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......@@ -34,4 +34,30 @@
publisher = {American Physical Society (APS)},
}
@Article{NitschkeVoigt_JoGaP_2022,
author = {Ingo Nitschke and Axel Voigt},
title = {Observer-invariant time derivatives on moving surfaces},
journal = {Journal of Geometry and Physics},
year = {2022},
volume = {173},
pages = {104428},
doi = {10.1016/j.geomphys.2021.104428},
publisher = {Elsevier {BV}},
}
@Misc{NitschkeSadikVoigt_A_2022,
author = {Nitschke, Ingo and Sadik, Souhayl and Voigt, Axel},
title = {Tangential Tensor Fields on Deformable Surfaces -- {H}ow to Derive Consistent {$L^2$}-Gradient Flows},
year = {2022},
doi = {10.48550/arXiv.2209.13272},
publisher = {arXiv},
}
@Article{BachiniKrauseNitschkeVoigt_2023,
author = {Bachini, Elena and Krause, Veit and Nitschke, Ingo and Voigt, Axel},
title = {Derivation and simulation of a two-phase fluid deformable surface model},
journal = {Journal of Fluid Mechanics},
year = {2023},
}
@Comment{jabref-meta: databaseType:bibtex;}
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......@@ -165,36 +165,75 @@ Stuff have been done.
\subsection{Notation and Mathematical Preliminaries}
\begin{itemize}
\item $ \Dmat $, $ \Dlow $, $ \Dupp $, $ \Djau $ (ref. \cite{NitschkeVoigt_2023})
\begin{align}
\Dlow\Rb &= \Djau\Rb + \Sb[\Vb]\Rb + \Rb\Sb[\Vb] \label{eq:Dlow_rel_to_Djau}
\end{align}
\item surface deformation gradient:
\begin{align}
\Gbcal[\Wb] \label{eq:Gbcal}
&:= \nablaC\Wb - \normal\nablaC\Wb\otimes\normal
= \Gb[\Wb] + \normal\otimes\left( \nabla\wnor + \shop\wb \right) - \left( \nabla\wnor + \shop\wb \right)\otimes\normal \\
\Gb[\Wb] \label{eq:Gb}
&:= \IdS\nablaC\Wb
= \nabla\wb - \wnor\shop
\end{align}
for the orthogonal decomposition $ \Wb=\wb+\wnor\normal\in\tangentR $, with $ \wb\in\tangentS $ and $ \wnor\in\tangentS[^0] $.
Its symmetric and skew-symmetric part is
\begin{align}
\Sb[\Wb] \label{eq:Sb}
&:= \frac{1}{2}\left( \Gbcal[\Wb] + \Gbcal^T[\Wb] \right)
= \frac{1}{2}\left( \Gb[\Wb] + \Gb^T[\Wb] \right)
= \frac{1}{2}\left( \nabla\wb + (\nabla\wb)^T \right) - \wnor\shop \formComma \\
\Abcal[\Wb] \label{eq:Abcal}
&:= \frac{1}{2}\left(\Gbcal[\Wb] - \Gbcal^{T}[\Wb]\right)
= \Ab[\Wb] + \normal\otimes\left( \nabla\wnor + \shop\wb \right) - \left( \nabla\wnor + \shop\wb \right)\otimes\normal \formComma \\
\Ab[\Wb] \label{eq:Ab}
&:= \frac{1}{2}\left(\Gb[\Wb] - \Gb^{T}[\Wb]\right)
= \frac{1}{2}\left( \nabla\wb - (\nabla\wb)^T \right)
= -\frac{\rot\wb}{2}\Eb
\end{align}
\end{itemize}
Since we strictly adhere to the notation and preliminaries in \cite{Nitschke2023}, in this section we provide only a for this paper necessary essential introduction.
For more details, we refer to \cite{Nitschke2023} and the literature based on it.
We assume a sufficiently smooth parameterizable moving surface $ \surf\subset\R^3 $ in space and time.
Building on this, we consider Euclidean-based $ n $-tensor fields in $ \tangentR[^n] $.
Important subtensor fields are tangential n-tensor fields in $ \tangentS[^n] < \tangentR[^n] $ and (biaxial) Q-tensor fields in $ \tangentQR < \tangentR[^2] $.
The latter space in turn comprises surface-conforming Q-tensor fields in $ \tangentCQR < \tangentQR $ and tangential (flat-degenerated) Q-tensor fields in $ \tangentQS < \tangentCQR $.
More constructive:
$ \tangentS[^n]= \{ \Rb\in\tangentR[^n] \mid \forall\sigma\in S_n : \normal\Rb^{T_{\sigma}} = \nullb \} $ for the set of n-permutations $ S_n $ and normal field $ \normal\bot\surf $;
$ \tangentQR = \{ \Rb\in\tangentR[^2] \mid \Rb^T = \Rb \text{ and } \Tr\Rb = 0 \} $;
$ \tangentCQR = \{ \Qb \in \tangentQR \mid \exists\lambda\in\tangentS[^0] : \Qb\normal = \lambda\normal \} $;
$ \tangentQS = \{ \rb\in\tangentS[^2] \mid \rb^T = \rb \text{ and } \Tr\rb = 0 \} = \{ \Qb\in\tangentCQR \mid \Qb\normal=\nullb \} $.
On tangential tensor fields we use the covariant derivative $ \nabla: \tangentS[^n] \rightarrow \tangentS[^{n+1}] $ and its common derived differential operators
like the covariant divergence $ \div=\Tr\circ\nabla=-\nabla^*: \tangentS[^n] \rightarrow \tangentS[^{n-1}] $.
On more general $ n $-tensor fields we use the componentwise surface derivative $ \nablaC: \tangentR[^n] \rightarrow \tangentR[^n]\otimes\tangentS $,
which is basically the scalar-valued covariant derivative on its Cartesian proxy component fields, see \cite{NitschkeSadikVoigt_A_2022,NitschkeVoigt_2023,BachiniKrauseNitschkeVoigt_2023, Nitschke2023}.
The componentwise trace-divergence is $ \DivC = \Tr\circ\nablaC: \tangentR[^n] \rightarrow \tangentR[^{n-1}] $.
Note that only on right-sided tangential $ n $-tensor fields $ \tangentR[^{n-1}]\otimes\tangentS $ holds the $ L^2 $-adjoint relation $ \DivC=-\nablaC^* $.
In our models this is always the case for stress tensor fields.
On the other hand we could define the adjoint componentwise gradient $ \GradC:=-\DivC^* $.
We use this operator solely for scalar fields $ f\in\tangentS[^0] $, where $\GradC f = \DivC (f\IdS) = \nabla f + \meanc f \normal$ holds, with mean curvature $ \meanc = \Tr\shop $,
(tensor-valued) second fundamental form\footnote{Aka (extended) Weingarten map or shape operator, depending on the context.} $ \shop = -\nablaC\normal \in\tangentS[^2] $,
and surface identity tensor $ \IdS\in\tangentS[^2] $, \ie\ it is $ \IdS\Wb $ the tangential part of the vector field $ \Wb\in\tangentR $.
Based on the derivative $ \nablaC $ on vector fields $ \Wb=\wb + \wnor\normal\in\tangentS\oplus(\tangentS[^0])\normal=\tangentR $, we introduce a few recurring quantities, which are
the surface
deformation\footnote{The origin of the naming arises from considering small surface deformations ``$ \surf + \eps\Wb $''.}
gradient and its tangential part
\begin{align}
\Gbcal[\Wb] \label{eq:Gbcal}
&:= \nablaC\Wb - \normal\nablaC\Wb\otimes\normal
= \Gb[\Wb] + \normal\otimes\left( \nabla\wnor + \shop\wb \right) - \left( \nabla\wnor + \shop\wb \right)\otimes\normal \formComma \\
\Gb[\Wb] \label{eq:Gb}
&:= \IdS\nablaC\Wb
= \nabla\wb - \wnor\shop \formPeriod
\end{align}
Their symmetric and skew-symmetric parts are
\begin{align}
\Sb[\Wb] \label{eq:Sb}
&:= \frac{1}{2}\left( \Gbcal[\Wb] + \Gbcal^T[\Wb] \right)
= \frac{1}{2}\left( \Gb[\Wb] + \Gb^T[\Wb] \right)
= \frac{1}{2}\left( \nabla\wb + (\nabla\wb)^T \right) - \wnor\shop \formComma \\
\Abcal[\Wb] \label{eq:Abcal}
&:= \frac{1}{2}\left(\Gbcal[\Wb] - \Gbcal^{T}[\Wb]\right)
= \Ab[\Wb] + \normal\otimes\left( \nabla\wnor + \shop\wb \right) - \left( \nabla\wnor + \shop\wb \right)\otimes\normal \formComma \\
\Ab[\Wb] \label{eq:Ab}
&:= \frac{1}{2}\left(\Gb[\Wb] - \Gb^{T}[\Wb]\right)
= \frac{1}{2}\left( \nabla\wb - (\nabla\wb)^T \right)
= -\frac{\rot\wb}{2}\Eb \formComma
\end{align}
where $ \Eb\in\tangentS[^2] $ is the Levi-Civita tensor, \ie\ $ -\Eb\wb= *\wb $ gives the tangential Hodge-dual of $ \wb $,
and $\rot\wb = - \Eb\dbdot\nabla\wb $ the curl of $ \wb $.
The kinematic of $ \surf $ can be characterized by the observer velocity $ \Vb_{\!\ofrak}\in\tangentR $ \wrt\ any valid surface observer, see \cite{NitschkeVoigt_JoGaP_2022}.
Within a spatial discretization this observer velocity could serve as the grid velocity for instance.
However, from a physical point of view we are only interested in the material velocity $ \Vb\in\tangentR $, which determines the motion of the material first orderly.
The only mandatory relation between observer and material velocity is $ \Vb_{\!\ofrak}\normal = \Vb\normal =: \vnor $, \ie\ the tangential part $ \vb_{\!\ofrak} = \IdS\Vb_{\!\ofrak} $ of the observer velocity is arbitrary.
Simple choices could be $\Vb_{\!\ofrak}=\Vb$ (material/Lagrangian perspective) or $ \vb_{\!\ofrak} = \nullb $ (transversal/tangential-Eulerian perspective).
For us the observer velocity is only import to determine local observer-invariant tensor rates sufficiently,
\eg\ $ \Dmat\Vb = \partial_t\Vb + (\nablaC\Vb)(\Vb-\Vb_{\!\ofrak}) $ is the material acceleration and
$ \Dmat\Rb = \partial_t\Rb + (\nablaC\Rb)(\Vb-\Vb_{\!\ofrak}) $ the material tensor rate of $ n $-tensor fields $ \Rb\in\tangentR[^2] $.
Other tensor rates can be derived from the material rate, \eg\
$ \Djau\Rb = \Dmat\Rb - \Abcal[\Vb]\Rb + \Rb\Abcal[\Vb] $
is the Jaumann/corotational tensor rate and
\begin{align}
\Dlow\Rb
&= \Dmat\Rb + \Gbcal^T[\Vb]\Rb + \Rb\Gbcal[\Vb]
= \Djau\Rb + \Sb[\Vb]\Rb + \Rb\Sb[\Vb] \label{eq:Dlow_rel_to_Djau}
\end{align}
the lower-convected rate, see \cite{NitschkeVoigt_2023} for more observer-invariant (sub-)tensor rates, relations between them and their orthogonal decompositions.
\subsection{Energetic Contributions}
......@@ -439,6 +478,30 @@ Beside the transition to the tangential calculus, we only determined the surface
\subsection{Energy Rate}\label{sec:energy_rate}
Following \cite{Nitschke2023}, the influence of an energy flux potential $ \fluxpotential_{\alpha} $ on the total energy rate is entirely determined by its acting generalized material forces
$ \Fb_{\alpha}\in\tangentR $ and $ \Hb_{\alpha}\in\tangentQR $, and the process variables $ \Vb\in\tangentR $ and $ \Dmat\Qb\in\tangentQR $,
through the quantity $ \dot{\energy}_{\alpha}:= \innerH{\tangentR}{\Fb_{\alpha}, \Vb} + \innerH{\tangentQR}{\Hb_{\alpha}, \Dmat\Qb} $.
In absence of a Q-tensor force $ \HbAC $ for activity, the fluid forces \eqref{eq:active_geometric_force} and \eqref{eq:active_nematic_force} yield
\begin{align*}
\dot{\energy}_{\AC}
&= \innerH{\tangentR}{\FbIA + \FbNA, \Vb}
= -\innerH{\tangentR[^2]}{\cIA\IdS + \cNA\IdS\Qb, \nablaC\Vb}
= - \energyAC \formPeriod
\end{align*}
As a consequence the rate of the total system energy $ \energyTOT = \energyK + \energyEL + \energyTH + \energyBE $,
comprising the kinetic and all considered potential energies,
is given by
\begin{align}\label{eq:energy_rate}
\ddt\energyTOT
&= -2\left( \energyIM + \energyNV \right) - \energyAC \formPeriod
\end{align}
Since the active energy flux $ \energyAC $ \eqref{eq:fluxAC_origin}/\eqref{eq:fluxAC} is not bounded from below,
active models following \eqref{eq:model_lagrange_multiplier} do not have to be dissipative.
We can only ensure that $ \ddt\energyTOT + \energyAC \le 0 $.
Note that \eqref{eq:energy_rate} holds in general without incorporating any constraints, including that for \eqref{eq:model_lagrange_multiplier} mandatory inextensibility $ \DivC\Vb=0 $.
However, inextensibility does not have any influence on the solutions of the active models within $ \tangentR\vert_{\DivC\Vb=0} $, and hence on the energy dynamics.
Therefore it is $ \energyAC = \energyNA $ \eqref{eq:active_nematic_force} valid for solutions of \eqref{eq:model_lagrange_multiplier} and the geometric activity can be neglected \wrt\ the energy rate.
%% The Appendices part is started with the command \appendix;
......
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