= Kinematic bubble transport in an exponentially narrowing vessel
{title2=$\phi_t+V_s(1-2\phi)\phi_z=2\beta V_s\phi(1-\phi)$}
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>, $\dot z=V_s(1-2\phi)$ and $\dot\phi=2\beta V_s\phi(1-\phi)$. Uniform initial concentration stays uniform in the uninterrupted bubbly region and follows this logistic law. Individual bubbles move at $V_s(1-\phi)$ instead: their rise distance there is $\eta=-(2\beta)^{-1}\log[\phi_0+(1-\phi_0)e^{-2\beta V_st}]$. Boundary interactions and dense-mixture backflow can invalidate the simplified closure.
Back to article page
