The column the chord rule cannot settle

A wave nothing in it travels with worked out the kinematic wave on two flux laws, a river and a road, and found that everything follows from one curve: disturbances travel at its slope, material at the chord to the origin, and a front at the chord between the states on either side. It closed by listing the places that picture fails, and the first was a flux curve whose curvature changes sign. There, it said, the solution can contain waves that are part shock and part fan, and the chord rule alone does not determine the answer.

A settling suspension is that case, and not an exotic one. Every water-treatment clarifier, every mineral thickener and every blood sample left standing in a tube is a column of particles whose flux curve bends both ways. This essay solves the simplest such column — a uniform suspension left to settle between a closed top and a closed floor — and shows what the extra condition does. It makes the top of the column behave exactly as the chord rule says and the bottom behave quite differently, and the difference decides how long settling takes: not a finite time, but for ever.

A hindered settling law bends the flux both ways #

A single sphere falls through a viscous liquid at its Stokes speed $v_0$, the speed whose limits How small is small enough measured. A crowd of them falls more slowly, because each particle’s descent pushes liquid up past its neighbours, and the Richardson–Zaki law gives the hindered speed as $v_0(1 - \varphi/\varphi_m)^n$ at a solids volume fraction $\varphi$. The downward flux of solids is then

$$f(\varphi) = v_0\,\varphi\,(1 - \varphi/\varphi_m)^n.$$

The conventions are these. The exponent is $n = 4.65$, the value measured for particle Reynolds numbers below about 0.2; the packing fraction at which particles touch and stop is $\varphi_m = 0.64$, random close packing of equal spheres; lengths are scaled on the column height $H$ and times on $H/v_0$, the time a lone particle takes to fall the whole column. In the figures height is measured up from the floor. For a dilute suspension the law gives $v \approx v_0(1 - 7.27\varphi)$; Batchelor’s calculation for randomly placed spheres gives a coefficient of 6.55, which is the same kind of many-particle correction that A viscosity made of particles computes for a suspension’s viscosity, and within about eleven per cent of it.

The curve rises from zero, peaks at $\varphi^* = \varphi_m/(n+1) = 0.113$ with a flux of 0.0458 of $v_0$, and falls back to zero at packing — arriving there with zero slope, because the exponent is greater than one. Between the peak and packing its curvature changes sign at $\varphi_i = 2\varphi_m/(n+1) = 0.227$: concave below, convex above. A river under Manning’s law has a curve that is convex everywhere, and a road under Greenshields’ law one that is concave everywhere. This one is both, and that is the whole of what is new.

The top is one chord; the bottom is a chord and a curve #

A column starting uniform at $\varphi_0 = 0.1$ has two jumps in it from the first instant: clear liquid above the suspension at the top, and the suspension against a floor where particles must pile up to packing. For a jump from a lower concentration above to a higher one below, the admissible solution is read off the lower convex envelope of the flux curve between the two states — Oleinik’s condition, the scalar version of the rule The only law that forbids it found selecting shocks in a gas. Where the envelope is a straight chord the jump is a shock moving at the chord’s slope. Where the envelope follows the curve the jump is a fan, spreading at the curve’s slope.

At the top the envelope from 0 to 0.1 is the chord from the origin, because the curve lies above that chord everywhere between. So the interface is a single sharp front, falling at the chord’s slope $f(\varphi_0)/\varphi_0$ — which is just the hindered settling speed, 0.454 of $v_0$. The chord rule works. The front is sharp from the first instant only because the suspension began as a jump against clear liquid; a smooth concentration profile would have had to steepen into one first, at a breaking time set by the curve’s slope, as Every compression becomes a shock in the end worked out for sound.

At the bottom it does not. The straight chord from $\varphi_0$ to packing rises at 0.084 and conserves solids exactly, but for most of its length it runs above the curve, and a jump into a denser state is admissible only when the curve lies above its chord — so it is not the envelope. The envelope is a chord from $\varphi_0$ that touches the curve tangentially at $\varphi_a = 0.317$, followed by the curve itself from there to packing. The bottom of the column is therefore a shock rising at 0.148 that jumps only as far as 0.317, with a fan of concentrations from 0.317 to packing beneath it. The tangency is the extra condition the conservation law does not contain, and it changes the shock’s speed by three quarters.

The tangency also says what kind of front this is. Lax’s condition for a stable shock asks that characteristics run into it from both sides. Above the sediment shock they do: the suspension’s characteristics rise more slowly than the shock. Below it, at the tangent point, the characteristic speed of 0.317 is exactly the shock’s speed, so the characteristics there neither run in nor pull away — they travel with it. That is why a fan can attach to its underside without a gap: the fan’s first characteristic is the shock’s own path. The fan is the expansion half of the asymmetry Turning the other way is free measured in a gas, where compressions steepen into shocks and expansions spread without cost, arriving here in a column with no gas and no thermodynamics.

The tangent point is found two ways that share nothing. One solves for the concentration at which the chord’s slope equals the curve’s; the other builds the lower convex hull of the curve from samples by a monotone chain and reads off its first vertex, never mentioning a tangent. From 0.05 the chord touches at 0.382 and the shock rises at 0.086; from 0.1, at 0.317 and 0.148; from 0.2, at 0.241 and 0.201. The two routes agree to within the sample spacing at every start tested, and the tangency itself holds to 10⁻⁷.

The column in height and time #

With both jumps resolved, the whole test can be drawn at once.

The interface falls in a straight line. The sediment shock rises in a straight line. Beneath the shock, the fan’s characteristics leave the floor at time zero, each carrying one concentration between 0.317 and packing at its own slope, and the denser ones are slower. The interface and the shock meet at $t = 1.661$ at a height of 0.246, and at that moment the uniform suspension is gone: every particle is either in the fan or packed. What had been three waves — the shock, fan and contact that one diaphragm in a gas produces have here lost their contact, because a single conserved quantity has no second family to carry one — becomes one front separating clear liquid from a graded layer.

From there the interface is still a shock, falling at the chord from the origin to whatever concentration the fan presents to it. That concentration rises as the interface descends into denser parts of the fan, so the chord flattens and the interface slows. It never stops.

Four zones, and the one that is graded #

The concentration profiles make the structure concrete.

At $t = 1$ the column has four zones. Clear liquid above 0.546. The original suspension at 0.1 down to 0.149. A jump there to 0.317. And below it, a layer graded all the way to packing at the floor. That last zone is the fan, and its concentration at each height is fixed by which characteristic reaches that height at that time: at a height of 0.10 the construction gives 0.367 and the solve 0.364, at 0.05 it gives 0.427 against 0.425, and at 0.01, 0.512 against 0.509. The shock at the top of the fan sits at 0.153 in the solve against 0.149 from the construction.

The graded layer is a prediction with a practical reading. A settled bed sampled shortly after a batch test is not at packing throughout; it is loose at the top and dense at the floor, and the gradient is not a sign of compression by the weight above — this model has no weight in it — but of characteristics that have not yet arrived.

The interface slows for ever #

The consequence for the question an operator asks — how long until it has settled — is sharp.

The chord-only solution — one sharp front straight to packing — puts the interface on its final height, 0.156 of the column, at $t = 1.86$ and stops it there. At that moment the admissible interface still has 53 per cent of the bed’s thickness to fall. It has 40 per cent still to come at $t = 3$, 23 per cent at 10, 10 per cent at 100, 5.1 per cent at 1000, 2.6 per cent at 10⁴ and 1.35 per cent at 10⁵. The settlement still to come falls as a power of time, and its local exponent over successive decades from 10 to 10⁵ is −0.353, −0.308, −0.292 and −0.281, closing on $-1/(n-1) = -0.274$.

That exponent comes from one fact about the flux law: it arrives at packing with zero slope. The characteristics carrying the last few per cent of concentration below packing travel at speeds proportional to the $(n-1)$th power of the remaining gap, so they crawl, and the interface waits for them. The slow tail is purely kinematic. Nothing in the model resists compression; the bed takes for ever to finish forming because the information that it has finished takes for ever to arrive.

In ordinary units the numbers are memorable. Particles with a Stokes speed of 1 mm/s in a 30 cm column: the interface falls at 0.454 mm/s and meets the rising front after 8.3 minutes, 7.4 cm above the floor; the chord-only answer would have declared the test over at 9.3 minutes with a 4.69 cm bed. By the admissible solution 22 per cent of the bed’s settlement is still to come after an hour, 7.4 per cent after a day, and 2.6 per cent after five weeks.

One settling curve reads the whole flux law #

The slow tail has a use. At any moment after the interface enters the fan, the concentration just below it is a single value $\varphi^+$, and a mass balance on the characteristic that carries it gives a construction older than any computer: draw the tangent to the settling curve at that moment and extend it back to the height axis at $H_t$. Then $\varphi^+ H_t = \varphi_0 H$, and the tangent’s slope is the hindered speed at $\varphi^+$. One batch test, read off a stopwatch and a ruler, therefore gives the flux curve at every concentration the fan passes through.

The construction checks against the solution. At $t = 2$ the tangent’s intercept gives $\varphi^+ = 0.3488$ against the fan’s 0.3486; at $t = 5$, 0.4406 against 0.4406; at $t = 30$, 0.5297 against 0.5296; at $t = 100$, 0.5634 against 0.5633. The flux read off the slope at $t = 5$ is 0.001946 of $v_0$ against the law’s 0.001947. This is how thickeners have been sized from cylinder tests since the 1950s, and it depends on exactly the structure this essay computes: without the fan beneath the interface there would be nothing for the tangent to read.

Which starting concentrations make a shock at all #

The construction depends on where the starting point sits on the curve.

From 0.05 the interface falls at 0.685, the sediment shock rises at 0.086 and the chord-only front would rise at 0.058; from 0.1, 0.454, 0.148 and 0.084; from 0.3, 0.053, 0.164 and 0.047. The admissible front always rises faster than the chord-only one, because it only has to reach the tangent concentration. And at the inflection, 0.227, the tangent point runs into the starting point itself. Above it the curve is convex all the way to packing, the envelope is the curve from the start, and there is no sediment shock at all: the bed grows as a pure fan whose leading edge rises at the curve’s slope at $\varphi_0$ — 0.198 from 0.25, 0.164 from 0.3. The front’s rise peaks near the inflection at about 0.20. The interface and the bed meet later as the suspension thickens — at $t$ = 1.30 from 0.05, 1.66 from 0.1, 2.66 from 0.2, 4.62 from 0.3 and 12.3 from 0.4 — because the interface slows faster than the bed speeds up.

How the column was solved, and why the easier scheme agrees #

The dots in the figures come from a finite-volume solve that knows nothing about tangents: 400 or 800 cells, a closed floor and a closed top, and at each cell face the exact Godunov flux — the least flux over the interval between the two neighbouring states when the concentration increases downward, the greatest when it decreases, which for this curve means checking whether the flux maximum lies between them. Solids are conserved to 10⁻¹⁶.

The interface error is 1.8 × 10⁻² of the column height at 100 cells, 1.0 × 10⁻² at 200, 6.6 × 10⁻³ at 400, 3.6 × 10⁻³ at 800 and 2.0 × 10⁻³ at 1,600: a slope of −0.81, near the first order a first-order scheme smearing a front over a few cells should approach. The sediment shock tracks the tangent’s slope to within 0.004 of the column height throughout its rise.

The first essay on this subject warned that a conservative scheme with the wrong dissipation can converge to the wrong weak solution, and the obvious candidate was run to see. The Murman–Roe flux takes the upwind value in the direction of the chord between two states and nothing else, which is exactly the scheme that produces a stationary expansion shock across a sonic point. Here it reproduces the Godunov column to 2 × 10⁻¹⁶. The reason is specific and worth knowing: the fan at the floor runs from 0.317 to packing, entirely above the flux maximum at 0.113, so the curve’s slope never changes sign inside it and the chord and the curve always agree on which way is upwind. The failure needs a fan that crosses the maximum, and a column settling from a uniform start never makes one. The tangent that the chord rule lacks is supplied here by the boundary and the characteristics, not by the scheme’s dissipation.

What the kinematic column cannot show #

Compression. A real bed near packing carries its own weight through particle contacts, and that effective stress slows consolidation by a mechanism with no place in a flux depending on concentration alone. Measured thickeners show a slow final settlement too, and much of it is that physics rather than this one; the kinematic tail is the part that is there even without it.

Anything that is not a single, equal, inert particle. Mixed sizes segregate and turn the scalar law into a system of several conserved concentrations with several families of waves; flocculating particles change size as they settle; inclined walls speed settling by the Boycott effect; and a suspension released from rest can form fingers and channels that no one-dimensional model contains.

The front’s thickness. The interface drawn as a line is, in a real tube, a zone a few particle diameters thick spread by particle diffusion and polydispersity, the same division of labour The discontinuity that has a thickness describes for a gas shock.

The reversed flow. Pushing liquid up through a bed at the hindered speed holds the particles still instead of letting them fall, and The bed that weighs itself is that fluidised state. The Richardson–Zaki law was first measured there, and the flux curve here is the same curve read in the frame of the particles rather than of the container.

Still open: the thickener that is fed while it settles #

The next calculation opens the column. A continuous thickener receives suspension part-way down at a steady feed flux and withdraws thickened slurry from the floor, and its steady states are set by a construction on the same flux curve: the operating line of the underflow drawn tangent to the batch curve, whose point of contact fixes the thickest underflow a thickener of given area can deliver. It is the tangent of this essay put to work, and computing it — the limiting flux, the steady concentration profile, and what happens to the stored bed when the feed exceeds the limit — would show whether the tangency that decides a batch test also decides the capacity of a plant.

Beside it is the case the Roe flux would get wrong: a dense suspension layered over a dilute one, where the fan must cross the flux maximum. It is gravitationally unstable as well as kinematically awkward, so a real column overturns before either question is settled, and deciding which instability wins is a calculation with a second dimension in it.