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 $\normal\bot\surf$;
$\tangentQR=\{\Rb\in\tangentR[^2]\mid\Rb^T =\Rb\text{ and }\Tr\Rb=0\}$;
$\tangentQS=\{\rb\in\tangentS[^2]\mid\rb^T =\rb\text{ and }\Tr\rb=0\}=\{\Qb\in\tangentCQR\mid\Qb\normal=\nullb\}$.
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 ``$\surf+\eps\Wb$''.}
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\
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}
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@@ -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
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 $\ddt\energyTOT+\energyAC\le0$.
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.
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