AnodeConsumption

The two domains are separated by an interface $\tilde{\Gamma}(t) = \overline{\Sigma_{anode}(t)} \cap \overline{\Sigma_{bath}(t)}$, see Fig. 1.

The mathematical model has been derived from equation (1), assuming the magnetic potential and the magnetic field to be zero. Given the unknown interface $\tilde{\Gamma}(t)$ between the anode and the bath, we search for the electric potential $V: \Sigma=\Sigma_{anode} \cup \Sigma_{bath} \times (0, T] \to \mathbb{R}$ such that,

$\nabla \cdot (\sigma \nabla V) = 0$ in $\Sigma$,

$\sigma \nabla V \cdot \mathbf{n} = \frac{I}{|\partial \Sigma_{in}|}$ on $\partial \Sigma_{in}$,

$\sigma \nabla V \cdot \mathbf{n} = -\frac{I}{|\partial \Sigma_{out}|}$ on $\partial \Sigma_{out}$,

$\sigma \nabla V \cdot \mathbf{n} = 0$ on $\partial \Sigma \setminus (\partial \Sigma_{in} \cup \partial \Sigma_{out})$,

${V}_{\tilde{\Gamma}} = 0$ on $\tilde{\Gamma}(t)$,

${\sigma \nabla V \cdot \mathbf{n}}_{\tilde{\Gamma}} = 0$ on $\tilde{\Gamma}(t)$.

The conductivity $\sigma$ has a constant value in the two media $\sigma_{anode}$ and $\sigma_{bath}$. The interface can be parameterized with the level set function $\tilde{\varphi}$ as $\tilde{\Gamma}(t) = {\mathbf{x} \in \Sigma : \tilde{\varphi}(\mathbf{x}, t) = 0}$. The conductivity is then given by:

$\sigma(\tilde{\varphi}) = \sigma_{bath} + (\sigma_{anode} - \sigma_{bath}) H(\tilde{\varphi})$.

The interface moves at speed $k \ \sigma \nabla V$ ($k>0$ is a given coefficient), $\tilde{\varphi}$ satisfies the level-set equation,

$\frac{\partial \tilde{\varphi}}{\partial t} + k \sigma \nabla V \cdot \nabla \tilde{\varphi} = 0$ in $\ \Sigma \times (0,T]$,

$\tilde{\varphi}(x, 0) = \tilde{\varphi}_0(x)$ in $\Sigma$,

$\tilde{\varphi}(x, t) = \tilde{\varphi}D$ on $\partial \Sigma{out} \times (0,T]$.

where $\tilde{\varphi}_D$ is a positive constant.