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Hindered settling is the reduction of the particle settling velocity by displaced-fluid backflow and interactions with other particles. In the specified model, the downward speed is $V_s$ at vanishing concentration and falls to zero at the maximum packing fraction ${\varphi }_{\max }$. The characteristic speed of a concentration disturbance will differ from this individual-particle speed.

Take $z$ positive upwards. For a locally uniform horizontal suspension without diffusion or bulk fluid motion, the upward particle flux isParticle volume conservation gives the scalar conservation lawThis is the quadratic hindered-settling flux. Along its characteristic curves,For a smooth initial profile ${\varphi }_0(\zeta )$, the map is $z=\zeta +c({\varphi }_0(\zeta ))t$. Its Jacobian is $1+(2V_s/{\varphi }_{\max }){\varphi }_0^{\prime }(\zeta )t$. A shock develops where this first reaches zero: the compressive condition is ${\varphi }_0^{\prime }<0$, andwhen the minimum is negative and no earlier boundary discontinuity is imposed. A discontinuous compressive initial state can instead contain a shock from $t=0$.

Let $z=s(t)$ separate states ${\varphi }_L$ below the shock and ${\varphi }_R$ above it. Integrating the conservation law across a moving thin interval gives the Rankine-Hugoniot conditionEquivalently, the flux relative to the shock agrees on both sides. For this quadratic flux,Since $j^{\prime \prime }=2V_s/{\varphi }_{\max }>0$, a physical compression shock has ${\varphi }_L>{\varphi }_R$ and satisfies $c({\varphi }_L)>\dot{s}>c({\varphi }_R)$: characteristic curves enter it from both sides. These conditions select the shock over a nonphysical expansive discontinuity.

Take $z$ upward and $u_s>0$ downward. The quadratic hindered-settling flux is $j(\varphi )=-u_s\varphi (1-\varphi )$. Particle mass conservation gives the kinematic sedimentation equationon a smooth characteristic curve. Integrate the scalar conservation law across a moving discontinuity: $V[\varphi ]=[j]$. The Rankine-Hugoniot condition gives the settling shock speedwhere $L$ is the lower side and $R$ the upper side. Since $j^{\prime \prime }=2u_s>0$, a compressive shock has ${\varphi }_L>{\varphi }_R$ and its neighboring characteristic curves point into it.

Put $a={\varphi }_1$, so ${\varphi }_2=2a$, with $0<a<1/4$. The two-layer sedimentation with a compression shock hasThese respectively separate upper suspension from clear fluid, lower suspension from upper suspension, and deposit from lower suspension. Their initial positions are $H,z_m,0$. The characteristic curves in the two suspended states have slopes $u_s(2a-1)$ and $u_s(4a-1)$; clear-fluid and deposit characteristic curves have slopes $-u_s$ and $u_s$.

For $z_m=H/2$, $V_2$ and $V_3$ meet first. Their intersection givesThe potential meeting of the upper two fronts would take $(H-z_m)/(2u_{s}a)$, which is later in the stated range. The new front separates particle volume fraction one from particle volume fraction $a$, soIt follows $z=h_{c1}+u_{s}a(t-t_{c1})$ until it meets $z=H-u_s(1-a)t$. The final mass-conserving intersection isFor $z_m=H/2$, these are $H(1-3a/2)/[u_s(1-a)]$ and $3aH/2$.

The PDF prints a plus sign before $a{z}_m/H$ in the stopping-time numerator. That sign contradicts both the top-front trajectory and particle mass conservation; the correct sign is minus, as derived here. Once all particles are deposited, the remaining $1$-to-$0$ front is stationary because both limiting fluxes vanish.

The first panel shows the two-front merger and the final intersection, with straight characteristic curves in each suspended layer. The second panel shows the tuned three-front intersection.

Put $a={\varphi }_1$, $b={\varphi }_2=a/2$, $z_m=4H/5$, $L=H-z_m$ 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,The two bounding straight shocks initially haveThe upper front's nominal meeting with the fan's upper edge occurs atThe lower front meets the fan's lower edge atThe 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 ${\dot{z}}_u=u_s(\varphi -1)$ at the clear-fluid front and ${\dot{z}}_d=u_s\varphi$ at the deposit front. Substitution of the fan particle volume fraction and matching to the straight sections givesThe 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 isThe last surviving characteristic of a settling fan joins $(z_m,0)$ to $(h_c,t_c)$. It has constant particle volume fractionIt is swallowed simultaneously by the clear-fluid and deposit shocks at final deposition. This requires $b\le {\varphi }_{\ast }\le a$, which in the given geometry is equivalent toEquality places the last characteristic curve on the upper fan edge. In the generic fan-intersection regime $a>a_{\ast }$, both nominal meetings and the two curved fronts are realized.

For $0<a<a_{\ast }$ there is early extinction of a settling rarefaction fan. The deposit shock reaches the upper fan edge before the clear-fluid front does, atIt then travels through the remaining uniform particle volume fraction $a$ with speed $u_{s}a$. 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 isIn 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. 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.

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.

The Rankine-Hugoniot condition for the quadratic hindered-settling flux gives $V=u_s({\varphi }_L+{\varphi }_R-1)$, with lower and upper states labelled $L,R$. It includes a downward clearing front (${\varphi }_R=0$), an upward deposition front (${\varphi }_L=1$), and a stationary final deposit/clear-fluid interface.