Granular media
I was almost going to choose this topic for my nonlinear physics course project. Though this is very interesting and exciting, I couldn’t find enough content to develop into a project.
Granular media, like sand, rice or cereal, also happen to form interesting patterns when vibrated on a membrane. This was first discovered and published in Nature by Umbanhowar, Melo & Swinney from the Center for Nonlinear Dynamics at the University of Texas at Austin. This was the same team that had created the very popular Faraday waves video posted on YouTube.
The physical phenomena is similar to Faraday waves seen in liquids almost a hundred and thirty years ago. The key difference is that while liquid faraday waves can be derived from a continnum model such as the Navier-Stokes equations, no such model exists for granular media as far as my limited knowledge goes. Therefore, the analytical derivations for the purposes of a project is limited.
The phase-plot is in terms of a dimensionless amplitude given by
where A is the driving amplitude, g is the acceleration due to gravity, and f is the driving frequency. Some of these have been experimentally verified by means of a molecular dynamics simulation. Once again, the computational part wasn’t sufficient to fulfill the requirements because the equations are “simple” non-linear coupled ordinary differential equations. Whereas systems we have studied in class were described by partial differential equations.
The term “oscillon” was coined by the paper mentioned above. In recent years, there has been a lot of research into these structures. The picture above is oscillons in a nonlinear field model and the author has some really cool videos on his site.