2024-01-25-17-09-44

Activity/NAC/NAC.tex

@@ -221,7 +221,7 @@ The variations $ \innerH{\tangentR}{\deltafrac{\energyIA}{\Vb},\Wb} =: -\innerH{
 		&= \cIA\DivC\IdS
 		 = \cIA\GradC 1
 		 = \cIA\meanc\normal \formComma \\
-	\FbNV \label{eq:active_nematic_force}
+	\FbNA \label{eq:active_nematic_force}
 		&= \cNA\DivC\left( \IdS\Qb\IdS \right) \formPeriod
 \end{align}
 
@@ -232,7 +232,7 @@ The variations $ \innerH{\tangentR}{\deltafrac{\energyIA}{\Vb},\Wb} =: -\innerH{
 }
 
 The active flux potential does not depend on any Q-tensor rate, which could serves as a process variable associated to $ \Qb $.
-As a consequence no active molecular forces $ \HbAC\in\tangentQR $ occurs in the Q-tensor equation, \resp\ we stipulate $ \Hb_{\IA} = \Hb_{\NV} = 0 $ in compliance with the setup in \cite{Nitschke2023}.
+As a consequence no active molecular forces $ \HbAC\in\tangentQR $ occurs in the Q-tensor equation, \resp\ we stipulate $ \Hb_{\IA} = \Hb_{\NA} = \nullb $ in compliance with the setup in \cite{Nitschke2023}.
 
 \subsubsection{Surface Conforming Q-Tensor}
 
@@ -246,12 +246,12 @@ and a tangential vector-valued surface non-conforming field $ \etab\in\tangentS
 see \cite{NitschkeVoigt_2023}.
 Substituting this decomposition into the active nematic force field  \eqref{eq:active_nematic_force} yields
 \begin{align*}
-	\FbNV
+	\FbNA
 		&= \cNA\left( \DivC\qb - \frac{1}{2}\GradC\beta \right)
 		 = \cNA\left(\div\qb - \frac{1}{2}\nabla\beta + \left( \qb\dbdot\shop - \frac{\meanc}{2}\beta \right)\normal\right) \formComma
 \end{align*}
 which does not depend on any surface non-conforming parts of the Q-tensor field.
-Therefore it holds $ \FbNV|_{\Qb\in\tangentQR} = \FbNV|_{\Qb\in\tangentCQR} $ \wrt\ the space of surface conforming Q-tensor fields
+Therefore it holds $ \FbNA|_{\Qb\in\tangentQR} = \FbNA|_{\Qb\in\tangentCQR} $ \wrt\ the space of surface conforming Q-tensor fields
 \begin{align*}
 	\tangentCQR 
 	&:= \left\{ \Qb \in \tangentQR \mid \exists\lambda\in\tangentS[^0] : \Qb\normal = \lambda\normal \right\} \formPeriod
@@ -260,6 +260,21 @@ Therefore it holds $ \FbNV|_{\Qb\in\tangentQR} = \FbNV|_{\Qb\in\tangentCQR} $ \w
 
 \subsection{Active Beris-Edwards Models}
 
+\begin{subequations}\label{eq:model_lagrange_multiplier}
+\begin{gather}
+    \rho\Dmat\Vb \label{eq:model_lagrange_multiplier_floweq}
+        =  \GradC\left(\pTH - p\right) 
+                    + \left(\fnorBE + \cIA\meanc\right)\normal
+                    + \DivC\left( \SigmabEL + \SigmabIM  + \SigmabNV + \cNA\IdS\Qb\IdS \right)
+                    + \sum_{\gamma\in\Cset} \Fb_{\gamma} \formComma\\
+    \widetilde{M} \Dt[\Phi]\Qb \label{eq:model_lagrange_multiplier_moleculareq}
+                 = \HbEL + \HbTH + \xi\HbNV^1 + \xi^2\widetilde{\Hb}^{2,\Phi}_{\NV} + \sum_{\gamma\in\Cset} \Hb_{\gamma} \formComma\\
+    0 = \DivC\Vb \formComma\\
+    \nullb = \Cb_{\gamma}, \quad \forall\gamma\in\Cset \label{eq:model_lagrange_multiplier_constraineq}
+\end{gather}
+\end{subequations}
+$ \SigmabNV = \SigmabNV^0 + \xi\SigmabNV^1 + \xi^2\SigmabNV^2 $
+
 \subsection{Energy Rate}\label{sec:energy_rate}
 
 %% The Appendices part is started with the command \appendix;