Past exam of the topics in fluid mechanics course of the University of Oxford / 2009 / 2 / ii

A spherical bubble displaces liquid volume $4\pi a^3/3$. Neglect its own weight compared with the liquid buoyancy, as required by the stated gas/liquid approximation. Equating $4\pi a^3{\rho }_{l}g/3$ with the quadratic drag givesRetaining a nonnegligible gas density would add the factor $1-{\rho }_g/{\rho }_l$. Let $V_s=\sqrt{4ag/(3C_D)}>0$, so the upward slip is $u_1-u_2=V_s$.

With constant phase densities, adding the two volume balances gives ${\partial }_x[\alpha u_1+(1-\alpha )u_2]=0$. The closed lower end fixes the total volume flux to zero. Solving it together with the slip relation yieldsIndividual bubbles move upward for $0\le \alpha <1$, while small phase-fraction disturbances move at the characteristic speed $q^{\prime }(\alpha )=V_s(1-2\alpha )$. The latter is negative when $\alpha >1/2$: wave speed and bubble speed are different quantities.

A cleared region starts at the closed bottom, with ${\alpha }_L=0$ and ${\alpha }_R=3/4$. Its Rankine-Hugoniot condition speed is $[q]/[\alpha ]=V_s/4$. The intended clearing solution is thereforeIt is an entropy shock for the concave conservation law flux, since $q^{\prime }(0)=V_s>V_s/4>q^{\prime }(3/4)=-V_s/2$. Its speed also equals the bubble speed on its uncleared side, so it follows the last layer of bubbles. Under this continuation the last bubble reaches the outlet whenafter which $\alpha =0$ throughout the pipe.

There is a boundary qualification for this dense initial mixture. The characteristics in the uncleared state point into the pipe from the top. To continue the displayed solution all the way to the stated clearance time, its incoming top trace must remain $\alpha (1^{-},t)=3/4$ until clearing. Initial characteristics alone cover the shock only until $t=4/(3V_s)$, when its backward right-state characteristic first meets the top boundary. Atmospheric pressure by itself is not a boundary condition for this scalar phase-fraction equation. Thus the displayed formula and printed clearance time give the intended maintained-trace branch; a different top phase-fraction closure could change the later evolution. This is the incoming boundary condition in dense bubble clearing, and is separate from the local derivation of the conservation law.