Plateau–Rayleigh instability

Key takeaways

Plateau–Rayleigh instability belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Plateau–Rayleigh instability to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Plateau–Rayleigh instability from memory before moving on to harder problems.

Reference excerpt

In fluid dynamics, the Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same total volume but less surface area per droplet. It is related to the Rayleigh–Taylor instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets. The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. Work has been done in the 1990s on the final pinching profile by attacking it with self-similar solutions. The instability is named for Joseph Plateau and Lord Rayleigh.

History The first to recognize the universal behaviour the break up of liquid jets into droplets and its wave behaviour was Félix Savart in 1833. Photography was not yet available, but he used stroboscopic light to study the phenomena. Joseph Plateau created an experimental set up (known as a Plateau tank) to study the formation of droplets in 1843 and identified the important role of surface tension in 1849. Gotthilf Hagen also studied the phenomena the same year but with reduced mathematical treatment. A debate on the nature of the phenomena started between Plateau and Hagen, whose letters where published in the Annalen der Physik. In 1873, Plateau found experimentally that a vertically falling stream of water will break up into drops if its length is greater than about 3.13 to 3.18 times its diameter, which he noted is close to π. In 1879, John William Strutt, 3rd Baron Rayleigh used linear stability analysis to describe the instability. In 1891, Rayleigh also published photographies of the phenomena.

Theory

The explanation of this instability begins with the existence of tiny perturbations in the stream. These are always present, no matter how smooth the stream is (for example, in the liquid jet nozzle, there is vibration on the liquid stream due to a friction between the nozzle and the liquid stream). If the perturbations are resolved into sinusoidal components, we find that some components grow with time, while others decay with time. Among those that grow with time, some grow at faster rates than others. Whether a component decays or grows, and how fast it grows is entirely a function of its wave number (a measure of how many peaks and troughs per unit length) and the radius of the original cylindrical stream. The diagram to the right shows an exaggeration of a single component. By assuming that all possible components exist initially in roughly equal (but minuscule) amplitudes, the size of the final drops can be predicted by determining by wave number which component grows the fastest. As time progresses, it is the component with the maximal growth rate that will come to dominate and will eventually be the one that pinches the stream into drops. Although a thorough understanding of how this happens requires a mathematical development (see references), the diagram can provide a conceptual understanding. Observe the two bands shown girdling the stream—one at a peak and the other at a trough of the wave. At the trough, the radius of the stream is smaller, hence according to the Young–Laplace equation the pressure due to surface tension is increased. Likewise at the peak the radius of the stream is greater and, by the same reasoning, pressure due to surface tension is reduced. If this were the only effect, we would expect that the higher pressure in the trough would squeeze liquid into the lower-pressure region in the peak. In this way we see how the wave grows in amplitude over time. But the Young–Laplace equation is influenced by two separate radius components. In this case one is the radius, already discussed, of the stream itself. The other is the radius of curvature of the wave itself. The fitted arcs in the diagram show these at a peak and at a trough. Observe that the radius of curvature at the trough is, in fact, negative, meaning that, according to Young–Laplace, it actually decreases the pressure in the trough. Likewise the radius of curvature at the peak is positive and increases the pressure in that region. The effect of these components is opposite the effects of the radius of the stream itself. The two effects, in general, do not exactly cancel. One of them will have greater magnitude than the other, depending upon wave number and the initial radius of the stream. When the wave number is such that the radius of curvature of the wave dominates that of the radius of the stream, such components will decay over time. When the effect of the radius of the stream dominates that of the curvature of the wave, such components grow exponentially with time.

Linear stability analysis By decomposing an arbitrary disturbance on the free surface into its constitutive harmonics/wavelengths one can derive a condition for the stability of the jet in terms of the perturbation:

ω 2 = σ k ρ a 2 I 1 ( k a ) I 0 ( k a ) ( 1 − k 2 a 2 ) , {\displaystyle \omega ^{2}={\frac {\sigma k}{\rho a^{2}}}{\frac {I_{1}(ka)}{I_{0}(ka)}}\left(1-k^{2}a^{2}\right),}

… excerpt ends here. Continue reading the full article.

Illustrations

Plateau–Rayleigh instability: Water dropping from a tap — Water dropping from a tap

Worked examples

Example 1 — a first encounter with Plateau–Rayleigh instability

Start with the simplest possible case. Write down what Plateau–Rayleigh instability claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Plateau–Rayleigh instability before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Plateau–Rayleigh instability ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Plateau–Rayleigh instability

In research: Plateau–Rayleigh instability appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Plateau–Rayleigh instability in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Plateau–Rayleigh instability is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fluid dynamic instabilities, Fluid dynamics, so understanding it makes those chapters shorter.
In everyday life: Look for Plateau–Rayleigh instability outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Plateau–Rayleigh instability in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Plateau–Rayleigh instability means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Plateau–Rayleigh instability out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Plateau–Rayleigh instability in simple terms?

In fluid dynamics, the Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same total volume but less surface area per droplet. It is related to the Rayleigh–Taylor instability and is part o…

Why does Plateau–Rayleigh instability matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Plateau–Rayleigh instability?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Plateau–Rayleigh instability.