= Bubble clearing in a cylindrical vessel
{title2=$\Phi(t)=\phi_0[1-V_s(1-\phi_0)t/H]_+$}
For initially uniform bubbles with upward speed $V_s(1-\phi)$, no new gas production and free escape at the top, gas conservation gives flux $V_s\phi(1-\phi)$. A clearing <shock> rises from the impermeable bottom with speed $V_s(1-\phi_0)$ 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.
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