The rod that tumbles for ever in a stream

Momentum going sideways found that viscosity is the transport of momentum across a flow, from faster layers to slower ones, and that in a steady shear — a liquid between two plates, one sliding past the other — the velocity rises in a straight line from one plate to the other. The drag a distant wall decides found that a sphere moving slowly through such a liquid has a drag that belongs to the sphere, set by its radius and the viscosity. Both are statements about the flow and its forces on a body that is either a layer of the fluid or a ball.

A rod is a different kind of body, because it has a direction. Put a short rod into a shearing liquid — a grain of rice in syrup between two sliding plates, a fibre in a polymer melt being extruded, a red blood cell in a capillary — and the question is not only how it moves but how it turns. The answer was given by George Barker Jeffery in 1922, as an exact solution of the slow-flow equations for an ellipsoid. It is one of the strangest results in fluid mechanics, and it is strange for the same reason the sphere’s drag is simple: in a flow too slow for inertia to matter, nothing has a memory except the geometry.

A sphere turns steadily, a rod in fits and starts #

In a steady shear the velocity is $u = \dot\gamma y$ along the flow direction, with $\dot\gamma$ the shear rate. That flow can be split into two parts. One is a pure rotation: the liquid turns as a whole at the rate $\dot\gamma/2$, the vorticity. The other is a pure strain, stretching the liquid along one diagonal and squeezing it along the other, at the same rate. A small sphere suspended in the flow, too symmetric to feel the strain, simply turns with the rotation: at half the shear rate, steadily, for ever.

A rod feels both. The rotation turns it at a steady rate, like the sphere. The strain tries to align it with the stretching diagonal and so pushes it toward the flow direction, more strongly the longer the rod. The two combine into a turning rate that depends on where the rod is pointing. For a spheroid of aspect ratio $r$ — length over width — lying in the plane of the shear, at an angle $\varphi$ from the flow direction, Jeffery found

$$\dot\varphi = -\frac{\dot\gamma}{r^2 + 1}\left(r^2\sin^2\varphi + \cos^2\varphi\right).$$

When the rod is across the flow, $\sin^2\varphi = 1$ and it turns at nearly the full shear rate. When it is along the flow, the bracket is $\cos^2\varphi = 1$ and it turns at only $\dot\gamma/(r^2 + 1)$ — for a rod twenty times as long as it is wide, four hundredths of a per cent of the rate across.

The consequence is a motion of long pauses and quick flips. A rod lying along the flow turns very slowly; the slow turning carries it gradually to a small angle, where the turning speeds up; it swings through the cross-flow direction in a rush and comes back to alignment the other way round, end over end. A rod twenty times as long as it is wide spends more than four-fifths of its time within ten degrees of the flow direction and the rest flipping. To an observer watching only for a while, it would seem to have lined up. Then it flips.

The flipping looks, at first sight, like the tumbling of the axis that will not hold, the book spun about its middle axis that turns end over end in the air with nothing touching it. The resemblance is only in the picture. A freely spinning body flips because its own angular momentum is conserved and rotation about the middle axis is unstable; its flips are a property of the body. A rod in a viscous stream has no angular momentum worth mentioning — its inertia is negligible against the liquid’s viscous torque — and it flips because the flow turns it at a rate that depends on its orientation. Stop the flow and the rod stops at once, in whatever orientation it has reached; stop the book’s spin and nothing is left to flip.

The period that does not depend on the orbit #

The time for a full turn is the integral of $1/|\dot\varphi|$ round the circle, and it comes out with no dependence on where the rod started:

$$\dot\gamma T = 2\pi\left(r + \frac{1}{r}\right).$$

The period is smallest for a sphere, $4\pi$ units of strain, which is a full turn at half the shear rate. It grows linearly with the aspect ratio for long rods and with its reciprocal for flat discs: a disc tumbles like a coin flipped by the flow, with the same period as a rod of the inverse shape. A typical paper fibre, a few hundred times as long as it is wide, takes thousands of units of strain to flip; in a shear of a hundred per second that is about ten seconds per flip, nearly all of it spent lying still along the stream.

So far the rod has been confined to the plane of the shear. In three dimensions its axis can point anywhere, and Jeffery found a second surprise there.

Each starting orientation puts the rod on a closed orbit, labelled by a number now called the orbit constant, and the rod follows that orbit for ever. A rod started in the plane of shear tumbles in it. A rod started along the vorticity axis — perpendicular to both the flow and the velocity gradient — spins slowly about its own length, rolling like a log on the stream. A rod started in between sweeps out a path that looks, from the side, like the stroke of a kayak paddle: its tip dips toward the plane of shear while it flips, and rises toward the vorticity axis while it lingers near alignment.

And every one of those orbits has the same period, $2\pi(r + 1/r)/\dot\gamma$. The integration behind the figure measures it for each orbit separately rather than assuming it, and the orbits close on themselves after exactly that time. A tumbling rod and a rolling one, given the same shape, return to their starting orientations together.

Nothing to choose an orbit #

The family of orbits is a family of neutral equilibria. The flow does not push a rod from one orbit to another, and it does not favour any. Whatever orbit a rod is put on, it stays on, and a small disturbance moves it to a neighbouring orbit, where it then stays.

That is a mathematical consequence of the slow-flow equations, which the drag a distant wall decides and the earlier essays on viscosity worked within. When the Reynolds number is small — when viscous forces dominate the inertia of the fluid and of the particle — the equations of motion are linear and contain no time derivative of the flow. The rod’s angular velocity at each instant is fixed entirely by its orientation and the shear at that instant. The motion is a flow on the sphere of orientations with no friction-like term to make it settle, and in a flow of that kind, closed orbits are what generically appears.

This is also why the rods in a pipe sort themselves differently from those between plates. In the parabolic flow the fourth power in a pipe described, the shear rate is zero on the axis and largest at the wall, so a rod near the axis barely turns at all while one near the wall flips quickly; each follows its own Jeffery orbit at its own local rate, and a population that entered the pipe isotropic leaves it with an orientation that depends on the radius at which each rod travelled. Red blood cells in a vessel, discs rather than rods, tumble and tank-tread in the same way, and their orientation, which decides how much they add to the blood’s resistance, follows the local shear across the vessel.

In practice orbits are selected all the time, by every effect the model leaves out. A rod with even a little inertia drifts toward the tumbling orbit in the plane of shear; in a viscoelastic liquid — the kind that climbs a rotating rod — it drifts toward the log-rolling orbit, aligned with the vorticity; Brownian motion, for a rod small enough, spreads it over all orbits with a distribution that depends on the ratio of the shear rate to its rotational diffusion. Which of these wins decides how fibres orient in a molten plastic as it is injected into a mould, and therefore how strong the moulded part is in each direction.

A suspension that cannot settle #

The consequence that is easiest to see is in a suspension. Take a dilute suspension of identical rods, spread evenly over every orientation in the plane of the shear, and shear it. Each rod follows its own Jeffery orbit. Because each has the same period, the population behaves as a single object: it bunches toward the flow direction as the rods linger there, flips in a crowd, and returns to its starting distribution after a full period — then does it again.

This is what makes a suspension one of the liquids the fluid that answers back found departing from Newton’s law of viscosity: its resistance depends on the history of the flow, through the arrangement of what is suspended in it, and not only on the shear rate at the moment. A rod adds to a liquid’s resistance to shear mostly when it is caught across the stretching direction, at forty-five degrees to the flow, where the strain must work hardest to move liquid round it; when it lies along the flow it adds very little. For slender rods the extra stress from each is proportional to $\sin^2\varphi\cos^2\varphi$. Averaged over the population, that factor starts at one-eighth, falls toward zero as the rods line up, rises again as they flip together, and repeats. The suspension’s viscosity oscillates in time under a perfectly steady shear, with a period of half a Jeffery orbit.

The oscillation was seen by Anczurowski and Mason in the 1960s, in suspensions of carefully made rods and discs, as a periodic variation of the orientation distribution that persisted for many periods before collisions and imperfections damped it. In practical suspensions — fibres in a polymer, clay particles in water, the starch grains of the suspension that seizes when pushed hard — the damping is quick, because the rods differ in length, bump into each other and diffuse, and the viscosity settles to a value that depends on all of those. The model’s oscillation shows what the settling has to undo: not a natural tendency to equilibrate, but a perfect memory of the starting state that only imperfections erase.

Shear it back #

The perfect memory has a second consequence that can be demonstrated on a desk.

Because the slow-flow equations contain no inertia, a rod’s motion depends only on the shear, not on how long it has been going or how fast. Reverse the shear and every rod’s angular velocity reverses. Shear forward by some amount, then backward by the same amount, and every rod retraces its path exactly and returns to where it began. The population, which had been bunched and flipped and bunched again, comes back to its starting distribution, one rod at a time.

This is G. I. Taylor’s demonstration of kinematic reversibility in a different form. Taylor filled the gap between two concentric cylinders with glycerine, injected a blob of dye, turned the inner cylinder several times until the dye was smeared into an invisible sheet round the annulus, and then turned it back the same number of times: the blob reassembled. The rods do the same thing with their orientations, and the explanation is the same. In creeping flow, time has no direction that the equations can tell; only the history of the boundary’s motion matters, and reversing it undoes everything. Diffusion is the exception, because it is irreversible in the sense the equation that only runs forwards described: a spreading blob of dye cannot be unspread by reversing anything, and in Taylor’s experiment the reassembled blob is slightly blurred by exactly the diffusion that happened during the stirring. Rods of a few micrometres suffer the rotational version — Brownian rotation, which spreads them over orbits and cannot be reversed — and their reversal is correspondingly imperfect. Mixing in a viscous liquid requires either inertia, diffusion, or a sequence of motions whose reversal is not itself one of the allowed motions — which is why microscopic swimmers cannot swim by any reciprocal stroke, and why a scallop opening and shutting its shell in syrup would go nowhere.

Where the orbits stop #

The rods are rigid and alone. Jeffery’s solution is for a single ellipsoid in an unbounded liquid. Rods within a few lengths of each other interact through the flow they disturb, and in suspensions above a volume fraction of about $1/r^2$ they touch. Both destroy the degeneracy of the orbits, and concentrated fibre suspensions in industry are dominated by those interactions. Flexible fibres bend as they flip, and long ones can buckle into loops.

The particle is neutrally buoyant and small. A rod much denser than the liquid sediments, and its settling couples to its turning; a rod comparable in size to the gap between the plates feels the walls, which slow its flip and can trap it near alignment.

The Reynolds number is zero. For a rod a millimetre long in glycerine, sheared at a few per second, the particle Reynolds number is a few thousandths, and Jeffery’s motion is a good description; the same rod in water is at a Reynolds number near one, and it is not. At Reynolds numbers of order one the rod’s own inertia and the liquid’s push it toward the tumbling orbit, and at a few tens to hundreds, depending on its shape, it stops tumbling altogether and settles at an angle: the fixed point the creeping-flow equations could not have.

The liquid is Newtonian. In a viscoelastic liquid, normal stresses push rods toward the vorticity axis, where they roll like logs instead of tumbling. That drift is what aligns fibres across a flow in some polymer processing, the opposite of the alignment along the flow that a Newtonian liquid’s long pauses suggest.

Still open: how fibres orient in a crowded flow #

Industrial fibre suspensions are rarely dilute. Glass and carbon fibres in moulded plastics are present at a tenth or more of the volume, long enough to touch each other many times per flip, and the orientation they end up with, frozen in when the plastic sets, decides the stiffness and strength of the part in each direction. The standard models for predicting it start from Jeffery’s equation and add a term for the effect of interactions — a rotational diffusion proportional to the shear rate, introduced by Folgar and Tucker in 1984 — with a coefficient that is fitted to experiments rather than derived.

That fitted term works well enough to be used in commercial moulding software and badly enough that a series of corrections have been added to it: slowing the rate at which orientation changes, accounting for fibre flexibility, distinguishing short-range contacts from long-range flow interactions. Whether a single physical model can be derived from Jeffery’s equation and the mechanics of fibre contacts, without adjustable coefficients, and predict the orientation of concentrated suspensions in complex moulds, has not been settled.

The single rod, by contrast, is completely understood, and it is a useful thing to keep in mind about every flow too slow for inertia. A creeping flow has no way to forget. It moves bodies according to where they are and nothing else, so every arrangement it produces can be undone by running it backwards, and every orbit it allows is as good as every other. Whatever settles a suspension — Brownian motion, collisions, the liquid’s elasticity — is something that has been added to the viscous flow, not something the viscous flow does by itself.