At second order the <Taylor-expanded no-slip boundary condition> is
$$
\psi_{2y}(0)=-\sin\theta\,\psi_{1yy}(0)=\sin^2\theta=\frac12(1-\cos2\theta),\qquad \psi_{2x}(0)=-\sin\theta\,\psi_{1xy}(0)=0,
$$
with $\nabla^4\psi_2=0$, $\psi_{2y}\to U_2$, and $\psi_{2x}\to0$ at infinity. Set the irrelevant boundary constant to zero. The general relevant solution contains a zero <Fourier mode> and a second harmonic:
$$
\psi_2=A+By+Cy^2+Dy^3+\operatorname{Re}\{[E e^{-2y}+F e^{2y}+y(G e^{-2y}+J e^{2y})]e^{2i\theta}\}.
$$
Bounded velocity eliminates $C,D$ and growing harmonics. Averaging the tangential boundary condition gives $B=\langle\sin^2\theta\rangle=1/2$. This is the <mean boundary velocity determines Taylor-sheet swimming speed> principle: the remaining mean velocity is uniform and equals the far-field velocity. Therefore
$$
\boxed{U_2=\frac12,\qquad \mathbf V_{\rm swim}=-\frac{\epsilon^2}{2}\mathbf e_x+O(\epsilon^4).}
$$
The absence of odd powers follows because changing the amplitude sign is just a half-wavelength translation. Although unnecessary for the speed, the oscillatory solution can also be written explicitly as $\psi_2=y/2-(y/2)e^{-2y}\cos2\theta$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 334 / 1 / c / Solution

{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.

Taylor-sheet swimming next to a rigid wall

A <Taylor expansion> of the <no-slip boundary condition> about the flat sheet makes the second-order mean tangential velocity $\langle u_2(0)\rangle=-\langle y_1\partial_yu_1(0)\rangle$. The mean mode of <Stokes flow> is linear in height when no pressure gradient is imposed. In an unbounded fluid, bounded velocity excludes mean shear; in a confined fluid, the <force-free> condition excludes it. The remaining mean velocity is uniform and equals the swimming speed in the sheet frame. Thus the leading speed can be found from the first-order field without solving the oscillatory second-order field.

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