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Only the axisymmetric and $\cos 2\theta$ modes are needed. The exterior biharmonic forms compatible with the far-field velocity areTerms $r^2\log r$ and $r^4\cos 2\theta$ are excluded because they would dominate the imposed shear.

The two conditions at $r=a$ applied separately to each Fourier mode giveHenceDirect differentiation gives $\psi ={\psi }_r=0$ at $r=a$. At infinity the remaining disturbance velocity is $O(r^{-1})$, so the imposed shear is recovered. This is the Biharmonic stream function for a fixed disk in planar shear.