Plateau–Rayleigh instability

A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions.

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.

[3][4] Later, Rayleigh showed theoretically that a vertically falling column of non-viscous liquid with a circular cross-section should break up into drops if its length exceeded its circumference, which is indeed π times its diameter.

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.

Likewise at the peak the radius of the stream is greater and, by the same reasoning, pressure due to surface tension is reduced.

When the effect of the radius of the stream dominates that of the curvature of the wave, such components grow exponentially with time.

The component that grows the fastest is the one whose wave number satisfies the equation[8] A special case of this is the formation of small droplets when water is dripping from a faucet/tap.

If the diameter of the faucet is big enough, the neck does not get sucked back in, and it undergoes a Plateau–Rayleigh instability and collapses into a small droplet.

[9][10] The stream of urine experiences instability after about 15 cm (6 inches), breaking into droplets, which causes significant splash-back on impacting a surface.

Three examples of droplet detachment for different fluids: (left) water, (center) glycerol, (right) a solution of PEG in water

Intermediate stage of a jet breaking into drops. Radii of curvature in the axial direction are shown. Equation for the radius of the stream is $R(z)=R_{0}+A_{k}\cos(kz)$ , where $R_{0}$ is the radius of the unperturbed stream, $A_{k}$ is the amplitude of the perturbation, $\scriptstyle z$ is distance along the axis of the stream, and $k$ is the wave number

Rain water flux from a canopy. Among the forces that govern drop formation: Plateau–Rayleigh instability, surface tension , cohesion , Van der Waals force .