\subsection{Prescribed rotation on fixed geometry}\label{sec:rotation-numerics}
We also impose a nonzero prescribed velocity on the same circular interface.
The disk rotates with angular velocity $\omega(t)=a(t)=1-e^{-t}$ while its
geometric boundary remains fixed. Write $r(x)=(-Y,X)$ and
$\sigma=X^2+Y^2$. Define the compactly supported streamfunction lift
\begin{align*}
 \psi_g&=-\tfrac12\sigma\chi(\sigma),\qquad g=\curl\psi_g,\\
 \chi(\sigma)&=1-2772\int_0^{\eta(\sigma)}q^5(1-q)^5\,dq,\\
 \eta(\sigma)&=\min\!\left\{1,\max\!\left\{0,
 \frac{\sigma-0.35^2}{0.47^2-0.35^2}\right\}\right\}.
\end{align*}
The cutoff is $C^5$, $g=r$ for $|x-c|\le0.35$, and $g=0$ for
$|x-c|\ge0.47$. With the nonzero-shear field $U$ from
\eqref{eq:manufactured}, the exact solution and ordinary bulk force are
\begin{equation}\label{eq:rotation-manufactured}
 u(x,t)=a(t)(g+U),\qquad
 f=e^{-t}(g+U)-\nu(1-e^{-t})(\Delta g+\Delta U).
\end{equation}
Here $\Delta U$ is evaluated separately on each side of the interface, as
in \eqref{eq:manufactured-force}. In particular, $u=a(t)r$ and
$f=e^{-t}r$ inside the disk. The manufactured field is not only the
prescribed lift: its centered part $aU$ and its interfacial shear are nonzero.
At $T=0.5$, the $L^2(\Omega)$ norm of the continuous force beyond that
generated by $a g$ alone is $1.82845$.

Let $g_h=\curl I_h\psi_g$ use only the common coarse Hermite space.
For all tested meshes,
$R+\eps+\sqrt2h<0.35$; hence every coarse cell meeting a kernel
support lies in the rigid region. Polynomial reproduction gives
$g_h=r$ on those cells, and also throughout the disk. In particular,
every regularization support has the exact rigid lift.
The discrete centered unknown is
$w_h^n=u_h^n-a(t_n)g_h$. With $M_h,K_h$ denoting the mass matrix and
the matrix for $(\nabla v,\nabla z)$, and $b_v,b_{\Delta v}$ denoting
the corresponding continuous-field load vectors, its right-hand side
uses the force vector
\begin{equation}\label{eq:rotation-centered-load}
 d_n(b_U+b_g-M_hg_h)
 -\nu(1-d_n)(b_{\Delta U}+b_{\Delta g})
 -\nu a(t_n)K_hg_h.
\end{equation}
The last coefficient is the endpoint value $a(t_n)$, rather than its
interval average. Thus both the interpolation mismatch of $g_h$ and the
backward Euler lifting term are retained. The force is obtained from
\eqref{eq:rotation-manufactured}, not manufactured from the discrete
matrices. The same exact mass pairing implements transfer of the
centered unknown between the two enrichment patterns.

For this extension only, kernel subdivision contains each master-grid
cut and its reflection about the evaluation point. The resulting
symmetric nonnegative Gaussian weights are normalized to mass one in
each coordinate. This ensures affine moment reproduction without
negative correction weights. Diagnostics retain the mass defect before
normalization, the normalized mass and first-moment defects, and the
assembled discrepancy between $C_hg_h$ and the rigid boundary moments.
The archived static solver and its results are unchanged. Lift load
and norm integrals are split at the circles of radii $0.2,0.35,0.47$;
they use order 16 curved quadrature. These are still quadrature
approximations to the exact-integral method in the theory.

Table~\ref{tab:rotation-convergence} uses $M=16,32,64$, $N=400$, and
switches the enrichment pattern at every step. The definitions of
$E_0,E_1$ now use the total field in \eqref{eq:rotation-manufactured},
including the body interior. To distinguish boundary error from time
reconstruction error, we report both
\[
 D_{\Gamma,\mathrm{cont}}=
 \norm{\gamma_\Gamma U_h-a(t)r}_{L^2(0,T;L^2(\Gamma))},\qquad
 D_{\Gamma,\mathrm{node}}^2=
 \sum_n\tau_n\norm{\gamma_\Gamma u_h^n-a(t_n)r}_{L^2(\Gamma)}^2.
\]
The latter equals the time-integrated sharp trace of $w_h^n$.
The former is evaluated with analytic exponential time integrals,
including the cross term with $r$; it does not replace $a(t)$ by
$a(t_n)$. The finite-step temporal contribution remains when the mesh
is refined at fixed $N$.

\begin{table}[tbp]
\centering
\caption{Fixed-geometry prescribed rotation, $N=400$, with a mesh change
at every step. Total-field errors are percentages; both sharp-slip
norms are absolute.}
\label{tab:rotation-convergence}
\input{numerics/rotation/results/rotation_table.tex}
\end{table}

\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{numerics/rotation/figures/rotation_convergence.pdf}
\caption{Total-velocity errors and sharp-interface slip for prescribed
rotation. The two slip curves distinguish the continuous prescribed
motion from its endpoint reconstruction on each time interval.}
\label{fig:rotation-convergence}
\end{figure}

The coarse $M=16$ lift resolves the narrow cutoff transition poorly;
its velocity interpolation error is not suppressed in this test.
The final relative total-velocity error decreases from $16.4591\%$ to
$3.05830\%$ and $0.479175\%$. At $M=64$, the centered final relative
$L^2$ error against $a(T)U$ is $1.41233\%$. In comparison, the
continuous field $a(t)g$ alone would have a $12.4361\%$ relative
total-velocity error, independent of resolution. These diagnostics
confirm that the computation resolves a nontrivial fluid response
beyond prescribing the rigid motion.

At fixed $M=32$, the cases with $0,20,80,400$ mesh changes have final
relative $L^2$ errors between $3.0582811\%$ and $3.0582964\%$.
The centered projection losses are nonzero when the patch changes.
Increasing kernel, surface, original cut-load, and lift-load orders
from $(16,10,24,16)$ to $(24,16,32,24)$ at $M=16$ changes the final
solution by $5.61\times10^{-9}$ in relative $L^2$ and
$8.95\times10^{-9}$ in relative $H^1$.

\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{numerics/rotation/figures/rotation_switching_balance.pdf}
\caption{Left: switching frequency at fixed discretization parameters.
Right: absolute residuals of the centered energy identity and the
full reconstructed-velocity identity, including constraint work.
The latter work is an algebraic constraint functional, not a reported
hydrodynamic torque.}
\label{fig:rotation-switching}
\end{figure}

Across these eight runs, the largest absolute residual of the centered
energy identity is $1.94\times10^{-17}$, and that of the reconstructed
full-energy identity is $5.14\times10^{-16}$. The affine constraint
residual is at most $8.10\times10^{-15}$. Before kernel normalization,
the maximum one-dimensional mass defect is $2.06\times10^{-8}$;
after normalization the mass defect is at most $4.45\times10^{-16}$
and the first centered moment is below $10^{-18}$.
Independent checks differentiate the analytic lift,
verify its rigid and clamped regions, and reconstruct the sharp trace
from saved coefficients. This verifies a prescribed nonzero boundary
velocity on fixed geometry; it does not test interface translation or
fluid-driven rigid-body motion. As in the stationary experiment,
finite-resolution trends and quadrature sensitivity do not establish
a new asymptotic rate or a quadrature-consistency theorem.