The core that does not move

A viscosity that depends on the question is the collection’s account of fluids whose apparent viscosity changes with how hard they are sheared. This essay is about the sharper case: a fluid that does not shear at all below a stress, which produces an exact threshold, an exactly rigid region, and a set of consequences that are cleaner than anything a Newtonian fluid offers.

It is also about what happens to all of that when the model is relaxed by one parameter, which is a good deal more than the model’s users expect.

The plug radius is a force balance #

Push a fluid down a pipe of radius $R$ with a pressure gradient $G$. Take a cylinder of radius $r$ inside it and balance the pressure force on its ends against the shear on its side:

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

That is the whole derivation, and it contains no fluid. It is true of water, of honey, of toothpaste, of a turbulent flow — of anything at all in a straight pipe at steady state, because it is a statement about forces and not about a constitutive law.

Now add the one thing that makes a Bingham fluid a Bingham fluid: it does not shear at all until the stress reaches $\tau_y$. Combine the two and the fluid is unyielded exactly inside

$$r_0 = \frac{2\tau_y}{G},$$

with nothing approximate about it. The plug radius is known before any equation is solved.

The profiles show what that does. Outside $r_0$ the fluid shears; inside, it moves as a solid body, and the computed velocity across the plug is constant to a spread of exactly zero — not to $10^{-15}$, to zero, because the profile there is a single number.

The plug never goes away. At twice the threshold gradient half the radius is rigid; at ten times, a tenth of it still is; and there is no pressure at which a yield-stress fluid in a pipe behaves like a Newtonian one, because the stress at the centreline is exactly zero whatever the pressure and the centre is therefore always unyielded.

That has a practical face. Near the threshold most of the section is a plug sliding along on a thin sheared film at the wall, so the flow is closer to a solid being pushed through a lubricated tube than to a liquid — and the wall is where every one of the transport, the heating and the mixing happens. It is everything happening in a thin layer with the layer made by a constitutive law rather than by a Reynolds number.

What “unyielded” is not #

Two readings of the plug get made and both are wrong, and disposing of them is quick.

It is not stationary. The plug moves — faster than anything else in the pipe, since the velocity is largest where the shear has finished accumulating. What it does not do is deform: every particle in it keeps its neighbours, and the strain rate there is exactly zero. A rigid body translating is still a moving body.

And it is not a different material. The plug is the same fluid as the annulus round it, at the same temperature and the same everything else. What distinguishes the two regions is the local stress, which is a property of the flow rather than of the fluid — so the plug boundary moves when the pressure changes, and a particle can be in the plug at one station and shearing at another if the pipe narrows.

Both of those are worth saying because the picture of a solid core surrounded by liquid invites a third mistake: that there is a surface between them with a jump across it. There is not. The velocity and the shear stress are both continuous at $r_0$; what is discontinuous is the second derivative, because that is where the constitutive law changes branch. It is a weak discontinuity, of the same kind as the one at the edge of a boundary layer, and like that one it is a feature of the model rather than of the fluid.

Buckingham and Reiner’s quartic #

Integrating the profile over the cross-section gives the flow rate. Writing $\xi = \tau_y/\tau_w = r_0/R$,

$$Q = \frac{\pi R^4 G}{8\mu}\left(1 - \tfrac{4}{3}\xi + \tfrac{1}{3}\xi^4\right),$$

which is the Newtonian answer times a bracket in one dimensionless number.

Checked rather than quoted, at five yield stresses, the two agree to seven parts in a hundred billion. That check matters more here than it looks: the bracket is a difference of terms that nearly cancel near $\xi = 1$, and a closed form that has been mis-transcribed produces a curve of the right general shape everywhere except where it is being used.

Exactly zero, and not nearly zero #

Below $\xi = 1$ — that is, below the pressure gradient at which the wall stress first reaches the yield stress — the bracket is negative and the flow rate is zero. Not small: zero, at any pressure below the threshold and for any length of time. Push at 99.9999 per cent of the threshold gradient for a century and nothing comes out.

That is an exactness of a different kind from most of the ones on this site. It is not a number that comes out clean; it is a statement about a set, and the set has an interior.

The way it leaves is worth having. Both the bracket and its first derivative vanish at $\xi = 1$ and the second derivative is four, so

$$\frac{Q}{Q_{\text{Newtonian}}} \to 2(1 - \xi)^2.$$

A square, with a leading constant of exactly two — not the cube a first guess produces, which is what this calculation returned before the expansion was done properly. The measured exponent is 2.0000068 and the measured constant 1.99996.

Which makes the yield stress worth least where it matters most #

The square is not a curiosity. It is the reason yield-stress fluids are hard to predict.

The elasticity — the percentage change in flow rate for a one per cent change in $\tau_y$ — is 0.15 far from the threshold and 197 just inside it. It diverges as the threshold is approached, because the flow is going to zero quadratically while the yield stress it depends on is not.

So a one per cent error in a measured yield stress is a 1.6 per cent error in flow at a wall stress twice the yield stress, and a 197 per cent error at a wall stress one per cent above it. The regime in which the number matters most is the regime in which it is worth least, which is a pattern the site meets elsewhere — it is the same shape as the number that cannot break a drop, where the answer is a saturation in one regime and a divergence in the other.

Yield stresses are measured, and they are measured badly: values for the same material from different rheometers differ by tens of per cent. Multiply that by 197.

What the threshold does to a pipeline #

The threshold has an engineering face worth stating, because it is where the exactness earns money.

A crude oil that has waxed in a shut-down pipeline is, to a good approximation, a Bingham fluid, and restarting the line means exceeding the threshold gradient over its whole length. The restart pressure needed is $2\tau_y L/R$, which for a yield stress of 20 pascals in a half-metre line a hundred kilometres long is 80 bar — a number set entirely by the gel and the geometry, with the pump and the oil’s viscosity nowhere in it.

Two things follow that are not obvious. The restart pressure does not depend on how fast the line is to be restarted, because at the threshold the flow rate is zero however much time is allowed. And it scales with length, so a long line cannot be restarted end to end and has to be broken into sections — which is a design consequence of a threshold being a threshold rather than a steep gradient.

The same arithmetic run the other way sets the minimum pipe radius for a given available pressure, and it is the reason a fluid with a yield stress is transported in a larger pipe than its viscosity alone would suggest.

And the exactness is the model’s #

Everything above is exact for a fluid that is rigid below yield. Real materials are not.

Replace the rigid plug with a very large but finite viscosity — the bi-viscous model, which is what every numerical code actually solves, because a rigid region is not something a solver can represent — and the fluid creeps at every pressure. There is no threshold at all; there is a knee.

Above the knee, the apparent yield stress a flow measurement infers is within two per cent of the true one, which is why the model is useful. Below it, the apparent yield stress is exactly proportional to the detection threshold: one decade of “what counts as no flow”, one decade of apparent yield stress, with a measured exponent of 1.0000000000000016. And the position of the knee is set by the viscosity ratio $\mu_0/\mu$, which is a parameter of the model that no experiment reports.

That is the honest summary of the whole subject. A yield stress is a number about a measurement protocol as much as about a material, and the exact threshold above is exact within a model whose one relaxation removes it. Every rheology paper that reports a yield stress reports, implicitly, the lowest shear rate its instrument could resolve.

The same fluid, in a rheometer #

There is a companion difficulty on the measurement side, and it is the reason the numbers going into the arithmetic above are as poor as they are.

A yield stress is measured in a rotational rheometer by ramping the stress and watching for motion, or by ramping the shear rate and extrapolating the stress to zero rate. The first is the detection problem above wearing different clothes: what counts as motion is an instrument setting. The second is worse, because it extrapolates a curve towards a region the instrument cannot enter — the same shape of mistake as extrapolating a damping curve to find a flutter speed, and with the same tendency to answer with more confidence than the data supports.

A third method avoids both: measure the flow rate through a pipe at several pressures and fit Buckingham and Reiner’s bracket. That has the advantage of using the whole curve rather than its endpoint, and the disadvantage that near the threshold — where the curve is most informative about $\tau_y$ — the flow rate is smallest and hardest to measure. The sensitivity figure above is the same statement read from the experimenter’s side: the measurement is best conditioned exactly where the signal is weakest.

There is no way round that. It is a property of a quantity defined by a vanishing, and the site meets it again wherever a threshold has to be located by watching something go to zero.

The same balance with nothing pushing #

The plug radius came from a force balance containing no fluid, plus one constitutive fact applied at the end. The same pair answers a question with no pipe in it: what such a material does when the only thing acting on it is its own weight.

A layer of thickness $h$ resting on a slope of angle $\theta$ carries a shear stress at its base of $\rho g h \sin\theta$. It flows only if that exceeds the yield stress — so there is a maximum thickness that can simply stand there, $\tau_y/(\rho g\sin\theta)$, and anything thinner is rigid permanently rather than slowly.

That single statement is why paint on a wall does not sag, why mortar stays on a trowel, and why drilling mud holds rock cuttings in suspension when the pumps stop — which is the property the mud is formulated for, and a Newtonian fluid of any viscosity whatever cannot supply it, because a viscous fluid only ever slows a settling particle down.

It also gives a measurement that avoids the whole detection-threshold difficulty. Let a cylinder of the material collapse under its own weight and measure how far it slumped: the final shape is a geometry rather than a vanishing rate, and inferring a yield stress from where something stopped asks nothing of an instrument’s lowest resolvable speed.

Why the model is worth keeping anyway #

Three reasons, and they are not “it is approximately true”.

The plug radius is not the model’s. $r_0 = 2\tau_y/G$ follows from the force balance and from the existence of some stress below which the material’s response changes character. It survives the bi-viscous relaxation as the radius where the two viscosities meet, and it survives any other regularisation too.

The bracket is a one-parameter family. Everything about the flow of a Bingham fluid in a pipe is a function of $\xi$ alone, so a single measured curve of flow against pressure collapses onto a single line, and departures from it are evidence about the material rather than about the pipe. That is the same argument a dimensionless group always makes, and it is exactly as strong here as anywhere.

The plug is a real prediction and it is observed. Magnetic resonance velocimetry of concentrated suspensions in pipes shows a flat velocity profile over the middle of the section, with the flat region growing as the pressure is reduced, and the measured plug radii track $2\tau_y/G$ over the range where the material’s yield stress is well characterised. It is not a modelling convenience; it is a thing that photographs.

And the threshold is where the design decisions are. A pipeline carrying a waxy crude that has gelled needs a restart pressure, and the restart pressure is the threshold. It is not a soft number in practice for the reason it is not soft in the model: the flow below it is not slow, it is unmeasurable, and a pump sized to deliver a slow flow delivers none.

Where this sits among the site’s other thresholds #

It is worth placing the Bingham threshold beside the others this collection has, because they are not all the same kind of thing.

A critical Reynolds number is not a threshold in this sense at all: nothing is exactly zero on either side of it, and the number that is not a number is about how badly it fails to be one. A critical Richardson number is a sufficient condition with an exact value and no exact converse. A choking condition is exact and is a statement about what a signal can do rather than about a material.

The yield stress is the cleanest of them in the model and the least clean in the laboratory, and the reason is the one this essay is about: it is defined by the absence of a response, and an absence is the hardest thing an instrument can certify. Every other threshold here is located by watching something change; this one is located by watching nothing happen, and how long the watching lasts is part of the answer.

What is not claimed #

Steady, fully developed, laminar, in a circular pipe. All four are load-bearing. A yield-stress fluid in a channel has a different bracket, in an annulus a different one again, and the entrance region — which for a plug flow is not the Newtonian one — is not treated here at all.

The plastic viscosity is constant. Real materials are usually Herschel–Bulkley — a yield stress and a power-law response above it — and the bracket for that case is a different function with a different exponent at the threshold. Everything qualitative above survives; none of the numbers does, which is why the essay quotes the exponent as a property of the Bingham bracket rather than of yield-stress fluids generally, and why a departure from the ordinary linear response has to be measured rather than assumed away.

No thixotropy. Real gelled materials have a yield stress that depends on how long they have been at rest and on their shear history, so a single $\tau_y$ is already a simplification before any of the above starts.

The pipe is smooth and the fluid does not slip on it. Concentrated suspensions frequently do slip at a wall, on a thin depleted layer, and apparent-viscosity measurements that ignore it report a material property that is a property of the wall — which is the instrument in the answer in a field where it is particularly hard to detect.

And the bi-viscous knee is a model of a model. The creep of a real yield-stress fluid is not a second Newtonian viscosity; it is a slow, often power-law, often ageing response. What the figure demonstrates is that any regularisation makes the inferred yield stress depend on the detection threshold, not that the dependence has this particular shape.