The thread that cannot stay a thread

Turn a tap on gently and watch the stream. Near the spout it is a smooth cylinder; a few centimetres down it develops a visible waviness; a little further it is a chain of separate drops. The drops are not all the same size but they are close, and the spacing between them is remarkably consistent.

The question worth asking is not why it breaks. It is why it breaks at that spacing — because nothing upstream specifies a spacing, and the answer is the same kind of answer as why a string sounds particular frequencies: a continuous system given arbitrary input returns a specific number.

Surface tension does the breaking #

The first thing to get straight is the direction of the effect, because the usual intuition has it backwards.

A cylinder of liquid of radius $R$ and length $L$ has volume $\pi R^2 L$ and surface $2\pi R L$. Now imagine that same volume as a row of spherical drops. Work the areas out and the row of drops has less surface than the cylinder, for any drop bigger than a certain size. Since surface costs energy, the cylinder is not the low-energy configuration. It is a system sitting somewhere it would rather not be, and surface tension is the thing that would rather it were elsewhere.

So surface tension does not hold a stream together against something else pulling it apart. Surface tension is what pulls it apart. The same $\gamma$ that makes a free drop round makes a thread break, and the sign flip comes entirely from the shape.

Why the circumference is the dividing line #

The mechanism is in the curvature. Take a thread and squeeze it slightly at one place and bulge it slightly at another — a sinusoidal perturbation of wavelength $\lambda$. At the neck, two curvatures act in opposite senses. The azimuthal curvature, going round the thread, is larger there because the radius is smaller, so it raises the pressure. The axial curvature, along the thread, has the neck curving the other way, so it lowers the pressure.

Which wins depends on the wavelength. A short-wavelength perturbation has strong axial curvature, the axial term dominates, the neck is at lower pressure than the bulge, and liquid flows from bulge to neck: the disturbance is filled in and dies. A long-wavelength perturbation has weak axial curvature, the azimuthal term dominates, the neck is at higher pressure, and liquid is squeezed out of the neck into the bulge: the disturbance grows and the thread pinches off.

The dividing wavelength is where the two exactly cancel, and it works out to $\lambda = 2\pi R$ — the thread’s own circumference. Anything longer grows; anything shorter decays. That criterion is Plateau’s, established in the 1870s from careful observation of liquid columns, and it says which disturbances are dangerous without saying which one wins.

The relation the whole argument runs on is a pressure difference set by curvature, with a radius in the denominator. So a variation in the thread’s radius is a variation in pressure — and pressure differences move liquid. Everything else is bookkeeping about which way it moves: toward the fat part if the axial curvature dominates, toward the thin part if the azimuthal one does, and the wavelength decides which.

Which one wins #

Every wavelength above the circumference grows, and a real disturbance contains many of them at once — a tap’s stream is perturbed by vibration, by the spout’s shape, by air. So the observed spacing is not the wavelength that was put in; it is the wavelength that grows fastest, because after enough growth it dominates everything else regardless of what the initial amplitudes were.

Rayleigh computed the growth rate in 1878 and it is the quantity the figure plots:

$$\omega^2 = \frac{\gamma}{\rho R^3}\, x(1 - x^2)\, \frac{I_1(x)}{I_0(x)}, \qquad x = kR = \frac{2\pi R}{\lambda}.$$

The $(1-x^2)$ factor is Plateau’s criterion made quantitative: it is positive for $x < 1$ and negative above, so growth stops exactly at the circumference. The Bessel ratio comes from solving the flow inside the thread. The prefactor sets the timescale and contains all the physical properties; the shape of the curve contains none of them.

The number, and what is not in it #

The maximum is at $x \approx 0.697$, giving

$$\lambda_{\max} = \frac{2\pi R}{0.697} \approx 9.02 R = 4.51\,(2R).$$

About four and a half diameters. That is the spacing of drops from a tap, the spacing of drops from a dripping icicle, the spacing of beads on a spider’s web thread, and the spacing of ink drops in an industrial printer.

What is striking is what the number does not contain. $\gamma$ cancels. $\rho$ cancels. $R$ enters only as the unit. The prefactor decides how fast the break-up happens and the ratio $\lambda_{\max}/R$ is a pure number that is the same for water, mercury, molten glass and liquid metal. It is a fact about cylinders, not about liquids.

That is the same species of result as the angle a rainbow has to be or the exponent in a falloff law: a number that looks as though it ought to depend on the substance and depends only on the arrangement.

How long it takes #

The growth rate has units, and putting numbers to them says whether any of this is observable.

The timescale is $\tau = \sqrt{\rho R^3 / \gamma}$, the only combination of the three quantities with the dimensions of time. For water at a one-millimetre radius that is about 3.7 milliseconds; the fastest mode grows as $e^{t/\tau'}$ with $\tau'$ a few times larger, so a disturbance amplifies by a factor of ten in something like ten milliseconds.

In that time a stream falling under gravity travels a few centimetres — which is exactly the distance from a tap at which the break-up is seen. The theory does not merely predict a spacing; it predicts the break-up length, and the fact that both come out right from one calculation is the strongest evidence that the mechanism is the one described.

The scaling is severe, because $R^{3/2}$ falls fast. A hundred-micron jet has a timescale of about 120 microseconds, so it breaks within a millimetre or two — which is why a fine jet appears to be drops almost immediately, and why photographing the intact portion needs microsecond exposures.

Before anything grows, the perturbation is just a wave on the surface with a wavelength and an amplitude. What makes an instability is that such a wave’s amplitude is multiplied rather than carried along — the shape stays put and grows, where an ordinary wave keeps its size and moves. That is the whole distinction between a wave and an instability, and it is visible in a pair of snapshots: one is displaced, the other is taller.

Selection out of noise #

The mechanism by which a specific wavelength emerges from an unspecific disturbance is worth stating on its own, because it recurs across the whole of physics.

Every mode grows exponentially, each at its own rate $\omega(k)$. Start with all of them at small and comparable amplitudes. After a time $t$, mode $k$ has been multiplied by $e^{\omega(k)t}$. Because the exponent differs between modes, the ratios between them grow exponentially too: a mode growing ten per cent faster than another is ahead by a factor of $e^{0.1\omega t}$, which becomes enormous while both are still small enough for the analysis to hold.

So the fastest mode does not merely win — it wins by a margin that grows without bound, and it does so before anything becomes large. The selection is complete long before the thread actually breaks. This is why the output is sharp even though the input is noise, and it is the same argument that picks a single wavelength in a great many pattern-forming systems — including the one that selects which frequencies a driven oscillator responds to, where a sharp output likewise comes out of a broad input.

A mixture does not stay a mixture, and the reason is that modes add linearly while they are small. A real disturbance is a superposition of many wavelengths, each growing at its own exponential rate — and a superposition whose components grow at different exponential rates stops being a mixture very quickly indeed. After ten e-foldings of the fastest mode, a component growing at half the rate is smaller by a factor of a hundred and fifty.

Satellites, and where the linear theory ends #

The analysis above is linear: it assumes the perturbation is small compared with the radius, so the equations can be expanded to first order. That is excellent for predicting which wavelength wins and useless for predicting what the drops look like, because the last stage of pinch-off is violently nonlinear — the neck radius goes to zero and the curvature goes to infinity.

The observable consequence is satellite drops. Between each pair of main drops a real jet leaves one or several much smaller ones, formed from the ligament of liquid stretched between two necks as they pinch. Nothing in the linear theory predicts them, they are a nuisance in inkjet printing, and suppressing them is a matter of driving the jet at a chosen amplitude and frequency rather than letting noise select.

That is the other half of the practical story: because the instability amplifies whatever it is given, a jet can be told what to do. Drive it at a wavelength near the fastest-growing one and the imposed mode dominates from the start, and the drops come off in perfect sequence, one per cycle, at a rate the driver chooses. That is exactly how a continuous-inkjet printer works, and it turns an instability from a failure mode into a metering device.

The same instability, in things that are not taps #

The mechanism needs only a cylinder of one fluid inside another and an interfacial tension between them, so it appears wherever those exist.

Coating a fibre. A layer of liquid on a thread is itself a cylindrical interface, and it breaks into a row of beads — which is why a spider’s web catches drops at regular intervals, and why some coatings must be cured before they have time to bead.

Molten metal jets. Powder for metal additive manufacturing is often made by breaking a molten stream deliberately, and the resulting spheres are round because a free drop minimises its surface.

Inside the body. The liquid lining a small airway is an annular film on a cylinder, and it is unstable in the same way. If it beads, it can bridge the airway and close it — a mechanism implicated in airway closure at low lung volumes, and one more place where surfactant is doing structural work.

In space. Without gravity to stretch it, a liquid bridge between two supports is a nearly perfect cylinder, and the Plateau limit becomes directly testable: a bridge longer than its circumference cannot exist. That experiment has been done on orbit and the criterion holds.

A cylinder is not a minimum of surface energy. It is a ridge: every wavelength above the circumference is a direction down, and only the wavelengths below it cost area to excite. So the question the calculation answers is not whether the thread breaks but which direction off the ridge is steepest — and the answer, about nine radii, is the one the drops end up spaced by.

Why a sheet does not do this #

The mechanism above needs the azimuthal curvature — the one that goes round the thread and gets stronger as the neck narrows. A flat film has no such curvature. Both of its principal curvatures are zero, and pinching it anywhere adds area at every wavelength, so a sheet of liquid is stable against exactly the disturbance that destroys a cylinder.

That is why the same surface tension produces three quite different fates according to shape. A drop is stable and stays a drop. A cylinder is unstable to everything longer than its circumference. A sheet is stable, and when it goes it goes by a completely different route: it thins by drainage until it is a few tens of nanometres thick, and then a hole nucleates somewhere in it.

Once a hole exists the sheet is finished, and the manner of the finishing is worth following. The rim of the hole retracts at a speed set by momentum rather than by energy — the rim collects the film it sweeps through, so it is accreting mass as it accelerates, and the steady result is $v = \sqrt{2\gamma/\rho h}$ for a film of thickness $h$. For a soap film about a micron thick that is some ten metres per second, which is why a bubble does not appear to pop so much as vanish.

And the rim is a cylinder. It is a ring of liquid, of a radius set by how much film it has already gathered, and it is subject to precisely the instability this essay is about — so it beads, at about four and a half of its own diameters, and the beads become drops. That is the spray a bursting bubble leaves behind, the scalloped edge of a splash, and the reason the sea puts salt into the air at all. The instability is the last thing that happens to a sheet as well as the only thing that happens to a jet.

The same analysis, run as an instrument #

Not every mode of a liquid cylinder grows. The calculation above was for the axisymmetric ones — the squeezes and bulges that keep the cross-section circular — and there is a second family in which the cross-section itself deforms, from a circle to an ellipse and back. Those modes have no way to lower the surface area, so they do not grow at any wavelength. They oscillate.

That makes them useful. Push liquid through an elliptical orifice and the jet leaves with an elliptical cross-section, which surface tension immediately tries to correct; it overshoots, and the jet proceeds downstream oscillating between two perpendicular ellipses with a wavelength that depends on $\gamma$, $\rho$, the jet radius and the jet speed. Measure the wavelength with a ruler and the surface tension follows.

The reason to want such an awkward instrument is timing. A jet a centimetre downstream at five metres per second has a surface one two-thousandth of a second old, and a surfactant molecule takes far longer than that to find its way from the bulk to a freshly created interface. So the $\gamma$ that a static method measures on a surface hours old is not the $\gamma$ that acted while a drop was forming — and for anything with surfactant in it the two can differ by a factor of two. The oscillating jet reads surface tension at an age of milliseconds, which is the age that matters for spraying, coating and breathing.

It is also where Niels Bohr began: his first published paper, in 1909, was a determination of surface tension by the vibration of a jet, correcting Rayleigh’s treatment for viscosity and for the finite amplitude of the oscillation. The same dispersion relation that predicts the break-up on one branch is a measuring instrument on the other.

What is conserved while it happens #

Two constraints hold throughout, and they are worth naming because they are what make the outcome computable at all.

Volume. The liquid is incompressible over any pressure this process generates, so material leaving a neck arrives in a bulge and the total is fixed. That is what turns “the surface would like to be smaller” into a definite prediction: the thread cannot simply shrink, it must redistribute, and the redistribution is what the drops are.

Momentum, once nothing external acts. In free fall the whole assembly is unforced apart from gravity acting equally on all of it, so the break-up happens in a frame where the centre of mass moves uniformly. This is why the analysis can be done in the thread’s own frame and why the drops emerge with very nearly the jet’s velocity rather than being flung sideways.

Those two together are why the figure’s growth curve can be trusted as a prediction rather than as an illustration. Nothing in it is fitted; the volume constraint and the pressure field from curvature are enough, and the only physical inputs are the ones that cancel out of the answer.

In the collection’s usual language the direction is a count. There are vastly more ways to arrange a given volume of liquid as a row of drops than as one cylinder of a precise radius, so the counting argument points the same way the energy one does. That is the ordinary situation and it is worth checking rather than assuming — the two arguments disagree often enough to be interesting when they do.

Where the model stops #

The perturbation is small. Everything above is first order in the amplitude, so it predicts the winning wavelength and not the drop sizes, the satellites or the pinch-off dynamics. The final singularity has its own self-similar theory and is a subject in itself.

The thread is not moving. The analysis is done in the frame of the fluid, which is fine for a jet in free fall over the short distance the break-up occupies, and wrong for a jet that is being stretched — where the radius is falling as the instability grows and the fastest wavelength is a moving target.

Viscosity is neglected in the form quoted. Rayleigh’s inviscid result is the one plotted. Viscosity slows the growth and shifts the maximum to longer wavelengths, without limit for a very viscous thread — which is why a stream of honey breaks into widely spaced blobs rather than closely spaced drops. Adding viscosity changes the number, and the mechanism not at all.

There is nothing outside the thread. A jet moving fast through air is also subject to aerodynamic break-up, which is a different instability with a different wavelength, and it takes over at high enough speed. That regime belongs to aerodynamics rather than to capillarity.

The interface is sharp. Two miscible fluids have no interfacial tension once they have mixed, and the instability fades as they do.

The history, and the two names on it #

Plateau established the geometric criterion in the 1870s by patient experiment on liquid columns supported in a density-matched bath — a way of switching gravity off before there was any other way to do it. He found the limit at $\lambda = 2\pi R$ and could go no further, because the criterion says which disturbances grow and not how fast.

Rayleigh supplied the rate in 1878, and with it the selection. The distinction between the two contributions is the distinction between a stability boundary and a dispersion relation, and it is worth keeping: the first says a system will not stay put, and only the second says what it will do instead.

The modern name attaches both, which is fair, and the physics has been re-derived for viscous threads, for threads in another fluid, for charged threads and for threads with surfactant on them. In every case the criterion survives untouched and the number moves.

The ladder from here #

Later rungs on this anchor: the viscous case, and the widely spaced blobs it gives. Pinch-off as a singularity, where the neck radius follows a power law in the time remaining and forgets its initial conditions. Satellite formation. Electrospraying, where charge on the surface adds a term and produces droplets orders of magnitude finer. And the same competition of growing modes in systems with nothing liquid in them, where a fastest-growing wavelength turns noise into a pattern with a spacing.