A viscosity that depends on the question

Ask what the viscosity of water is and the answer is a number: about a millipascal-second at room temperature, and it will still be that number tomorrow in a different apparatus.

Ask what the viscosity of paint is and the honest answer is a question. At what shear rate? Paint that is being brushed is being sheared at ten thousand reciprocal seconds and behaves like a thin liquid; paint that is sitting on a wall is being sheared at a hundredth of that and behaves like a soft solid. Those are not two states of the paint. They are two values of the same function, and the function is what the fluid actually has.

The simplest law that is not Newton’s #

The constitutive relation used throughout this essay is the power law,

$$\tau = K|\dot\gamma|^{n-1}\dot\gamma,$$

with $n$ the flow index and $K$ the consistency. At $n = 1$ it is Newton’s law with $K = \mu$. Below one the fluid is shear-thinning, above one shear-thickening.

It is the simplest departure from Newton’s law that is still a law, and it is chosen here for exactly that reason. It is also wrong at both ends of any real fluid’s range: a power law with $n < 1$ has infinite viscosity at zero shear rate and zero viscosity at infinite shear rate, and no material does either. Where it is used it is used over a decade or two of shear rate, with the exponent fitted there, and it is a description rather than a mechanism.

What it buys is a closed-form pipe flow, which is enough to make the point this essay is about.

The wall stress that does not know the fluid #

Integrating the momentum equation over a cylinder of radius $r$ inside a pipe gives

$$\tau(r) = \frac{G r}{2},$$

with $G$ the pressure gradient — and that is the end of it. The stress distribution in a pipe is fixed by the pressure gradient and the geometry and does not depend on the fluid at all. It is a force balance; the constitutive law has not been used.

Everything rheological is in what strain rate that stress produces. Inverting the power law gives

$$u(r) = \frac{n}{n+1}\left(\frac{G}{2K}\right)^{1/n}\left(R^{(n+1)/n} - r^{(n+1)/n}\right),$$

and at $n = 1$ that collapses to the parabola, as it must.

The energy balance closes both ways, which is the check this collection insists on: $\int\Phi\,dV$ with $\Phi = K|\dot\gamma|^{n+1}$ over the section equals $G$ times the flow rate, exactly, for every index. Two integrals that share no arithmetic, agreeing to a part in ten million on a four-thousand-point quadrature.

What flattening actually does #

The profile at $n = 0.2$ is nearly flat across most of the pipe and turns sharply near the wall. That shape is universally described as “plug-like”, and the description carries a suggestion that is false: that the fluid is moving more freely.

What has happened is that all of the shearing has been pushed into a thin annulus at the wall. By the argument in the essay that priced a gradient, that is the expensive place to put it — the cost goes as the square of the gradient, and a thinner layer raises the gradient faster than it removes fluid.

That concentration is the reason a shear-thinning fluid’s heat is a problem in an extruder and not in a pipe. The same total dissipation is being made in a tenth of the volume, and the temperature rise that follows from it — which contains no gap at all — is not the average one.

The number an instrument reports #

Here is where the essay’s title earns itself.

A capillary viscometer pushes fluid through a tube, measures a flow rate and a pressure drop, and divides. From those it forms a wall shear stress, $\tau_w = GR/2$, which is right; and a wall shear rate, which it takes as $4Q/\pi R^3$, which is right only for a Newtonian fluid.

The true wall shear rate is larger for a shear-thinning fluid, because the profile is steeper at the wall than a parabola carrying the same flux. The ratio is

$$\frac{\dot\gamma_{w,\text{true}}}{\dot\gamma_{w,\text{apparent}}} = \frac{3n+1}{4n},$$

which is Rabinowitsch’s correction. For $n = 1$ it is one. For a drilling mud at $n = 0.35$ it is 1.46.

The uncomfortable part is that the correction needs $n$, and $n$ is what the instrument is trying to measure. In practice the log-log slope of the apparent curve gives an $n$, that $n$ gives a correction, and the process is iterated — which works, and which means the reported viscosity of a non-Newtonian fluid is a derived quantity that depends on a model having been assumed.

The one number that survives #

If a shear-thinning fluid has no viscosity, what does a pipe designer use?

The answer that the industry settled on is the generalised Reynolds number, and it is worth looking at because it is an honest piece of dimensional bookkeeping rather than a fudge. Form the apparent viscosity at the wall — the true wall stress divided by the true wall shear rate — and build a Reynolds number on it. The laminar friction factor then comes out as 16 over that number, exactly as for a Newtonian fluid, and the transition to turbulence happens at roughly the same value.

That is a real result and it is not a coincidence. It works because the laminar friction factor depends on the fluid only through the relation between flow rate and pressure drop, and the generalised Reynolds number is constructed from precisely that relation. What it buys is a single chart; what it costs is that the viscosity in it is not a property of the fluid but of the pipe the fluid is in. Change the pipe’s diameter and the wall shear rate changes, so the apparent viscosity changes, so the number is built on a different value of a material function.

A designer who understands that can use the chart with confidence and can also see why extrapolating from a laboratory tube to a production line is not a matter of scaling a Reynolds number.

Why it is not one number, in three sentences #

The deeper point is not about the correction; it is about what a viscosity is.

For a Newtonian fluid the stress is proportional to the strain rate, so a single measurement at one shear rate determines the behaviour at every other. For anything else it does not: a measurement at one shear rate determines the behaviour at that shear rate. Quoting one number for such a fluid is therefore not an approximation but a category error, and the number quoted is a property of the instrument’s operating point rather than of the material.

That is why rheology’s standard output is a curve rather than a value, and why “the viscosity of blood” is a question that needs a shear rate attached before it means anything. Blood in a large artery at a few hundred reciprocal seconds behaves nearly Newtonianly; blood in a capillary at ten does not, and the difference is a factor of three.

What is being left out #

Two things, and both are larger than the shear-thinning being described.

A yield stress. Many of the materials this essay is about do not flow at all below a threshold stress: toothpaste, mayonnaise, drilling mud, fresh concrete. A power law has no such threshold, and the Bingham and Herschel–Bulkley models that do produce a pipe flow with a genuinely unsheared plug in the middle — a region with zero rate of strain and therefore, by the dissipation function, zero cost. That is a real plug, unlike the merely flat profile at $n = 0.2$, and it changes the answer qualitatively.

Elasticity. A polymer solution stores energy as well as destroying it, because its molecules are being stretched and can pull back. Nothing in this essay’s arithmetic allows for a stress that depends on the flow’s history rather than on its present strain rate, and the consequences of that — rod climbing, die swell, a jet that pulls itself upstream — are qualitatively unlike anything a power law can produce.

Both are visible in ordinary materials and neither is here. What is here is the smallest possible departure from Newton’s law, and it is enough to break a viscometer.

The third exclusion, which is about time rather than rate #

There is a category between the two omissions above, and it is the one that most of this essay’s own examples actually belong to. A power law says the stress depends on the present strain rate. Elasticity says it depends on the strain history. Thixotropy says it depends on the shear-rate history — a fluid whose apparent viscosity at a fixed shear rate keeps falling for as long as the shearing continues, and then recovers when it stops.

The mechanism is a structure rather than a molecule. Many of these materials contain a loose network — flocculated particles, a colloidal gel, a clay platelet stack — that shearing breaks apart and that attraction and Brownian motion rebuild. Breaking is fast; rebuilding is usually far slower, taking seconds to hours, and the two timescales are separate material properties.

That makes the flow curve of such a fluid not a curve. Ramp the shear rate up and then back down in a viscometer and the two paths do not coincide: the descending branch lies below the ascending one, because the structure broken on the way up has not had time to rebuild. The enclosed area is the standard index of how thixotropic a material is, and its dependence on the ramp rate is the honest statement that there was never a single curve to measure.

And it is the property the essay’s own examples are sold for. A paint must thin under the brush at once, then rebuild slowly enough to flow out level and quickly enough not to run down the wall — two different timescales, which no fluid whose viscosity depends on the instantaneous shear rate alone can possibly satisfy. A drilling mud must gel within seconds of circulation stopping, so that rock cuttings stay suspended in a stationary well, and must break again when the pumps restart; its gel strength is specified at ten seconds and again at ten minutes, which is a two-point measurement of a rebuild curve.

So the material function this essay has been careful to insist on is itself an idealisation. For a great many working fluids there is not one, and what there is instead is a surface over shear rate and time.

Where the effect is largest, and it is not in a pipe #

The pipe is where the arithmetic is cleanest and it understates the effect, because a pipe’s range of shear rates is only a factor of a few. The places where a shear-thinning fluid does something a Newtonian one cannot are the places where the shear rate varies by orders of magnitude within one piece of apparatus.

A paintbrush. Under the brush the shear rate is $10^4$ s⁻¹ and the paint is thin; on the wall it is nearly zero and the paint is thick enough not to sag. A Newtonian paint with the brushing viscosity would run; one with the sagging viscosity could not be brushed. The whole product is the existence of a material function rather than a value.

A blood vessel that narrows. Blood entering a capillary is sheared far harder than blood in the aorta, and thins accordingly — which is a large part of why the pressure drop through the microcirculation is not the catastrophe that a constant-viscosity calculation on those diameters predicts.

And a drilling mud, which is designed around it. The mud must be thin enough to pump down the pipe at a high shear rate and thick enough to hold rock cuttings in suspension when circulation stops and the shear rate is zero. Those two requirements are contradictory for any Newtonian fluid and are routine for a shear-thinning one with a yield stress.

In each of those the useful quantity is not the viscosity at any particular shear rate but the range over which it varies, which is what the flow index measures: the smaller $n$, the wider the spread. A fluid at $n = 0.35$ has an apparent viscosity a hundred and forty times larger at one reciprocal second than at ten thousand.

What the picture cannot show #

The flow index is an input. Nothing in this essay measures an $n$; the values in the viscometer table are from the rheological literature and are quoted as measurements. What is computed is the consequence of an index, which is a different kind of statement.

The flow is laminar, fully developed and isothermal. Turbulent flow of a shear-thinning fluid is a subject of its own, in which the thinning suppresses turbulence near the wall and the friction factor is below the Newtonian one at the same Reynolds number — the opposite of what the laminar arithmetic here would suggest.

And a power law diverges at both ends. At zero shear rate the apparent viscosity of a shear-thinning power-law fluid is infinite and no real fluid’s is; at infinite shear rate it is zero and no real fluid’s is. Real materials have plateaux at both ends and the power law is the straight part in between, on a log-log plot, over a stated range.

A note on the word “apparent” #

The vocabulary of this subject is unusually careful and it repays reading literally.

An apparent viscosity is a stress divided by a strain rate at a stated condition. It is a number with the dimensions of a viscosity that is not a material constant, and the adjective is a warning rather than a hedge.

An effective viscosity is something else again: a number chosen so that a Newtonian calculation reproduces a measured result. The one this collection uses for a suspension is of that kind, and so is the eddy viscosity of a turbulence closure — both are fitted quantities standing in for a mechanism, and both are useful exactly as far as the calculation they were fitted to.

A consistency — the $K$ in the power law — has dimensions that depend on the flow index, which is the clearest possible signal that it is a fitted coefficient and not a property. Pa·s^n is not a unit anything can have; it is a unit a curve fit can have.

Keeping the three apart is most of what it takes to avoid the errors this essay is about, and the literature that does not keep them apart is the literature in which a viscosity gets quoted without a shear rate.

Who found it, and when #

Rabinowitsch published the correction in 1929 and Mooney independently in 1931, both working on rubber solutions where the effect is enormous. Ostwald and de Waele had given the power law a few years earlier. The recognition that a “viscosity” measured in a capillary is a derived quantity rather than a material property came out of that work and has been standard in rheology ever since, and is still routinely lost in every other field that borrows a viscometer.

The surprising connection is with a result about pipes that is not rheological at all. The wall stress $GR/2$ is independent of the fluid, which means two completely different fluids driven through the same pipe at the same pressure gradient exert exactly the same force on the wall — the same drag, the same pumping load per unit area — while carrying quite different flow rates. That is not true of a body in an external flow, where the drag depends on the fluid in every way. The difference is that a pipe is a confined flow whose momentum balance closes on the geometry, and it is the same reason a confined flow’s energy balance closes locally when an external one’s does not.

Where the ladder goes next #

Beside this rung is the other way a viscosity stops being a number — a suspension, which has a perfectly good viscosity that is not the liquid’s — and the film that heats itself, which is a third way, since a viscosity that depends on temperature depends on the flow that is heating it.

Below it is what a flow is, where the continuum assumption that lets a stress be related to a strain rate at all is made, and the price of a gradient, which is the accounting every number here was checked against.