Head-drag correction to planar flagellar propulsion

For a small-slope inextensible filament with prescribed transverse coordinate $Y(s,t)$, tangential-to-normal resistance ratio $\gamma$, and head drag parameter $\delta$, transverse force balance gives $V={\int }_0^L{Y}_{t}ds/[L(1+\delta )]$. The mean axial swimming speed is $-(1-\gamma )⟨{\int }_0^L{Y}_s(Y_t-V)ds⟩/[L(\delta +\gamma )]$. The mean material longitudinal deformation velocity vanishes for a periodic inextensible stroke anchored longitudinally at its base. A moving base offset must be distinguished from a coordinate measured relative to the actual attachment point.