 Solution

Use the uniformly filled narrowing vessel of the preceding part. A bubble within its uniform bubbly region seesIts material rise speed is $\dot{\eta }=V_s(1-\varphi (t))$, not the concentration-characteristic speed. Integrating from $\eta (0)=0$ givesThe cylindrical limit $\beta \to 0$ is $\eta =V_s(1-{\varphi }_0)t$, as required. The bottom clearing front follows this same position by the local Rankine-Hugoniot condition. In the narrowing vessel the formula applies until the bubble leaves at $z_0+\eta =H$, or a boundary interaction changes the uniform state. Its formal large-time limit is ${\eta }_{\infty }=\log (1/{\varphi }_0)/(2\beta )$, while $\varphi \to 1$. Crowding and liquid backflow invalidate the assumed dilute, negligible-liquid-velocity closure before this limit is a reliable prediction of a real tankard.