Momentum going sideways

Stir honey and it resists. Stir water and it resists much less. The quantity being measured has a name everybody knows and a mechanism almost everybody has wrong, and the mechanism turns out to be one this collection has already built.

The usual account says the fluid is in layers, and the layers rub. It gets the answer right for the picture above and it predicts nothing else, because there are no layers — the division into layers is drawn by whoever is describing the flow, not by the fluid.

The definition, which is a measurement #

Take two parallel plates a distance $h$ apart with fluid between them, hold one still and drag the other at speed $U$. A force is needed. Measure it per unit area of plate and call it $\tau$. What is found is

$$\tau = \mu \frac{U}{h},$$

with $\mu$ a property of the fluid. Written pointwise, with $u(y)$ the velocity at height $y$:

$$\tau = \mu \frac{\mathrm{d}u}{\mathrm{d}y}.$$

Stress proportional to velocity gradient, with the constant of proportionality the dynamic viscosity. A fluid obeying this is called Newtonian, and water, air, most oils and most solvents obey it well over enormous ranges. Many important fluids do not, and that is a later rung.

The units of $\mu$ are pascal-seconds. Water is about $1.0 \times 10^{-3}$ Pa·s at room temperature, air about $1.8 \times 10^{-5}$, honey about 10, and pitch about $2 \times 10^8$ — a range of eleven orders of magnitude across substances that all pour.

The units that give the mechanism away #

Now divide by the density:

$$\nu = \frac{\mu}{\rho}, \qquad [\nu] = \frac{\text{Pa}\cdot\text{s}}{\text{kg}\,\text{m}^{-3}} = \frac{\text{m}^2}{\text{s}}.$$

Metres squared per second. That is not the unit of any friction coefficient. It is the unit of a diffusivity — the same unit as the $D$ in the diffusion equation, and the same unit as a thermal diffusivity.

Units are not a proof, but they are a strong hint, and here the hint is exact. What $\nu$ measures is how fast momentum spreads sideways through a fluid, and for water it is $1.0 \times 10^{-6}$ m²/s, or 1.0 mm²/s. Salt in water diffuses at about $10^{-9}$ m²/s; heat in water at $1.4 \times 10^{-7}$. Momentum diffuses through water a thousand times faster than salt does.

The experiment that shows it #

The steady picture at the top hides the mechanism because nothing is changing. Start the plate suddenly instead and watch how the motion spreads.

The solution is

$$u(y,t) = U \left[ 1 - \operatorname{erf}\!\left( \frac{y}{\sqrt{4\nu t}} \right) \right],$$

which is exactly the error-function solution of the diffusion equation with $\nu$ in the place of $D$. The distance the motion has penetrated grows as $\sqrt{\nu t}$, not as $\nu t$ — so quadrupling the time doubles the depth, and there is no well-defined front, only a profile that fades.

That is the same statement made about ink in the diffusion essay, and the two figures are the same shape because they are the same mathematics.

Why momentum diffuses at all #

The molecular picture is short and it is where the two temperature dependences come from.

In a gas, molecules travel in straight lines between collisions, over a mean free path. A molecule that starts in a fast-moving region and ends up in a slow one carries its extra momentum with it and delivers it on collision. That transport is the viscosity. Estimating it gives

$$\mu \sim \tfrac13 \rho \bar{v} \lambda,$$

with $\bar{v}$ the mean molecular speed and $\lambda$ the mean free path — a result Maxwell obtained in 1860 and which contains a prediction so odd he initially disbelieved it. Since $\lambda \propto 1/\rho$, the density cancels: a gas’s viscosity does not depend on its pressure. Halve the amount of air in a box and its viscosity is unchanged, because there are half as many carriers each going twice as far. Maxwell tested it experimentally and found it true over a wide range.

The same expression predicts $\mu \propto \bar{v} \propto \sqrt{T}$: a gas gets more viscous when heated, because its carriers move faster.

Liquids work the other way. Molecules are in permanent contact, and momentum transfer is limited not by how far a molecule travels but by how readily it can slip past its neighbours — an activated process, with a rate going as $e^{-E/kT}$. So a liquid’s viscosity falls steeply with temperature: water’s halves between 20 °C and 55 °C, and heavy oil’s changes by orders of magnitude across an engine’s operating range.

Two opposite temperature dependences from one word is the strongest evidence that “stickiness” is not the mechanism. It is the carrier that matters, and gases and liquids have different ones.

Why momentum diffuses at all is a question about molecules crossing a plane. A molecule that wanders from a fast layer into a slow one carries its old momentum with it and gives it up in the next collision, and a molecule going the other way does the reverse. The net effect is a transfer of momentum down the velocity gradient, carried out by molecules whose own speeds are enormously larger than any flow speed involved. Viscosity is that transfer, and it is why a gas’s viscosity rises with temperature — faster molecules cross the plane more often — where a liquid’s falls.

The number that says which transport wins #

Because $\nu$ is a diffusivity, it can be compared directly with the other diffusivities a fluid has, and the comparisons are informative rather than decorative.

The ratio of momentum diffusivity to thermal diffusivity is the Prandtl number. For water it is about 7 — momentum spreads seven times faster than heat. For air it is 0.7, so the two are comparable. For liquid metals it is around 0.01, so heat outruns momentum by a hundredfold, which is why sodium is used to cool reactors and why a mercury thermometer responds as it does. For heavy oils it reaches thousands.

The ratio of momentum diffusivity to mass diffusivity is the Schmidt number, and for water it is around a thousand: momentum spreads a thousand times faster than dissolved salt. That is why stirring is such an effective way to mix — it is not that stirring diffuses anything faster, but that it folds the fluid so the slow diffusion has less distance to cover.

None of these numbers has units. Each is one diffusivity against another, which is only a meaningful comparison because all three quantities are diffusivities in the first place — the thing the friction picture cannot say.

The number that says which transport wins is a ratio rather than a rate, and this is where dimensionless groups earn their keep. Momentum and heat both diffuse, each with its own diffusivity, and which of them spreads faster in a given fluid is decided by the ratio of the two — a pure number with no units to argue about. Comparing a diffusivity against a velocity would be comparing incommensurable things; comparing it against another diffusivity is not.

The condition at the wall #

Everything above needs a boundary condition, and the one that is used is that the fluid in contact with a solid surface moves with that surface. Zero relative velocity. It is called the no-slip condition and it is why the figure draws the fluid at the fixed plate as stationary.

It is worth being clear that this is an experimental fact rather than a derived one. Nothing in the definition of a fluid requires it; a fluid that slipped at walls is conceivable and was seriously proposed in the nineteenth century. The evidence is indirect but overwhelming: pipe-flow rates, drag on spheres and the damping of oscillating plates all agree with theory built on no-slip and disagree with theory built on slip.

It does fail in known circumstances. In a sufficiently rarefied gas — where the mean free path is comparable with the apparatus — molecules arriving at a wall carry information from a mean free path away, and a slip velocity appears. On some engineered surfaces liquids slip by tens of nanometres’ worth. But over the ordinary range it holds, and essentially all of fluid mechanics rests on it.

The consequence is large. Because the fluid at a surface is stationary, any relative motion between a fluid and a surface must be accommodated by a velocity gradient near it, and gradients cost stress. All resistance to motion through a fluid begins here.

What the stress is doing, thermodynamically #

Dragging the plate does work, at a rate $\tau U$ per unit area, and the fluid is not speeding up in the steady state. So the work goes somewhere: it goes into heat, at exactly the rate it is supplied.

This is worth stating because it makes viscosity’s status clear. Diffusion of momentum is a dissipative process, in the strict sense: it converts organised motion into disorganised motion and cannot be run backwards. A velocity difference smoothed out by viscosity does not spontaneously re-sharpen, for the same reason ink does not gather and for the same reason as everything else with a direction in it.

The heating is usually negligible and occasionally decisive. In a journal bearing, in a high-speed extrusion die, or in the mantle of a planet, viscous dissipation is a primary heat source. In a stirred cup of tea it raises the temperature by microkelvin.

The asymmetry is worth stating in the collection’s usual terms. The plate can put organised motion into the fluid at any rate the operator chooses; the fluid returns none of it as organised motion, at any rate at all. That is not a limitation of the apparatus but the definition of a dissipative process, and it is why viscosity appears in the equations with a sign that fixes a direction in time while every other term in them would run equally well backwards.

The eleven orders of magnitude #

The range of viscosities among ordinary substances is worth dwelling on, because it is larger than the range of almost any other everyday material property and it is why the word covers such different-looking behaviour.

Air is $1.8 \times 10^{-5}$ Pa·s. Water is fifty-five times more. Olive oil is about eighty times water; glycerol about a thousand; honey about ten thousand. Bitumen is around $10^8$ times water, which is why the Queensland pitch-drop experiment — running since 1927 — has produced nine drops in a century and is still a liquid by every test that matters.

At the far end, glass at room temperature is sometimes said to be a liquid of enormous viscosity, and the familiar claim that old cathedral windows are thicker at the bottom because they have flowed is false: the viscosity of window glass at 20 °C is around $10^{40}$ Pa·s, which gives a flow of far less than an atomic diameter over the age of the universe. The thickness variation is how the glass was made.

What lets one quantity span eleven orders is that it is a rate of a molecular process, and rates depend exponentially on an activation energy. A modest change in how tightly molecules are held becomes an enormous change in how readily they slip past each other, which is the same exponential sensitivity that makes other activated processes span comparable ranges.

Eleven orders of magnitude is the span kinematic viscosity covers between the thinnest gas and the thickest usable liquid, and it is worth pausing on because it is larger than most physical ranges anyone meets. Nothing about the mechanism changes across it. What changes is the density of scatterers and the strength of what holds them together, and the same two-line argument about molecules crossing a plane applies at both ends.

The claim written as an equation #

Everything above can be compressed into one line of the equation of motion for a fluid, and it is worth writing because the compression makes the claim exact rather than suggestive.

Newton’s second law for a parcel of Newtonian fluid, per unit mass, is

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\frac{\nabla p}{\rho} + \nu \nabla^2 \mathbf{u}.$$

The last term is the viscous one, and it is $\nu$ times a Laplacian — which is exactly the right-hand side of the diffusion equation, applied to each component of the velocity. There is no analogy being drawn here and no approximation being made: the viscous term is the diffusion term, with velocity in the place of concentration, and the error-function solution earlier is that equation solved with everything else set to zero.

What distinguishes a fluid from a diffusing dye is the other two terms. The pressure gradient enforces incompressibility, and $(\mathbf{u}\cdot\nabla)\mathbf{u}$ — momentum carried by the flow rather than across it — is quadratic in the unknown. Every difficulty in fluid mechanics is in that one nonlinear term, and everything this essay is about is in the linear one beside it.

When the nonlinear term is small, the flow runs backwards #

Delete the advective term, as is legitimate when it is small compared with the viscous one, and something startling follows: the remaining equation has no time derivative and is linear, so reversing the driving reverses the flow exactly. The fluid retraces its path.

That sounds like a contradiction of the dissipation argument two sections above, and it is not. G. I. Taylor made the demonstration famous: a blob of dye in glycerine held between two cylinders, the inner one turned four times until the dye is drawn into a smear that looks thoroughly mixed, and then turned four times back — whereupon the smear gathers itself into a blob again.

Nothing has been undone thermodynamically. The work done against viscosity was dissipated as heat on the way out and dissipated again on the way back, and the fluid is warmer at the end than at the start by twice the amount. What was reversible is the kinematics — which parcel of fluid is where — and that is a different question from whether the process created entropy. The blob does not reassemble perfectly either, and what spoils it is molecular diffusion of the dye, the one genuinely irreversible thing happening.

The same linearity has a consequence for anything small enough to live in this regime. A swimmer whose stroke is a reciprocating motion — out and back along the same path — goes precisely nowhere, since the return stroke undoes the outward one exactly. That is Purcell’s scallop theorem, and it is why bacteria swim with rotating helices and beating flagella rather than with anything that opens and shuts.

The other viscosity #

There is a second viscous coefficient, and the section below that says viscosity has nothing to do with compression is right about the modulus and incomplete about the coefficient.

A fluid resists being compressed by an amount, which is the bulk modulus, and it also resists being compressed at a rate, which is a viscosity — the bulk viscosity $\zeta$. Shear viscosity opposes a change of shape; bulk viscosity opposes a change of volume; and a complete Newtonian stress law carries both.

Stokes assumed $\zeta = 0$ for convenience in 1845, and the assumption is exactly right only for a dilute monatomic gas. For everything else it is wrong, sometimes enormously. The mechanism is that a compression raises the translational temperature immediately and the rotational and vibrational temperatures only after a delay, so energy sloshes between the internal degrees of freedom and the translational ones out of step with the compression, and the lag is a dissipation.

Sound is where this is measured, because a sound wave is a rapid compression and nothing else. Computing the attenuation of sound in a polyatomic gas from shear viscosity and heat conduction alone gives an answer far too small — for carbon dioxide, by a factor of order a thousand — and the deficit is the bulk viscosity. It is also why the absorption of sound in air depends so strongly on humidity: water molecules catalyse the vibrational relaxation of nitrogen and oxygen, which moves the relaxation frequency straight through the audible band.

What it is not #

Two distinctions are worth drawing explicitly, because both are commonly blurred.

Viscosity is not thickness in the sense of density. Mercury is thirteen times denser than water and about one and a half times as viscous. Air is a thousand times less dense than water and only about fifty-five times less viscous, so its kinematic viscosity is fifteen times larger — momentum diffuses through air faster than through water, which is not what anybody expects.

Viscosity is not what makes a fluid resist being compressed. That is the bulk modulus, and it is a separate property with its own coefficient. Shear viscosity resists a change of shape at constant volume; a fluid can have a large one and be effectively incompressible, which describes water.

The oscillating case, and a depth that has a number #

There is a second exact solution worth having, because it is the one that turns up in practice most often: a plate that oscillates in its own plane rather than starting suddenly.

The fluid’s response is a wave that travels away from the plate and dies as it goes, with amplitude falling by a factor of $e$ over a distance

$$\delta = \sqrt{\frac{2\nu}{\omega}},$$

the Stokes penetration depth. For water at 1 Hz it is about 0.6 mm; at 1 kHz about 0.02 mm. So a surface vibrating quickly drags a very thin layer with it and leaves the rest of the fluid undisturbed.

The consequences are practical. It is why an ultrasonic cleaning bath acts at surfaces rather than in the bulk. It is why a quartz crystal microbalance can weigh a film in liquid — the crystal senses only what is within a fraction of a millimetre of it. And the same $\sqrt{\nu/\omega}$ governs how deep a seasonal temperature swing reaches into the ground, with the thermal diffusivity in place of $\nu$, because it is the same equation with a different quantity diffusing.

That last identity is the payoff of the whole page. Once viscosity is understood as a diffusivity, results about it transfer intact to heat and to concentration, and every solution obtained once is three solutions.

Where the model stops #

Stress is proportional to strain rate. This is the Newtonian assumption, and paint, blood, cornflour suspensions and polymer melts all break it. The next-but-one rung is what happens when they do.

The fluid has no memory. A Newtonian fluid’s stress depends on the strain rate now. Polymeric liquids remember their recent deformation and are viscoelastic — partly liquid, partly solid, with a relaxation time.

The flow is slow enough to stay orderly. Everything here describes a flow whose layers do not overturn. Where the flow becomes disorderly, momentum is transported by the flow’s own eddies far more effectively than by molecules, and the effective viscosity can exceed the molecular one by orders of magnitude. That regime, and the number that predicts it, belong to the study of turbulence rather than to this essay.

The continuum holds. Momentum diffusion is a coarse-grained description of molecular transport, so it needs many molecules per gradient length — the same condition that the mean free path makes precise.

The ladder from here #

Later rungs on this anchor: flow in a pipe, and the fourth power that comes out of it. The oscillating plate, and the penetration depth that shrinks as the square root of frequency. Viscoelasticity and relaxation times. Suspensions, where a viscosity is an effective property of a mixture. The temperature dependence of liquids in detail, and why lubricants are formulated to have as little of it as possible. And viscous heating, where the dissipation stops being negligible and starts changing the viscosity that produced it.