Fixed-speed energy optimum for a planar flagellum (${\beta }_{\ast }=B/(B+\sqrt{B^2+AB})$)

In resistive-force theory with $0<\gamma <1$, head drag $\delta \ge 0$ and the ideal constant-inclination condition ${\alpha }^2=\beta$, put $r=1-\gamma$, $d=1+\delta$, $A=r({\delta }^2-\gamma )$ and $B=\gamma d^2$. At prescribed swimming speed $U$, eliminating wave speed using linear resistive-force propulsion of a travelling filament givesIts derivative has numerator $A{\beta }^2+2B\beta -B$. This numerator is strictly increasing on $(0,1)$, is negative at zero, and equals $(\delta +\gamma {)}^2$ at one. Thus its unique zero is the displayed global minimum. When $A=0$ the formula gives ${\beta }_{\ast }=1/2$; when $\delta =0$ it gives $1/(1+\sqrt{\gamma })$. A planar sawtooth realizes constant inclination in the ideal model; smooth rounding and elastic costs change that idealization.