A rotating triangle in a granular medium
Channel: Nils Berglund
Date: 02/09/2023

This simulation of a granular medium features a rotating triangular obstacle, as was suggested in comments to a previous video with gravity and a fixed triangle. Here there is no gravity, so that the motion of the medium is primarily a consequence of the rotating triangle.
The triangle starts at rest, to give the medium a chance to reach an equilibrium configuration. Then its angular speed increases, before decreasing again. The angle is given in degrees, and the angular speed in radians per simulation second. 
The particles in this simulation interact via a Lennard-Jones potential, and are subject to a viscous damping term and to a one-sided harmonic repulsion from the obstacle. There are periodic boundary conditions. 
This simulation has two parts, showing the evolution with two different color schemes:
Initial y coordinate: 0:00
Lattice of neighbors: 2:24
 In the first part, each particle is given a color at the beginning of the simulation, which is a function of its y-coordinate. The particle then keeps the same color throughout the simulation.In the second part, the particles are not drawn at all, but the lines joining neighboring particles are. Here neighboring is defined as being within a fixed distance. The color of the lines changes slightly with their length.
To save on computation time, particles are placed into a "hash grid", each cell of which contains between 3 and 10 particles. Then only the influence of other particles in the same or neighboring cells is taken into account for each particle. 
The Lennard-Jones potential is strongly repulsive at short distance, and mildly attracting at long distance. It is widely used as a simple yet realistic model for the motion of electrically neutral molecules. The force results from the repulsion between electrons due to Pauli's exclusion principle, while the attractive part is a more subtle effect appearing in a multipole expansion. For more details, see https://en.wikipedia.org/wiki/Lennard-Jones_potential

Render time: 39 minutes 39 seconds
Color scheme: Part 1 - HSV/Jet
Part 2 - Twilight by Bastian Bechtoldhttps://github.com/bastibe/twilight

Music: "Powerup!" by Jeremy Blake@RedMeansRecording

Current version of the C code used to make these animations: 
https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)

#granular_material #lennardjones #LennardJones #molecular_dynamics #phonon