= Solution
Put $a=\phi_1$, $b=\phi_2=a/2$, $z_m=4H/5$, $L=H-z_m$ and $D=(1-b)z_m$. The <quadratic hindered-settling flux> is convex. At the initial internal interface its lower <particle volume fraction> is smaller than its upper <particle volume fraction>, so <characteristic curves> diverge: the <entropy solution> has a <settling rarefaction fan>, not a compression <shock>. Within this fan,
$$
\boxed{\phi(z,t)=\frac12\left(1+\frac{z-z_m}{u_st}\right),\qquad z_m+u_s(2b-1)t<z<z_m+u_s(2a-1)t.}
$$
The two bounding straight <shocks> initially have
$$
\boxed{V_1=-u_s(1-a),\qquad V_2=u_sb=\frac{u_sa}{2}.}
$$
The upper front's nominal meeting with the fan's upper edge occurs at
$$
\boxed{t_{c1}=\frac{L}{u_sa}=\frac{H}{5u_sa},\qquad h_{c1}=H-\frac{(1-a)L}{a}=\frac{H(6a-1)}{5a}.}
$$
The lower front meets the fan's lower edge at
$$
\boxed{t_{c2}=\frac{z_m}{u_s(1-b)}=\frac{4H}{5u_s(1-a/2)},\qquad h_{c2}=\frac{bz_m}{1-b}=\frac{4aH}{5(2-a)}.}
$$
The lower meeting is always physical in the stated range. The upper meeting need not happen before the suspension disappears; this distinction matters for sufficiently small $a$.
For the <curved settling fronts within a rarefaction fan>, the <Rankine-Hugoniot condition> gives $\dot z_u=u_s(\phi-1)$ at the clear-fluid front and $\dot z_d=u_s\phi$ at the deposit front. Substitution of the fan <particle volume fraction> and matching to the straight sections gives
$$
\boxed{z_u=z_m-u_st+2\sqrt{u_saLt},\qquad z_d=z_m+u_st-2\sqrt{u_sDt}.}
$$
The first curve applies after $t_{c1}$ if that meeting occurs, and the second after $t_{c2}$ until it leaves the fan or deposition ends. If both fronts meet inside the fan, their intersection is
$$
\boxed{t_c=\frac{(\sqrt{aL}+\sqrt D)^2}{u_s},\qquad h_c=aL+bz_m=\frac35aH.}
$$
The <last surviving characteristic of a settling fan> joins $(z_m,0)$ to $(h_c,t_c)$. It has constant <particle volume fraction>
$$
\boxed{\phi_* =\frac{\sqrt{aL}}{\sqrt{aL}+\sqrt D},\qquad\frac{h_c-z_m}{t_c}=u_s(2\phi_*-1).}
$$
It is swallowed simultaneously by the clear-fluid and deposit <shocks> at final deposition. This requires $b\leq\phi_*\leq a$, which in the given geometry is equivalent to
$$
\boxed{a\geq a_*=1-\sqrt{\frac23}.}
$$
Equality places the last <characteristic curve> on the upper fan edge. In the generic fan-intersection regime $a>a_*$, both nominal meetings and the two curved fronts are realized.
For $0<a<a_*$ there is <early extinction of a settling rarefaction fan>. The deposit <shock> reaches the upper fan edge before the clear-fluid front does, at
$$
t_e=\frac{D}{u_s(1-a)^2},\qquad z_e=z_m+u_s(2a-1)t_e.
$$
It then travels through the remaining uniform <particle volume fraction> $a$ with speed $u_sa$. The upper front retains $V_1$ until the final intersection. Since the final bed height is still $h_c=3aH/5$, the actual stopping time is
$$
\boxed{t_c=\frac{H-h_c}{u_s(1-a)}=\frac{H(1-3a/5)}{u_s(1-a)},\qquad0<a<a_*.}
$$
In this regime $(h_{c1},t_{c1})$ is only an extrapolated intersection; for $a<1/6$ its extrapolated height is even negative. The line joining the initial internal interface to the final point would have a <particle volume fraction> greater than $a$, so it is not a <characteristic curve> emitted by that fan. \b[The single fan-<characteristic curve> stopping-time construction does not apply to the entire printed interval $0<a<1/2$; the missing subregime must be included.]
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-330-settling-fans.png]
{title=Settling rarefaction fan with curved fronts, and the small-concentration regime where the fan disappears early}
Both panels label the actual final deposit height and time; the second shows the deposit <shock> leaving the fan while the upper front remains straight. Finally, the <density inversion in a settling suspension> puts more particles, hence denser fluid, above lighter fluid. <Rayleigh-Taylor instability> and overturning can accompany this profile in a real container. They are excluded by the one-dimensional quiescent <kinematic sedimentation> model used for these calculations.
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