Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 39C / c / iii / Solution

A rigid body obeys the no-slip boundary condition. Its outer potential flow generally has an $O(U)$ tangential slip velocity at the surface, so the boundary layer must make an $O(U)$ velocity correction across thickness $\delta$. Its strain is therefore $O(U/\delta )$, and in two dimensions its dissipation per unit axial length iswhich exceeds the $O(\mu U^2)$ outer dissipation by $a/\delta \sim {\mathrm{Re}}^{1/2}$.

The dominant drag consequently depends on the detailed boundary-layer velocity field and possibly on boundary-layer separation. It cannot be obtained by integrating the outer irrotational strain alone.

Solved by gpt-5.6-sol high.