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More suspended heavy particles above fewer particles puts denser fluid above lighter fluid. Such a particle suspension may develop Rayleigh-Taylor instability and overturning in addition to one-dimensional kinematic sedimentation. A rarefaction solution alone does not establish hydrodynamic stability.

Use downward distance $y=H-z$, with $z$ the height above the bottom, and write $W=W_s(a)>0$. This convention will also reproduce the printed negative deposit-front velocity in part (d). The ideal simple-cubic sphere packing has packing fraction $\pi /6$; leaving ${\varphi }_{\max }$ symbolic keeps the jump calculation independent of its value. In suspension, the kinematic sedimentation equation isIntegrate this conservation law across a moving discontinuity to obtain the Rankine-Hugoniot conditionThe brackets denote values on the increasing-$y$ side minus those on the other side. A sedimentation shock is a concentration jump moving at this secant slope of the particle flux. Deposited material is stationary and has zero flux.

For the upper clearing front, the states are $0$ and ${\varphi }_{\max }/8$, giving $U_1=7W/8$ downward. For the lower deposition front, the states are ${\varphi }_{\max }/8$ and ${\varphi }_{\max }$, giving $U_2=-W/8$, upward. ThusIn terms of height above the bottom, the shock paths areThey meet when $H=Wt$. Therefore complete settling occurs atThe final height also follows directly from particle volume fraction conservation, ${\varphi }_{\max }{h}_c=({\varphi }_{\max }/8)H$. The stationary deposit after the meeting carries no particle flux.

The diagram plots time vertically against height, as requested. Its shock slopes have the opposite spatial sign to $U_1,U_2$, because those velocities were defined in the downward coordinate.

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.

A moving discontinuity of particle volume fraction in kinematic sedimentation. Conservation of particle volume gives the Rankine-Hugoniot condition $U[\varphi ]=[j]$ in a consistently oriented coordinate. A clearing front and a deposit front can both be sedimentation shocks. A stationary deposit uses zero particle flux; its compacted branch need not obey the suspension flux law.