= Boundary expression for transverse filament thrust
The anisotropic-drag contribution to axial force from a small-slope transverse motion is $F=(c_\perp-c_\parallel)\int y_xy_t\,dx$. Using the <small-slope elastohydrodynamic filament equation> and <integration by parts> gives
$$
F=B\left(1-\frac{c_\parallel}{c_\perp}\right)\left[\frac12y_{xx}^2-y_xy_{xxx}\right]_0^L.
$$
Thus this propulsive contribution is determined by endpoint slope, bending moment, and shear. A separately imposed axial translation adds ordinary longitudinal drag. Exact second-order longitudinal motion required by <inextensibility> also contributes to instantaneous total drag, but its time-derivative contribution averages to zero over a deformation cycle.
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