In this section, we will express A-V Formulation on geometry in axisymmetric.
First, we suppose that \(B\) is independent from \(\theta\) (of cylindric coordinates \((r,\theta,z)\)) and \(V\) only depends on \(\theta\).
So we have \(\mathbf{B} = \begin{pmatrix} B_r(r,z) \\ 0 \\ B_z(r,z) \end{pmatrix}_{cyl}\) and as \(\mathbf{B} = \nabla \times \mathbf{A}\), \(\mathbf{A}\) can be written \(\mathbf{A} = \begin{pmatrix} 0 \\ A_{\theta}(r,z) \\ 0 \end{pmatrix}_{cyl}\).
We can now calculate \(\nabla\times(\nabla\times \mathbf{A})\) :
\[ \nabla\times\mathbf{A} = \begin{pmatrix} -\partial_z A_{\theta} \\ 0 \\ \frac{1}{r} \partial_r (r A_{\theta}) \end{pmatrix}_{cyl}\]
\[ \nabla \times \left( \nabla \times A \right) = \begin{pmatrix} 0 \\ -\frac{\partial^2 A_{\theta}}{\partial z^2} - \frac{\partial^2 A_{\theta}}{\partial r^2} - \frac{\partial\frac{A_\theta}{r}}{\partial r} \\ 0 \end{pmatrix}_{cyl}\]
Therefore, with \(\nabla V = \begin{pmatrix} 0 \\ \frac{1}{r} \frac{\partial V}{\partial \theta} \\ 0 \end{pmatrix}\), we have :
\[ -\frac{1}{\mu}\frac{\partial^2 A_{\theta}}{\partial z^2} - \frac{1}{\mu}\frac{\partial^2 A_{\theta}}{\partial r^2} - \frac{1}{\mu}\frac{\partial\frac{A_\theta}{r}}{\partial r} + \sigma \frac{\partial A_\theta}{\partial t} + \frac{\sigma}{r} \frac{\partial V}{ \partial\theta}=0\]
The equation can be rewritten as :
\[ -\frac{1}{\mu}\frac{\partial^2 A_{\theta}}{\partial z^2} - \frac{1}{\mu r} \frac{\partial \left( r \frac{\partial A_\theta}{\partial r}\right)}{\partial r} +\frac{1}{\mu r}\frac{\partial A_\theta}{\partial r} - \frac{1}{\mu r}\frac{\partial A_\theta}{\partial r} + \frac{1}{\mu r^2}A\theta + \sigma \frac{\partial A_\theta}{\partial t} + \frac{\sigma}{r} \frac{\partial V}{ \partial\theta}=0\]
With \(\Delta A_\theta = \frac{\partial^2 A_\theta}{\partial z^2} + \frac{1}{r} \frac{\partial \left( r \frac{\partial A_\theta}{\partial r} \right)}{\partial r} \), it becomes :
\[ -\frac{1}{\mu}\Delta A_\theta + \frac{1}{\mu r^2}A_\theta + \sigma \frac{\partial A_\theta}{\partial t} + \frac{\sigma}{r} \frac{\partial V}{ \partial\theta}=0 \hspace{2cm} \text{(AV-1 Axis)}\]
On the other hand, the equation (AV-2) becomes :
\[ \nabla \cdot \left( \sigma \begin{pmatrix} 0 \\ \frac{1}{r} \frac{\partial V}{\partial \theta} \\ 0 \end{pmatrix} + \sigma \begin{pmatrix} 0 \\ \frac{\partial A_{\theta}}{\partial t} \\ 0 \end{pmatrix} \right) = 0 \hspace{2cm} \text{(AV-2)}\]
\[\begin{eqnarray*}
\frac{\sigma}{r^2} \frac{\partial^2 V}{\partial \theta^2} &=& 0 \\
\frac{\partial^2 V}{\partial \theta^2} &=& 0 \hspace{2cm} \text{(AV-2 Axis)}
\end{eqnarray*}\]
\(V\) becomes a polynomial of the second degree on \(\theta\).
To conclude, the A-V Formulation becomes :
A-V Formulation in axisymmetric coordinates
\[\text{(AV Axis)}
\left\{ \begin{matrix}
-\frac{1}{\mu}\Delta A_\theta + \frac{1}{\mu r^2}A_\theta + \sigma \frac{\partial A_\theta}{\partial t} + \frac{\sigma}{r} \frac{\partial V}{ \partial\theta}=0 & \text{ on } \Omega^{axis} & \text{(AV-1 Axis)} \\
\frac{\partial^2 V}{\partial \theta^2} = 0 & \text{ on } \Omega_c^{axis} & \text{(AV-2 Axis)} \\
A_{\theta} = 0 & \text{ on } \Gamma_D^{axis} & \text{(D Axis)} \\
\frac{\partial A_{\theta}}{\partial \mathbf{n}^{axis}} = 0 & \text{ on } \Gamma_N^{axis} & \text{(N Axis)}
\end{matrix} \right.\]
With \(\Delta A_\theta = \frac{\partial^2 A_\theta}{\partial z^2} + \frac{1}{r} \frac{\partial \left( r \frac{\partial A_\theta}{\partial r} \right)}{\partial r} \)
The weak formulation of equation (AV Axis) in two dimensions can be expressed as follows :
We multiply the equation (AV-1 Axis) by \(\phi \in H^1(\Omega)\) and integrate it on \(\Omega\) :
\[ \int_{\Omega}{ \left( - \frac{1}{\mu} \Delta A_\theta + \frac{1}{\mu r^2}A_\theta + \sigma \frac{\partial A_\theta}{\partial t} \right) \cdot \phi \ dxdydz} + \int_{\Omega_c}{ \frac{\sigma}{r} \frac{\partial V}{\partial \theta} \cdot \phi \ dxdydz} = 0 \hspace{1cm} \text{(Weak AV Axis)}\]
\[\scriptsize{
\int_{\Omega}{ \frac{1}{\mu} \nabla A_\theta \cdot \nabla \phi \ dxdydz } - \int_{\Gamma}{ \frac{1}{\mu} \frac{\partial A_\theta}{\partial \mathbf{n}} \cdot \phi \ d\Gamma} + \int_{\Omega}{\frac{1}{\mu r^2} A_\theta \cdot \phi \ dxdydz} +\int_{\Omega}{ \sigma \frac{\partial A_\theta}{\partial t} \cdot \phi \ dxdydz} + \int_{\Omega_c}{ \frac{\sigma}{r} \frac{\partial V}{\partial \theta} \cdot\phi \ dxdydz } = 0 \hspace{1cm} \text{(Disc AV Axis)}
}\]
\[\scriptsize{
\int_{\Omega^{axis}}{ \frac{1}{\mu} \tilde\nabla A_\theta \cdot \tilde\nabla \phi \ rdrdz } - \int_{\Gamma^{axis}}{ \frac{1}{\mu} (\tilde\nabla A_\theta\cdot n^{axis}) \cdot \phi \ d\Gamma} + \int_{\Omega^{axis}}{\frac{1}{\mu r^2} A_\theta \cdot \phi \ rdrdz} +\int_{\Omega^{axis}}{ \sigma \frac{\partial A_\theta}{\partial t} \cdot \phi \ rdrdz} + \int_{\Omega^{axis}_c}{ \frac{\sigma}{r} \frac{\partial V}{\partial \theta} \cdot\phi \ rdrdz } = 0
}\]
With \(\tilde{\nabla} = \begin{pmatrix} \partial_r \\ \partial_{\theta} \\ \partial_z \end{pmatrix}_{cyl}\)
\[\scriptsize{
\int_{\Omega^{axis}}{ \frac{r}{\mu} \tilde\nabla A_\theta \cdot \tilde\nabla \phi \ drdz } - \int_{\Gamma_D^{axis}}{ \frac{r}{\mu} (\tilde\nabla A_\theta\cdot n^{axis}) \cdot \phi \ d\Gamma}- \int_{\Gamma_N^{axis}}{ \frac{r}{\mu} (\tilde\nabla A_\theta\cdot n^{axis}) \cdot \phi \ d\Gamma} + \int_{\Omega^{axis}}{\frac{1}{\mu r} A_\theta \cdot \phi \ drdz} +\int_{\Omega^{axis}}{ \sigma r \frac{\partial A_\theta}{\partial t} \cdot \phi \ drdz} + \int_{\Omega^{axis}_c}{ \sigma \frac{\partial V}{\partial \theta} \cdot\phi \ rdrdz } = 0
}\]
We impose the boundary conditions :
So, the weak formulation is :
Weak formulation of A-V Formulation in axisymmetric coordinates
\[\scriptsize{
\text{(Weak AV Axis)} \\
\left\{ \begin{eqnarray*}
\int_{\Omega^{axis}}{ \frac{r}{\mu} \tilde\nabla A_\theta \cdot \tilde\nabla \phi \ drdz } + \int_{\Omega^{axis}}{\frac{1}{\mu r} A_\theta \cdot \phi \ drdz} +\int_{\Omega^{axis}}{ \sigma r \frac{\partial A_\theta}{\partial t} \cdot \phi \ drdz} + \int_{\Omega^{axis}_c}{ \sigma \frac{\partial V}{\partial \theta} \cdot\phi \ rdrdz } = 0 \\
\text{for } \phi \in H^1(\Omega)
\end{eqnarray*} \right.
}\]