Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 329 / 1 / c / Solution

When $\mathbf{B}=0$, part (b) gives $\mathbf{V}=2\mathbf{A}/3$ and $\mathbf{\omega }=0$. The total boundary velocity is thereforeThe decaying velocity potentialis harmonic for $r>a$, andhas exactly this value at $r=a$. It is a force-free potential-dipole field and decays as $r^{-3}$.

When $\mathbf{A}=0$, define the degree-three harmonic polynomialAn appropriate decaying harmonic potential isIndeed, the tangential boundary value is proportional towhich combines the prescribed slip with the rigid rotation found in part (b). Since $H_3/r^7=O(r^{-4})$ and multiplication of its gradient by $\mathbf{x}$ preserves that order, the exterior velocity decays as