= Solution
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 $4\pi a^3\rho_lg/3$ with the quadratic drag gives
$$
\boxed{(u_1-u_2)|u_1-u_2|=\frac{4ag}{3C_D}.}
$$
Retaining 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 yields
$$
\boxed{u_1=V_s(1-\alpha),\quad u_2=-V_s\alpha,\quad\alpha_t+V_s[\alpha(1-\alpha)]_x=0.}
$$
Individual bubbles move upward for $0\leq\alpha<1$, while small phase-fraction disturbances move at the <characteristic speed> $q'(\alpha)=V_s(1-2\alpha)$. 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 therefore
$$
\boxed{\alpha(x,t)=\begin{cases}0,&0<x<V_st/4,\\3/4,&V_st/4<x<1,\end{cases}\qquad0<t<4/V_s.}
$$
It is an <entropy shock> for the concave <conservation law flux>, since $q'(0)=V_s>V_s/4>q'(3/4)=-V_s/2$. 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 when
$$
\boxed{t_{\rm clear}=\frac4{V_s}=\sqrt{\frac{12C_D}{ag}},}
$$
after 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>.
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