= Nonlinear resistive-force balance for a travelling filament
{title2=$\mathbf w=c\mathbf t-(V-U)\mathbf i$}
For an inextensible planar filament with tailward tangent $\mathbf t=(X_s,Y_s)$, a travelling shape has material velocity $(V-U)\mathbf i-c\mathbf t$ in stationary fluid. Thus the relative fluid velocity is the displayed vector. With $\mathbf n=(-Y_s,X_s)$ and $q=V-U>0$, a local inertial normal <drag> $K_N|w_n|w_n\mathbf n$ has axial component $-K_Nq^2|Y_s|^3$. A positive tangential slip $w_t=c-qX_s$ gives the opposite axial force. A prescribed tangential law $K_Tw_t^{3/2}h(s)\mathbf t$ therefore gives the <force-free> balance $K_Nq^2\int|Y_s|^3ds=K_T\int(c-qX_s)^{3/2}X_sh(s)ds$. The weight $h(s)$ is part of the constitutive approximation and must be justified separately from its speed exponent.
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