= Bubble rise near a distant stress-free free surface
{title2=$U_b/U_\infty=1-a/(2d)+o(a/d)$}
For a clean inviscid bubble a distance $d\gg a$ below a gravity-restored approximately flat <free surface>, the surface displacement scales as $F/(\rho gd^2)=O(a^3/d^2)$. Zero normal <velocity> and zero tangential traction are imposed to leading order by an opposite normal image <Stokeslet> at distance $2d$ from the bubble. Its incident <velocity> is $-F/(8\pi\mu d)$, which gives the displayed correction at fixed <buoyancy>. The image can be represented by a same-radius inviscid drop of density $2\rho$ in liquid of density $\rho$. Its deformation-producing strain is $O(F/(\mu d^2))$, so the leading fractional capillary deformation is $O(\mathrm{Bo}(a/d)^2)$; small <Bond number> is the conventional stronger sufficient criterion.
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