The stress a pipe knows

A capillary viscometer measures two things and reports a third. It measures the pressure drop along a tube and the flow rate through it, and from those it reports a viscosity.

The first half of that inference is exact for every fluid there is. The second is not exact for any fluid except one, and the gap between them is this collection’s recurring statement in a piece of laboratory glassware.

The half that is a force balance #

Take a cylinder of fluid of radius $r$ inside a tube, in fully developed flow. The pressure force on its ends is $\pi r^2\,\Delta p$; the shear force on its curved surface is $2\pi r L\,\tau(r)$; the momentum flux in equals the momentum flux out because nothing is accelerating. So

$$\tau(r) = \frac{\Delta p}{L}\frac{r}{2}, \qquad \tau_w = \frac{\Delta p}{L}\frac{R}{2}.$$

No constitutive law appears anywhere in that derivation. It holds for water, for a polymer melt, for a slurry, for toothpaste, for a fluid nobody has a model for. The stress profile is linear in the radius and the wall stress is the pressure gradient, exactly, and a pressure measurement is therefore a stress measurement with no assumptions in it.

The check on that is worth doing rather than asserting, because it is easy to believe a statement of that generality and hard to notice when it is being used with an extra assumption smuggled in. Four fluids are put in the tube — Newtonian, shear-thinning, Carreau and a Bingham plastic with a yield stress — their velocity profiles are computed by integrating each one’s own flow curve inwards from the wall, and the stress each is carrying at each radius is recovered from its local shear rate. All four lie on the same straight line.

The profiles are nothing alike. A Newtonian fluid answers the linear stress with a parabola; a shear-thinning one with a blunter profile, because it shears easily where the stress is high and stiffly where it is low; a shear-thickening one with a sharper one; and a Bingham plastic with a plug, because inside the radius where the stress falls below its yield point nothing shears at all — the core that does not move.

The Bingham case is the sharpest form of the general point. Over the whole plug the velocity profile is flat and says nothing whatever about the fluid, and the stress there is still exactly linear in the radius, because the force balance does not care whether anything is shearing.

There is a second reason to labour the force balance, and it is about what a viscometer is for. The instrument exists because the constitutive law of the fluid is unknown — that is the whole point of measuring it. So every step of the inference has to be valid without one, or the measurement is assuming what it is meant to determine. The stress step is; it is the only step that is.

The same argument disposes of a natural objection. Somebody might reasonably ask whether the linear stress profile is a consequence of the flow being laminar and fully developed rather than of the force balance alone, and it is: those are the two hypotheses, and both are geometric rather than material. Laminar means no turbulent momentum transport across the radius; fully developed means nothing is changing along the tube. Given those, the balance closes for any fluid, and neither hypothesis is a statement about what the fluid is made of.

The half that is not #

What the instrument can compute from the flow rate is

$$\Gamma = \frac{4Q}{\pi R^3},$$

which for a Newtonian fluid is the wall shear rate. For anything else it is not, and it has a name of its own — the apparent wall shear rate — precisely because it keeps being mistaken for one.

For a power-law fluid the true wall shear rate is larger by $(3n+1)/4n$: a factor of two at $n = 0.2$, 1.58 at 0.3, 1.25 at 0.5, exactly one at 1, and below one for a shear-thickening fluid. The measured ratio from the computed profiles reproduces the closed form to six figures, which is the check that the quadrature is doing what the algebra says.

A viscosity computed as $\tau_w/\Gamma$ is therefore wrong by exactly that factor: fifty per cent low at $n = 0.2$, twenty-seven at 0.4, ten at 0.7, nine per cent high at $n = 1.5$. The only case in which it is right is the case the uncorrected formula was derived for.

Where the factor comes from is worth a sentence, because it makes the size of the error predictable. The flow rate is $\int u\,2\pi r\,dr$, which after an integration by parts is $\pi\int \dot\gamma\, r^2\,dr$: the apparent shear rate is an $r^2$-weighted average of the actual shear rate across the tube. For a Newtonian fluid the shear rate is linear in $r$ and that average lands exactly on the wall value. For a shear-thinning fluid the shear rate is concentrated near the wall — the profile is blunt in the middle — so the weighted average sits below the wall value, and the apparent rate under-reads.

Which is why the direction of the error is never in doubt. Shear-thinning fluids always have a true wall shear rate above the apparent one, and therefore a true viscosity below the reported one. An uncorrected capillary measurement of a shear-thinning fluid always reports the fluid as thicker than it is, by a factor that grows as the thinning gets stronger.

What the correction needs #

The general repair is the Weissenberg–Rabinowitsch–Mooney relation,

$$\dot\gamma_w = \frac{\Gamma}{4}\left(3 + \frac{d\ln\Gamma}{d\ln\tau_w}\right),$$

and the important thing about it is not the algebra but what it asks for: a derivative of the whole flow curve. The slope of the log of the apparent shear rate against the log of the wall stress is not available in any single run, at any accuracy, however carefully that run is made.

Taken from three runs at wall stresses five per cent apart — which is what the instrument is actually used to do — the slope comes out at $1/n$ to six figures, and the corrected shear rate is exact to a part in a billion.

So one point of data gives the stress exactly and gives the shear rate not at all. Getting the second requires neighbouring measurements, which is to say it requires the experiment the instrument was bought to avoid: a rheometer measures a curve, and a single-point viscosity is a curve of one point with its slope assumed.

It is worth noticing what makes the three-run measurement work as well as it does. The relation asks for a logarithmic derivative, and logarithmic derivatives are the numerically pleasant kind: the quantity being differenced is the same order of magnitude at all three stresses, the spacing is relative rather than absolute, and there is no cancellation. Five per cent apart in stress is enough to recover the slope to six figures, which means the correction costs two extra runs rather than a sweep.

That is a happier situation than the general one this collection has been describing. Reaching a local quantity from integral data is normally hopeless; here it is easy, because the missing information — how the flow rate responds to a change in stress — is exactly what a second run supplies. The instrument can measure the derivative. It simply has to be asked to.

A fluid with no exponent #

A Carreau fluid is Newtonian at low shear and thinning at high, so no single $n$ describes it and the power-law correction factor does not apply anywhere. The general relation still works, because the slope it asks for is local: the recovered shear rate is exact to a part in ten thousand at every wall stress tried, while the uncorrected reading is out by nothing at the low-shear end and by twenty-four per cent at the fast one.

That is the practically important case, because it is the shape almost every real fluid has. A polymer solution has a low-shear plateau, a thinning region and a high-shear plateau, and its apparent viscosity from a single-point measurement is wrong by an amount that depends on which part of the curve the measurement happened to land in.

Why this is the same statement as the rest #

The pattern is the one this collection keeps arriving at. An exact constraint fixes a total and does not reach a local quantity.

The force balance is the constraint. What it fixes is the stress — everywhere, exactly, for any fluid. The shear rate at the wall is a different functional of the velocity profile: it is a derivative at a point, and that is precisely the class of quantity integral constraints cannot reach. The flow rate is a second constraint, an integral of the profile weighted by the radius, and it is not enough either — one integral does not determine one derivative.

What closes the gap is not a better measurement of the same kind. It is the smoothness of the flow curve, supplied from outside as an assumption that neighbouring stresses give neighbouring rates, and then differentiated.

What the profile would have told, if it could be seen #

There is an obvious alternative that is worth pricing, because its unavailability is the reason the correction exists.

If the velocity profile across the tube could be measured directly, the wall shear rate would follow from its slope and no correction would be needed at all. Modern optical and magnetic-resonance methods can do exactly that, and where they are available the argument on this page is a historical one.

They are not available inside a capillary of half a millimetre bore at high pressure, which is where most of this measuring is done, and they were not available at all when the correction was derived. So the practical position is that the instrument sees two integrals of the profile — the flow rate, weighted by $r$, and the pressure drop, which is the stress — and has to reconstruct a derivative from them.

Seen that way the correction is a piece of inverse reasoning rather than a fudge factor: it uses the fact that the stress profile is known exactly to convert a family of flow-rate measurements into a family of shear-rate measurements, one at each wall stress. Every point of the recovered flow curve is obtained from a different run, and the curve is assembled rather than measured.

Three practical consequences #

A single-point viscosity of a non-Newtonian fluid is not a viscosity. It is a wall stress divided by a number that would be the wall shear rate if the fluid were Newtonian, and the two differ by up to a factor of two over the ordinary range of shear-thinning behaviour. Reported without the correction, it is a quantity with an instrument in its definition.

The correction cannot be applied afterwards to a single measurement. A published apparent viscosity from one run cannot be corrected by a later reader, because the slope that would correct it was never measured. That is a different situation from a missing calibration constant, and it is why the raw quantities — the pressure drop and the flow rate — are what should be published.

And the geometry does not help. Changing the tube’s radius or length changes the stress and the flow rate together, so a second tube at the same wall stress gives the same apparent shear rate. What is needed is a second wall stress, which means a second pressure drop, which is the same experiment run again.

The same shape in two other instruments #

Two other measurements in this collection have exactly this structure, and putting them beside this one shows that it is a property of what is being asked rather than of rheometry.

A flowmeter reads a pressure difference and infers a flow rate, and the pressure difference is exact while the inference needs the approach profile’s kinetic-energy coefficient — a functional of a profile the instrument cannot see. There the uncertainty is a few per cent for a turbulent approach and eight for a laminar one, which is the same arithmetic in a different disguise: an integral is known and a shape is not.

A wake survey measures a drag exactly and the wake not at all. There the constraint and the wanted quantity coincide, which is the lucky case; here they do not.

And there is a third case worth naming because it goes the other way. The permanent pressure loss of a metering device is computed from a control volume with no viscosity in it at all, and it is exact for the same reason the stress here is exact: momentum does not ask where the energy went. The pattern in all four is that the balances are exact and the profiles are not, and which of the two a measurement needs decides whether it can be trusted from one run.

Why the fluid is the difficult part rather than the flow #

It is worth being explicit about what makes this harder than the Newtonian case, because the difficulty is not where a reader might expect it.

Nothing about the flow is complicated. It is steady, laminar, one-dimensional, fully developed and axisymmetric, with a linear stress profile and no separation, no instability and no turbulence anywhere in it. This is the simplest flow in the subject.

What is complicated is that the relation between stress and shear rate is the unknown. In a Newtonian fluid that relation is one number, so the flow rate determines it and the measurement closes in one run. In a non-Newtonian fluid it is a function, and determining a function from integrals of it is the situation this collection has already measured the difficulty of.

So the honest description of a capillary viscometer is that it is an instrument for measuring a function by a sequence of integral measurements, and that a viscosity depends on the question being asked of it in a stronger sense than usual: not only does the fluid have a different viscosity at each shear rate, but the shear rate at which a given run measured it is not directly observable either.

That also explains why the correction is invisible in most textbook treatments. They introduce the capillary viscometer while the fluid is still Newtonian, where the apparent and the true rate coincide identically, and the distinction has nowhere to appear. By the time the fluid stops being Newtonian the formula has already been learned.

A note on why the plug is the clearest case #

Of the four fluids, the Bingham plastic is the one worth remembering, because it makes the whole argument visible without any arithmetic.

Over the inner thirty per cent of the tube its velocity profile is flat. There is no shear, no gradient, and nothing about the fluid’s constitutive behaviour is being exercised — a probe traversing that region would learn nothing at all about what the fluid is. And the stress there is still exactly linear in the radius, rising from zero at the axis to the yield value at the plug’s edge, because the force balance is a statement about forces rather than about motion.

So the two halves of the inference are as far apart as they can be made: an exact stress profile across a region where the velocity carries no information whatever. Anybody who has accepted that picture has accepted the general statement, and the power-law arithmetic afterwards is only putting a number on it.

It also explains a practical difficulty with such fluids. The flow rate through a tube with a plug is dominated by the plug moving as a body, so the relation between flow rate and pressure drop is insensitive to the fluid’s behaviour at low shear — the very region a rheologist most wants to characterise. Measuring it requires either a much smaller tube, so that the plug occupies less of the section, or an instrument that shears the fluid uniformly.

What the two halves cost, in the laboratory #

It is worth closing with the practical arithmetic, because the asymmetry between the two halves of the inference has a price and the price is small.

Measuring the stress requires one pressure drop, which is a single reading of a single transducer. Measuring the shear rate requires three flow rates at three pressure drops, which is three runs instead of one — a factor of three in time, on an instrument whose runs take minutes.

Against that, the error avoided is up to fifty per cent, and it is a systematic error rather than a random one, so it does not average away over repeats. Running one point three times gives three copies of the same wrong number; running three points once gives the right one.

That is not a difficult trade, and the reason it is worth stating is that the wrong practice is the default rather than a shortcut somebody chose. A single-point measurement is what an instrument returns when it is used the obvious way, the number it prints is called a viscosity, and nothing about the reading announces that it is a viscosity at a shear rate the instrument did not measure.

The general habit follows. When an instrument reports a quantity it inferred rather than measured, find out which of its readings were measurements and which were assumptions — and if an assumption is a derivative, expect to need neighbouring points.

What is not claimed #

The correction is old and is not being discovered here. Rabinowitsch published it in 1929 and Mooney independently in 1931, and any rheology text states it. What is computed here is its size on four constitutive laws, and the demonstration that the stress half of the inference is genuinely assumption-free while the rate half is genuinely not.

Fully developed flow is assumed throughout. Entrance effects, which for a shear-thinning fluid can persist for a long way, are a separate correction and are not in any of these numbers.

Wall slip is not modelled. Many of the fluids this correction is applied to — suspensions, pastes, filled polymers — slip at a smooth wall, which changes the flow rate at a given stress and is diagnosed by running tubes of different radii. Where slip is present the correction above is necessary and not sufficient.

And the Bingham stress check is limited by its own arithmetic. The profile has a corner at the yield radius and a three-point difference straddling that corner reads a slope that is neither side’s, which is why the residual there is two parts in ten thousand rather than at round-off like the others.