A viscosity the flow cannot decide

A viscosity made of particles derives Einstein’s five halves and then lists what it does not depend on: the particles’ size, their density, the shear rate, the fluid. A dilute suspension of rigid spheres is Newtonian, with a viscosity decided by its volume fraction alone. The list is complete for spheres because a sphere has nothing else it could depend on. It has no orientation.

A rod does. Stir fibres, needles or elongated crystals into a liquid and shear it, and each particle turns in the flow, and the stress it adds depends on which way it points while it turns. The question this essay answers is whether that makes the viscosity merely harder to compute or leaves it undetermined. The answer is the second, and the reason is one of the strangest properties of slow viscous flow.

A rod tumbles round an orbit it never leaves #

George Jeffery solved the problem in 1922 for a rigid spheroid of aspect ratio $r$, force-free and torque-free, in a simple shear flow $u = \dot\gamma y$. The particle’s axis $\mathbf{p}$ turns according to

$$\dot{\mathbf{p}} = \boldsymbol\Omega\cdot\mathbf{p} + \lambda\left(\mathbf{E}\cdot\mathbf{p} - (\mathbf{p}\cdot\mathbf{E}\cdot\mathbf{p})\,\mathbf{p}\right), \qquad \lambda = \frac{r^2 - 1}{r^2 + 1},$$

where $\boldsymbol\Omega$ and $\mathbf{E}$ are the flow’s rotation and rate of strain. The rotation turns every particle at the same rate; the strain turns an elongated one towards the stretching direction, and $\lambda$ says how much it cares. A sphere has $\lambda = 0$ and simply spins with the fluid.

Every solution is periodic. The axis traces a closed curve on the unit sphere, and the curves form a family labelled by a constant $C$ that the motion never changes: $C = \tan\theta\,\sqrt{r^2\cos^2\phi + \sin^2\phi}\,/\,r$, with $\theta$ measured from the vorticity axis and $\phi$ from the gradient direction in the plane of shear. At $C = 0$ the rod lies along the vorticity axis and rolls on it like a log. As $C$ grows, the rod’s tip sweeps a wider loop, and at $C = \infty$ it tumbles end over end in the plane of shear.

Integrated for ten periods with a fourth-order Runge–Kutta scheme, a rod’s orbit constant stays fixed to $10^{-7}$, the worst of three orbits, and it returns to its starting angle to the same precision. The flow knows about the orbit and conserves it.

Lying down, and flipping in a hurry #

The motion within an orbit is as distinctive as the orbit. The rate at which a rod turns in the plane of shear is $\dot\gamma(r^2\cos^2\phi + \sin^2\phi)/(r^2 + 1)$: fast when it lies across the flow, slow — by a factor of $r^2$ — when it lies along it. So a long rod spends almost all its time lying down in the stream and occasionally flips over, rapidly, to lie down the other way round. The period of a whole cycle is $2\pi(r + 1/r)/\dot\gamma$, which is 63.5 shear times for a rod ten times as long as it is thick, and the same on every orbit.

That last fact matters later. Every rod of a given shape has the same period no matter which orbit it is on, so a population of identical rods that start in step stays in step for ever.

The stress an orbit carries #

A rod adds stress to the suspension because it resists being stretched. The flow tries to stretch the fluid along the strain’s extensional diagonal; a rigid rod lying along that diagonal refuses, and the fluid around it has to shear harder to make up the difference. For a slender body the result, due to Batchelor in 1970, is that the rod’s contribution to the stress is proportional to $(\mathbf{pp}:\mathbf{E})\, \mathbf{pp}$ — to how hard the flow is trying to stretch it along its own length — with a coefficient $r^2/(\ln 2r - 3/2)$ per unit volume fraction. In a simple shear that makes the orientation part of the intrinsic viscosity, the extra viscosity per unit volume fraction,

$$[\eta]_p = \frac{r^2}{\ln 2r - 3/2}\,\langle p_x^2\,p_y^2\rangle,$$

averaged over time along each rod’s orbit and over the rods present. The product $p_x p_y$ is the component of the axis along the stretching diagonal, squared once for the stretch and once for the stress it produces.

A rod rolling on the vorticity axis has $p_x = p_y = 0$ at every instant. It lies along the one direction the shear does not stretch at all, and it adds nothing. A rod tumbling in the plane of shear adds the most, though still less than a rod held at forty-five degrees would, because it spends most of its time lying along the flow where $p_y$ is small and passes through the stretching diagonal only during its flips. For aspect ratio ten the orientation part runs from zero on the vorticity axis through 1.79 at $C = 1$ to 2.76 in the plane. At aspect ratio twenty the range is from zero to more than five.

The viscosity is a function of the orbit. That is a complication, not yet a paradox.

Nothing in slow flow moves a rod between orbits #

The paradox is that the orbit is never decided. Slow viscous flow is linear and reversible: run the shear backwards and every rod retraces its path exactly, which is why nothing swims by reciprocal strokes and why a dye streak sheared in a Couette cell can be unmixed. A process that could move a rod from one orbit to another systematically — towards the vorticity axis, say — would have to move it back when the shear reversed, and a drift that reverses with the flow averages to nothing. The Stokes equations have no mechanism for selecting an orbit, and Jeffery’s solution confirms it: every orbit is a solution and each is neutral.

So the suspension’s viscosity is a functional of the distribution of orbit constants among its rods, and that distribution is whatever it was when the shear began. It is a memory, and nothing in the equations of motion ever erases it. Rods poured in from a bottle and stirred have one distribution; rods extruded through a nozzle, lying along the flow in the plane of shear, have another; rods that have previously been sheared in an oscillating flow that pushed them towards the vorticity axis have a third. Each is a legitimate, permanent, steady state of the Stokes problem, and each has a different viscosity.

This is not the dependence on history that a viscoelastic fluid has. A polymer solution remembers its recent deformation and forgets it over a relaxation time. A dilute rod suspension in slow flow, taken literally, never forgets anything: its relaxation time is infinite, and its steady viscosity is not a property of the material.

The smallest noise decides, and its size does not matter #

Real suspensions do settle to a definite viscosity, so something outside the model decides the orbits. The cleanest candidate is rotary Brownian motion: thermal agitation turns every rod randomly at a rate set by a rotary diffusivity $D_r$. If $D_r$ is comparable with the shear rate, the rods are simply jostled towards an isotropic distribution and Jeffery’s orbits are irrelevant. The interesting case is $D_r$ far smaller than the shear rate — so small that within any one orbit the noise does nothing noticeable.

The noise still matters, because it acts for ever and the orbits are neutral. Over a time of order $1/D_r$ it moves rods across orbits, and the distribution it builds is decided by a balance between diffusion across the orbits and the way the Jeffery motion weights different parts of each orbit — a balance in which $D_r$ appears on both sides and cancels. The limiting distribution is independent of how weak the noise is. Leal and Hinch computed it in 1971.

The simulation here does it by brute force: three hundred rods of aspect ratio five, started isotropic, turned by Jeffery’s equation and kicked by a seeded random rotation for three diffusion times, at two diffusivities, three and one thousandths of the shear rate. The weak limit needs $D_r$ well below $\dot\gamma/r^3$, which for these rods is about $8 \times 10^{-3}$. The two runs select the same $\langle p_x^2 p_y^2\rangle$ to 0.1 per cent — 0.03925 and 0.03929 — and the second run’s direct time average, which includes every kick within every orbit, agrees with the value reconstructed from its orbits to half a per cent. A thousandth of the shear rate is weak enough.

What makes this a singular limit rather than a small correction is the order of the operations. With no noise at all, the viscosity is undetermined. With noise of any strength at all, below a threshold, it is determined, and it is the same number. The viscosity of the limit is not the limit of the viscosities, because at zero there is no viscosity to take a limit of.

How weak is weak, for real rods #

The rotary diffusivity of a rod falls as the cube of its length, which makes the weak-noise regime the ordinary one for anything visible. For a rod of length $L$ and aspect ratio $r$ in a liquid of viscosity $\mu$, $D_r \approx 3k_BT\ln r/(\pi\mu L^3)$. In water at room temperature a rod ten microns long and a micron thick turns diffusively at about $10^{-2}$ per second; a rod of the same shape a hundred microns long, a thousand times more slowly.

The weak limit needs $D_r$ below the shear rate divided by $r^3$. For the ten-micron rod that is a shear rate above about ten per second — gentle stirring. For the hundred-micron rod it is a shear rate above a hundredth of a second, which is no condition at all, and the millimetre-long glass and wood-pulp fibres of industry, longer and thinner still, clear it by a wider margin. So the fibres of paper-making and fibre-reinforced moulding are always in the regime in which slow flow cannot decide their orbits and something negligible must, while colloidal rods a micron long cross from the strong-noise regime to the weak one as the shear rate rises through the values a rheometer spans. The second case is where the shear-thinning in the next figure is actually measured.

The same arithmetic says how long the deciding takes. A rod’s orbit distribution relaxes over a few diffusion times, and for the hundred-micron rod in water that is several days. Any process shorter than that — a pass through a mould, a few seconds in a pump — never reaches the limit, and its viscosity is the viscosity of whatever orbits the fibres entered with.

One suspension, five viscosities #

For rods of aspect ratio five, the orientation part of the intrinsic viscosity is zero with every rod on the vorticity axis, 2.16 with every rod tumbling in the plane of shear, and anything between under some other preparation. An isotropic start that the flow then keeps gives 1.24. Weak Brownian motion gives 1.22, close to the isotropic start by coincidence of this particular average rather than of the distributions, which the figure above shows are visibly different: the noise empties both ends, taking rods off the vorticity axis and out of the plane of shear alike.

Strong Brownian motion, fast enough to keep the rods isotropic against the shear, gives 2.08 — the isotropic average $\langle p_x^2 p_y^2\rangle = 1/15$ times the coefficient. The fall from 2.08 at slow shear to 1.22 at fast shear is a shear-thinning that comes from orientation alone, with a Newtonian liquid and rigid particles. It is the same shape of behaviour a polymer solution’s shear-dependent viscosity shows, produced by a completely different mechanism, and its high-shear plateau is the number the weak-noise limit decides.

Noise is not the only candidate, and each gives a different answer #

The argument that fixed the weak-noise limit applies to any weak effect that breaks the Stokes equations’ reversibility, and there are several. Particle inertia, once the particle Reynolds number is not quite zero, makes elongated rods drift towards tumbling in the plane of shear. Weak elasticity in the suspending liquid makes them drift the other way, towards the vorticity axis, which is how a viscoelastic liquid lines fibres up across a flow. Hydrodynamic interactions between rods in a suspension that is not quite dilute act like a noise whose strength scales with the shear rate itself, the effect that fibre-composite moulding models represent with an interaction coefficient.

Each of these is weak, and each decides the orbit distribution completely, because it acts on a neutral family for as long as the flow runs. And they do not decide it the same way. The viscosity of a dilute rod suspension is set by whichever of several negligible effects is least negligible, and a suspension in which two of them are comparable has a viscosity that depends on their ratio, not on their size. That is the practical content of the paradox, and the reason fibre-suspension rheology is measured rather than predicted.

A suspension that rings #

The equal periods of all orbits have a visible consequence. Start a suspension of identical rods all lying along the flow — as they would after flowing through a contraction — and switch on the shear. Every rod dwells, then every rod flips at the same moment, every half-period. The stress spikes at each collective flip and falls back between them, and in slow flow the spikes never decay, because nothing dephases rods that share a period.

Real rods are not identical. With aspect ratios spread ten per cent either side of ten, the periods spread by about the same fraction, the flips drift out of step, and within four or five periods the stress settles to a steady value. Damped oscillations of exactly this kind were measured in the viscosity of rod suspensions started from alignment by Ivanov, van de Ven and Mason in 1982, and the damping rate is a measure of the spread of shapes. It is the one place where the collective memory of the orbits appears directly in a rheometer’s reading, and it is the reason a fibre suspension’s start-up transient is a record of how it was loaded rather than of what it is.

The numbers, and what they were checked against #

The orbits and the period are the foundation, and both were checked against a long integration: the orbit constant drifts by $10^{-7}$ over ten periods and the period from the orbit quadrature agrees with $2\pi(r + 1/r) = 63.4602$ to six figures. The viscosity along each orbit is computed by quadrature in the angle, with the time weighting $d\phi/\dot\phi$ taken from Jeffery’s rate, which avoids integrating through the fast flip.

What the picture cannot show #

Slender-body stress to leading order. The coefficient $r^2/(\ln 2r - 3/2)$ is the first term of an expansion in $1/\ln 2r$, and the part of the stress that does not depend on orientation, which is of order one, is dropped. At aspect ratio five the neglected terms are not small; the figures are right about which histories give more and which less, and approximate about the absolute values.

Dilute and non-interacting. Every rod turns in the undisturbed shear. Rods whose length is comparable with their spacing — a volume fraction above about $1/r^2$ — interact, and the interaction itself becomes the noise that picks the orbits.

Rigid, neutrally buoyant, far from walls. Real fibres bend when long and thin enough, and a bending fibre follows orbits of its own; a wall within a rod length changes the flow the rod turns in.

Stokes flow. No inertia in the fluid or the particle, which is exactly the assumption whose failure, at small but finite Reynolds number, is one of the effects that decides the orbits.

The convention the numbers depend on #

The intrinsic viscosity is the increase in viscosity divided by the viscosity of the liquid and by the volume fraction of particles, in the limit of small volume fraction; only its orientation-dependent part is shown. The aspect ratio is length over thickness. The orbit constant is reported as $C/(1 + C)$, which runs from zero for a rod on the vorticity axis to one for a rod in the plane of shear. Time is in units of one over the shear rate, and the rotary diffusivity in units of the shear rate.

Who found it, and when #

Jeffery published the orbits in 1922 and suggested that some slow effect would eventually carry every particle to the orbit of least dissipation; G. I. Taylor watched ellipsoids turning in a viscous shear the following year. Mason and his collaborators spent the 1950s and 1960s measuring orbit distributions in sheared suspensions and found them drifting slowly over many periods, which was the sign that something outside the Stokes problem was acting. Batchelor gave the slender-body stress in 1970. Leal and Hinch, in 1971 and 1972, showed that weak Brownian motion selects a unique distribution independent of its strength and computed the resulting viscosity. The damped oscillations of start-up flows were measured by Ivanov, van de Ven and Mason in 1982.

It belongs beside the reciprocal theorem and the world with no inertia as another consequence of slow flow’s reversibility, and beside Einstein’s spheres as the case in which reversibility stops being a convenience and becomes an obstacle.

Still open: which negligible effect wins #

The weak-noise limit is one of at least three candidates, and the calculation that follows computes the others on the same footing. Particle inertia at small Reynolds number drifts rods towards the plane of shear; weak elasticity drifts them towards the vorticity axis; a small Brownian diffusivity spreads them. Each has its own limiting distribution when acting alone, and when two act together the answer depends on the ratio of their strengths — a one-parameter family of viscosities between two limits, decided by a number no rheometer measures directly. Computing that family for a rod of aspect ratio ten, and asking at what particle size the inertial drift overtakes Brownian motion in water, would say which fibre suspensions have a viscosity that can be predicted at all. Beside it is the same question for particles that are not quite carried by the flow, where the inertia is the particle’s own rather than the fluid’s.