 Solution

The radius law integrates to $r(z)=r_0{e}^{-\beta z}$, so the cross-sectional area is $A(z)=\pi r_0^2{e}^{-2\beta z}$. Conserving gas volume in a horizontal slice yieldsor the kinematic bubble transport in an exponentially narrowing vessel equationThe geometrical term increases the concentration as a given gas flux enters a smaller area. On concentration characteristic curves,Put $c=2\beta V_s$. A characteristic starting at $(z_0,0)$ with ${\varphi }_i=\varphi (z_0,0)$ hasThe first formula follows by integrating $d\varphi /[\varphi (1-\varphi )]=cdt$; inserting it into $dz/dt$ gives the second. For initially uniform ${\varphi }_0$, the uninterrupted bubbly region remains spatially uniform, with this time-dependent concentration. These characteristic curves carry concentration information; they are not individual bubble trajectories, whose speed is $V_s(1-\varphi )$.