Terminal-drift density law for compressible bubbles (${\rho }_g/{\rho }_0=[{\alpha }_0^2(1-{\alpha }_0)/({\alpha }^2(1-\alpha )){]}^{3/5}$)

For steady upward bubbles in a stationary liquid, assume fixed bubble mass, quadratic drag force, negligible gas weight and inertia, and negligible wall force. Then $a\propto {\rho }_g^{-1/3}$. Hydrostatic mixture support and terminal drag force balance give $v^2\propto a(1-\alpha )$. The gas mass flux $J=\alpha {\rho }_{g}v$ is constant, hence $J^2\propto {\alpha }^2(1-\alpha ){\rho }_g^{5/3}$. Comparing at a reference point proves the displayed relation. The approximation requires a small gas/liquid mass density ratio and macroscopic variation scales; it need not hold near a gas-dominated endpoint or rapid acceleration region.