Optimal pitch of a helical microswimmer ($\tan \varphi =\sqrt{2},U_{\max }=\omega b/(2\sqrt{2})$)

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 $\omega$.