As far as I know, no one has written a paper on the physics of lava lamps. This is a pity since lava lamps may be telling us something about the Earth. Of course, the physics of the lava lamp would have to be simplified: “Consider a spherical lava lamp…”.
A geophysicist does not take a lava lamp apart to understand how it works. She observes it from outside and infers the physics. R.U. Shure, after hours of observation, concludes the following: The active material in the lamp appears to be a wax that is denser than the oil it is in when it is cold but buoyant when it is hot. The thermal expansion of the two fluids must be quite different so that there is a density cross-over when density is plotted against temperature. The wax rises to the surface when it is hot, cools off, becomes dense, and then sinks to the bottom where it is heated by the underlying lamp. The wax is therefore stable at the top or the bottom of the system, presumably because it is much denser or much lighter than the surrounding oil. It therefore does not have time to mix, even though the fluid in the lamp is clearly convening. (Surface tension may be important, as it was in the Benard experiments, of Rayleigh-Benard fame. The theoretical basis was provided by Lord Rayleigh, who ignored surface tension.)