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Take $z=0$ at the base and assume gas leaves freely at the liquid surface, without being retained as foam. There is no new bubble production after filling. Neglecting the liquid velocity, gas-volume conservation is a scalar conservation lawAn impermeable base has no incoming bubbles. Thus a clear region develops below the uniform bubbly region. The clearing discontinuity satisfies the Rankine-Hugoniot condition, with states $0$ and ${\varphi }_0$:For the dilute regime $0<{\varphi }_0<1/2$, the characteristic speeds obey $j^{\prime }(0)>s>j^{\prime }({\varphi }_0)$, so this is the admissible compressive shock. Before it reaches the surface,The remaining bubbles move upwards at $s$ and escape at the top. Averaging the void fraction over the entire cylindrical volume givesThis bubble clearing in a cylindrical vessel describes bubbles in the liquid, not a hypothetical retained gas layer above it. A closed surface or long-lived foam would require a different upper boundary condition.