= Solution
The <force> scale is $F\sim\rho ga^3$. At the distant <free surface>, the Stokeslet's <pressure> and viscous normal-stress scales are $F/d^2$. Gravity restores a displaced surface by a <pressure> change $\rho gh$, so
$$
\boxed{h\sim\frac F{\rho gd^2}=O\!\left(\frac{a^3}{d^2}\right).}
$$
Thus $h/d=O((a/d)^3)\ll1$, consistent with treating the boundary as approximately flat and impermeable at leading order. This uses the stipulated gravity-dominated restoring balance.
The liquid-air interface has negligible tangential traction. The <stress-free image of a normal Stokeslet> places an opposite point <force> at the reflection point, a distance $2d$ above the bubble centre. Across the flat surface the combined normal <velocity> is odd and the tangential <velocity> is even, so the normal <velocity> and tangential stress vanish there. A same-radius inviscid drop of density $2\rho$ has downward excess weight $4\pi\rho ga^3/3$, exactly opposite the bubble's upward <buoyancy>. Its exterior Stokeslet therefore supplies this image, giving the asserted full-space comparison to leading order in $a/d$.
The image's incident axial <velocity> at the bubble is
$$
u_{\rm image}=-\frac{F}{4\pi\mu(2d)}=-\frac{F}{8\pi\mu d}.
$$
To first reflection, the bubble translates with this locally uniform incident <velocity> plus its isolated rise speed $F/(4\pi\mu a)$. Hence the <bubble rise near a distant stress-free free surface> is
$$
\boxed{U_b=\frac{F}{4\pi\mu a}-\frac{F}{8\pi\mu d}
=U\left(1-\frac a{2d}+o(a/d)\right).}
$$
Only the first relative correction is retained; the image's spatial variation and additional reflections affect higher terms.
The customary capillary-controlled small-deformation condition is a small <capillary number>,
$$
\boxed{\mathrm{Ca}=\frac{\mu U}{\gamma}=\frac{\rho ga^2}{3\gamma}\ll1,}
$$
equivalently a small <Bond number> $\mathrm{Bo}=\rho ga^2/\gamma$. This is a sufficient condition, not a necessary condition for the special isolated spherical solution in part (b). More precisely, the image's strain is $O(F/(\mu d^2))$, so its deformation-producing normal stress compared with $\gamma/a$ is $O(\mathrm{Bo}(a/d)^2)$. When exploiting the exact isolated solution, that smaller combination controls the leading boundary-induced distortion. If capillarity at the distant free surface is also retained, the gravity-dominated deflection scaling requires $\gamma/(\rho gd^2)\ll1$; $a\ll\sqrt{\gamma/(\rho g)}\ll d$ allows both the usual spherical-bubble criterion and gravity-dominated surface restoration.
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