Similarity solution for tangentially forced Stokes wedge
= Similarity solution for tangentially forced Stokes wedge
For $\psi=r^2f(\theta)$, biharmonicity gives
$f^{(4)}+4f''=0$. In the wedge $-\alpha<\theta<0$, no slip at the lower wall and no penetration plus tangential stress $S$ at the upper surface give
$$
f(-\alpha)=f'(-\alpha)=f(0)=0,
\qquad \mu f''(0)=S.
$$
The resulting surface speed is
$$
U(r)=\frac{Sr}{\mu}
\frac{1-\cos2\alpha-\alpha\sin2\alpha}
{\sin2\alpha-2\alpha\cos2\alpha}.
$$