Void fraction ($\varphi =V_{\mathrm{gas}}/V_{\mathrm{mixture}}$)

The void fraction is the fraction of a mixture's local volume occupied by gas. For a horizontally uniform bubbly liquid it is the transported concentration in a one-dimensional gas-volume continuity equation. It is distinct from gas mass fraction because gas and liquid have different mass densities.

If the vessel radius is proportional to $e^{-\beta z}$, its area is proportional to $e^{-2\beta z}$. Gas-volume conservation gives the displayed equation. Along concentration characteristic curves, $\dot{z}=V_s(1-2\varphi )$ and $\dot{\varphi }=2\beta V_s\varphi (1-\varphi )$. Uniform initial concentration stays uniform in the uninterrupted bubbly region and follows this logistic law. Individual bubbles move at $V_s(1-\varphi )$ instead: their rise distance there is $\eta =-(2\beta {)}^{-1}\log [{\varphi }_0+(1-{\varphi }_0)e^{-2\beta V_{s}t}]$. Boundary interactions and dense-mixture backflow can invalidate the simplified closure.

For initially uniform bubbles with upward speed $V_s(1-\varphi )$, no new gas production and free escape at the top, gas conservation gives flux $V_s\varphi (1-\varphi )$. A clearing shock rises from the impermeable bottom with speed $V_s(1-{\varphi }_0)$ by the Rankine-Hugoniot condition. The displayed mean void fraction averages the remaining uniform region over the whole liquid depth $H$. This assumes the dilute regime and no retained foam layer.