Two slow things make a fast one

Inject a drop of dye into a slow flow of water in a millimetre tube and it arrives at the far end smeared over a length of pipe. Two mechanisms are available to explain the smearing and neither of them works on its own.

Shear does not mix. The parabolic profile pulls the drop into a long paraboloid — the middle runs at twice the mean speed and the wall does not move at all — and nothing has been mixed, because the process is exactly reversible: run the flow backwards and the drop reassembles.

Molecular diffusion is far too slow. Its coefficient in water is about 10⁻⁹ m²/s, so it takes a quarter of an hour to cross a millimetre and thirty years to cross a metre.

Together they give something neither has.

What the pair does that neither can #

A molecule in the pipe is carried at the speed of its own streamline, and diffusion moves it between streamlines. So over time it samples the fast middle and the slow wall, its average speed over a long journey is the mean speed of the flow, and its position relative to the centre of the slug performs a random walk.

That random walk is a diffusion, and the step length is the transverse crossing distance while the step time is the time to cross the section. The result — Taylor’s, from 1953 — is that the cross-sectionally averaged concentration obeys an ordinary diffusion equation in a frame moving at the mean speed, with

$$D_{\text{eff}} = D\left(1 + \frac{\mathrm{Pe}^2}{48}\right), \qquad \mathrm{Pe} = \frac{Ua}{D}$$

for a round tube of radius a.

Irreversibility is the whole difference. Advection is reversible and diffusion is not, so a process that is 99.9 per cent advection inherits the irreversibility of the remaining tenth of a per cent and multiplies it. That is a general shape worth carrying: a small irreversible step in a large reversible process makes the whole thing irreversible, and the rate is set by the large process rather than the small one.

The constant is a computation, not a quotation #

The 48 comes from a cell problem: the concentration adjusts to a shape across the section that satisfies a Poisson equation with the velocity deviation as its source, and the effective diffusivity is the average of that shape against the velocity. It is solved here numerically on a radial grid, and the constant that comes out is 48.0000 for a tube and 52.5 for a plane channel — which is Aris’ 2/105, written the other way up.

That matters more than it might seem. The constant is different for every geometry, and it is frequently transcribed from one to another: a channel result quoted with a tube’s 48 is wrong by ten per cent, and a bundle-of-tubes model of a packed bed with the wrong constant is wrong by whatever the pore shape decides. Computing it rather than looking it up costs twenty lines.

The result that makes people check the algebra #

At large Péclet number the D in the numerator cancels and

$$D_{\text{eff}} \to \frac{U^2a^2}{48\,D},$$

so increasing the molecular diffusivity decreases the dispersion. The mechanism explains it without any algebra: dispersion happens because molecules are carried at different speeds for a while, and a molecule that crosses the section quickly is carried at the wrong speed for less time. A tracer that diffused infinitely fast would be perfectly mixed across the section at every instant, would travel at the mean speed exactly, and would not disperse at all.

This is the same structure as a bearing’s friction or a bed’s permeability: a transport coefficient that is a property of the combination rather than of the material, and whose dependence on the material can run either way.

Where the answer begins to apply #

Taylor’s result is an asymptotic one: it holds after diffusion has had time to sample the section, which takes $a^2/D$ and is a thousand seconds for a millimetre of water. Before that the spreading is not diffusive at all — it is the reversible shear picture with a diffusive correction — and the variance grows differently.

The regime where the answer is simple begins further downstream than most pipes are long, and that observation is the reason chromatography looks the way it does. A packed column cuts the diffusion distance from the tube radius to the particle radius, which reduces the settling time by the square of the ratio: from a thousand seconds to a fraction of one. The same reasoning explains the migration to capillary columns, and it is one of the few places where an asymptotic condition drives the design of an entire instrument.

Péclet is Reynolds times Schmidt, and that is why liquids are extreme #

The Péclet number can be split, and the split explains why this effect is so much larger in liquids than in gases:

$$\mathrm{Pe} = \frac{Ua}{D} = \frac{Ua}{\nu}\cdot\frac{\nu}{D} = \mathrm{Re}\cdot\mathrm{Sc}.$$

The Schmidt number ν/D compares how fast momentum diffuses with how fast the tracer does. For a gas it is about one — molecules carry both, so both diffuse at the same rate — and for a small molecule in water it is about a thousand, because momentum is transmitted by a network of collisions and the molecule itself has to physically move.

So the millimetre tube in these figures, running at a centimetre a second, has a Reynolds number of ten — an unmistakably creeping, laminar, gentle flow — and a Péclet number of ten thousand. The flow is slow and the transport is violently advective at the same time, and that combination is exactly what makes Taylor dispersion a liquid phenomenon. A gas line has a Reynolds number in the hundreds of thousands and a Péclet number no larger.

How wide a slug actually gets #

The number worth carrying out of the whole subject is not the diffusivity but the spread it produces, and that is one line: a diffusive process gives a variance growing as $\sigma^2 = 2D_{\text{eff}}t$.

For the millimetre tube at a centimetre a second, ten metres of pipe takes a thousand seconds, and

$$\sigma = \sqrt{2 \times 2.08\times10^{-3} \times 1000} = 2.0\ \text{metres}.$$

A dye injection that started as a drop is two metres long by the time it has gone ten. Molecular diffusion alone over the same time would have spread it by 1.4 millimetres. That factor of fifteen hundred in length — the square root of the two million in diffusivity — is what an experimenter actually meets, and it is why the residence time in a laminar tube is a distribution rather than a number.

The consequence for anybody measuring anything in a tube is unwelcome and general: a laminar pipe is a bad delay line. A pulse of tracer sent down one arrives smeared over a length comparable with the distance it travelled, so the arrival time carries much less information than the injection did. Turbulent pipes are far better, packed columns better still, and both for the same reason — they cut the time a molecule spends on the wrong streamline.

What the slug looks like before the answer applies #

“The variance grows differently” is an unsatisfying way to leave the pre-asymptotic regime, because the difference is computable and the shape it produces is not the one the formula predicts.

Before diffusion has sampled the section, a molecule stays on its own streamline, so its displacement relative to the mean is simply its velocity deviation multiplied by the time. The variance is then

$$\sigma^2 = \left\langle (u - U)^2 \right\rangle t^2,$$

growing as $t^2$ rather than $t$ — ballistic rather than diffusive. For Poiseuille flow in a tube the mean square deviation is $U^2/3$, so $\sigma = Ut/\sqrt3$ exactly, and setting that equal to the asymptotic $2D_{\text{eff}}t$ puts the crossover at $t = a^2/8D$ — an eighth of the section’s own diffusion time, which is the same condition the settling figure draws.

The shape is the more interesting half, and for a round tube it can be written down. Equal areas of the cross-section correspond to equal intervals of $r^2$, and Poiseuille’s velocity is linear in $r^2$, so the tracer’s velocity is distributed uniformly between zero and twice the mean. The early slug is therefore not a spreading Gaussian at all: it is a perfect top hat, of length $2Ut$ in the frame moving with the mean, with a sharp edge at each end.

That is a coincidence of the tube’s geometry, and the plane channel does not share it. There the velocity is quadratic in a coordinate that is uniformly distributed, so the fluid piles up near the fast end of its own velocity range: the early slug has a steep leading front and a long trailing tail, and it is skewed from the outset.

Both then relax to a Gaussian, and slowly. The approach is algebraic rather than exponential — the skewness of the distribution decays only as the inverse square root of time — so a measurement taken several settling times downstream still has a visibly asymmetric arrival curve while its variance is already obeying Taylor’s formula to a few per cent.

Which matters wherever the arrival curve itself is the measurement. A chromatographic peak that has not reached the asymptotic regime is not Gaussian, and fitting a Gaussian to it misplaces the centre as well as the width — reporting a retention time biased towards the tail and a resolution better than the column has. Peak tailing is routinely attributed to adsorption on the stationary phase, and some of it is simply this: a column not yet long enough for its own dispersion to have become diffusive.

The same mechanism, without a pipe #

Taylor’s argument needs only two ingredients — a velocity that varies across the flow, and something that moves tracer across it — so it applies well beyond tubes, and the general name for it is shear dispersion.

In a river, the velocity varies from bank to bank and turbulence does the cross-stream mixing. The same derivation gives an effective longitudinal diffusivity hundreds of times the turbulent one, and it is the standard tool for predicting how a spill will have spread by the time it reaches a town downstream. The cross-stream mixing time is the width squared over the turbulent diffusivity, which for a fifty-metre river is hours — so the settling condition is the first thing to check there too.

In a porous medium, the velocity varies from pore to pore and molecular diffusion mixes across streamlines. The same structure appears with the pore size in place of the tube radius, and the result is that a bed’s dispersion coefficient is much larger than its molecular one at any reasonable flow — which is why the residence time in a packed reactor is a distribution that has to be designed against.

In the atmosphere, wind shear plus turbulent mixing across the boundary layer disperses a plume along the wind far faster than the turbulence alone would. It is the same equation with different numbers, and the derivation is Taylor’s.

What it costs to measure a flow with a tracer #

The dispersion is also the reason a tracer measurement in a laminar tube is harder than it looks, and the arithmetic is worth one paragraph because it is the practical half of the whole essay.

A pulse injected at the inlet arrives at a station a distance L downstream centred at the mean transit time L/U and spread over $\sigma = \sqrt{2D_{\text{eff}}L/U}$. The relative spread is therefore

$$\frac{\sigma}{L} = \sqrt{\frac{2D_{\text{eff}}}{UL}},$$

which for the tube in these figures is 20 per cent at ten metres and still 6 per cent at a hundred. It falls only as the square root of the distance, so a longer pipe gives a sharper relative answer and a much wider absolute one — the pulse never becomes narrow, it only becomes narrow compared with how far it has come.

The transit time is therefore knowable and the arrival time is not, which is the distinction that matters for anybody using a tracer to measure a flow rate. Averaging over the arriving pulse recovers the mean speed to whatever precision the concentration measurement allows; picking a single arrival event — the first appearance of dye, say — measures the fastest streamline instead, and in a laminar tube that is exactly twice the mean. The two answers differ by a factor of two and both are repeatable, which is the most expensive kind of measurement error there is.

Two mechanisms, and which one a design should attack #

Because the effective diffusivity is a product of a shear and a diffusion, there are two ways to change it, and they pull in opposite directions.

Reduce the shear and the dispersion falls: a flatter profile disperses less, which is why a turbulent pipe — whose profile is far blunter than a parabola — disperses less per unit length than the laminar formula would suggest, and why a packed bed’s dispersion is dominated by pore-to-pore velocity differences rather than by the profile within a pore.

Reduce the distance and the dispersion falls too, and much faster, because the effective diffusivity goes as the square of the transverse length. This is the lever every separation technology pulls: a capillary column instead of a packed one, a smaller particle in the packing, a narrower channel in a chip.

The two are not symmetrical. Halving the shear halves the dispersion; halving the width quarters it, and also cuts the settling time by four. Geometry beats flow control, which is why the history of chromatography is a history of columns getting narrower and particles getting smaller rather than of flows being made more uniform.

What the model does not contain #

No turbulence. Everything here is laminar Poiseuille flow. Turbulent pipes disperse too, and much more, but the mechanism is different — the cross-stream transport is turbulent rather than molecular, and the effective diffusivity comes out proportional to $U a$ rather than to $U^2a^2/D$. The famous result for a turbulent pipe is $D_{\text{eff}} \approx 10.1\,a\,u_\tau$, and that constant is a measurement rather than anything computed here.

No secondary flow. A curved pipe has a secondary circulation in its cross-section, which transports tracer across streamlines much faster than molecules do and cuts the dispersion by orders of magnitude. Coiled columns exploit this deliberately, and nothing here computes it.

A passive tracer. The dye is assumed not to change the density, the viscosity or the flow. A buoyant or viscous contaminant does all three.

One dimension out. The result is for the cross-sectionally averaged concentration, and it says nothing about the shape within the section — which is exactly what the cell problem computes and what the averaging then throws away.

No wall interaction. Adsorption on the wall adds a whole extra mechanism, and in chromatography it is the point rather than a nuisance: the interesting dispersion there is the competition between this one and the exchange with the stationary phase.

Who found it, and when #

Geoffrey Taylor published the tube result in 1953, from an experiment with a millimetre glass tube and potassium permanganate, and the paper’s method is worth noting: he found the effective diffusivity by measuring how a slug spread, and the theory in the same paper is short.

Rudolf Aris generalised it in 1956 by the method of moments — tracking the variance of the distribution rather than solving for its shape — which removed Taylor’s restrictions, supplied the +D that Taylor’s leading-order result lacked, and gave the constants for other cross-sections. That is why the pair of names travels together, and the two contributions are of different kinds: Taylor found the mechanism and Aris found the machinery.

The result has since become the standard model for solute transport in porous media, in blood vessels, and in rivers, where its most-quoted application is the estimation of how far a pollutant will have spread by the time it reaches a town.

Where the ladder goes next #

Everything above assumes the flow is steady. Make it oscillate and a new number appears — one that decides whether the profile has time to become Poiseuille’s at all — and above about ten the middle of the pipe moves as a plug with the fastest fluid in a ring near the wall, which is what the flow in an artery actually looks like.