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

Let $a$ denote the bubble size, $U$ its speed, and $\mathrm{Re}=Ua/\nu \gg 1$ the Reynolds number. Outside a thin boundary layer, inertia dominates and the flow is an irrotational flow with velocity scale $U$ and strain scale $U/a$. Its viscous dissipation has orderin three dimensions.

The boundary-layer thickness isFor a clean bubble the stress-free boundary condition prescribes tangential stress rather than tangential velocity. The outer flow already has tangential strain of order $U/a$, so cancelling that stress requires a velocity correction of onlynot a correction of order $U$. Its gradient remains $O(U/a)$, and its boundary-layer dissipation iswhich is asymptotically smaller.

In steady translation, the drag power $DU$ balances viscous dissipation. The leading estimate therefore follows from the known outer flow alone:in three dimensions. This is the dissipation estimate for high-Reynolds-number bubble drag.