The fluid that has not finished its last deformation

A viscosity that depends on the question establishes that a non-Newtonian fluid’s viscosity is not a number: it depends on the shear rate, so quoting one requires quoting the rate it was measured at.

This essay is about a fluid for which even that is not enough, because its stress does not depend on the shear rate either. It depends on the shear rate and every shear rate before it, and the fluid is therefore stressed at moments when nothing at all is happening.

The constitutive equation is a convolution #

The simplest fluid with a memory has a stress

$$\tau(t) = \int_{-\infty}^{t} G(t-s)\,\dot\gamma(s)\,ds,$$

a weighted integral of the strain rate over the fluid’s whole past, with the weight $G$ falling off as the fluid forgets. A Newtonian fluid is the case where $G$ is a delta function: the stress is the present rate times a viscosity and nothing else.

For a single relaxation mode the kernel is an exponential, $G(r) = (\eta/\lambda)e^{-r/\lambda}$, and $\lambda$ is the relaxation time: the age past which the fluid has essentially forgotten.

That is a different species of memory from the diffusive ones elsewhere in this collection. An exponential kernel has a time constant, so there is a definite age past which the past does not matter — where the wall kernel in the wall the fluid is listening to has a heavy tail and no mean at all.

The stress that outlives the flow #

Shear the fluid at a constant rate for two seconds and then stop. The stress rises towards its steady value, and at the moment the flow stops it starts to fall — exponentially, with the relaxation time: 60.7 per cent of it is still there half a relaxation later, 36.8 per cent after one, 13.5 after two, 4.98 after three and 0.674 after five. In absolute terms the stress goes 5.244, 3.181, 1.170, 0.4305, 0.0583.

One relaxation time after the flow has stopped, 36.8 per cent of the peak stress is still there. Two relaxation times: 13.5 per cent. Five: 0.67 per cent. Those are exactly the powers of $1/e$, which is what a single-mode kernel means and is the one property of this model a real fluid does not have.

The physical picture is molecular and worth having. The polymer chains have been stretched and aligned by the flow; when the flow stops they are still stretched, and they relax by Brownian motion back to their coiled state. The stress is the chains pulling, and it lasts as long as the coiling takes.

Where the viscosity went #

There is a book-keeping question worth answering, because it explains what a viscosity is for a fluid with a memory.

Shear the fluid at a constant rate for a long time and the stress settles. Its settled value is the strain rate times the integral of the kernel over all delays — and for the exponential kernel that integral is exactly $\eta$. So the steady viscosity of a viscoelastic fluid is the area under its memory, and nothing else.

That has two consequences. The first is that a viscosity measured in steady shear says nothing at all about the shape of the kernel: a fluid with a short intense memory and one with a long weak one can have identical viscosities and behave completely differently in anything unsteady. The second is that the time it takes to reach that steady value is the relaxation time, so a viscometer that has not run for several relaxation times is reporting a transient.

Both are the same statement made about a mean elsewhere in this collection: what a mean profile cannot tell anybody finds that an integral of a distribution does not determine the distribution, and a viscosity is an integral of a kernel.

Same total strain, two different states #

The sharper demonstration is a comparison rather than a decay, and it is the one that shows the fluid’s state is genuinely a functional of its history.

Two histories impose exactly the same total strain of two: one over a fifth of a second at a rate of 10, one over four seconds at a rate of 0.5. The fast one peaks at a stress of 18.13 and the slow one at 4.908, a factor of 3.7; five seconds after the start the fast one has fallen to 0.1492 and the slow one is at 1.806, a factor of 12.1 the other way. A Newtonian fluid would not distinguish them at all — its stress is instantaneous, so after both have stopped both are at zero, and nothing about the fluid records which happened.

The viscoelastic fluid reaches a peak stress of 18.1 in the fast history and 4.9 in the slow one, a factor of 3.7. And long after both have finished — five seconds in, when neither is moving — the slow one has left twelve times more stress behind, because it finished more recently.

That reversal is worth stopping on. The fast history produced the larger stress and forgot it sooner; the slow one produced less and is still holding it. The state of the fluid is not a function of what was done to it but of when.

The one number that decides how much of this matters #

The comparison that decides everything is between the fluid’s relaxation time and the flow’s own, which is the Deborah number that a solid if it is not given time is about.

The Deborah number across five processes makes the span concrete: a slow pour at 0.01, stirring by hand at 0.2, the flow computed here at 0.5, an extruder die at 10, a fibre spinline at 100 — four decades, on one fluid with one relaxation time. Below about a hundredth, the flow is so slow that the fluid relaxes completely between one moment and the next: the convolution collapses to the present strain rate times $\int G$, which is a viscosity, and the fluid is Newtonian. Above about one the fluid has no time to relax at all and behaves as an elastic solid being deformed.

The interesting band is the two decades in between, and it is where every process that uses these fluids sits — an extruder die, a fibre spinline, a coating flow, an inkjet. The band is narrow and almost everything of industrial interest is inside it, which is not a coincidence: a fluid whose memory was irrelevant would be replaced by a cheaper one.

What the solver computed, and how it was checked #

The convolution is evaluated directly at four thousand steps, with a single-mode exponential kernel and prescribed strain-rate histories. Nothing here is a flow solution — there is no momentum equation, no geometry and no boundary — and that is deliberate: the essay is about what a constitutive equation carries, and putting it in a geometry would confuse a property of the fluid with a property of the apparatus.

Three checks. That the stress one relaxation time after cessation is exactly $1/e$ of the peak, to two per cent, which tests the kernel and the quadrature at once. That it is below two per cent after five, so the memory does end. And that two histories of equal total strain differ by at least a factor of three in peak stress, and that the slow one leaves more behind at late times — the second half of which is the counter-intuitive part and is the one worth asserting.

What “fading memory” has to mean #

The convolution above is not an arbitrary choice of model, and it is worth saying why, because the reasoning constrains every constitutive equation anybody can write.

Three requirements settle nearly all of it. The stress must depend on the deformation history and not on anything else — that is the principle of determinism. It must not depend on what is happening elsewhere in the fluid — local action. And the influence of the past must decrease as the past recedes — fading memory, which is the assumption that makes the integral converge.

A linear functional of the history satisfying those is exactly a convolution with a kernel that decays, which is Boltzmann’s form. So the model here is not one theory among many; it is the general linear theory, and everything else is either nonlinear or a special case of it.

What the requirements do not fix is the kernel, and that is where the physics of the particular fluid lives. A polymer’s kernel comes from its chain dynamics, a suspension’s from the microstructure’s rearrangement, and a gel’s from a network’s rearranging bonds. The framework is universal and the number of decades of memory is the material.

Where the single mode is wrong #

The model has one relaxation time and no real fluid has one. That is the largest simplification here and it is worth being specific about what it costs.

A polymer solution has a spectrum of relaxation times, spanning several decades, because the chains have modes of many lengths and the long-wavelength modes relax slowest. The kernel is then a sum of exponentials, and a sum of exponentials with a broad enough spread of time constants is indistinguishable from a power law over any finite window.

So a real fluid’s memory looks like the diffusive ones after all — no single time constant, a slow algebraic decay — and the exponential above is what one mode of it does. The measured consequence is that a real relaxation experiment does not give a straight line on a logarithmic plot, and fitting one returns a number that depends on the window fitted over, which is the range a real Reynolds number does not have’s complaint in a different subject.

The practical repair is to fit several modes, and the honest statement is that the number of modes is chosen by the fitter rather than by the fluid.

What a memory does to a flow #

Everything above is about a fluid on its own. Put it in a flow and the memory produces effects that no amount of adjusting a viscosity reproduces, and three of them are worth naming because they are how viscoelasticity is recognised in practice.

Die swell. A polymer emerging from a die expands, sometimes to several times the die diameter, because the chains were stretched inside and recoil once the constraint is removed. The fluid is remembering the die.

Rod climbing. A rotating rod in a viscoelastic fluid draws the fluid up the rod rather than throwing it outwards. The stretched chains form hoops under tension around the axis, and the hoops squeeze inwards and force fluid up. There is nothing in a Newtonian description that can produce it.

Extensional thickening. A viscoelastic fluid pulled apart resists far more than its shear viscosity suggests, because a stretching flow keeps stretching the same chains rather than tumbling them. That is why these fluids are used in fibre spinning and in drag reduction, and it is invisible in every shear measurement — a limitation of the same kind the stress a pipe knows records for a capillary viscometer.

And elastic turbulence. At Reynolds numbers far below any transition, a viscoelastic flow can become chaotic, because the stress feeds back on the flow that produced it with a delay — which is a destabilising loop of exactly the kind a lag creates in a control system. That is the lag that makes flutter possible’s mechanism with its sign reversed, and it is why a memory is not always stabilising. The same reversal appears in two lifts at one incidence, where a lagged separation point feeds energy into a wing rather than damping it.

How the memory is measured #

Since the kernel is the whole of the fluid’s mechanical description, measuring it is what rheology is for, and there are three standard routes with three different weaknesses.

Step strain, and watch the stress relax. This gives the kernel directly — the relaxation modulus is the kernel — and it is the cleanest in principle. Its weakness is the step: a real instrument takes milliseconds to impose one, so everything faster than that is lost, and the fastest modes are exactly the ones that carry the largest stresses.

Small-amplitude oscillation, swept in frequency. This gives the kernel’s Fourier transform, as a storage and a loss modulus, which is the pair of curves the previous figure shows. It reaches shorter times than a step does and it is restricted to small amplitudes by construction, so it says nothing about the nonlinearity.

Steady shear at many rates. This gives one number per rate and, as the section above says, only the area under the kernel. It is the easiest measurement and the least informative.

The three are complementary rather than alternatives, and a fluid characterised by only the third is a fluid characterised by an integral of what matters.

What the picture cannot show #

The stress drawn here is a single scalar, and a viscoelastic stress is a tensor with normal components that a Newtonian shear flow does not have. The first normal stress difference is what causes both die swell and rod climbing, and it is entirely absent from the linear model computed here.

The kernel figure also cannot show the thing that matters most about a real one, which is that it is a sum rather than a single exponential. Three curves at three relaxation times are drawn; a real fluid is their sum, and the sum looks like none of them.

The two kinds of memory, side by side #

This collection has now computed both species of fluid memory, and putting them together is worth a section because the difference decides how each is handled numerically and what each can do.

A relaxation memory has a time constant. Its kernel is an exponential, it can be replaced exactly by one extra state variable per mode — the differential form of the constitutive equation — and a computation carrying it pays a fixed cost per step. There is a definite age past which the past is gone.

A diffusive memory has none. Its kernel is a power law, it cannot be replaced by any finite set of state variables, and a computation carrying it either evaluates a convolution or solves a field equation that holds the history implicitly. Nothing is ever fully forgotten.

The practical difference is enormous and it is why viscoelastic codes are tractable and history-force codes are not: an exponential kernel turns a convolution into an ordinary differential equation, and a power-law kernel does not. It is also why the standard trick in the second case is to approximate the power law by a sum of exponentials — borrowing the first kind’s tractability at the price of a longest time that the physics does not have.

And it explains why a real polymer, with a broad spectrum of modes, sits between them: enough exponentials with a wide enough spread look like a power law over any window anybody measures.

Who found it, and when #

Maxwell wrote the differential form in 1867, as a model for a gas rather than a polymer, and Boltzmann gave the integral form — the convolution above — in 1874, as the general statement for a material with a fading memory. That is the equation used here.

The Deborah number is Reiner’s, from 1964, and its name comes from the Song of Deborah: the mountains flowed before the Lord, which is a statement about time scales rather than about mountains. The polymer-physics account of where the relaxation times come from is Rouse’s and Zimm’s, from the 1950s, and it is what turns the kernel from a fitted curve into a prediction.

Limits recorded rather than smoothed over #

Linear viscoelasticity only. The convolution above is linear in the strain rate, so it cannot produce shear thinning, normal stresses or any of the effects the previous section names. Real constitutive models — upper-convected Maxwell, Giesekus, FENE-P — are nonlinear, and their nonlinearity is where most of the interesting behaviour is.

One mode. Stated at length above. Every number quoted here is that mode’s.

The kernel is prescribed, not derived. Where a polymer’s relaxation times come from is a question for polymer physics rather than for fluid mechanics, and nothing here derives one. What this essay computes is what a kernel does, given one.

No flow. There is no momentum equation anywhere in this essay. The strain-rate histories are imposed, so nothing here says what a viscoelastic fluid would actually do in a geometry — only what its stress records.

And the fluid is taken to have been at rest for ever before the start. A convolution over an infinite past needs an initial condition, and “at rest since the beginning of time” is the one used. Any real measurement inherits whatever was done to the sample before it was loaded, which is why rheologists rest their samples and why the resting time is reported.