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):
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.