 Solution

On the upper half-filament, write $\mathbf{r}(s)=(h(s),s)$ with $0\le s\le L$. To first order in slope, resistive-force theory supplies the uniform transverse load $f_x={\zeta }_{\perp }U$, and Euler--Bernoulli balance givesMidpoint symmetry and clamping impose $h(0)=h^{\prime }(0)=0$; the free-end force and moment conditions are $h^{\prime \prime \prime }(L)=h^{\prime \prime }(L)=0$. With $H=h/L$ and $\xi =s/L$,Four integrations giveIn particular, $H(1)=\mathrm{Sp}/8$ and $H^{\prime }(1)=\mathrm{Sp}/6$. The small-slope approximation first fails when the end slope is order one, giving the estimate $\mathrm{Sp}\sim 6$; parametrically, breakdown occurs at Sperm number of order one.