Kinematic sedimentation (${\varphi }_t+{\partial }_{z}j(\varphi )=0$)

A one-dimensional particle suspension settling without particle diffusion or overturning obeys the scalar conservation law ${\varphi }_t+{\partial }_{z}j(\varphi )=0$. The upward flux is negative for downward settling. Characteristic curves, compressive shocks and rarefaction waves describe concentration transport rather than individual particle paths.

A particle suspension contains two particle species with separate particle volume fractions. In an independent-settling model, each species obeys its own flux conservation equation, while a shared stationary deposit can couple the jump conditions through its maximum total packing fraction. Such independence is an imposed model; general hindered settling need not decouple species.

Initially, separate clearing fronts of fast and slow species coexist with a mixed deposition front. After the fast clearing front meets that deposit, the remaining suspension contains only the slow species. Its later sedimentation shock builds a pure-slow layer above the earlier mixed deposit. Final layer thicknesses and composition follow from separate particle-volume conservation.

For a bidisperse kinematic sedimentation model with normalized suspension concentrations $q_i$, isolated speeds $W_i$, and a stationary deposit with $p_1+p_2=1$, the upward deposit-front speed magnitude obeys $s(p_i-q_i)=W_i{q}_i(1-q_i)$. Adding gives $s={\sum }_i{W}_i{q}_i(1-q_i)/(1-{\sum }_i{q}_i)$; substitution determines each deposited fraction. These are Rankine-Hugoniot conditions with a stationary compacted branch.

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.

Particle mass conservation gives $h_c=[{\varphi }_1(H-z_m)+{\varphi }_2{z}_m]/{\varphi }_{\max }$. Averaging concentration over the entire original container gives $\bar{\varphi }={\varphi }_{\max }{h}_c/H$. This average differs from the concentration inside the deposit itself.

With upper concentration $a$ and lower concentration $2a$, the clearing, internal and deposit fronts have speeds $-u_s(1-a)$, $-u_s(1-3a)$ and $2u_{s}a$. If the lower two merge, the resulting deposit/upper-suspension front has speed $u_{s}a$. Final deposit thickness is $a(H+z_m)$ by mass conservation.

For initial concentrations $a$ above $z_m$ and $2a$ below it, all three straight shocks meet when $z_m=H(1-a)/(1+a)$. The common time is $H/[u_s(1+a)]$ and height is $2aH/(1+a)$. Afterwards only a stationary deposit/clear-fluid boundary remains.

For downward settling speed $u_s(1-\varphi )$ in upward coordinate $z$, particle flux is $j=-u_s\varphi (1-\varphi )$. The concentration-characteristic speed is $j^{\prime }=u_s(2\varphi -1)$ and differs from particle speed. Since $j^{\prime \prime }=2u_s>0$, decreasing concentration with height creates a compressive shock, while increasing concentration creates a rarefaction wave.

If lower initial concentration $b$ is less than upper concentration $a$, the entropy solution expands into a fan with $\varphi =[1+(z-z_m)/(u_{s}t)]/2$. Its characteristic speeds range from $u_s(2b-1)$ to $u_s(2a-1)$. The fan can be clipped by a clearing or deposition shock.

For upper concentration $a$, lower $a/2$ and interface height $4H/5$, both curved fronts meet inside the fan only if $a\ge 1-\sqrt{2/3}$. At smaller $a$, the deposition front reaches the fan upper edge first, then advances through uniform concentration $a$. The clearing front remains straight and final time is $H(1-3a/5)/[u_s(1-a)]$.

For upper concentration $a$, lower concentration $b<a$ and $L=H-z_m$, define $A=aL$ and $D=(1-b)z_m$. Once inside the settling rarefaction fan, clearing and deposition shocks obey $z_u=z_m-u_{s}t+2\sqrt{u_{s}At}$ and $z_d=z_m+u_{s}t-2\sqrt{u_{s}Dt}$. These curves apply only while their neighboring concentration remains within the fan.

If both shocks meet inside a settling rarefaction fan, $t_c=(\sqrt{A}+\sqrt{D}{)}^2/u_s$ and the last characteristic has concentration ${\varphi }_{\ast }=\sqrt{A}/(\sqrt{A}+\sqrt{D})$. It runs from the initial internal interface to the final deposit point. This construction requires $b\le {\varphi }_{\ast }\le a$; otherwise a front exits the fan before final deposition.

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.