Kinematic bubble transport in an exponentially narrowing vessel (${\varphi }_t+V_s(1-2\varphi ){\varphi }_z=2\beta V_s\varphi (1-\varphi )$)

If the vessel radius is proportional to $e^{-\beta z}$, its area is proportional to $e^{-2\beta z}$. Gas-volume conservation gives the displayed equation. Along concentration characteristic curves, $\dot{z}=V_s(1-2\varphi )$ and $\dot{\varphi }=2\beta V_s\varphi (1-\varphi )$. Uniform initial concentration stays uniform in the uninterrupted bubbly region and follows this logistic law. Individual bubbles move at $V_s(1-\varphi )$ instead: their rise distance there is $\eta =-(2\beta {)}^{-1}\log [{\varphi }_0+(1-{\varphi }_0)e^{-2\beta V_{s}t}]$. Boundary interactions and dense-mixture backflow can invalidate the simplified closure.