Advection-Diffusiuon: weak form

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