Clean-bubble Stokes drag ($\mathbf{F}=-4\pi \mu a\mathbf{U}$)

A spherical clean bubble with negligible internal dynamic viscosity has zero tangential traction and permits tangential slip. In the creeping-flow, negligible-deformation limit its drag is $-4\pi \mu a\mathbf{U}$, versus $-6\pi \mu a\mathbf{U}$ for a no-slip rigid sphere.

A spherical clean bubble with constant gas mass density and prescribed dissolution law $\dot{M}=-4\pi aD{\rho }_a$ obeys $\dot{a}=-D/a$. With negligible gas dynamic viscosity, instantaneous terminal velocity is $\dot{z}=\Delta \rho g{a}^2/(3\mu )$. Eliminating time yields the displayed radius-height relation and maximum rise $z_{\mathrm{diss}}=\Delta \rho g{a}_0^4/(12\mu D)$. This uses an instantaneous force balance; a literally constant speed during shrinkage instead gives $a^2$ linear in height. The Reynolds number must remain small for the Stokes flow drag model to apply.

For a clean inviscid bubble a distance $d\gg a$ below a gravity-restored approximately flat free surface, the surface displacement scales as $F/(\rho g{d}^2)=O(a^3/d^2)$. 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.

A buoyantly rising inviscid bubble in otherwise quiescent unbounded liquid has an exterior Stokeslet with $F=4\pi \mu aU=4\pi \rho g{a}^3/3$. Its tangential traction vanishes on the sphere. The dynamic normal traction is $-3F\cos \theta /(4\pi a^2)$, while the hydrostatic normal-traction variation is $\rho ga\cos \theta$. Their dipoles cancel at the terminal velocity, leaving a constant total normal stress compatible with uniform gas pressure even when surface tension is zero. This is an exact spherical solution in the stated Stokes model, not a proof of shape stability under arbitrary perturbations.