What a fluid takes out of a swing

A tuning fork rings for several seconds in air and for a fraction of a second in water. A pendulum in a vacuum bell swings for hours and in air for minutes. A quartz watch crystal is sealed, and the seal is not to keep dust out.

In every case the fluid is taking energy out of an oscillation, and the question is at what rate. The obvious answer — use the steady drag at the instantaneous velocity — is wrong for almost everything, and it is wrong in a direction and by an amount that this essay can put numbers on.

The wall, where the accounting is clean #

Stokes’ second problem gives the velocity above an oscillating wall as $U_0 e^{-y/\delta}\cos(\omega t - y/\delta)$, with $\delta = \sqrt{2\nu/\omega}$.

The mean dissipation per unit area, integrated across the whole layer, works out at

$$\bar{W} = \frac{\mu U_0^2}{2\delta},$$

and the same number arrives from the wall’s own stress: $\langle\tau_w u_{\text{wall}}\rangle$. Two routes, one an integral through the fluid and the other a product at a plane, agreeing to a part in a thousand on a four-hundred-cycle average. That is the check this site requires of every dissipation it quotes.

What the formula says is that the cost is inversely proportional to the layer thickness. The fluid’s viscosity has not changed and the wall’s speed has not changed; what has changed is how much fluid is being asked to accommodate the difference.

The sphere, and how early it goes wrong #

A wall is infinite and has no size of its own to compare the layer against. A body does, and that comparison is the whole story.

Landau’s solution for a sphere oscillating in a viscous fluid gives a force with two parts — one in phase with the velocity, which damps, and one in phase with the acceleration, which is an added mass:

$$c = 6\pi\mu a\left(1 + \frac{a}{\delta}\right), \qquad m_{\text{add}} = \tfrac23\pi a^3\rho\left(1 + \frac{9\delta}{2a}\right).$$

The damping is Stokes’ answer multiplied by $(1 + a/\delta)$. At low frequency $\delta$ is large, the correction vanishes, and the steady drag is recovered. At high frequency $\delta$ is small and the damping rises as $\sqrt{\omega}$.

The question is where the crossover is, and the answer is startlingly low.

For a millimetre sphere in air, $\delta$ is 2.2 mm at one hertz. The layer is bigger than the body, which is the quasi-steady regime — and only just. By ten hertz the layer is 0.7 mm and the body is no longer inside it.

The frequency at which a body is quasi-steady is set by its own size against the layer, not by the frequency sounding small. For anything smaller than a centimetre in air, that frequency is below a hertz.

Where the energy actually goes #

The dissipation lives in the layer, and the layer is thin.

At a kilohertz in air, $\delta$ is 70 µm. A millimetre body oscillating at that frequency is dissipating in a shell a fifteenth of its own diameter thick, and everything further away is being carried along without being sheared — which is the added-mass term, and it stores energy rather than destroying it.

That separation is unusually clean. In a steady flow the near field and the wake are both dissipating and both are part of the same disturbance. In an oscillating one the fluid divides sharply into a thin dissipative shell and an outer region that is purely inertial, and the two enter the force as a real and an imaginary part of the same coefficient.

What it does to a quality factor #

The quantity a designer cares about is $Q$ — how many radians of oscillation before the energy falls by a factor of $e$ — and for a resonator of mass $m$ it is $\omega(m + m_{\text{add}})/c$.

The trend along that list is the one to take away, and it runs the opposite way to intuition. A larger body has a higher air-limited $Q$, because the damping grows as $a^2/\delta$ while the mass grows as $a^3$. So miniaturising a resonator makes its fluid damping relatively worse, at a rate of one power of the size — which is a fundamental obstacle to micromechanical resonators and is the reason they are almost all packaged under vacuum.

The frequency trend goes the other way. Damping grows as $\sqrt{\omega}$ and the stored energy grows as $\omega$, so $Q$ improves as $\sqrt{\omega}$: a high-frequency resonator of a given size is better off than a low-frequency one. A quartz watch crystal at 32 kHz would have an air-limited $Q$ in the tens of thousands, and it is sealed anyway, because its material $Q$ is a hundred times that.

The other half of the force, which is not a loss #

The added mass deserves a paragraph, because it is the part of the force that is usually confused with the damping.

It is in phase with the acceleration, so it does no net work over a cycle — it stores energy on one half and returns it on the other. What it does is lower the resonant frequency, by adding to the effective mass, and for a light body in a dense fluid the effect is enormous: a hollow sphere oscillating in water has an added mass comparable with its own.

The two halves come from the same solution and have completely different characters, and the low frequency limit shows it. The damping tends to Stokes’ constant. The added mass diverges, as $\delta/a$, which is the history force — the fact that a slowly oscillating body is dragging along a region of fluid whose size grows without limit as the frequency falls. That divergence is real and it is why unsteady Stokes flow at very low frequency needs the Basset integral rather than a coefficient.

A frequency-dependent coefficient is a memory #

The damping in this essay depends on $\omega$, and that is a stronger statement than it looks. A force proportional to the velocity with a constant of proportionality is an instantaneous force; a force whose coefficient varies with frequency cannot be. By the ordinary correspondence between the two domains, a coefficient that depends on frequency is a convolution in time — the force at this instant depends on what the body was doing at every earlier one.

That is the Basset history force, and it is the same object as everything computed above, written the other way round. For a sphere it is

$$F_{\text{hist}} = 6\pi\mu a^2\!\int_{-\infty}^{t}\frac{\dot v(\tau)}{\sqrt{\pi\nu(t-\tau)}}\,\mathrm d\tau,$$

a weighted integral over the body’s past acceleration with a kernel falling as the inverse square root of the elapsed time. Transform that kernel and the $\sqrt{\,\cdot\,}$ becomes the $\sqrt\omega$ this essay’s damping rises with, and the $\delta/a$ its added mass diverges by. One expression in the time domain, two coefficients in the frequency domain, and no new physics between them.

The mechanism is diffusion, and it is worth naming because it explains why the memory is so long. What a body leaves behind when it accelerates is vorticity, generated at its surface and then diffusing outward, and the force it feels now depends on where that vorticity has got to. Diffusion has no finite speed of propagation, so nothing the body has ever done is entirely forgotten — and the $t^{-1/2}$ tail is the slowest decay a convergent memory can have.

Which is why the low-frequency divergence noted above is not an artefact. A slowly oscillating body has had time to grow a disturbance many radii across, so the fluid it is dragging has no bounded size, and no coefficient exists to summarise it.

And it is why the term is routinely dropped and should not always be. In particle tracking, the history integral costs an entire stored trajectory per particle, against a handful of numbers for everything else, so it is the first thing left out of a simulation. That is defensible for a heavy particle in a gas, where the fluid’s inertia is a thousandth of the particle’s and every term carrying $\rho_f$ is negligible — the same comparison that decides whether the added-mass and pressure-gradient terms matter. For a bubble, a neutrally buoyant tracer or anything in a liquid, the history force is the same order as the drag, and a code that has dropped it is not solving the problem it claims to.

The number a resonator designer works with #

There is a compact way to see all of this at once, and it is the ratio the whole design turns on.

Write the damping as $6\pi\mu a(1 + a/\delta)$ and the stored energy as $\tfrac12 m\omega^2 X^2$ with $X$ the displacement amplitude. In the high-frequency regime, where $a \gg \delta$, the air-limited quality factor comes out proportional to

$$\frac{\rho_{\text{body}}}{\rho_{\text{fluid}}}\cdot\frac{a}{\delta},$$

a density ratio times a size ratio.

Both factors say the same thing in different words: a resonator is quiet in a fluid to the extent that it is heavy compared with the fluid it displaces and large compared with the layer it drags. Steel in air scores about six thousand on the first factor and something between one and a hundred on the second. Steel in water scores eight on the first, which is why nothing rings in water.

The design consequences follow immediately and are all uncomfortable for small devices. Shrinking a resonator hurts the second factor. Making it lighter — which is what a micromachined structure is — hurts the first. Raising the frequency helps, as the square root, which is why every micromechanical resonator that works is a high-frequency one.

And none of it is avoidable by choosing a better material or a cleverer shape. It is a property of the fluid the device is sitting in, which is why the answer in practice is always the same: take the fluid away.

Why a tuning fork is not a sphere #

Everything above is for a sphere, and the objects it is being applied to are not spheres. That is a real limitation and it is worth being clear about its size.

For a body oscillating at high frequency — layer much thinner than the body — the damping becomes a surface quantity: it is $\mu U_0^2/2\delta$ per unit area, integrated over the surface, whatever the shape. So in that regime the sphere calculation generalises exactly, with the sphere’s area replaced by the real one, and the error is a geometric factor of order one arising from the tangential velocity varying over the surface.

At low frequency the shape matters much more, because the disturbance extends far beyond the body and its form depends on the whole geometry. A tuning fork tine is a slender beam and its damping per unit length is closer to a cylinder’s than to a sphere’s, which differs by a logarithm.

The figures here use spheres of the objects’ own sizes, and say so. What is being computed is a scale, not a specification.

What the picture cannot show #

The amplitude is small. Everything here is a linear solution, valid while the displacement amplitude is small compared with the layer thickness. A body swinging further than that generates steady streaming — a second-order mean flow with no mean anywhere in the driving — which carries energy away by a completely different route and is not in these numbers.

The fluid is unbounded. A resonator in a package a few layer thicknesses across is squeezing the gas rather than shearing it, and the damping is a squeeze film’s rather than this one’s. That is the dominant mechanism in most micromechanical devices and it is larger than the free-space value by orders of magnitude.

The body is rigid. A resonator that flexes is presenting a surface whose own shape is changing, and the damping then depends on the mode shape rather than only on the size — which is why a tuning fork’s two tines damp differently from a single beam of the same dimensions.

And the gas is a continuum. A resonator in a low-pressure package is in a regime where the mean free path is comparable with the layer, the damping stops following this law, and it becomes proportional to the pressure rather than to the square root of the frequency — which is exactly how a vacuum gauge of the viscosity type works.

The same layer, three times, doing three things #

It is worth collecting what this collection has now asked of one solution, because the Stokes layer turns out to be doing three unrelated jobs.

It sets a thickness. The wall that shakes is about the profile itself — an exponential envelope, a linear phase lag, and a depth √(2ν/ω) beyond which the fluid does not know the wall is moving. That is a kinematic statement and needs no energy at all.

It sets a damping. This essay, which is the same profile integrated for its dissipation, and the answer depends on the layer’s thickness because the gradient does.

And it sets a mean flow. An oscillation with somewhere to go is the second-order mean of the same solution, which produces a steady circulation from a driving that has no mean anywhere — and the coefficient in it, three-quarters, comes from integrating the nonlinear term across exactly this layer.

Three results, one profile, and the second of them is the one that decides whether a machine works. That is a fair summary of why an exact solution of a simple problem is worth more than it looks: the profile took Stokes an afternoon and has been paying out for a hundred and seventy years.

One measurement this makes possible #

A last consequence, and it turns the essay round: if the damping is a known function of the fluid’s properties, then measuring the damping measures the fluid.

The quantity that appears is $\sqrt{\rho\mu}$ — the density–viscosity product — because the layer thickness carries $\sqrt{\nu}$ and the stress carries $\mu$. So an oscillating body immersed in an unknown fluid reports one number, and it is a geometric mean of the two properties rather than either of them.

That is the basis of the vibrating-wire and torsional-crystal viscometers, which are the standard instruments for viscosity at high pressure and high temperature — conditions in which no capillary or falling-body method survives, and in which the corrections a capillary needs would be unmeasurable anyway. They are calibrated in a known fluid, they measure a resonance width, and they return the product to a fraction of a per cent.

The limitation is built into the physics: one resonance cannot separate the density from the viscosity. Instruments that need both measure the frequency shift as well, which gives the added mass and therefore the density, and divide. The real and imaginary parts of one coefficient, used as two independent measurements — which works because the two halves of the force have genuinely different physical origins, one a shear in a thin layer and the other an acceleration of a large region.

Who found it, and when #

Stokes did the whole of it in 1851, in the paper that also contains the pendulum experiments the theory was written to explain — the damping of a swinging pendulum in air was, at the time, one of the few precise measurements available of any fluid property. Basset added the history integral in 1888.

The surprising connection is with a measurement technique that is not mechanical. A quartz crystal microbalance measures the mass of an adsorbed film by watching a resonator’s frequency shift, and its damping shift measures something else: the viscosity–density product of whatever the crystal is immersed in, by exactly the formula above. So a single oscillating crystal is simultaneously a balance, through its added mass, and a viscometer, through its damping — and the two readings come from the real and imaginary parts of one complex coefficient. The same solution that says a tuning fork goes quiet in water is what lets a five-megahertz disc weigh a monolayer.

Where the ladder goes next #

Below this rung is the wall that shakes, which is the layer this whole essay is about, and the price of a gradient, which supplies the integral.

Beside it is an oscillation with somewhere to go, which is what the same layer does at second order, and the last of the oil, which is the damping mechanism that takes over when the fluid is confined rather than free.