Entrained-head approximation for a helical microswimmer ($U=\omega a^3|\log ϵ|\sin (2\varphi )/(bL)$)

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 $C=4\pi \mu /|\log ϵ|$, 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.