The drift a sound leaves behind

The depth an alternating shear reaches finds that an alternating shear does not reach far. A plate sliding back and forth in its own plane drags the fluid above it — by conducting momentum sideways, which is what a viscosity does — to a depth $\sqrt{2\nu/\omega}$ and no further — seventy micrometres in air at a kilohertz — and beyond that depth the fluid does not know the plate exists.

That is a first-order result. It is the answer to the equations with every product of two small quantities thrown away, and for the motion itself that is entirely adequate: the fluid goes back and forth, and it comes back.

The terms thrown away do not come back. A product of two oscillating quantities has an average over a cycle that is not zero unless they happen to be a quarter-cycle apart, and inside the boundary layer they are not: the phase of the motion varies with depth, which is exactly what that solution computes. So there is a steady force per unit volume inside the layer, it does not average away, and over many cycles it drives a flow.

What does not average to zero #

Take the acoustic velocity just outside the layer to be $U(x)\cos\omega t$, a standing wave with $U(x) = U_0\sin kx$. Inside the layer the velocity is that, damped and phase-shifted with depth, exactly as that solution gives it.

The nonlinear term in the equation of motion is $(\mathbf{u}\cdot\nabla)\mathbf{u}$, a product of the velocity with its own gradient. Average it over a cycle. The horizontal velocity oscillates as $\cos\omega t$ and its vertical partner — supplied by continuity, because a horizontal flow whose strength varies along the channel must have somewhere to go — oscillates with a phase that depends on depth. The two are not in quadrature, so their product has a mean.

Rayleigh worked the average through in 1884 and got a result of one line. The steady velocity the layer imposes on the fluid just outside it is

$$u_{\text{slip}} = -\frac{3}{4\omega}\,U\frac{dU}{dx},$$

which for the standing wave above is $-\tfrac38 (U_0^2/c)\sin 2kx$.

Three things in that expression are worth separating, because each is a surprise.

It is quadratic in the amplitude. A drift proportional to $U_0^2$ is invisible at ordinary loudness and unmissable at high intensity. A very loud tone — a hundred and forty-six decibels, an acoustic velocity of a metre a second in air — gives a drift of about a millimetre a second. Ordinary conversation, sixty decibels, gives an acoustic velocity twenty thousand times smaller and a drift four hundred million times smaller than that. There is no drift in a room, and there is a visible circulation in an ultrasonic bath, and the two statements are one square law apart.

It is zero where the sound is largest. The drift depends on the product of the velocity and its gradient, and the gradient vanishes where the velocity peaks. So the circulation is driven from the quarter-points, the places where nothing about the acoustic field is remarkable. This is a general feature of second-order effects and it is a reliable way of misdiagnosing them: the strongest driving of the mean flow is not where the oscillation is strongest.

And there is no viscosity in it.

The one thing the coefficient is worth knowing for #

The three quarters is not the interesting part of the expression and it is worth saying which part is.

Writing the layer’s horizontal velocity as the outer flow minus a correction that dies with depth, continuity supplies a vertical partner whose every term carries $dU/dx$, and the cycle average of $u\,\partial u/\partial x + v\,\partial u/\partial y$ is a set of elementary integrals over depth of $e^{-y/\delta}$ against $e^{-y/\delta}\cos(y/\delta)$ and $e^{-y/\delta}\sin(y/\delta)$. Working them out gives the coefficient, and it is a half from one term and a quarter from the other — which is why it is not something obvious like one.

What matters is that $\delta$ divides out of every one of those integrals, and that is not a coincidence discovered at the end. It follows from the exponential being a function of $y/\delta$ alone, so each integral over depth produces one factor of $\delta$ and each derivative across it produces one factor of $1/\delta$. Every appearance of the viscosity in the problem is inside $\delta$, and the cancellation is structural rather than arithmetic.

That is the observation the next section turns into a statement about limits, and it is available without evaluating a single integral.

The viscosity that cancels out #

This is the part of the result that matters most and is easiest to pass over.

The streaming is generated entirely inside the boundary layer. The layer exists because the fluid is viscous, its thickness is $\sqrt{2\nu/\omega}$, and in a fluid ten times thinner the layer is about three times thinner, and how thin a fluid is depends on whether it is a gas or a liquid in a way that has no common sign. The Reynolds stress that drives the flow lives in that layer and nowhere else.

And the flow it produces is the same. Make the fluid less viscous and the generating region shrinks; the stress inside it grows in exactly the compensating proportion, because the velocity must fall from its outer value to zero at the wall whatever the thickness over which it does so. The product is independent of $\nu$.

The consequence is a statement about limits rather than about fluids. Setting the viscosity to zero in the equations and then solving gives no streaming at all: there is no boundary layer, no phase shift with depth, no Reynolds stress and no mean flow. Solving with viscosity and then letting it go to zero gives the streaming above, undiminished.

The two orders of operation disagree, and the disagreement does not shrink as the viscosity does. That is what a singular limit is, and this is one of the cleanest examples of one in fluid mechanics. A fluid with a small viscosity is not a fluid with none; it is a fluid with a thin layer in which everything happens, and shrinking the layer does not reduce what happens in it.

The same shape recurs whenever a small parameter multiplies the highest derivative in an equation, and the general lesson is worth carrying out of fluids entirely. If a quantity is generated in a region whose size is set by a small parameter, check what happens to the intensity in that region before concluding that the quantity is small. The two usually trade, and they sometimes trade exactly.

Where the cells come from, and why there are four #

The slip velocity is a boundary condition on the outer fluid, and the outer fluid then does what a viscous fluid does with a boundary condition: it flows.

In a channel of half-height $h$ much larger than the layer, the outer steady flow is fixed by three requirements. It must match the slip velocity at both walls, it must have no net flow through any cross-section, and it must be a solution of the steady equations, which at these speeds is the same slow-flow problem a pipe poses. Those three give a velocity varying as $3\eta^2 - 1$ across the channel, where $\eta$ is the height divided by $h$ — outward along the walls, back along the middle, with a reversal at $\eta = 1/\sqrt3$.

Along the channel the sign alternates with $\sin 2kx$, so the pattern repeats every half acoustic wavelength rather than every wavelength. That is the count: two cells above the centreline and two below in each wavelength, four in all, turning in opposite senses.

Nothing about the fluid enters. The cells are the same size in air and in water at the same frequency in the same tube, because their size is half an acoustic wavelength, and their shape is the same because the outer solution has no material constant in it. The pattern was first seen in the dust figures Kundt used in the 1860s to measure the speed of sound, where two waves meeting make the standing pattern the dust reports — the dust does not merely settle at the nodes, it is gathered there by a circulation nobody could then account for.

A pattern seen sixty years before it could be explained #

The streaming was visible long before it had a name, and the history is unusually clean about who saw what.

Faraday in 1831, studying the patterns fine powder makes on a vibrating plate, noticed that light powder did not behave like heavy powder. Heavy grains gathered at the nodes, as everybody expected — a grain sitting where nothing moves stays there. Light ones gathered at the antinodes instead, which no account of a vibrating plate could produce. Faraday attributed it correctly to currents in the air, and could go no further.

Kundt in 1866 turned the same phenomenon into an instrument. A tube of gas with a piston at one end and cork dust inside it: drive the piston at resonance and the dust arranges itself into ridges half a wavelength apart, so the wavelength can be measured with a ruler and the speed of sound in the gas follows from the driving frequency. It was for decades the standard way of measuring the speed of sound in a gas one has only a little of.

What was noticed and not explained is that the ridges have structure. The dust does not merely fail to be blown away from the nodes; it is gathered there, it arrives from both sides, and the ridges develop striations across them at a spacing that has nothing to do with the wavelength.

Rayleigh’s calculation of 1884 accounts for all of it. The gathering is the outer circulation, which runs along the walls toward the nodes. The striations are the inner cells, which are as wide as the layer is thick. And the reason the effect had resisted explanation for half a century is that it is second order: everything in the first-order description of a standing wave is symmetric in time, and nothing in it can produce a one-way transport of anything.

That is a general obstacle rather than a historical accident. A description accurate to first order in an amplitude is time-reversible whatever the system, so no amount of refining it will produce a net transport. Transport in an oscillating system is always a second-order effect, which is why it is always weak, always quadratic in the driving, and always invisible to the linear theory that otherwise works perfectly. The same statement covers acoustic radiation pressure, the drift of a particle in a non-uniform oscillating field, and the slow transport of water by surface waves.

What it is used for, and the comparison that decides it #

A steady circulation of a millimetre a second is not impressive next to anything mechanical. It is impressive next to diffusion.

The crossing exists because the two laws have different powers. A diffusion time is $L^2/2D$ — the relation between a length and a time that every diffusion problem shares — and a transport time is $L/u$, so the ratio of the two is $Lu/2D$ — a Péclet number — and it passes one at $L = 2D/u$ whatever the numbers are. Below that width, waiting is faster than stirring. Above it, stirring wins by a margin that grows without limit.

For a small molecule in water, $D$ is about two thousandths of a square millimetre a second — the same coefficient a wandering particle measures — so at a drift of a quarter of a millimetre a second the crossing is at sixteen micrometres. Channels in microfluidic devices are tens to hundreds of micrometres wide, which puts them on the wrong side of the crossing for diffusion and the right side for a drift — and a drift is exactly what is hard to arrange in a channel too small for any moving part.

That is what acoustic mixing is bought for, and the reason it is bought rather than a stirrer is geometric: the sound is applied from outside, there is nothing in the channel, and the frequency is high enough that the oscillation itself displaces a parcel by less than the channel width while the drift carries it clear across.

It is also why the technique is described wrongly as often as it is. The oscillation does not mix. A parcel of water in a one-metre-per-second field at a kilohertz swings back and forth through a hundred and sixty micrometres and returns to where it began, and repeating that a million times mixes nothing at all. What mixes is the second-order flow, it is a thousand times slower than the oscillation, and it is the only part of the motion that goes anywhere.

A thin layer, a small amplitude, and no attenuation anywhere #

The outer solution assumes the layer is thin. Everything above requires $\delta \ll h$, and there are two other regimes. When the layer is comparable to the channel the inner and outer flows are not separable and the cell structure changes character; when the channel is very much larger than a wavelength the flow is not confined at all and the streaming becomes a jet rather than a circulation. The figures here are drawn in the thin-layer regime and say what their ratio is.

Rayleigh’s expression is the low-amplitude limit of a second-order theory. It is second order in the acoustic amplitude, and the neglected fourth-order terms matter once the streaming Reynolds number — the drift speed times the channel width over the kinematic viscosity — approaches one. In the air case above that number is about one, so the figure is at the edge of its own validity, and in a real ultrasonic bath it is far past it and the flow is turbulent. The neglect is the one every expansion in an amplitude makes, and it fails the way the vacuum’s own linearity does: not gradually, but at a scale that can be computed.

Attenuation has been left out entirely. A standing wave in a real tube is not perfectly standing, because the sound is absorbed as it goes, and an absorbed wave delivers momentum. That produces a second kind of streaming — a bulk flow driven throughout the volume rather than at the walls, in the direction the sound travels — which is what carries the quartz wind from an ultrasonic transducer. Near a wall the boundary-driven flow dominates; far from one, the absorption-driven flow does; and in most real apparatus both are present.

And the whole construction assumes a Newtonian fluid. In a liquid with any structure to it, the layer contains shear rates of thousands per second and a single viscosity stops being enough to describe what is happening there. The streaming from a viscoelastic fluid is not given by Rayleigh’s formula and can change sign.

The inner cells, a parcel’s path, and why the dust chooses the nodes #

They cannot show the inner flow. Inside the boundary layer itself there is a second set of cells, turning opposite to the outer ones, and they are as real as the ones drawn. They are omitted because the layer is under a per cent of the channel in both figures and drawing them would require the vertical axis to be stretched five hundredfold — at which point the outer cells, which are the subject, would be off the page.

Nor can they show a particle’s path. A streamline of the mean flow is not the path of a fluid parcel: the parcel oscillates through a distance many times larger than the drift it accumulates in one cycle, so its actual trajectory is a tight spiral creeping along the streamline. The two coincide only after averaging, and anything that depends on the instantaneous position — a particle settling, a reaction between two species brought together — sees the spiral and not the line.

And they cannot show why the dust ends up at the nodes rather than being swept around with the fluid. A particle in a sound field feels a force of its own, which depends on how its density and compressibility compare with the fluid’s, and it competes with the drag from the streaming. Which of the two wins depends on the particle’s size, with the crossover around a micrometre in water at a megahertz — so a suspension of mixed sizes is sorted rather than gathered, and that sorting is a technique in its own right.

Still open: whether the boundary-driven and bulk-driven flows can be separated in practice #

The two mechanisms — a slip velocity imposed at a wall and a body force distributed through the volume — produce flows with different geometry and different scaling, and in a device of a few hundred micrometres both are present at comparable strength. Which dominates depends on the ratio of the channel to the attenuation length and on the sharpness of the corners, and the honest position is that most published measurements in microfluidic devices cannot separate them.

It matters because the two respond differently to the things a designer controls. The boundary-driven flow scales with the square of the amplitude and not with frequency; the bulk-driven flow scales with the square of the amplitude and with the square of the frequency, since attenuation does. A measurement over a decade of frequency should separate them, and the measurements that have been attempted are complicated by resonances of the device itself moving the amplitude at the same time.

The habit worth carrying away is the one about limits. An effect generated inside a region whose thickness is set by a small parameter need not be small, and setting that parameter to zero from the start may remove the effect entirely. Streaming is the clearest case: the fluid’s viscosity builds the layer, the layer builds the flow, and the flow does not care how viscous the fluid was.