ROUGH DRAFT authorea.com/76048

# Constitutive relationships: Retardation factor due to adsorption

\begin{aligned} R(u) = 1 + \frac{\rho}{n} \times \frac{\partial S}{\partial u}\end{aligned}

## Freundlich isotherm:

\begin{aligned} S = k_p u^b \\ \frac{\partial S}{\partial u} = b k_p u^{b-1}\end{aligned}

## Fluid velocity field:

\begin{aligned} \vec{v} = \frac{\vec{v_e}}{R(u)}\end{aligned}

## Diffusivity:

\begin{aligned} D = \frac{D_l}{R(u)}\end{aligned}

Constants: $$b$$, $$k_p$$, $$D_l$$

Velocity field: $$\vec{v_e}$$

# Partial differential equation

Mass conservation law, assuming that $$\vec{v}$$ and $$D$$ are functions of $$u$$.

\begin{aligned} \frac{\partial u}{\partial t} = - \nabla \cdot \left( \vec{v} u \right) + \nabla \cdot \left( D \nabla u \right) \\\end{aligned}

Where $$u$$ is concentration and $$t$$ is time.

# Weak formulation

Integrate on volume $$\Omega$$ and multiply by the test function $$s$$.

\begin{aligned} \int_{\Omega}\frac{\partial u}{\partial t} s d\Omega + \int_{\Omega}\nabla \cdot \left(\vec{v} u \right) s d\Omega - \int_{\Omega} \nabla \cdot \left( D \nabla u \right) s d\Omega = 0\end{aligned}

Integrate by parts the last term of the left-had side.

\begin{aligned} \int_{\Omega} \nabla \cdot \left( D \nabla u \right) s d\Omega = \int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega - \int_{\Omega} D \nabla u \nabla s d\Omega \end{aligned}

Apply Gauss divergence theorem on the first part of the left-hand side.

\begin{aligned} \int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega = \int_{\Gamma_{N} \cup \Gamma_{D}} s D \nabla u \cdot n d\Gamma\end{aligned}

Because $$s = 0$$ on $$\Gamma_{D}$$.

\begin{aligned} \int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega = \int_{\Gamma_{N}} s D \nabla u \cdot n d\Gamma\end{aligned}

Put the pieces together. \begin{aligned} \int_{\Omega}\frac{\partial u}{\partial t} s d\Omega + \int_{\Omega}\nabla \cdot \left(\vec{v} u \right) s d\Omega - \int_{\Gamma_{N}} s D \nabla u \cdot n d\Gamma + \int_{\Omega} D \nabla u \nabla s d\Omega = 0\end{aligned}