Fourier decomposition in the azimuthal direction and approximates the problem in a meridian section with Lagrange finite elements. Due to the $$\theta$$-periodicity, the domain of computation $$\Omega$$ must be axisymmetric.

## Domain decomposition and simply connected insulating sub-domain hypothesis

The domain of computation $$\Omega$$ is splitted into the three following sub-domains:

1. A conducting fluid domain, denoted by $$\Omega_{c,f}$$, where the conductivity, permeability, viscosity and density of the fluid are constant and positive.
2. 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 $$i$$.
3. 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.

The insulating sub-domain $$\Omega_v$$ is assumed to be simply connected so the magnetic field can be 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:

1. The Navier-stokes equations are only approximated in the fluid domain $$\Omega_{c,f}$$.
2. 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 $$I$$.
3. The magnetic field $$\bH^c$$ is approximated in $$\Omega_c=\Omega_{c,f} \cup (\underset{i\in I}{\cup}\Omega_{c,s}^i)$$.
4. The scalar potential $$\phi$$ is approximated in $$\Omega_v$$.

## SFEMaNS's possibilities

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

1. Hydrodynamic. The Navier-Stokes equations are approximated. Thermal effect can also be considered when solving the temperature equation.
2. Magnetic. The Maxwell equations are approximated with a given velocity field.
3. Magnetohydrodynamic (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.
4. Ferrohydrodynamic (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 is being implemented. The Maxwell equations are also being adapted for such problems.

The following extensions have been implemented in SFEMaNS:

1. Stabilization method, called entropy viscosity, for problems with large kinetic Reynolds numbers.
2. Non axisymmetric geometry.
3. Multiphase flow problem with variable density, dynamical viscosity and electrical conductivity.
4. 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.
5. Quasi-static approximation of the MHD equations.

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