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Commit e1dea6da authored by Nitschke, Ingo's avatar Nitschke, Ingo
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2024-01-25-17-09-44

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......@@ -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;
......
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