Plateau–Rayleigh instability
In
The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions.[1][2]
History
The Plateau–Rayleigh instability is named for
Theory
The explanation of this instability begins with the existence of tiny perturbations in the stream.
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.[8]
Although a thorough understanding of how this happens requires a mathematical development (see references[6][8]), 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.
When all the maths is done, it is found that unstable components (that is, components that grow over time) are only those where the product of the wave number with the initial radius is less than unity (). The component that grows the fastest is the one whose wave number satisfies the equation[8]
Examples
Water dripping from a faucet/tap
A special case of this is the formation of small
Urination
Another everyday example of Plateau–Rayleigh instability occurs in urination, particularly standing male urination.[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. By contrast, if the stream contacts a surface while still in a stable state – such as by urinating directly against a urinal or wall – splash-back is almost completely eliminated.
Inkjet printing
Continuous inkjet printers (as opposed to drop-on-demand inkjet printers) generate a cylindrical stream of ink that breaks up into droplets prior to staining printer paper. By adjusting the size of the droplets using tunable temperature or pressure perturbations and imparting electrical charge to the ink, inkjet printers then steer the stream of droplets using electrostatics to form specific patterns on printer paper[11]
Notes
- ^ doi:10.1063/1.868540.
- ^ .
- ^ Plateau, J. (1873). Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires [Experimental and theoretical statics of liquids subject to only molecular forces] (in French). Vol. 2. Paris, France: Gauthier-Villars. p. 261. From p. 261: "On peut donc affirmer, abstraction faite de tout résultat théorique, que la limite de la stabilité du cylindre est comprise entre les valeurs 3,13 et 3,18, … " (It can thus be affirmed, apart from any theoretical result, that the limit of the stability of the cylinder lies between the values 3.13 and 3.18, … )
- ^ Retardation of Plateau–Rayleigh Instability: A Distinguishing Characteristic Among Perfectly Wetting Fluids by John McCuan. Retrieved 1/19/2007.
- ^ Luo, Yun (2005) "Functional nanostructures by ordered porous templates" Ph.D. dissertation, Martin Luther University (Halle-Wittenberg, Germany), Chapter 2, p.23. Retrieved 1/19/2007.
- ^ ISBN 978-0-387-00592-8.
- ISBN 978-0-442-29401-4.
- ^ a b c John W. M. Bush (May 2004). "MIT Lecture Notes on Surface Tension, lecture 5" (PDF). Massachusetts Institute of Technology. Retrieved April 1, 2007.
- ^ Urinal Dynamics: a tactical guide, Splash Lab.
- ^ University physicists study urine splash-back and offer best tactics for men (w/ video), Bob Yirka, Phys.org, Nov 07, 2013.
- ^ [1]"Inkjet printing - the physics of manipulating liquid jets and drops", Graham D Martin, Stephen D Hoath and Ian M Hutchings, 2008, J. Phys.: Conf. Ser