Settling velocity ($W$)

The settling velocity is a particle's vertical velocity relative to a stated frame. With height increasing upward, a sedimenting particle has $W<0$.

For spheres with the same density contrast, balance submerged weight, proportional to $r^3$, against Stokes drag law, proportional to $r{W}_s$, to obtain $W_s\propto r^2$. With constant drag coefficient, quadratic drag is proportional to $r^2{W}_s^2$, giving $W_s\propto r^{1/2}$. The two radius laws require their corresponding particle Reynolds number limits.

A Reynolds number based on particle diameter and relative speed in the carrier fluid. Small values justify Stokes drag law; a high-Reynolds-number regime with nearly constant drag coefficient instead gives quadratic drag. The drag law must be evaluated in the appropriate regime for every species.

For downward settling velocity magnitude $W_s$ and near-bed particle volume fraction ${\varphi }_b$, a horizontal absorbing bed receives particle volume per area per time $J_s=W_s{\varphi }_b$. A bed inclined at angle $\vartheta$ to the horizontal receives $W_s{\varphi }_b\cos \vartheta$ per unit actual area, in the absence of resuspension or additional normal fluid motion.

Take downward as positive and let gravity have magnitude $g$. For a rigid sphere of density ${\rho }_p$ in otherwise stationary fluid of density ${\rho }_f$, its weight minus buoyancy is $\frac{4}{3}\pi a^3({\rho }_p-{\rho }_f)g$. At terminal velocity, this balances the Stokes drag law:This signed downward settling velocity is negative for a buoyantly rising sphere and zero for equal densities. For fixed density contrast and dynamic viscosity, the submerged weight grows as $a^3$ while linear drag at fixed speed grows as $a$; their balance gives the settling-velocity radius scaling $U_s\propto a^2$. Meanwhile,Thus doubling the radius multiplies the predicted Stokes settling speed by four and the particle Reynolds number by eight. This makes the formula useful for separating particles by size, while also providing its own check against extrapolation to large particles.

For a computed water example, take ${\rho }_f=1000\mathrm{kg}{\mathrm{m}}^{-3}$, $\mu =1.00\times 10^{-3}\mathrm{Pa}\mathrm{s}$, ${\rho }_p=2500\mathrm{kg}{\mathrm{m}}^{-3}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, and $a=10.0\mu \mathrm{m}$. These are specified rounded inputs rather than a claim of exact water properties at every temperature. They giveThe diameter is $20.0\mu \mathrm{m}$, and both the net downward weight and the upward drag force are $6.16\times 10^{-11}\mathrm{N}$. At this terminal velocity, falling $1.00\mathrm{mm}$ takes $3.06\mathrm{s}$. At $a=20.0\mu \mathrm{m}$ the prediction is $1.308\mathrm{mm}{\mathrm{s}}^{-1}$ with ${\mathrm{Re}}_d=0.05232$; at $a=50.0\mu \mathrm{m}$ it is $8.175\mathrm{mm}{\mathrm{s}}^{-1}$ with ${\mathrm{Re}}_d=0.8175$, which is no longer a convincing small-Reynolds number case. The figure labels these as predictions of the Stokes settling velocity formula and shows where its inertia check fails.

The calculation describes the eventual terminal velocity, not the release transient governed by particle acceleration and possibly the Unsteady Stokes equation. Walls, hydrodynamic interactions, departures from a rigid sphere, or a different surface condition require a different resistance. For sufficiently small particles, Brownian motion can obscure individual settling trajectories; the formula still specifies the gravitational drift within its assumptions. The balancing argument already appears in article 43 of ^[ref].

Hindered settling is the reduction or modification of particle settling by the displaced carrier-fluid backflow and by interactions with other particles. A simple dilute mixture model gives every species the same volume-averaged backflow.

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.

For hindered settling with normalized particle volume fraction $s$, upward coordinate and downward isolated settling velocity magnitude $V_0$, the scalar conservation law has conservation law flux $f(s)=-V_{0}s(1-s{)}^{\alpha }$. For $\alpha >0$, its convexity changes at $s=2/(\alpha +1)$ when this lies in the physical interval. This change distinguishes spreading rarefaction waves from compressing characteristic curves; a flux with an inflection can produce both a shock wave and a rarefaction wave in one transition. At $\alpha =0$ the flux is linear and the profile simply translates.

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.

Consider a column $0<z<H$ initially filled with a uniform bidisperse particle suspension. In an independent-settling idealization, normalized concentrations $q_i$ obey fluxes $j_i=-W_i{q}_i(1-q_i)$, with $W_i>0$. A stationary packed deposit has fractions $p_1+p_2=1$. The deposit composition from sedimentation jump conditions determines its upward front speed $s$ and the fractions $p_i$.

Suppose species 1 has the faster downward clearing front: $v_1=W_1(1-q_1)>v_2=W_2(1-q_2)$. Before interaction, the fast clearing front is at $H-v_{1}t$, while the deposit front is at $st$. They first meet atAll species 1 has then entered the earlier mixed deposit. Above it, only species 2 remains suspended; its later sedimentation shock adds a pure-species-2 layer. Separate particle-volume conservation laws giveHere $h_2$ is the final pure-species-2 layer thickness. This two-stage picture follows from the stated independent fluxes and packed stationary branch; general coupled hindered settling can have a different wave structure.

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.

Use the quadratic hindered-settling flux $j(c)=-u_{s}c(1-c)$ in upward coordinate $z$, where $c$ is particle volume fraction normalized by the packed value. Initially $c=2a$ for $0<z<z_m$ and $c=a$ for $z_m<z<H$, with $0<a<1/2$. Clear fluid lies above, and a stationary packed deposit with $c=1$ forms below.

The Rankine-Hugoniot condition gives the clearing, internal and deposit-front trajectories before interactions:The internal jump is compressive because the lower concentration is larger. If the internal and deposit fronts meet first, the remaining deposit/upper-suspension jump joins $c=1$ to $c=a$ and advances at $u_{s}a$. If the clearing and internal fronts meet first, their replacement joins $c=2a$ to clear fluid. Either ordering conserves the same particle volume:Thus final packed thickness does not depend on which pair of fronts interacts first, although the intermediate wave pattern does.

For two-layer sedimentation with a compression shock, the clearing, internal and deposit fronts can all meet at one event. The concentration is normalized by the packed value: initially it is $a$ above $z_m$ and $2a$ below, with $0<a<1/2$, in a column of height $H$.

Equating the clearing and deposit trajectories givesRequiring the internal front to pass through the same point givesThis also satisfies particle-volume mass conservation, $h_c=a(H+z_m)$. At $t_c$ the entire particle population has entered the stationary packed deposit and the remaining interface separates that deposit from clear fluid. The special value of $z_m$ is the boundary between the two possible orders of pairwise sedimentation shock interactions.

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.

Let $c=\varphi /{\varphi }_{\max }$ be particle volume fraction normalized by the packed value, and use the quadratic hindered-settling flux $j(c)=-u_{s}c(1-c)$. Takewith clear fluid above $h$. To isolate breaking of this smooth profile, do not impose an additional packed-bed discontinuity. A characteristic curve labelled by its initial height $\xi$ carries constant concentration and followsThe mapping first loses invertibility whenThe earliest time occurs at $\xi =h$, giving $t_c=3h/(4u_s)$ and $z_c=h/4$. This explicitly shows why the time uses normalized concentration and the isolated settling-speed scale $u_s$; the physical flux for $\varphi$ is $-u_s\varphi (1-\varphi /{\varphi }_{\max })$.

The new sedimentation shock joins concentrated fluid below to clear fluid above. Its signed speed is $u_s(c_L-1)$, initially $-u_s$ as $c_L\to 0$. As the state behind it becomes more concentrated, its downward speed decreases until a boundary interaction changes the problem. This is the onset of characteristic breaking, not the onset of a separately imposed deposition front.

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.

Consider normalized initial concentration $a$ above $z_m=4H/5$ and $b=a/2$ below it, with $0<a<1$, in the quadratic hindered-settling flux model. The final packed height is $h_c=3aH/5$. The last surviving characteristic of a settling fan lies inside the fan only ifThe upper inequality gives $a\ge a_{\ast }=1-\sqrt{2/3}$; the lower gives $a\le 2/3$. Thus both front curves can be used through the final meeting only for $a_{\ast }\le a\le 2/3$.

If $a<a_{\ast }$, the deposition front leaves the fan through its upper edge before meeting the clearing front. It then travels through uniform concentration $a$ at speed $u_{s}a$; the clearing front stays straight. If $a>2/3$, the clearing front instead leaves the lower edge first, while the deposition front stays at speed $u_{s}b$. Matching the appropriate straight and curved pieces givesThe expressions agree at the transition values. The bounds prevent an algebraic intersection of extrapolated curves from being mistaken for a physical meeting inside the settling rarefaction fan.

For the quadratic hindered-settling flux, let the initial normalized concentration be $b$ below $z_m$ and $a>b$ above it, within a column $0<z<H$. The internal settling rarefaction fan hasBefore entering the fan, the upper clearing front follows $H-u_s(1-a)t$ and the lower deposition front follows $u_{s}bt$. They reach the fan at respective times $t_u=(H-z_m)/(u_{s}a)$ and $t_d=z_m/[u_s(1-b)]$.

Inside the fan, the Rankine-Hugoniot condition givesIntegrating these linear ordinary differential equations and matching to the entry events yieldswhere $A=a(H-z_m)$ and $D=(1-b)z_m$. Each curve applies only after that front enters the fan and while its adjacent concentration lies between $b$ and $a$. If a front leaves the fan, its subsequent speed is set by the uniform state it encounters, rather than by extrapolating these curves.

Suppose both curved settling fronts within a rarefaction fan meet while still inside the fan. Write $A=a(H-z_m)$ and $D=(1-b)z_m$, where $0<b<a<1$ are the initial normalized concentrations below and above $z_m$. Setting $z_u=z_d$ givesAt that point the fan concentration isThe final surviving characteristic curve runs from the initial internal interface to the meeting point. Its concentration $c_{\ast }$ must satisfy $b\le c_{\ast }\le a$. Failure of either bound means that a front leaves the fan before final deposition, requiring another branch of the entropy solution.

The meeting height reduces to $z_m+A-D=a(H-z_m)+b{z}_m$, agreeing independently with particle-volume mass conservation for a unit-normalized packed deposit.

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.