= Solution
{c}
Given an admissible <velocity field>, recover the <pressure gradient> from the <Stokes equation>, $\nabla p=\mu\nabla^2\mathbf v$, and integrate it, fixing the additive constant by matching to the bath <pressure> near the open ends. The <biharmonic equation> ensures the compatibility of this integration. For a freely moving hinge, the laboratory <velocity field> also contains the unknown rigid translation and rotation. Solve the interior and exterior <Stokes flows> with these wall <velocities>, then impose zero total external <force> and <torque> on the complete apparatus if its motor supplies only internal actuation. The pinned solution alone cannot determine that free motion.
At a fixed angle, the actuation <velocity> scale is $U\sim L|\dot\alpha|$. The <Stokes equation> gives $\Delta p\sim\mu U/L\sim\mu|\dot\alpha|$. Integrating normal <traction> over a plate of <length> $L$ gives a <force> per unit width of scale
$$
\boxed{F=\mu L\dot\alpha\,C(\alpha),}
$$
with a dimensionless signed coefficient depending on geometry and exterior matching. Normal <viscous stress> has the same dimensional scale and is included in $C$. The <force> reverses when the actuation reverses. A narrow gap can make $|C|$ large.
More specifically, the corrected local <streamfunction> gives, up to a reference <pressure>,
$$
p=\mu\dot\alpha\,K(\alpha)\log(r/r_*)+\text{constant},\qquad
K(\alpha)=\frac{2\cos\alpha}{\sin\alpha-\alpha\cos\alpha}.
$$
Indeed, inserting $\mathbf v=(-\dot\alpha rG'/2,\dot\alpha rG)$ into the vector <Laplacian> gives $p_r=-\mu\dot\alpha(G'''+4G')/(2r)=\mu\dot\alpha K/r$ and $p_\theta=0$. For $\alpha<\pi/2$, opening produces reduced <pressure> toward the hinge when matched at $r_*\sim L$. The logarithmic singularity is integrable: $\int_0^L\log(L/r)\,dr=L$, consistent with the stated <force> scale. A real hinge regularizes the singularity; the local formula is not a full end-matched solution.
The <scallop theorem> applies to a single periodically actuated hinge in a stationary <Newtonian fluid> with negligible fluid and body <inertia>. At each shape, <Linearity of Stokes flow> makes its rigid <velocity> proportional to $\dot\alpha$. Closing retraces the sequence of shapes and reverses the corresponding motion, irrespective of different opening and closing rates. The hinge may move during each half-cycle, but
$$
\boxed{\text{net rigid displacement and rotation over a reciprocal cycle}=0.}
$$
A motor can generate instantaneous <forces> and consume <energy> through <viscous dissipation> without producing sustained swimming. Additional independently controlled shape changes, appreciable <inertia>, or a fluid with memory would require a different analysis.
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