Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 76 / 1

Take $z=0$ at the base and assume gas leaves freely at the liquid surface, without being retained as foam. There is no new bubble production after filling. Neglecting the liquid velocity, gas-volume conservation is a scalar conservation lawAn impermeable base has no incoming bubbles. Thus a clear region develops below the uniform bubbly region. The clearing discontinuity satisfies the Rankine-Hugoniot condition, with states $0$ and ${\varphi }_0$:For the dilute regime $0<{\varphi }_0<1/2$, the characteristic speeds obey $j^{\prime }(0)>s>j^{\prime }({\varphi }_0)$, so this is the admissible compressive shock. Before it reaches the surface,The remaining bubbles move upwards at $s$ and escape at the top. Averaging the void fraction over the entire cylindrical volume givesThis bubble clearing in a cylindrical vessel describes bubbles in the liquid, not a hypothetical retained gas layer above it. A closed surface or long-lived foam would require a different upper boundary condition.

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 $V_s(1-\varphi )$.

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.