Bubble clearing in a cylindrical vessel ($\Phi (t)={\varphi }_0[1-V_s(1-{\varphi }_0)t/H{]}_{+}$)

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.