No randomness, and it mixes anyway

Stirring a fluid and mixing it are different things, and the difference has a theorem in it.

A steady two-dimensional flow moves fluid around energetically and cannot mix it at all. Its pathlines coincide with its streamlines — which is only true in steady flow — and a streamline is a curve. A particle released on one is confined to that curve for the rest of time, whatever the flow’s speed, whatever its Reynolds number, and however complicated the curve looks.

Take that flow and switch it, and everything changes.

The simplest unsteady flow there is #

Aref’s blinking vortex, from 1984, is two point vortices at $\pm d$ switched on alternately, each for a time $T$. It is chosen here for a reason that matters for the arithmetic: each half-period is an exact rotation, so the map from one period to the next can be written down in closed form and there is no time-stepping error anywhere in this essay. In a subject about the amplification of tiny differences, that is not a nicety.

One dimensionless number controls it:

$$\mu = \frac{\Gamma T}{2\pi d^2},$$

which is the angle, in radians, through which a vortex turns fluid sitting at its partner’s distance. Below about 0.1 the motion is regular. By 0.5 there is a chaotic sea with islands in it. By 1 the islands are small.

Nothing in that map asks whether the velocity field solves anything. This essay is kinematics: the question is what a given field does to the points it carries, and the answer would be the same if the field came from an impeller, a pair of rollers, or a random-number generator. That is why chaotic advection is not a turbulence phenomenon — it happens in flows that are laminar, slow and entirely predictable as fields.

Some of the fluid is stirred and some is not #

That picture is the one worth sitting with. The closed curves are invariant tori: fluid inside one is stirred round and round and is never mixed with fluid outside it. The scattered points belong to particles in the chaotic sea, which visit everywhere the sea reaches.

Two grains of dye a millimetre apart can therefore have entirely different fates. There is no single number describing how well this flow mixes, because mixing is a property of the orbit rather than of the flow — and an engineer who measures a mixing time by tracking one tracer particle has measured one orbit.

What the exponent means in fluid #

$\lambda = 1.23$ per period means neighbouring particles separate by a factor of $e$ every period. Two consequences, both arithmetic:

A factor of a thousand in 5.6 periods. Dye grains a micron apart are a millimetre apart in six switches of the vortices. In the steady flow, the same pair separates linearly and takes several thousand periods to do the same thing.

Predictability has a budget. Double-precision arithmetic knows a position to sixteen digits, and the exponent eats $\lambda/\ln 10 = 0.53$ digits per period, so thirty periods of prediction is all sixteen digits buy.

That figure is the cleanest statement of what determinism buys and does not. The equations are exact, the map is invertible, the flow is laminar — and the information about where the particle started has been destroyed by rounding. The steady flow, drawn beside it, returns to its starting point to the last bit for as long as anybody cares to run it.

Measuring the stretching is harder than measuring the exponent #

The second route to the exponent is to release a short line of dye and watch its length. It should grow at the same rate a pair of particles separates, and it does — but the measurement is much harder than the pair’s, in a way that is worth being explicit about because the difficulty is the subject.

Following a line that lengthens by $e^{1.23} = 3.4$ times per period needs a marker count multiplying by 3.4 per period. Forty thousand markers therefore buy six periods, and two hundred thousand buy seven; past that the polyline stops following the curve and reports a length that is an underestimate rather than a measurement. The fitted rate over the resolved range is 0.69 per period at forty thousand markers and 1.07 at two hundred thousand, climbing towards the 1.23 the particle pair gives.

A resolution limit and a physical effect look identical in one run and differ completely across two, which is the same test the material loop needed and the same lesson: a Lagrangian method loses resolution at exactly the rate the flow stretches material.

Stretching, folding, and why no dye is ever destroyed #

The filament in that figure has never crossed itself and never will. The map is invertible, so two particles cannot arrive at the same place, and a material line is a material line for ever.

Mixing in a flow with no diffusion is therefore not the destruction of structure but its refinement. The line is stretched, folded back on itself, stretched again; after $n$ periods it is $e^{\lambda n}$ times longer and packed into the same region, so the typical distance between adjacent strands falls as $e^{-\lambda n}$. Nothing has blurred; the structure has simply become finer than any instrument.

That is where molecular diffusion finally enters, and its role is worth stating precisely. Diffusion is hopeless at smoothing a large blob — it takes a metre-scale flow hours to diffuse — and unbeatable once the striations are microns apart, because its time scale goes as the square of the distance. The advection does the transporting and the diffusion does the last micron, and the mixing time is set by how fast the first process delivers the second its opportunity.

The same division of labour appears in the dispersion of a slug of dye along a pipe, where shear and diffusion between them give an effective diffusivity a million times the molecular one.

The steady flow’s proof, in one line #

The claim that a steady two-dimensional flow cannot mix deserves its proof written out, because it is short and because it says exactly where the exception is.

An incompressible plane flow has a stream function ψ with $u = \partial\psi/\partial y$ and $v = -\partial\psi/\partial x$. A particle’s position therefore obeys

$$\dot{x} = \frac{\partial\psi}{\partial y}, \qquad \dot{y} = -\frac{\partial\psi}{\partial x},$$

which is Hamilton’s equations with ψ as the Hamiltonian and x and y as the conjugate pair. A Hamiltonian with one degree of freedom that does not depend on time is conserved along every trajectory, so every particle stays on its own level set of ψ — its streamline — for ever.

That is the whole proof, and every word in it is load-bearing. Two-dimensional, because only a plane incompressible flow has a stream function. Steady, because a time-dependent ψ is a Hamiltonian with one and a half degrees of freedom, which is the smallest system that can be chaotic. A steady three-dimensional flow has no such constraint and mixes perfectly well, which is why a steady helical pipe and the ABC flow both have chaotic trajectories, and it is the most important limitation of this essay’s headline.

How to measure something that is never destroyed #

The observation that advection refines structure rather than destroying it has a consequence for anybody trying to put a number on how well a flow mixes, and it is more awkward than it first looks.

With no diffusion the concentration field is merely rearranged. Every point that was pure dye is still pure dye and every point that was clear is still clear; the set of values the field takes has not changed at all, and neither has any statistic computed from those values. The variance of the exact concentration field is constant for ever, in the steady flow and in the chaotic one alike, so a measure of mixedness built on it reports that no mixing has ever occurred — in a flow whose dye is visibly spread across the whole region.

So every usable measure of mixedness has a length scale in it, stated or otherwise. Coarse-grain the field at some resolution and then take the variance, and the number falls as the striations become finer than the filter. That is what an eye does, what a photograph does, and what a concentration probe of finite volume does — and each of them is reporting the flow through its own aperture. Two laboratories measuring the same beaker with different probes get different mixing times, and neither is wrong.

The chemical-engineering literature has carried the distinction since Danckwerts, under two names that deserve to be better known outside it. The intensity of segregation is how different the concentrations still are; the scale of segregation is how far apart the differences are. Pure advection leaves the first untouched for ever and drives the second down exponentially, and it is only the pair together that describes what has happened.

The measure that behaves best under this essay’s mechanism is one that weights fine structure less than coarse — a mix-norm, technically a negative Sobolev norm of the concentration field. Because the striation spacing falls as $e^{-\lambda n}$, such a norm decays exponentially at a rate set by the same $\lambda$ the particles gave, so it converts the stretching exponent directly into a mixing rate, and it does so without any arbitrary filter width having to be nominated.

Which puts this essay’s own figures in their place. The picture of a folded filament is a picture at a resolution, and the fact that it looks mixed after six periods and unmixed after two is a statement about how many strands fit in a pixel. The flow has done exactly one thing throughout — stretched a line at a constant exponential rate — and everything about how mixed the result looks is a property of the instrument. That is the same conclusion this collection reaches about every averaged quantity it examines, arriving here in the one case where nothing has been averaged at all.

How many periods a beaker needs #

The two mechanisms together give an estimate of the mixing time that neither gives alone, and it is worth doing because the answer is surprisingly small.

Start with a centimetre of dye in water, molecular diffusivity $10^{-9}$ m²/s, and a stirring period of a second. After $n$ periods the striations are $10^{-2}e^{-\lambda n}$ metres apart, and diffusion crosses a gap of size $s$ in about $s^2/D$. Mixing finishes when diffusion can cross a striation within one period:

$$\frac{(10^{-2}e^{-1.23n})^2}{10^{-9}} \approx 1 \quad\Longrightarrow\quad n \approx 5.$$

Five periods. Now the same arithmetic for the steady flow, where the striation thickness falls only as $1/1.2n$ rather than exponentially: the condition becomes $n^2 \approx 7\times10^4$, so $n \approx 260$.

A factor of fifty in the time, from a change that costs nothing — switching between two flows instead of running one. And the reason the number is small in the chaotic case is that the exponential is inside a logarithm: multiplying the beaker’s size by a thousand adds only $\ln(1000)/1.23 \approx 6$ periods.

Where the islands go, and why a stirrer has a schedule #

The practical consequence of the island structure is not obvious and is worth drawing out.

A mixing vessel with a steadily rotating impeller has islands, and no amount of extra power removes them: raising the speed changes their size and position and leaves them islands. The way to destroy an island is to make the flow depend on time in a way that moves it — reversing the impeller periodically, alternating between two impellers, or using a time-dependent boundary. Industrial static mixers do it in space instead: alternating elements which each produce a steady flow, so a particle passing through experiences a time-dependent one.

The blinking vortex is the laboratory version of exactly that, and the reason it is the standard model is that it is the least a flow can do and still mix.

Why a laminar mixer is worth building #

The engineering reading of all this is that mixing without turbulence is possible, and sometimes preferable.

Turbulence mixes well and costs energy — a stirred tank spends its power on a cascade whose small scales do the final work. Chaotic advection mixes at the same exponential rate with a laminar flow and a fraction of the power, because the stretching is done by the mean flow rather than by fluctuations. Where the fluid cannot tolerate turbulence — a shear-sensitive biological suspension, a polymer that degrades, a flow whose Reynolds number is simply too low to become turbulent — it is the only option.

The design rules follow from this essay’s figures rather than from a correlation. Time dependence is mandatory: a steady laminar mixer in a plane cannot work at all. Islands must be moved rather than stirred harder: raising the strength rearranges them. And the exponent, not the velocity, is the figure of merit, because the mixing time goes as ln(scale)/λ and therefore depends only logarithmically on how big the vessel is.

Microfluidic mixers are the clearest example: Reynolds numbers of order one, no possibility of turbulence, and mixing achieved entirely by making the channel’s geometry impose a time-dependent flow on the fluid passing through it.

What the model does not contain #

No diffusion at all. Every figure here is pure advection: a line stays a line, a patch keeps its area, and nothing ever blurs. Real mixing finishes with diffusion, and the ratio of the two — the Péclet number — decides where the striations stop getting finer. Nothing here computes it.

Two dimensions. The theorem that a steady flow cannot mix is two-dimensional and false in three: a steady three-dimensional flow can have chaotic streamlines, because a trajectory is no longer confined to a curve by having a stream function. That is why the ABC flow and a steady helical pipe mix, and it is the single most important limitation of this essay’s central claim.

Point vortices, which are singular. The blinking flow is not a solution of anything: it is a prescribed field. Two real vortices would move each other, decay, and eventually merge, and the point-vortex idealisation has no core.

A single exponent, from a single orbit. The exponent quoted is that orbit’s, in that region, at that strength. A proper characterisation would be a distribution of finite-time exponents, and it would be broad.

No walls, no free surface, no third vortex. All of which change the island structure entirely, because the island structure is a property of the map and not of the physics.

Who found it, and when #

Hassan Aref named chaotic advection in 1984 and the blinking vortex is his. The result was startling to the mixing community because it separated two things that had been assumed to travel together: complexity of the flow field and complexity of the trajectories. A very simple field, with two singularities and one parameter, produces trajectories nobody can predict.

The mathematics was older. Poincaré’s work on the three-body problem in the 1890s established that a deterministic system with few degrees of freedom can have unpredictable orbits, and the KAM theorem of the 1950s and 60s explained the islands: invariant tori survive a small perturbation, which is exactly why a weakly blinking vortex still has most of its fluid on closed curves.

What Aref supplied is that fluid particles in a two-dimensional incompressible flow are a Hamiltonian system — the stream function is the Hamiltonian, and x and y are the conjugate pair — so every result about Hamiltonian chaos applies directly to dye in a beaker. That correspondence is exact, and it is the reason this essay’s figures look like a mechanics textbook’s.

Where the ladder goes next #

Everything here is about where particles go. The other kinematic question is about the pattern itself — the arrangement of stagnation points, saddles and centres in a flow field — and it turns out that pattern is not free to be anything. The number of critical points of each kind on a body obeys a count that no flow, viscous or ideal, steady or not, is allowed to break.