 Small-slope elastohydrodynamic filament equation

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 $-B{y}_{xxxx}$, while the induced filament tension is higher order. Their instantaneous balance yields $c_{\perp }{y}_t=-B{y}_{xxxx}$. It is an overdamped fourth-order diffusion equation: short-wavelength bends relax much faster than long ones. Wiggins and Goldstein's flexive-propulsion paper develops the elastic-wave mechanism for low-Reynolds-number propulsion.