= Small-slope elastohydrodynamic filament equation
{title2=$c_\perp y_t=-By_{xxxx}$}
For a straight, unloaded <inextensible filament> with small transverse displacement, <resistive-force theory> gives viscous force density $-c_\perp y_t$. Variation of the bending energy gives $-By_{xxxx}$, while the induced <filament tension> is higher order. Their instantaneous balance yields $c_\perp y_t=-By_{xxxx}$. It is an overdamped fourth-order diffusion equation: short-wavelength bends relax much faster than long ones. https://arxiv.org/abs/cond-mat/9707346[Wiggins and Goldstein's flexive-propulsion paper] develops the elastic-wave mechanism for low-Reynolds-number propulsion.
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