= Solution
In the exterior domain $r>a$, the streamfunction obeys
$$
\boxed{\nabla^4\psi=0}.
$$
Using the <Streamfunction in polar coordinates> convention
$$
u_r=\frac1r\psi_\theta,
\qquad
u_\theta=-\psi_r,
$$
the no-slip condition on the stationary disk is
$$
u_r(a,\theta)=u_\theta(a,\theta)=0.
$$
Thus $\psi$ is constant on the disk and $\psi_r=0$ there. Choose the arbitrary constant to be zero:
$$
\boxed{\psi(a,\theta)=0,
\qquad
\psi_r(a,\theta)=0}.
$$
The imposed shear $\mathbf u_\infty=\gamma y\,\mathbf e_x$ has streamfunction
$$
\psi_\infty=\frac\gamma2y^2
=\frac{\gamma r^2}{4}(1-\cos2\theta).
$$
The far-field condition is therefore
$$
\boxed{\nabla\psi-\nabla\psi_\infty\longrightarrow0
\quad(r\to\infty)}.
$$
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