= Dissipation estimate for high-Reynolds-number bubble drag
For a clean bubble of size $a$ and speed $U$ at large <Reynolds number>, the outer <potential flow> violates the stress-free condition by a strain of order $U/a$. A boundary layer of thickness $\delta\sim a\operatorname{Re}^{-1/2}$ needs a velocity correction only of order $U\delta/a$, so its dissipation is smaller than the outer-flow dissipation by order $\delta/a$. The leading drag power is therefore obtained from
$$
DU\sim 2\mu\int_{\rm outer}e:e\,dV.
$$
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