= Dissolution-limited bubble rise
{title2=$a^4(z)=a_0^4-12\mu D z/(\Delta\rho g)$}
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 $\dot z=\Delta\rho ga^2/(3\mu)$. Eliminating time yields the displayed radius-height relation and maximum rise $z_{\mathrm{diss}}=\Delta\rho ga_0^4/(12\mu D)$. 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.
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