Molecular simulation of the formation of a cusp at the free surface of a fluid
    The problem is whether a cusp can form at the free surface of a liquid at the molecular scale. One can investigate the problem by placing some fluid in a box, and driving the flow with two solid rollers. Using this setup, Koplik and Banavar found a negative result (Phys. of Fluids 1994 Volume 6, Issue 2, pp. 480-488). However, this was because they studied the interface between two fluids of similar viscosity. I found that if one repeats the simulation with a fluid-vacuum interface, a cusp does appear. The simulation is tricky because the temperature must be carefully controlled, since the work done by the rollers dissipates into heat, that warms up the fluid. If the temperature is too high, the interface becomes too diffuse to harbor any cusps. In the right range of temperature, we observe cusps as shown in the animated gif below (be patient, the animation loads very slowly):
animated gif 14.1MB
On the other hand, if the temperature is too low, the liquid freezes, and of course the cusp does not appear (see below). This example shows how important is the temperature control.
animated gif 25.5MB
I did the above simulations using a classical approximation. The fluid is modeled with a Leenard-Jones potential (a crude model of liquid argon and other noble gases), using the Verlet algorithm to solve the Newton equations of motion, and the cells method to organize the atoms efficiently. The temperature is controlled by assigning a Maxwellian distribution to the velocities of the atoms that bounce at the lateral walls. In this way, the effective boundary condition is constant temperature at the walls, and heat is dissipated via conduction towards the walls of the container.