Rayleigh-Benard Convection

I found this nice paper about RB convection. However I am confused by what is going on page 6. In particular why we are suddenly using Helmholtz equation to find spatially periodic solutions. Aren't we working with convection, so why are we looking at it from a wave like point of view? Or maybe I'm just missing the point all together.

Furthermore I would like to run a quick experiment to collect data. I was planning on using a heating place, a small glass tube filled with olive oil and a thermometer. Any tips? Suggestions?

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What exactly are you looking for? If you can get your hands on a copy of any of Drazin's books (tinyurl.com/d22r476 or tinyurl.com/cxr5hq6) there is a good and detailed treatment of the maths involved. Not sure what you expect to find through a statistical thermal dynamics lens... – Jaime Dec 2 '12 at 4:43
Alrighty sorry for the vague question. Now I am looking for information about why the RB convection is still being studied, a qualitative description of the process, which is backed up by a mathematical explanation. Also a mathematical explanation of where the critical Rayleigh number comes from would be nice. And on the side: how can I create an at home experiment to convince myself of the convection cells. @Qmechanic thanks for the edit! – kuantumbro Dec 2 '12 at 22:37
To add to my answer: A layer of oil being heated on a non-stick pan shows the formation of these Benard cells. Be careful.. I think it would be safer to use canola/vegetable oils instead of olive oil since olive oil smokes in a shorter period of time. – drN Mar 3 '13 at 23:56

Hope it's not a little too late for this answer!!!

Why are we looking at this from a "wave-point-of-view"

Rayleigh's experiments were on spermacetti (whale oil) in a cylindrical container with an aspect ratio of depth to radius $h_0/r\lt\lt 1$. This can be treated as a membrane with a certain stiffness. Hence, it is common to find a wave . The instabilities you see in RB/MB convection are called short wavelength instabilities and these wavelengths can be calculated using linear/non-linear stability analysis and can be faithfully captured with well controlled experiments.

Short note on (non)linear stability analysis

In case you aren't aware of what this is. It is simply perturbing a system with a wave with a disturbance of the form $A \sin (k x)$ and with some clever maxima/minima differential calculus and algebra, figuring out the most destructive wavelength that emerges.

($A$ is the amplitude of the wave and is generally $0.01 h_0$, $k$ is the wavenumber $k = 2 \pi x/L$ and $L$ is the domain size).

Some useful references/papers

Besides Killercam's reference to Drazin, if you are really interested in RB or MB (Marangoni Benard) convection, I would suggest that you read Rayleigh's 1916 paper and Thomson's 1855 paper in addition to Kundu's textbook , particularly chapter 12.

Running experiments with olive oil

As for running experiments with olive oil in a cylinder: be careful; the dimensions of the cylinder (aspect ratio) can change the structures from the hexagonal RB/MB cells to perhaps a Rayleigh-Taylor type instability.

Extra fun stuff

If you find that your interest is roused by RB/MB convection, eventually you look at long wave instabilities that affect liquid film.

References

Thomson, J. On certain curious motions observable at the surfaces of wine and other alcoholic liquors Phil. Mag. Ser., 1855, 10, 330-333

Rayleigh On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. Philo. Mag. Series, 1916, 32, 529-546

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Dropbox links now dead. – Qmechanic Mar 1 at 22:16