Use the <Cartesian streamfunction> convention $u=\psi_y$, $v=-\psi_x$. Taking the <curl> of the incompressible <Stokes equation> eliminates pressure and gives the <biharmonic stream function for planar Stokes flow> equation
$$
\boxed{\nabla^4\psi=0\quad\hbox{for }y>\epsilon\sin\theta,\qquad \theta=x-t.}
$$
The material points have no horizontal velocity in the swimming frame and have vertical velocity $-\epsilon\cos\theta$. The <no-slip boundary condition> is therefore
$$
\boxed{\psi_y(x,\epsilon\sin\theta,t)=0,\qquad \psi_x(x,\epsilon\sin\theta,t)=\epsilon\cos\theta.}
$$
At infinity the laboratory fluid is at rest, so in this translating frame it moves with $U\mathbf e_x$:
$$
\boxed{\psi_y\to U,\qquad \psi_x\to0\quad(y\to\infty).}
$$
Velocity and its perturbations are periodic in $x$, with period $2\pi$, and the pressure has no imposed mean gradient. The <Taylor swimming sheet> is <force-free>; the unbounded problem's bounded far-field velocity enforces the absence of mean shear. An additive constant in the streamfunction has no physical effect.

Solution

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

{c}

= Taylor's swimming sheet
{c}
{synonym}

The Taylor swimming sheet is an infinite two-dimensional <microswimmer> whose prescribed traveling deformation drives <Stokes flow>. A transverse wave $y_s=\epsilon\sin(x-t)$ of small amplitude swims opposite to its direction of propagation, with speed $U=\epsilon^2/2+O(\epsilon^4)$ when lengths are scaled by inverse wavenumber and velocities by wave speed. Its <biharmonic stream function for planar Stokes flow> has first-order part $\psi_1=(1+y)e^{-y}\sin(x-t)$.

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