= Solution
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.
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