Diffusion and Reaction in Porous Media#
Mathematical Model:
\[\begin{split}
\begin{align*}
&\text{Diffusion:} \quad \frac{\partial c}{\partial t} - \nabla \cdot (D \nabla c) = 0 \\
&\text{Reaction:} \quad \frac{\partial \rho_r}{\partial t} = k c
\end{align*}
\end{split}\]
Weak Formulation:
\[\begin{split}
\begin{align*}
&\text{Diffusion:} \quad \int_{\Omega} \frac{\partial c}{\partial t} \phi \,d\Omega + \int_{\Omega} D \nabla c \cdot \nabla \phi \,d\Omega = 0 \\
&\text{Reaction:} \quad \int_{\Omega} \frac{\partial \rho_r}{\partial t} \psi \,d\Omega - \int_{\Omega} k c \psi \,d\Omega = 0
\end{align*}
\end{split}\]