Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 77 / 1 / d / Solution

The instantaneous work on the fluid equals the integral of $\mathbf{u}\cdot \mathbf{Du}$. For the oscillating rigid rod in resistive-force theory, $\mathbf{u}\cdot \mathbf{t}=\dot{x}\sin \theta$ and $\mathbf{u}\cdot \mathbf{n}=\dot{x}\cos \theta +s\dot{\theta }$. Odd terms in $s$ integrate to zero, givingThus the leading power of a rocking rod isBoth terms are quadratic in $ϵ$. Small angle does not make rotation a higher-order contribution: points away from the centre have rotational speed $s\dot{\theta }=O(ϵ\omega L)$, just as the translational speed is $O(ϵ\omega a)$. Their ratio is $L^2/(12a^2)$, controlled by geometry rather than by $ϵ$.