= Entrained-head approximation for a helical microswimmer
{title2=$U=\omega a^3|\log\epsilon|\sin(2\phi)/(bL)$}
Suppose a small spherical head co-translates with the large-scale flagellar flow and rotates at $\Omega-\omega$, while the local ambient rotation is approximated by $\Omega$. Its force is neglected and its torque on the fluid is $-8\pi\mu a^3\omega$. Balance this against the <axial resistance matrix of a slender helix>, with local coefficient $C=4\pi\mu/|\log\epsilon|$, to obtain the displayed swimming speed and motor power $8\pi\mu a^3\omega^2$. This is an entrainment approximation, distinct from treating the head as an isolated sphere in stationary fluid.
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