Two strainings, and the order they came in

A blob of dye in a stirred tank gets longer. The rate at which it is being drawn out at any instant is a property of the velocity gradient there and then, and the stretching rate that is not one number is this collection’s account of how to read that rate off the gradient: it depends on which way the line is pointing, and the largest of the available rates is an eigenvalue of the rate-of-strain tensor.

That essay leaves a question standing which looks like arithmetic and is not. Given the rate at every instant along a parcel’s path, how long is the line at the end?

The obvious answer is that the total stretch is the integral of the rate — that stretching, like distance, accumulates. It is wrong, and the size of the error is not small. This essay builds two histories out of the same two pieces of straining, applies them in the two possible orders, and finds lengths differing by a factor of 2.16 from integrated rates that are identical to zero difference.

The two pieces #

The first piece is a simple shear at rate 2: the fluid slides in one direction at a speed proportional to how far across the flow it sits. The second is a pure straining flow at rate 1, which pulls the fluid apart along one axis and squeezes it in along the other at the same rate. Each lasts one unit of time. Both are incompressible and both are steady while they last.

Each is applied for one unit of time, and the whole history is two units long. There are two ways to order them, and the question is whether the order matters.

The pieces are not chosen at random. They are chosen so that the comparison is fair in a specific and demanding sense: their rate-of-strain tensors have the same magnitude. The symmetric part of the shear’s velocity gradient has Frobenius norm root two; so does the pure strain’s. Whatever the fluid is doing in the first unit of time, it is being strained exactly as hard as in the second, by the measure that a rate-of-strain tensor supports.

So the integral of the strain rate over the whole history is 2.82843 in both orders — root two from the shear and root two from the strain, each acting for one unit of time — and the difference between the two integrals is 0.000000000 — not a small number, not a tolerance, but the number zero, because the same two quantities are being added in a different sequence.

What a deformation gradient does that a rate does not #

The object that carries a material line from where it started to where it is now is the deformation gradient: a matrix which, applied to the initial direction and length of a small material vector, returns its present direction and length. It is what longer, with nothing pulling it is about, and its defining property is the one this essay turns on.

Deformation gradients compose by multiplication. A parcel that has been through one motion and then another has the deformation gradient of the second times the deformation gradient of the first, in that order, with the later motion on the left. Rates add; gradients multiply.

Matrix multiplication does not commute. The product of the shear’s gradient with the strain’s is not the product of the strain’s with the shear’s, and no amount of care about how long each piece lasted will make them equal. The two orders are two different matrices, and the stretch is a property of the matrix rather than of the pieces that went into it.

That is the whole mechanism, and everything below is a measurement of how much it costs.

The measurement #

The stretch of the most-stretched material direction is the largest singular value of the deformation gradient — the longest semi-axis of the ellipse a unit circle of material directions is carried into. Both ellipses have exactly the area of the circle they came from, because both motions are incompressible and the determinant of each product is one to a part in 10¹⁵. What differs is their shape.

Shear first, then strain, gives a stretch of 6.087. Strain first, then shear, gives 2.818. The ratio is 2.16, and it is produced by two histories which no measurement of the instantaneous rate can tell apart.

The reason is visible in the picture and is worth stating in words. The shear stretches material lines that lie near its own sliding direction and does nothing to lines across it. The pure strain stretches along a fixed axis. Applied in one order, the shear leaves the material lines lying close to the strain’s stretching axis, so the strain finds them already aligned and pulls them out further. In the other order the strain acts first, on lines that are not yet aligned with anything, and the shear that follows then works on a set of directions the first motion has already fixed.

Stretching is not a quantity of work done to the fluid. It is a matter of what the fluid was pointing at when the work arrived.

The commutator, which is the whole of the difference #

The failure of two matrices to commute is itself a matrix — their commutator, the difference between the two products. For these two pieces it is not small: its largest entry is −4, against a shear rate of 2 and a strain rate of 1, which is what a factor of two in the answer requires. Shear first then strain gives a stretch of 6.087; strain first then shear gives 2.818. The ratio is 2.16, and the relative gap is 53.7 per cent of the larger.

The commutator has a physical reading. A shear carries the straining axes round with it, rotating the directions a subsequent strain will act along; a strain compresses the plane a subsequent shear slides in. Each motion changes the geometry the other one will meet. The commutator is exactly the size of that mutual interference, and when it vanishes — for two pure strains sharing an axis, say — the order genuinely does not matter and integrating the rate genuinely does work.

This is the practical test. The order matters exactly to the extent that the pieces of the history fail to commute, which is a computable matrix and not a matter of judgement.

What the solver computed, and how it was checked #

The two deformation gradients are built from matrix exponentials by scaling and squaring, which is the same routine used for the fine interleaving below and is therefore exercised at step sizes spanning four thousand to one — from one interval to 4,096 of them, over which the interleaved stretch converges on 3.602 with a residual exponent of 1.0000. Three things are checked before the comparison is allowed to mean anything.

That the two histories really do share a strain rate. The Frobenius norms of the two symmetric parts are compared and must agree exactly, not to a tolerance — they are the same two numbers in a different order, so any difference at all would be a bug in the norm rather than a fact about the flow. The check refuses a difference above 10⁻¹⁵ and reads zero.

That neither history has quietly compressed anything. The determinant of each product is required to be one to a part in 10¹². A stretch obtained by squashing is not the effect this essay is about, and incompressible is not a property of the fluid is a reminder that the assumption has to be checked rather than assumed from the fluid’s name.

And that the stretch is the stretch and not an artefact of the basis. The largest singular value is computed from the eigenvalues of the product of the gradient with its own transpose, which is basis-independent by construction, and the two singular values are checked to multiply to the determinant — 1.000000000 in both orders, since both motions are incompressible.

The refusals are exercised too: the ordering check is offered a claim that the two stretches differ by a factor of ten and refuses it, the volume check is offered a tolerance of zero and refuses that, and the interleaving check is asked to certify an exponent to no tolerance at all. A check that has never rejected anything is not evidence, which is why this collection runs its assertions against input they must refuse.

Why a material line does not simply align and stay aligned #

An objection worth answering: over a long history, does the line not settle into the most-stretching direction and stay there, after which the total is the integral of the rate along that one direction?

It does not, and the stretching rate that is not one number gives the reason. The stretching direction is a property of the velocity gradient at a point, and a parcel travelling through a flow meets a different gradient every instant, whose eigenvectors point somewhere else. Alignment is a race between the rate at which the strain pulls the line towards the current stretching axis and the rate at which that axis moves away, and in a general flow neither wins outright.

In the special case of a steady flow with fixed axes the line does align, exponentially, and the long-time stretching rate does become the largest eigenvalue. That is the case in which the naive integral is asymptotically right, and it is the case almost never met: a stirred tank, a boundary layer, a turbulent patch and the flow round a bend all present a parcel with a gradient that turns.

How finely the two have to be interleaved before the order stops mattering #

There is a limit in which the difference does vanish, and it comes with a rate.

Take the same total history and cut it finer: shear for half a unit, strain for half a unit, shear again, strain again. Then quarters. Then eighths. In the limit of infinitely fine alternation both orders converge to the same answer, which is the deformation gradient of the sum of the two rate tensors applied for the whole time — the Trotter product limit, the same statement that underlies operator splitting in every numerical solver that advances diffusion and advection in alternate steps.

The gap between the two orders falls as the first power of the step. The measured local exponent is 1.000000017. Cutting the history into 4,096 alternations reduces a gap of 3.27 in stretch to 0.00075, which is four thousand times smaller for four thousand times as many pieces.

That number is the honest answer to “when can the rate be integrated”. It can be integrated when the history is fine-grained compared with the time over which the straining directions turn — and the error of doing so is first order in the ratio of those two times, so it is a controlled approximation rather than a wrong one. What it is not is exact, and what it never becomes is exact for a history with two or three distinct phases in it, which is what most real flows present.

The same shape, elsewhere in the collection #

An answer that depends on the sequence rather than on the totals is not a curiosity of this one calculation, and three other essays here are the same statement in different clothes.

A parcel’s motion is a product of what happened to it, one step at a time. What a parcel does in the first instant splits the local motion into a translation, a rotation and a strain, and it is careful to say that the split is local and instantaneous. This essay is what happens when those instants are strung together: the pieces do not add, because the rotation in each one moves the axes the next strain will use.

A flow map is a time-ordered object, and area is the one thing that survives it. The area that must not move shows a numerical scheme whose Jacobian is exactly one at every step size, so the volume statement is robust in a way the stretch statement is not. That contrast is the general one: the invariants of a deformation survive the ordering and its magnitudes do not.

And chaotic advection is this effect run for a long time. Steady, three-dimensional, and mixing anyway is the demonstration that a flow with no randomness in it separates neighbouring parcels exponentially. The exponential is a product of many non-commuting gradients, and the exponent it grows at is not the average strain rate — it is smaller, sometimes much smaller, because the alignment that the ordering keeps disturbing is what would have made the two equal.

Where this changes an answer #

Mixing. The reason a stirrer is designed rather than sized is exactly this. Two impellers delivering the same power into the same tank strain the fluid at the same average rate and produce completely different mixing, because mixing is set by how much material interface is created and that is a product of deformation gradients along each parcel’s path. No randomness, and it mixes anyway is the collection’s account of how a perfectly deterministic flow achieves it, and the ordering here is the mechanism underneath: a stirring protocol that alternates two motions which do not commute stretches exponentially, and one that repeats a single motion does not.

Rheology. A polymer’s stress depends on how far its molecules have been stretched, so a fluid put through two deformations in one order can be stiffer afterwards than the same fluid put through them in the other. Steady-shear viscometry measures a number that describes neither.

And any model that carries a scalar “accumulated strain”. Damage models, dispersion models and some turbulence closures carry an integrated strain rate as a state variable, on the reasoning that it summarises the deformation history. It summarises a projection of it. Two parcels arriving at the same value of that scalar can differ by a factor of two in how far their material lines have been drawn out, which is precisely the quantity the scalar is standing in for.

What the two orders still agree about #

It is worth being exact about how much survives, because the answer is more than nothing.

Volume. Both deformation gradients have determinant one to a part in 10¹⁵. Neither history compresses anything, so whatever else the ordering changed, it did not change how much fluid there is.

The sum of the squared stretches, weighted correctly, is not conserved — but the product of the two singular values is, and it is one. So a factor of 2.16 in the long axis is exactly a factor of 2.16 in the thinness of the short one. What the ordering buys in length, it pays for in a thinner sheet, and the thinness is what makes diffusion finish the job.

And the strain rate at every instant. The rate is a property of the field, and every measure built out of it — the local dissipation, the enstrophy production, the second invariant — is identical between the two histories at every moment. The whole of the difference is in an integral that is not an integral.

Who found it, and where it sits #

The non-commutativity of finite deformations is older than fluid mechanics and belongs to the kinematics of continua: it is why the deformation gradient is written as a two-point tensor with a reference configuration attached, and why finite-strain theory keeps the whole history rather than a scalar. Trotter’s product formula, which is the limit measured above, arrived from operator theory in 1959 and reached fluid dynamics through numerical analysis, where every fractional-step method depends on it and every one of them carries the same first-order error.

The fluid-mechanical form of the result is due to the mixing literature of the 1980s, which found that the effectiveness of a stirring protocol depends on its sequence rather than its energy, and gave the field its standing example: two motions applied alternately produce exponential stretching where either alone produces linear.

Limits recorded rather than smoothed over #

Two dimensions and two pieces. The demonstration is a pair of two-by-two matrices, which is the smallest arrangement in which the effect exists. Three dimensions make it larger, not smaller, because there are more ways for two rotations of the straining axes to disagree.

The pieces are steady while they last. Real histories have gradients that vary continuously, so the exact ordering statement becomes a time-ordered exponential rather than a product of two terms. The time-ordered exponential is what the flow map is, and it has no closed form in general — which is the reason the flow map is computed rather than evaluated everywhere in this collection.

The strain rates are matched by Frobenius norm. Another norm would match them differently and change the numbers slightly. It would not change the zero: whatever the norm, both histories contain the same two pieces, so any integral of any instantaneous quantity is identical between them.

And the 2.16 is for these two pieces at these durations. Longer pieces make it larger and pieces that nearly commute make it smaller. The claim is not that the factor is two; it is that no integral of the rate contains it.