Fluid Mechanics

Fluid Mechanics

A wonderful example of Raleigh-Taylor instability, modeled by a computer-

http://www.psc.edu/science/2007/turbulence/images/raleigh4.png

Fluid mechanics provides the mathematics that gives credence to the behavior of fluids, be they liquid or gas. In the case of lava lamps, certain instabilities arise in the system that primarily have to do with surface tension between wax and water. Surface tension is the cohesive force between similar molecules in a liquid, it can be represented as a force or an area of energy. When these instabilities propagate that is when the beauty of the lava lamp comes forth, the great plumes of wax and the emergence of droplets with each other, all have to do with fluid mechanics.

Rayleigh-Taylor Instability

The interaction between two fluids of different densities- http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021999106005511-gr28.jpg

A Rayleigh-Taylor instability is when two fluids of different densities are in contact and suddenly the lighter of the two ends up on the bottom of the system. This happens in a lava lamp because the wax goes from being heavier than water to being lighter thanks to the heating element. When this inversion takes place the fluids can become unstable, partially because the gravitational potential of the system goes from a minimum to a maximum. If the surface free energy (intrinsically connected to surface tension) is less than that of the gravitational potential the system is unstable.

The base state of a Rayleigh-Taylor instability, gravity points in the downward direction-http://upload.wikimedia.org/wikipedia/en/a/a9/Rti_base.png

So as the more dense fluid (water in this case) pushes down on the less dense fluid (wax) in begins to 'pinch' the peaks of wax waves into pillars. As the instability propagates more and more columns appear and they make their way to the top of the system. Once all the wax is at the top of the lamp the system is stable again. Of course since the wax's density is so close to that of waters, falling just a little bit higher or lower whether its heated or not, by the time it reaches the top of the lamp it cools enough to be more dense than water again. Thus another Rayleigh-Taylor instability forms, only now the lighter fluid is water, and the pillars of wax drop down only to be heated up once again. So the system continues this cycle of trying to find stability indefinitely.

A fluid undergoing a Rayleigh-Taylor instability, note that when the wax reaches the top the system is again stable- http://www.comsol.com/blogs/wp-content/uploads/2012/07/Lava-Lamp.gif

Rayleigh-Plateau Instability

An animation of a Rayleigh-Plateau Instability-

http://www2.rikkyo.ac.jp/web/umpei/download/SD/SurfaceDiff4D.gif

Another common phenomenon found in lava lamps is Rayleigh-Plateau instability. If you've ever seen a stream of water break up into little tiny droplets this is the instability you're looking at. With respect to lava lamps, when the column of wax begins to rise it usually ends up breaking into large droplets. Mathematically this does make sense, since a sphere has less surface area than a cylinder and the fluid is being pushed on from all sides equally, the water pressure confines the column into spherical shapes.