= 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 $V=\int_0^L Y_tds/[L(1+\delta)]$. The mean axial swimming speed is $-(1-\gamma)\langle\int_0^L Y_s(Y_t-V)ds\rangle/[L(\delta+\gamma)]$. 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.
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