Biharmonic stream function for a fixed disk in planar shear
= Biharmonic stream function for a fixed disk in planar shear
{c}
For a stationary disk of radius $a$ in the far-field shear $u_\infty=\gamma y e_x$, using $u_r=\psi_\theta/r$ and $u_\theta=-\psi_r$, the exterior solution is
$$
\psi=\frac\gamma4\left[
r^2-a^2-2a^2\log\frac ra
-\left(r^2-2a^2+\frac{a^4}{r^2}\right)\cos2\theta
\right].
$$
It satisfies no slip at $r=a$ and approaches the imposed shear in velocity.