A larger sphere overtakes its neighbour and leaves on the line it came in on

Two equal spheres settling together move with one velocity, whatever their slant and separation, and so keep their shape for as long as they fall. A pair five radii apart and slanted at forty-five degrees leans its path three degrees from the vertical and never stops leaning. That essay found the reason in a symmetry of the pair and the absence of inertia, and ended by asking what becomes of the pair when the symmetry is spoiled a little — one sphere a few per cent larger than the other — and whether the faster sphere passes on or is captured.

It passes on, always. But the way it passes is more particular than overtaking. The larger sphere sinks past the smaller round a path that bulges outward as it goes by, and then comes back to exactly the line it arrived on. The smaller sphere, meanwhile, is carried sideways several radii while the larger is above it, and carried back by the same distance once the larger is below. Nothing about the meeting is left over when it is finished. The symmetry that kept an equal pair rigid has been replaced by a weaker one, and the weaker one is still enough to undo the whole encounter.

Two speeds and one coupling #

Two spheres of the same material, of radii $a$ and $\lambda a$, settle alone at speeds in the ratio $\lambda^2$, because the weight less the buoyancy grows as the volume and the Stokes drag as the radius. A sphere two per cent larger settles four per cent faster. Put the two near each other and each is carried by the flow the other’s weight makes. That flow is the same function of position for both spheres — the coupling between them is symmetric, which is the reciprocal theorem again — but the weights that drive it are not equal, and the larger weight belongs to the faster sphere.

So the velocity of the larger sphere relative to the smaller is the difference of their isolated speeds, less a correction made of the coupling times the difference of their weights:

$$\mathbf{V} = (\lambda^2 - 1)\,U_0\,\hat{\mathbf{g}} \;-\; (\lambda^3 - 1)\,6\pi\mu a\,\mathsf{M}(\mathbf{r})\cdot U_0\,\hat{\mathbf{g}},$$

with $U_0$ the smaller sphere’s Stokes speed and $\mathsf{M}$ the coupling. The second term has the opposite sign to the first. Near the smaller sphere, the larger one sinks through the smaller one’s downwash, which helps it; but the smaller one sinks faster still in the larger one’s stronger downwash, and the net effect is that the larger sphere gains on the smaller one more slowly than it would alone. For the coupling this essay uses the Rotne–Prager–Yamakawa form for spheres of different radii, the point force and its finite-size correction, which reduces for equal spheres to the far-field series the equal-pair essay used and agrees with it here to rounding.

The slowing is strong and long-ranged. Along the line of centres, at contact, the overtaking runs at a sixteenth of the mismatch; side by side, at about a third. Five radii apart the larger sphere still gains at only 56 per cent of its free rate in line, and at forty radii — twenty diameters — at 94 per cent. The reason it is so long-ranged is the one that made the equal pair’s speed-up long-ranged: the coupling falls off as one over the distance, the slowest decay a point force in a viscous liquid has.

What the slowing never does is reverse. Along the line of centres the share is $2 - 9/2\rho + 3/\rho^3$ over two, in the far-field coupling and for small mismatches, and it stays positive down to contact: the point-force part alone would bring it to zero at 2.25 radii, just outside touching, and the finite-size part lifts it back. There is no separation at which the two spheres settle at the same speed, so there is no stationary configuration for the pair to be captured into. The question the equal-pair essay ended on has a short answer: in this model a mismatched pair cannot be held.

A path that is its own reflection #

The relative path of the larger sphere is a streamline of the relative velocity field, and that field has a symmetry the equal pair did not need. Reflect the whole configuration in the horizontal plane through the smaller sphere. The coupling depends on the line of centres only through products of its components in pairs, so the reflection leaves the coupling’s vertical action unchanged and reverses its horizontal action: the relative velocity’s horizontal part is an odd function of height and its vertical part an even one.

Read as a statement about paths, that says something definite. Take the part of a path above the smaller sphere and reflect it in the horizontal plane, reversing the direction of travel: the reflected curve is also a solution, and since two solutions through the same point are the same, it is the continuation of the path below. Every relative path is its own mirror image. The larger sphere arrives from above at an offset $b$, bulges outward as the smaller sphere’s downwash pushes it aside, and leaves below at the offset $b$ exactly.

Underneath the geometry is the absence of inertia. The reflection turns the weights from downward to upward, and it is only because reversing every force in Stokes flow reverses every velocity that the reflected problem can be turned back into the original one. A sphere settling at a Reynolds number of a tenth drags a wake that is not the same in front and behind, the field loses its parity, and the larger sphere leaves on a different line from the one it came in on — the order at which the Reynolds number stops being small enough is the order of the displacement.

The computation follows the paths from forty radii above to forty below, with fourth-order steps no longer than a hundredth of the separation, and the exit offsets match the entry offsets to a few parts in ten thousand. The figure’s innermost path, from 0.85 radii, comes within 0.25 radii of contact; paths starting closer in come within a tenth of a radius of touching, where the far-field coupling cannot be trusted with the details. The symmetry argument does not need the details. It holds for the exact two-sphere mobility functions as it does for the far-field ones, because the exact functions have the same dependence on the line of centres; what the far-field coupling cannot supply near contact is how close the path comes, not where it leaves.

The swing an equal pair never finishes #

The equal pair’s sideways lean is still there while the encounter lasts, and it is larger than it looks. Release a pair five radii apart at a slant of forty-five degrees with the upper sphere two per cent larger. The smaller sphere — the lower — falls in the larger one’s downwash, and because the line of centres is slanted, that downwash has a sideways component: it is the same lean that drifted the equal pair three degrees from the vertical. For an equal pair it lasts for ever. For the mismatched pair it lasts until the larger sphere has drawn level, and then reverses, because once the larger sphere is below, the slant leans the other way.

At forty-five degrees the swing reaches seven radii after 158 radii of fall, which is the equal pair’s lean of three degrees acting for that distance and then fading as the larger sphere comes round. By 317 radii of fall the larger sphere is as far below as it began above, and the smaller one is back within five hundredths of a radius of the vertical line it started on — the residue is the integration stopping a step late, not a displacement. At thirty degrees, where the larger sphere starts nearer the vertical and takes longer to come round, the swing is twelve radii; at sixty degrees, three.

The swing is a transient whose size and duration both grow without limit as the mismatch shrinks, and the equal pair is the limit in which it never ends. That joins the two essays without strain: the equal pair’s permanent lean is the swing of an encounter whose other half never arrives.

One over the mismatch #

Both numbers go as one over the mismatch, and the reason is short. The sideways speed is set by the coupling, and does not care about the mismatch at all; the time the encounter takes is the separation over the overtaking speed, and that is proportional to the mismatch. So the swing is a fixed sideways speed for a time inversely proportional to the mismatch, and the fall over which it is undone is the settling speed for the same time.

That turns the equal-pair essay’s laboratory demonstration into a question of manufacturing tolerance. Precision steel balls a millimetre across are ground to diameters within a few tenths of a micrometre, a mismatch of a few parts in ten thousand. For such a pair the swing would be hundreds of radii and the fall that undoes it thousands — metres for millimetre balls — so in any column a laboratory has, a pair of bearing balls is rigid, and the lean of the equal pair is what is seen. Sand grains, sieved to a size band, differ by ten or twenty per cent, and a pair of grains settling together swings a radius or two and is undone within thirty to sixty-five radii of fall: in a suspension of natural grains, every pair is an encounter rather than a pair.

What touching does #

The mirror argument has one escape, and it is contact. A path that starts close to the axis heads for the smaller sphere and, in the exact hydrodynamics, never quite reaches it: lubrication between the surfaces slows the approach as the gap closes, and the gap is never zero. Real surfaces are rough. A sphere carrying asperities a hundredth of its radius high touches its neighbour when the smooth surfaces are still that far apart, and from then on the pair is held at that separation while the flow slides the larger sphere round the smaller. The contact can only push; when the flow would carry the spheres apart, they part.

That breaks the mirror. Contacts are made on the approach, above the smaller sphere, and broken on the retreat, below it, and the place where the path leaves the contact surface is fixed by where the flow first pulls the spheres apart — the same point for every path that touched. So every path that touches leaves on one streamline.

With a roughness of a hundredth of a radius, every larger sphere that arrives within 0.534 radii of the axis touches, slides round, and leaves at 0.534 radii. Arriving at 0.1 it is displaced 0.43 radii; arriving at 0.5, 0.03; arriving beyond 0.534 it misses contact, and the mirror holds again. A rougher sphere, a twentieth of a radius, gathers its touching paths onto 0.589. The displacement is always outward and always onto one line, and the line depends only weakly on the roughness — doubling the asperities moves it by a few hundredths of a radius. Most of the displacement is made by the fact of contact, not by its size.

This is the only way two spheres in Stokes flow can be displaced by meeting, and it is irreversible in the most literal sense: a path that came in at 0.1 and a path that came in at 0.4 leave on the same line, so running the motion backwards cannot tell which was which. Information about the offset has been destroyed at the contact, and destroying information is what a random-looking spread needs. That connects the encounter to what a viscosity made of particles needs, because the same irreversibility, at contacts in a sheared suspension, is what makes a suspension’s microstructure, and so its stress, depend on the direction of shear.

Why a cloud of settling grains spreads anyway #

A cloud of particles settling through a still liquid does spread: watched from the side, its particles wander sideways as they fall, by amounts that grow with time like a random walk’s. The encounter computed here cannot be the cause, for smooth spheres, because each encounter gives back what it took. Three things can be.

The first is roughness, as above: every touching encounter moves the larger sphere outward onto a fixed line, and a sequence of such moves, each at a different azimuth round a different neighbour, is a random walk with a step of a few tenths of a radius. The second is a third sphere. Three spheres settling together rearrange even when they are equal, and the three-body problem of settling is chaotic, as it is for point vortices: a pair’s encounter is interrupted by a third particle before it has finished undoing itself, and the undoing is lost. In a dilute suspension the interruption is the rule once the mean spacing is shorter than the distance an encounter needs, and for a mismatch of ten per cent at five radii that distance is sixty-five radii of fall. The third is the particles’ own inertia, which turns a particle into a low-pass filter of the flow round it and spoils the reversal the mirror depends on at the order of the particle Reynolds number.

A measured lateral spread in a dilute settling suspension is therefore never a record of pair encounters as such. It is a record of how often encounters are cut short, how rough the particles are and how fast they fall — three things a careful experiment can vary one at a time, which is how experiments separate them. The spreading itself is the same kind of quantity as how far a parcel gets in a turbulent flow, a dispersion, but in a settling suspension the randomness has to be put in by the particles because the fluid’s equations contain none.

What the far-field coupling cannot say #

The convention throughout: lengths are in radii of the smaller sphere, speeds in units of its Stokes speed, and the mismatch is the larger sphere’s excess radius as a fraction, both spheres of one material, so that the larger sphere’s isolated speed is $\lambda^2$ and its weight $\lambda^3$ times the smaller’s. Every number is for Stokes flow, at zero particle Reynolds number, in an unbounded liquid.

The coupling is the Rotne–Prager–Yamakawa tensor, a point force with its finite-size correction. It is accurate to a few parts in a thousand at three radii and beyond and wrong in detail inside a radius of contact, where lubrication forces it does not contain take over: the true overtaking speed in line falls to zero at contact rather than to a sixteenth. Smooth paths are drawn only while they stay a tenth of a radius or more from touching; the exit offsets they report are guaranteed by the symmetry rather than by the coupling’s accuracy there. The roughness model is the simplest one — the spheres are held at a fixed separation while the flow presses them together and released when it does not — and it says nothing about friction between the asperities, which would add a rotation the model has not got.

Spheres that differ in density as well as in size can be matched in isolated speed while differing in weight. Then the mismatch term vanishes and the relative motion is made by the coupling alone; its paths run from far away straight into near-contact, where the far-field coupling has nothing to say, so that case is left out rather than drawn from a model that cannot carry it. The figures also leave out the walls of any container, whose image flows add their own drift, and Brownian motion, which for spheres below a few micrometres in water undoes the encounter’s tidy geometry long before the mirror does.

Who saw the encounter whole #

Smoluchowski estimated the interaction of settling particles in 1911 and Stimson and Jeffery solved the equal pair along its line of centres in 1926. Wacholder and Sather computed two unequal spheres settling in 1974, and Batchelor’s papers of 1982, the second with Wen, set out the relative trajectories of a polydisperse pair and their symmetry as the basis of a theory of polydisperse sedimentation. The tensor used here is Rotne and Prager’s of 1969 and Yamakawa’s of 1970, generalised to unequal spheres. Davis and co-workers, from the 1980s, studied how roughness and contact make the encounters of settling spheres irreversible, and the roughness-induced displacement in shear flows is da Cunha and Hinch’s of 1996.

How it was checked #

With no mismatch the relative velocity is zero at every position tried, which is the equal pair’s rigidity recovered. The relative field’s divergence, computed with the azimuthal term the axisymmetry adds, is zero to the differencing error, as a point force and its dipole require. For equal spheres the coupling reproduces the far-field mobility series the equal-pair essay used to rounding at four, six and ten radii. Smooth encounters from five offsets leave within $5 \times 10^{-4}$ radii of their entry, and the smaller sphere’s swing returns within a part in a hundred of its peak. The model refuses a negative size ratio, a negative roughness, a separation inside contact, two identical spheres asked for an encounter they will never have, and a negative offset.

Still open: the third sphere that interrupts #

Every encounter here is between two spheres alone, and its whole content is that it undoes itself if it is allowed to finish. In a suspension it is not allowed to: a third sphere arrives within the sixty-five radii of fall that a ten-per-cent mismatch needs, and the undoing is interrupted part-way. The next calculation would set a third sphere on the larger one’s path at a chosen distance and offset, follow the three with the same coupling, and ask how much of the pair’s swing is left as a permanent displacement when the encounter is cut off at a given fraction of its length — so that a dilute suspension’s lateral spread can be written as the rate at which encounters are interrupted, times what each interruption keeps.