Helical microswimmer with a spherical head

For a flagellum with axial resistance matrix of a slender helix and a head with resistances $A_0=6\pi \mu a$, $D_0=8\pi \mu a^3$, neglect interactions between the parts. A motor with relative angular velocity $\omega$ givesThe head rotates at $\Omega -\omega$. A large head creates excessive translational resistance; a small head supplies too little rotational resistance, allowing the head to counterrotate while the flagellum barely moves relative to the fluid.

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.

When head translation is weakly resisted compared with the flagellum but head rotation is strongly resisted, $A_0\ll A$ and $D_0\gg D$. Then $U\simeq -B\omega /A=\omega b\sin \varphi \cos \varphi /(1+{\cos }^2\varphi )$. Maximizing this expression gives the displayed pitch and speed for the chosen handedness and positive $\omega$.