Serge-Étienne Parent edited section_Constitutive_relationships_Retardation_factor__.tex  over 8 years ago

Commit id: a09039ead84be8c63da06acd3215dfa267de51bd

deletions | additions      

       

Apply Gauss divergence theorem on the first part of the left-hand side.  \begin{align}  \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{align}  Because $s = 0$ on $\Gamma_{D}$.  \begin{align}  \int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega = \int_{\Gamma_{N}} s D  \nabla u \cdot n d\Gamma \end{align}  Put the pieces together.  \begin{align}  \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{align}