The shear that only reaches so far

Momentum going sideways makes viscosity a diffusion: a moving wall’s momentum spreads into the fluid, and given long enough it spreads without limit. Drag a plate steadily through a still fluid and eventually the whole container is in motion.

Reverse the plate and start again, and the picture changes completely. The momentum that went out has to come back before it has got far, and the fluid settles into a state where the wall’s influence extends a fixed distance and stops.

A depth, from two numbers #

The distance is $\delta = \sqrt{2\nu/\omega}$, and it can be obtained without solving anything.

Viscous diffusion spreads momentum a distance $\sqrt{\nu t}$ in time $t$ — that is the content of the diffusion equation, and it is why the steady problem has no length in it. An oscillation supplies a time: half a period, after which the driving reverses and the momentum already sent out is recalled. Put $t \sim 1/\omega$ into $\sqrt{\nu t}$ and the depth appears.

A diffusion has no length scale until something supplies a time, and that observation is worth more than the particular answer. It is why the same square root turns up in the depth an alternating current reaches into a metal — how far a field gets into metal is the identical calculation with the magnetic diffusivity in place of the kinematic viscosity — and in how far a day’s warmth penetrates the ground, and in how thick an alternating-field chemical reaction layer is.

The phase lag comes with it, and it is the tell that distinguishes this from a mere decay. Fluid one skin depth from the plate is fastest a full radian after the plate is, and two skin depths out it is a quarter of a cycle behind. Some of the fluid is always moving the wrong way.

The numbers, in real fluids #

The numbers are small, and their smallness is what makes the effect matter.

In air at a kilohertz the layer is seventy micrometres. That is why a narrow wind-instrument bore is lossy where a wide one is not: the loss is confined to a shell of that thickness against the wall, so it scales with the surface area while the useful energy scales with the volume, and halving the bore doubles the fractional loss at every frequency. The scaling with $\sqrt{\omega}$ is why the loss rises with pitch and why a small instrument is duller than a large one playing the same note.

In water at a hundred hertz it is sixty micrometres, which is smaller than most things anyone puts in water. That is the practical statement that at ordinary frequencies water behaves as though it had no viscosity except within a hair’s breadth of a surface — the observation the whole boundary-layer idea is built on.

Kinematic viscosity is the relevant quantity, not dynamic viscosity, and the ordering it produces is not the intuitive one. Air is far less viscous than water in the ordinary sense and its kinematic viscosity is fifteen times larger, because it is so much less dense. Momentum diffuses faster through air than through water, and boundary layers in air are correspondingly thicker.

Why it is a wave, and a very bad one #

The solution is worth naming properly, because calling it a decaying profile undersells what it is.

It is a wave. The combination exp(−y/δ) cos(ωt − y/δ) is a disturbance travelling into the fluid at speed $\omega\delta$, and every part of it moves outward — the crests, the zeros, the whole pattern. What makes it unfamiliar is that its wavelength and its decay length are the same number, so it dies within one cycle of travel and never looks like a wave.

That equality is not an accident of this problem. It happens whenever the governing equation is a diffusion equation rather than a wave equation, because a diffusion equation has one derivative in time against two in space, and the dispersion relation it produces has equal real and imaginary parts in the wavenumber. A diffusive wave is critically damped by construction, and no material can be found in which it propagates further.

The same object is the temperature wave that carries a summer into the ground, where the period is a year and the depth is a couple of metres — which is why a cellar is cool in August and, more strikingly, why it is at its warmest in about January, a phase lag of half a cycle at three or four skin depths down. Every feature the first figure draws is present in a hole in the ground, and it was measured there long before anybody wrote down a boundary layer.

Diffusion is the same equation wherever it appears, and the honest reason to draw a fluid rather than soil or a metal is that the fluid version has a force attached to it.

The force that is half a mass #

The force on the oscillating plate is the part of this with the most consequences, and it does something a drag is not supposed to do.

An ordinary viscous drag opposes the velocity and is in phase with it. This one is not. It leads by forty-five degrees, which decomposes into equal parts along the velocity and along the acceleration. The first is a genuine damping and removes energy from the plate. The second is indistinguishable from extra inertia: the plate behaves as though it were heavier, by the mass of a layer of fluid of order one skin depth thick.

That the split is exactly even is a property of the diffusion equation rather than of any fluid. It comes from the square root of $i$, which has equal real and imaginary parts, and there is no material or frequency that moves it.

The consequences are everywhere something small vibrates in a fluid. A quartz crystal microbalance measures the mass deposited on it by watching its resonant frequency; immerse it in a liquid and both halves appear, the added mass shifting the frequency and the damping broadening the resonance, and the ratio of the two shifts is how the liquid’s viscosity and density are separated. A vibrating-wire viscometer is the same measurement made deliberately.

When the pipe forgets its parabola #

The pipe version has a name and a dimensionless number, and the number is simply the radius measured in skin depths: $\alpha = R\sqrt{\omega/\nu}$.

Below one, the skin depth exceeds the radius, viscosity reaches the axis within a cycle, and the flow is quasi-steady — at every instant it looks like the Poiseuille profile the fourth power in a pipe computes, with the pressure gradient of that instant. The flow is in phase with the pressure and the resistance is the steady one.

Above about five, it does not. The core of the pipe never learns the wall is there; it responds to the pressure gradient as an inviscid slug of fluid would, which means it lags by ninety degrees rather than being in phase, and the velocity profile is flat across most of the pipe with all the shear crammed into a thin layer at the wall.

The overshoot, and what it means #

The annular overshoot in the second of those is the feature worth dwelling on, because it is genuinely counter-intuitive and it is computed rather than described.

At high Womersley number the fastest fluid is not in the middle. It is in a ring near the wall, and the middle is slower. The reason is the phase lag: the core, responding as an inviscid mass, lags the pressure by ninety degrees; the fluid very near the wall is held by viscosity and follows the pressure more closely; and in between there is a layer whose phase is such that it happens to be at its peak when the core is not. Add them at one instant and the profile has a bump.

Blood in the larger arteries sits in this regime. The aorta at a resting heart rate has a Womersley number of about fifteen, so the flow is plug-like, the shear is confined to the wall, and the resistance is not the Poiseuille resistance at all. That matters clinically: wall shear stress is what the endothelium responds to, and it is set by the thin layer rather than by the average flow, so a calculation that assumed a parabola would be wrong by the ratio of the radius to the skin depth.

A dimensionless number that is a ratio of two lengths usually announces which of two physical regimes applies, and the Womersley number is a clean example: it is the radius over the skin depth, and the two limits are the two things a pipe can be.

The steady case, for comparison #

It is worth setting the steady problem beside this one, because the contrast shows where the length came from.

Start a plate moving at constant speed in a still fluid and the profile is an error function whose width grows as $\sqrt{\nu t}$ for ever. There is no steady state and no depth: wait long enough and any given depth is moving. The problem has no length scale because nothing in its statement has the dimensions of one — a viscosity is a length squared over a time, a speed is a length over a time, and no combination of them is a length.

Put the fluid between two plates instead and a length appears, but it is supplied by the apparatus: the gap. The steady answer is then the straight-line profile of Couette flow, reached after a time of order the gap squared over the viscosity, and every subsequent question is about that geometry.

The oscillating case supplies the length from the driving rather than from the geometry, which is why it works in an unbounded fluid and why the answer contains no dimension of any container. That is unusual and useful: a measurement made with an oscillating surface in a large volume of fluid returns a property of the fluid, with no calibration against a cell geometry, which is exactly what makes vibrating-wire and torsional-crystal viscometers primary instruments rather than comparative ones.

Where a length scale comes from is worth tracking, and there are only three sources — the geometry, the driving, and the material — with the third being far rarer than the first two.

Where the model stops #

The fluid is Newtonian and incompressible. Blood is neither — it is shear-thinning, and its apparent viscosity falls in vessels narrower than about a third of a millimetre — so the Womersley calculation applied to an artery is a first approximation whose corrections are of order tens of per cent.

The motion is linear in the amplitude. At large amplitude the convective term matters, steady streaming appears — a slow drift superposed on the oscillation, driven by the nonlinearity within the layer — and the profile acquires harmonics. Acoustic streaming is that effect and it is what makes an ultrasonic bath work.

The wall is flat and rigid. A compliant wall changes the problem, an arterial wall is compliant, and the resulting wave propagation is a separate subject entirely.

And the layer is assumed laminar. At high enough amplitude the oscillating boundary layer becomes turbulent within part of each cycle, which happens under ocean waves and in the aorta at exercise, and none of the profiles here survive it.

What it is used to measure #

The oscillating layer is one of the more heavily instrumented pieces of fluid mechanics, and the reason is the clean split in the last figure but one.

A quartz crystal microbalance is a thin resonator whose frequency shifts in proportion to the mass on its face. In vacuum that is a mass measurement of monolayer sensitivity. Immersed in a liquid, the added mass of the entrained layer shifts the frequency and the damping broadens the resonance, and the two shifts are independent combinations of the density and the viscosity — so one instrument returns both, and the difference between a rigid film and a soft one shows up as a change in the ratio.

A vibrating-wire viscometer drives a wire in a magnetic field, using the induction of the field that makes the other and measures the current it draws. The in-phase part is the damping and the out-of-phase part is the added mass, and because both come from the same skin depth the instrument is self-calibrating in a way a capillary viscometer is not. These are the reference instruments for viscosity at high pressure, where no capillary would survive.

An oscillating-cup viscometer is the oldest of the family and the most direct: a cup of liquid on a torsion fibre, twisted and released, with the viscosity read from how quickly the oscillation dies. Its analysis is exactly the Womersley problem with the pipe replaced by a cup.

All three measure the same thing three ways, and the reason they agree is that the split between damping and added mass is fixed at forty-five degrees by the mathematics rather than by any property of the liquid. An instrument built on a fixed ratio is an instrument that cannot drift.

What the pictures cannot show #

The profile figure draws snapshots and the interesting quantity is the phase, which a snapshot cannot show. Four curves at four phases is the least misleading available substitute, and the reader has to supply the motion — the correct mental image is a wave travelling into the fluid at speed $\omega\delta$ while decaying, not a set of profiles that appear one after another.

The pipe figure normalises each profile to its own peak so the shapes can be compared, which throws away the fact that the profiles have wildly different amplitudes for the same pressure gradient. At Womersley number twelve the flow for a given pressure amplitude is far larger than at one, and the drawing suggests otherwise.

A last thing outside the drawings is where the energy goes. The damping half of the stress does work on the fluid at a rate proportional to the square of the plate’s speed, and every joule of it is dissipated inside the skin depth and appears as heat there. In an ultrasonic cleaner that heating is the point; in a wind instrument it is the loss that shortens the resonance; in an artery it is negligible. The figures draw velocities and stresses and never the dissipation, which is the quantity most of the applications are actually about — and it is the one place where the thickness that goes both ways matters here, since a fluid whose viscosity falls as it warms will thin its own boundary layer as it dissipates in it.

Where the ladder goes next #

The viscosity ladder began with momentum going sideways, which makes shear a diffusion, continued with the fourth power in a pipe, and reached the thickness that goes both ways, where a gas and a liquid respond to temperature in opposite directions. This rung asks what happens when the driving alternates and finds a length where there had been none. The rungs after it: steady streaming, the slow drift the nonlinearity leaves behind; the growing boundary layer on a body moving through a fluid, where the time is supplied by the transit rather than by a frequency; and the transition to turbulence within the layer, which is where the laminar picture ends.

The habit worth carrying away is that a problem with no length scale is a problem waiting for one. Diffusion supplies a relation between length and time and nothing more, and whichever of the two the situation supplies, the other follows — which is why the same square root turns up in a wind instrument, a metal, an artery and a summer’s warmth in the soil.