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Commit 4d93c2bd authored by Nitschke, Ingo's avatar Nitschke, Ingo
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2024-01-24-16-56-14

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Activity/NAC/NAC.pdf LFS
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Activity/NAC/NAC.tex
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......@@ -169,7 +169,8 @@ For this purpose we define the flux potential
&:= \frac{\cAC}{2}\int_{\surf} \Tr(\Dlow\IbxiQ)\dS \label{eq:fluxAC_origin}
\end{align}
where $ \cAC\in\R $.
Since the Jaumann derivative is $ \R^3 $-metric compatible, \ie\ $ \Djau\Id=\nullb $, it holds $ \Dlow\IbxiQ = -\xi\Djau\Qb + \Sb[\Vb]\IbxiQ + \IbxiQ\Sb[\Vb] $ by \eqref{eq:Dlow_rel_to_Djau}.
Since the Jaumann derivative is $ \R^3 $-metric compatible, \ie\ $ \Djau\Id=\nullb $, it holds
$ \Dlow\IbxiQ = -\xi\Djau\Qb + \Sb[\Vb]\IbxiQ + \IbxiQ\Sb[\Vb] $ by \eqref{eq:Dlow_rel_to_Djau}. %MProved
The space of Q-tensor fields $ \tangentQR $ is closed by the Jaumann derivative \cite{NitschkeVoigt_2023}, \ie\ $ \Tr(\Djau\Qb) = 0 $ particularly.
Therefore the flux potential \eqref{eq:fluxAC_origin} results in
\begin{align}
......@@ -177,7 +178,7 @@ Therefore the flux potential \eqref{eq:fluxAC_origin} results in
&= \cAC \innerH{\tangentSymR}{\IbxiQ,\Sb[\Vb]}
= \cAC \innerH{\tangentR[^2]}{\IdS - \xi \IdS\Qb,\nablaC\Vb} \formPeriod
\label{eq:fluxAC}
\end{align}
\end{align}%MProved
Contrarily to the genuine flux potentials in \cite{Nitschke2023}, this potential is neither quadratic, nor generally positive at all.
Moreover it transfers one-to-one into energy exchange with an sink/source, \eg\ the surrounded environment, as we see in Section \ref{sec:energy_rate}.
Therefore, \eqref{eq:fluxAC} does not stipulate a passive mechanism, hence we call it the active flux potential.
......@@ -205,7 +206,7 @@ For a better classification we decompose \eqref{eq:fluxAC} into an isotropic and
\end{align}
with $ \cIA=\cAC $, $ \cNA = -\xi\cAC $, and $\energyAC = \energyIA + \energyNA$.
The active isotropic flux potential $ \energyIA $ solely depends on the surface extensibility $ \DivC\Vb $,
and the active nematic flux potential $ \energyNA $ on the deviatoric stress $ \projQR\Sb[\Vb] = \frac{1}{2}(\nablaC\Vb + \nablaC^T\Vb) - \frac{\DivC\Vb}{3}\Id $ in direction of $ \Qb $.
and the active nematic flux potential $ \energyNA $ on the deviatoric stress $ \projQR\Sb[\Vb] = \Sb[\Vb] - \frac{\DivC\Vb}{3}\Id $ in direction of $ \Qb $.
Hence the active isotropic flux potential vanishes for inextensible fluids.
However, it cannot be neglected if one ensure $ \DivC\Vb=0 $ by the Lagrange-multiplier method, as we did in \cite{Nitschke2023}.
Eventually, Section \ref{sec:active_forces} leads to an additional pressure-like force, which has not any impact on divergence-free solutions $ \Vb $.
......@@ -220,14 +221,41 @@ The variations $ \innerH{\tangentR}{\deltafrac{\energyIA}{\Vb},\Wb} =: -\innerH{
&= \cIA\DivC\IdS
= \cIA\GradC 1
= \cIA\meanc\normal \formComma \\
\FbNV
\FbNV \label{eq:active_nematic_force}
&= \cNA\DivC\left( \IdS\Qb\IdS \right) \formPeriod
\end{align}
\rednote{
\begin{itemize}
\item The isotropic force field, \resp\ its stress tensor field $ \cIA\IdS $, corresponds to the isotropic part of the ansatz in \cite{Al-IzziAlexander_PRR_2023}
\end{itemize}
}
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}.
\subsubsection{Surface Conforming Q-Tensor}
Any Q-tensor $ \Qb\in\tangentQR $ field is uniquely orthogonal decomposable into a tangential (flat degenerated) Q-tensor field $ \qb\in\tangentQS $,
a scalar-valued normal eigenvector field $ \beta\in\tangentS[^0] $,
and a tangential vector-valued surface non-conforming field $ \etab\in\tangentS $, by
\begin{align}
\Qb &= \qb + \etab\otimes\normal + \normal\otimes\etab + \beta\left( \normal\otimes\normal -\frac{1}{2}\IdS \right) \formComma
\label{eq:qtensor_decomposition}
\end{align}
see \cite{NitschkeVoigt_2023}.
Substituting this decomposition into the active nematic force field \eqref{eq:active_nematic_force} yields
\begin{align*}
\FbNV
&= \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
\begin{align*}
\tangentCQR
&:= \left\{ \Qb \in \tangentQR \mid \exists\lambda\in\tangentS[^0] : \Qb\normal = \lambda\normal \right\} \formPeriod
\end{align*}
\subsection{Active Beris-Edwards Models}
......
Activity/mathematica/nematicMetric.nb 0 → 100644
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