= Linear resistive-force propulsion of a travelling filament
{title2=$U/V=(1-\gamma)(1-\beta)/(\delta+1-(1-\gamma)\beta)$}
For a periodic <inextensible filament>, let $q=\mathbf t\cdot\mathbf i$, $\alpha=\langle q\rangle=V/c$, $\beta=\langle q^2\rangle$ and $\gamma=\kappa_T/\kappa_N$. The material velocity is $(V-U)\mathbf i-c\mathbf t$. Applying the linear local drag tensor $\kappa_T\mathbf t\mathbf t+\kappa_N(I-\mathbf t\mathbf t)$ gives thrust $T=\kappa_NL\{V(1-\gamma)(1-\beta)-U[1-(1-\gamma)\beta]\}$. Equating this to head drag $\delta\kappa_NLU$ gives the displayed speed ratio. Whole spatial periods, or period averages, are needed for $c\int q\,ds=VL$. <Cauchy-Schwarz> gives $\alpha^2\leq\beta$, with equality only for constant axial tangent projection almost everywhere.
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