= 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 $\omega$ gives
$$
U=-\frac{BD_0\omega}{(A_0+A)(D_0+D)-B^2},\qquad
\Omega=\frac{(A_0+A)D_0\omega}{(A_0+A)(D_0+D)-B^2}.
$$
The 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.
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