= Taylor-sheet swimming next to a rigid wall
{c}
For a transverse <Taylor swimming sheet> below a flat <no-slip boundary condition> at height $d$, the first-order amplitude $f(y)$ satisfies $(\partial_y^2-1)^2f=0$, $f(0)=1$, $f'(0)=f(d)=f'(d)=0$. Writing $\Delta=\sinh^2d-d^2$ gives
$$
f(y)=\cosh y+\frac{d+\sinh d\cosh d}{\Delta}(y\cosh y-\sinh y)-\frac{\sinh^2d}{\Delta}y\sinh y.
$$
The <mean boundary velocity determines Taylor-sheet swimming speed>, yielding
$$
\overline U_2=-\frac12f''(0)=\frac{\sinh^2d+d^2}{2(\sinh^2d-d^2)}.
$$
This is greater than the unbounded value $1/2$ for every $d>0$. In a narrow gap it scales as $3/d^2$, with the small-amplitude calculation requiring $\epsilon\ll d$ as well as $\epsilon\ll1$. https://arxiv.org/html/2410.02278v1[Taylor's swimming sheet near a soft boundary] recovers this rigid-wall limit while studying how compliance changes propulsion.
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