The fluid that answers back

The two previous rungs assumed something without dwelling on it: that the stress needed to shear a fluid is proportional to how fast it is being sheared. Water obeys that assumption over a range of shear rates spanning many orders of magnitude. A great many fluids that matter do not obey it at all.

Once the proportionality fails, the question “what is this fluid’s viscosity?” stops having an answer, and the honest reply is another question: at what shear rate?

What the assumption actually was #

Newton’s proposal — in the Principia, as a hypothesis rather than a result — was that the resistance of a fluid to being sheared is proportional to the rate at which its parts are separated. In modern terms

$$\tau = \mu \dot{\gamma},$$

with $\dot{\gamma}$ the shear rate and $\mu$ a constant. The important word is constant: independent of $\dot{\gamma}$, independent of how long the shearing has gone on, independent of what happened before.

Each of those independences can fail separately, and the three failures name three families:

• Stress not proportional to rate: shear-thinning and shear-thickening fluids.
• No flow at all below a threshold: yield-stress fluids.
• Stress depending on the deformation history: viscoelastic and thixotropic fluids.

A relation between stress and deformation is called a constitutive law, and choosing one is a modelling decision rather than a derivation — which is why this whole area has a name of its own.

Thinning, which is the common case #

Most complex fluids get easier to shear the faster they are sheared. A power law captures much of it:

$$\tau = k \dot{\gamma}^{\,n}, \qquad n < 1.$$

Dividing by $\dot{\gamma}$ gives an apparent viscosity $k\dot{\gamma}^{\,n-1}$, which falls as the rate rises.

The mechanism is usually structural. A polymer solution contains long tangled molecules; shearing aligns them with the flow, in the way an applied field aligns anything with an orientation, and aligned molecules slide past one another more easily than tangled ones. A suspension of irregular particles has them tumbling randomly at rest and oriented in the flow when sheared. In both cases the fluid has microstructure, and shearing destroys some of it.

The engineering consequences are large and mostly deliberate:

Paint. Brushing is a high shear rate, so the paint thins and spreads. Once brushed, the rate falls to whatever gravity supplies, the viscosity climbs back, and the paint stays where it was put instead of running. Both requirements are met by one fluid because they are requirements at different shear rates.

Blood. Red cells aggregate at rest and disaggregate in flow, so blood is markedly shear-thinning. At the low rates in veins it is several times more viscous than at the high rates in arteries, which matters for the pipe-flow law that is often applied to it uncorrected, and which compounds with the fourth-power sensitivity to vessel radius rather than offsetting it.

Drilling mud. It must carry rock cuttings up a borehole while circulating and hold them suspended when circulation stops. A shear-thinning fluid with a yield stress does both.

Thickening, which is rarer and stranger #

A few fluids go the other way, and the best known is cornstarch in water at high concentration.

At low shear rates the suspension flows like a thick liquid. Increase the rate and the apparent viscosity rises, sharply — at high enough rates the material behaves as a solid, cracking rather than flowing. A person can run across a pool of it and will sink if they stand still.

The mechanism at moderate rates is that the particles are forced into clusters that cannot pass one another without dilating the packing, which the surrounding liquid must be drawn into. At the extreme it is frictional: particles are pushed into direct contact and the suspension jams into a temporary solid network.

The practical uses turn on that abruptness. Some body armour uses a shear-thickening fluid impregnated into fabric: soft and flexible at walking speeds, rigid at impact speeds. The material does not decide to be hard; the impact supplies a shear rate at which it already is.

The yield stress, which is a different failure #

The Bingham fluid in the figure does not start at the origin. Below a threshold stress it does not flow at all — it deforms elastically and returns, like a solid — and above it flows with a roughly constant differential viscosity.

That is not a large viscosity. It is a qualitative difference: the substance is a solid until pushed hard enough, and then a liquid.

Toothpaste holds a ridge on a brush because gravity’s stress is below its yield point, and comes out of the tube because squeezing exceeds it. Ketchup is the domestic case everybody has argued with, and striking the bottle works by supplying, briefly, a stress above the yield point. Wet concrete can be heaped, and a slump test is a measurement of yield stress dressed as a site procedure — the height a cone of it settles by is a direct reading of the stress at which it stops behaving like a fluid that finds its own level. Mayonnaise, gels and greases all live here.

Because a yield stress means the material is solid at low stress, it also means such fluids can hold particles suspended indefinitely — which is why a paint’s pigment does not settle, and why a drilling mud does not drop its cuttings when the pumps stop.

Fluids that remember #

The third failure is the strangest, because it puts a time into a fluid’s description.

A viscoelastic fluid responds partly like a liquid and partly like a solid, with a relaxation time $\lambda$ separating the two regimes. Deform it faster than $\lambda$ and it behaves elastically, storing energy and springing back. Deform it slower and it flows.

The consequences look like conjuring tricks and are all one property. A ball of silicone polymer bounces when thrown and spreads into a puddle when left overnight. Bread dough pulls back when stretched quickly and sags when left. A polymer solution being poured can be cut with scissors and the falling part will climb back up. And a rotating rod in a polymer solution makes the liquid climb up the rod rather than being flung outward, because the stretched molecules wrap it and squeeze.

The number that decides which behaviour appears is the ratio of the fluid’s relaxation time to the timescale of what is being done to it. Fast enough, everything is elastic; slow enough, everything flows. Even window glass is on this spectrum, at a relaxation time so enormous that no experiment can reach it — which is the honest version of the claim that glass is a very slow liquid, and it is not the same claim as the one about old windows.

The elastic half of the behaviour is the part a Newtonian fluid does not have. A viscoelastic fluid stores energy and dissipates it at the same time, and which one dominates is a question about how fast it is being asked — slow enough and it flows, fast enough and it bounces. That is why the same material can be poured and shattered, and why the phrase “is it a solid or a liquid” has no answer without a timescale attached.

Why the pipe law changes shape #

Everything on the previous rung assumed a Newtonian fluid, and it is worth seeing what a shear-thinning one does to it.

In a pipe the shear rate is zero on the axis and greatest at the wall. For a shear-thinning fluid that means the viscosity is highest on the axis and lowest at the wall — so the fluid near the middle moves nearly as one plug while the shearing is concentrated in a thin region near the wall.

The profile is blunter than a parabola, and in the limit of strong thinning it approaches plug flow. The flow no longer goes as $R^4$; for a power-law fluid the exponent is $(3n+1)/n$, which is 4 for $n = 1$ and rises as $n$ falls — 7 for $n = 0.5$. So a shear-thinning fluid is more sensitive to pipe radius than water, not less.

That result matters for anybody pumping a slurry, and it is a good example of the general lesson: replacing a constitutive law does not merely change a constant, it changes exponents.

The parabola is what a constant viscosity produces in a pipe, and a shear-thinning fluid does not give one. Its viscosity is lowest where the shear is highest — at the wall — so the profile flattens in the middle and steepens near the edge, approaching plug flow. That change of shape is why the fourth-power law fails for such fluids, and why pumping them is a different engineering problem rather than the same one with a different number.

Where the microstructure comes from #

It is worth being explicit about what all these fluids have in common, because it is the thing water does not have.

Every non-Newtonian fluid on this page contains something with a structure at an intermediate scale — long molecules, particles, droplets, cells, aggregates. Water is molecules and nothing else, so shearing it has nothing to rearrange, and its response cannot depend on the rate at which it is asked. Add anything with a shape and a size, and shearing starts to compete with whatever process restores that shape.

That competition is what sets the rate at which the behaviour changes. A polymer coil relaxes back to a random shape in a characteristic time; a suspension’s particles are randomised by the same molecular bombardment that makes small particles wander. Shear faster than that restoring process and the structure stays disturbed; shear slower and it keeps up. So the crossover rate is not arbitrary — it is the reciprocal of a relaxation time belonging to the microstructure.

This also explains why non-Newtonian behaviour is so common among things that matter and so rare among simple liquids. Foods, biological fluids, paints, muds, cosmetics and molten plastics are all structured materials. Solvents, molten metals and gases are not, and they are all Newtonian to excellent accuracy.

The restoring process is what makes the whole subject a competition. In a suspension the particles are randomised continuously by molecular bombardment, and whether a flow can hold them out of their random arrangement is a race between the shear rate and the diffusion rate. The ratio of the two is the Péclet number, and the shear-thinning curve is that ratio passing through one.

Measuring something that is not a number #

If viscosity depends on shear rate, a measurement must control the shear rate rather than merely apply a force. That is what a rheometer does: it imposes a known rate, measures the resulting stress, and sweeps the rate over decades to trace out the curves in the figures.

Two things follow. First, a single-point measurement of a non-Newtonian fluid is nearly meaningless unless the rate is stated — and a great many published viscosities are exactly that. Second, the shape of the curve is the material’s identity: two fluids can have identical apparent viscosities at one rate and behave completely differently everywhere else.

There is a further complication for fluids that remember. Their response depends on how long they have been sheared, so the measurement must specify a history as well as a rate. Thixotropic fluids — which thin over time under constant shear and recover at rest — will give a different answer on a rising sweep than on a falling one, and the loop between the two is itself the measurement.

One fluid, several names, depending on the question #

A theme worth drawing out before the limits: the same substance appears in different families depending on which timescale it is asked about, and none of the labels is wrong.

Blood is shear-thinning, and it is also viscoelastic, and in the finest capillaries it is not a continuum at all. Which description to use is set by the vessel and the heartbeat, not by the blood. Wet sand is a yield-stress material at slow deformation and a shear-thickening one under impact — which is why a beach is firm underfoot and why a hard enough blow shatters rather than compresses it. Molten polymer is viscoelastic on the timescale of an extrusion die and a simple thinning liquid on the timescale of a slow flow down a channel.

The general statement is that a constitutive law is a description of a material in a range, in exactly the sense that every model in this collection is a description in a range. What is unusual about rheology is that the range is so narrow and the departures so visible that nobody can pretend otherwise: a fluid that bounces when thrown and puddles overnight makes the point without any apparatus.

That has a practical corollary. Asking for a material’s rheology is not asking for a number or even a curve — it is asking for a curve over the rates and times that the application will actually produce. A paint measured only at brushing rates says nothing about whether it will sag, and one measured only at rest says nothing about whether it will brush.

A relation that holds excellently up to a point, departs gradually, and is badly wrong well beyond is the general shape of an approximation with a range — and Newtonian viscosity is one. The useful question is never whether a fluid is Newtonian but over what range of shear rates it behaves as though it were, which is why the same material carries several names in several industries and each name is right within its own range.

The two numbers that decide which fluid it is #

The essay has twice appealed to a ratio between the fluid’s own time and the time of whatever is being done to it. That ratio has a name, and so does a second one that is constantly confused with it — and keeping them apart settles most arguments about what a fluid “is”.

The Deborah number is the fluid’s relaxation time divided by the time over which the observation is made, $\mathrm{De} = \lambda/T$. It asks whether the material has had a chance to relax while it was being watched. A silicone ball thrown at the floor is observed over a millisecond and relaxes in a second, so $\mathrm{De}$ is a thousand and it behaves as a solid; left overnight, $\mathrm{De}$ is $10^{-5}$ and it is a puddle. Reiner, who introduced it in 1964, took the name from the line in the song of Deborah that the mountains flowed — his point being that nothing is a solid except relative to how long anybody waits.

The Weissenberg number is the relaxation time multiplied by the shear rate, $\mathrm{Wi} = \lambda\dot{\gamma}$. It asks something different: how far the microstructure has been stretched from its resting shape by the flow it is in.

The distinction matters because a flow can score high on one and zero on the other. A polymer solution sheared steadily in a rheometer is not changing with time at all, so its Deborah number is zero — and if the rate is high its Weissenberg number is large, the molecules are strongly stretched, and the fluid does things no viscosity curve predicts. The rod-climbing described above is exactly that case: a steady flow, nothing unsteady anywhere, and an elastic effect large enough to defeat the centrifugal one.

So the honest question about a fluid is not which family it belongs to but which of the two numbers the application makes large.

The additive that a viscosity curve cannot see #

There is one industrial use of a non-Newtonian fluid that is worth stating because it defeats everything on this page, including the measurements.

Dissolve a few tens of parts per million of a very long-chain polymer in water and pump it turbulently through a pipe. The friction falls — by as much as seventy or eighty per cent, at the same flow rate. Toms found the effect in 1948 while looking for something else, and it has been in commercial use for decades: the Trans-Alaska pipeline raised its throughput with it rather than by adding pumping stations, and New York used it in the 1970s to get more water through a fire hose than the hose could otherwise carry.

The awkward part is that the solution’s shear viscosity is essentially unchanged. At those concentrations every curve drawn in this essay would be indistinguishable from water’s, and a rheometer sweeping shear rate over four decades would report that nothing had been added.

What the polymer does is act on the flow’s own structures rather than on the bulk stress. Near a wall, a turbulent flow maintains itself through a cycle of stretching events, and a long molecule caught in one is pulled out straight, absorbs energy elastically and returns it out of phase — enough to interrupt the cycle. The mechanism belongs to the study of turbulent drag, and the lesson belongs here: a fluid’s constitutive behaviour can be dominated by a component too dilute to appear in any steady measurement, and there is a limit to the reduction that no polymer beats, which is a sign that the flow rather than the additive is setting the answer.

Where the model stops #

The power law is a fit. $\tau = k\dot{\gamma}^n$ is a description of a range of rates, not a mechanism. Real fluids have Newtonian plateaus at both ends — very low and very high rates — with the power-law behaviour in between, and extrapolating outside the fitted range is unsafe.

A yield stress may not exist. Whether any fluid has a true yield stress, or merely an extremely high viscosity at low rates, has been argued for decades. On any practical timescale the distinction does not matter; over geological ones it does, which is why mountains flow and why the mantle convects at all despite being rock.

Only shear is described. Everything here is a response to shearing. Extensional flows — a fluid being stretched rather than sheared — are a separate response with its own coefficient, and polymer solutions can be enormously more resistant to extension than to shear.

The fluid is homogeneous. Suspensions can separate under flow, with particles migrating away from walls, so the material in the apparatus is not the material that was put in.

Surface effects are ignored. In a fine channel or a thin film the material’s structure is comparable with the geometry, and interfacial energy starts to compete with the bulk response.

No temperature. Every curve here is at fixed temperature, and all of these fluids are strongly temperature-dependent — often more so than they are rate-dependent.

The ladder from here #

Later rungs on this anchor: viscoelasticity properly, with storage and loss moduli and the frequency sweep that separates them. Thixotropy and the hysteresis loop. Extensional rheology. Suspension mechanics and the jamming transition. Granular flow, where a heap of dry grains behaves as a yield-stress fluid with no liquid in it at all. And the polymer physics underneath shear thinning, where the alignment of long molecules explains why the exponent is what it is rather than merely fitting it.