RayleighTomotika Instability
This capillary instability was first studied by Rayleigh in 1879 for a "perfect" (zeroviscosity) incompressible fluid, neglecting the effect of the surrounding fluid. In 1935, Tomotika extended and generalized the results for a real (viscous) fluid, surrounded by another viscous fluid. In the movie above, a 3D lattice Boltzmann method (LBM) is used to simulate the growth of perturbations in a liquid fiber, surrounded by vapor phase. The Shan and Chen (1993) model is used to simulate multiphase conditions. The initial condition is an almost cylindrical fiber with a small sinusoidal perturbation. Interfacial forces cause this perturbation to grow and cause necking and pinching. The final stable configuration is an array of spherical drops, with the same total volume as the original cylindrical fiber, but lower surface area. 
The Ghostly Hand

This simulation begins with a STL representation of the (nondeformed) hand. The signed distance function field is then calculated outside and inside the hand surface. The animation essentially shows the various distance functions (implicit surfaces) beginning with a surface outside the original hand and ending with a surface inside the original hand. Refer to the presentation below about CARTGEN++ to get more information about signed distance functions and how to calculate them. 
2D Vortex Shedding

This is one of the classic examples of external fluid flow around a cylindrical object. The flow (from left to right) becomes unstable after a certain point and the cylinder begins shedding vortices in the wake region.
Refer to the presentation below about PRATHAM to get more information about how this problem was used as one of the validation test cases. 
One fine day, a very long time ago, I started reading a book called The Fractal Geometry of Nature by Benoit B. Mandelbrot. As you can imagine, I was obliged to set aside whatever I happened to be doing and get lost in the wonderful world of fractal geometry. Very dangerous, these sort of books. Especially if you are a student working on your Masters thesis.
As I might have remarked somewhere else, QBASIC was my vehicle of choice for spending hours and hours on computer graphics  including all the fractals on this page.
Clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth, nor does lightning travel in a straight line. Benoit Mandelbrot, The Fractal Geometry of Nature. 
The JavaScript designs on this page are a tribute to Benoit Mandelbrot.
One of the most famous cellular automata is Conway's game of life. It was invented by the British mathematician John Horton Conway in 1970. Click on the image to go to the JavaScript animation. Check out the wiki for additional details. 

Langton's Ant (wiki) is another simple automaton that exhibits a rather curious and surprising behavior. After appearing to move around randomly for a little while, the ant ultimately builds a highway and escapes! 