= Terminal-drift density law for compressible bubbles
{title2=$\rho_g/\rho_0=[\alpha_0^2(1-\alpha_0)/(\alpha^2(1-\alpha))]^{3/5}$}
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> $J=\alpha\rho_gv$ 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.
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