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.

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.

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.

For ${\alpha }_t+[V_s\alpha (1-\alpha ){]}_x=0$ in a finite upward pipe, bubbles move upward at $V_s(1-\alpha )$, but dense-mixture phase-fraction characteristics move downward. Hence an open upper end may be an incoming boundary for the scalar conservation law even though the bubbles themselves leave through it. Atmospheric pressure alone does not prescribe that incoming phase-fraction trace.

The bottom-clearing shock from zero to a uniform initial ${\alpha }_0$ has speed $V_s(1-{\alpha }_0)$. Continuing that constant right state to the outlet gives clearance time $L/[V_s(1-{\alpha }_0)]$, but at ${\alpha }_0>1/2$ it requires a compatible maintained incoming trace. For ${\alpha }_0=3/4$, the shock is $x=V_{s}t/4$, and a right-state characteristic traced backward from it has initial position $3V_{s}t/4$. It reaches the upper boundary at time $4L/(3V_s)$, before the formal clearance time $4L/V_s$. Later evolution therefore depends on the top closure. This boundary issue is distinct from the validity of the local slip and conservation derivation.