= Solution
Use the uniformly filled narrowing vessel of the preceding part. A bubble within its uniform bubbly region sees
$$
\phi(t)=\frac{\phi_0e^{ct}}{1-\phi_0+\phi_0e^{ct}},\qquad c=2\beta V_s.
$$
Its material rise speed is $\dot\eta=V_s(1-\phi(t))$, not the concentration-characteristic speed. Integrating from $\eta(0)=0$ gives
$$
\boxed{\eta(t)=V_st-\frac1{2\beta}\log\left(1-\phi_0+\phi_0e^{2\beta V_st}\right)
=-\frac1{2\beta}\log\left[\phi_0+(1-\phi_0)e^{-2\beta V_st}\right].}
$$
The cylindrical limit $\beta\to0$ is $\eta=V_s(1-\phi_0)t$, 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 $\eta_\infty=\log(1/\phi_0)/(2\beta)$, while $\phi\to1$. Crowding and liquid backflow invalidate the assumed dilute, negligible-liquid-velocity closure before this limit is a reliable prediction of a real tankard.
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