 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.