Shock wave of a narrow wedge moving through a Lennard-Jones gas
Channel: Nils Berglund
Date: 03/04/2022

This new version of a shock wave shows a triangular obstacle accelerating through a Lennard-Jones gas composed of 2178 particles, seen from the frame of reference of the obstacle. The forward angle of the obstacle has been chosen smaller than in the simulation https://youtu.be/dhWjggUPCUk and the region in which the temperature is controlled by a thermostat is located further to the right, and kept at a smaller temperature of 200 degrees (measured in terms of mean kinetic energy per particle).
The other particles, by contrast, evolve only according to Newton's equation. The temperature shown at the top left is the measured temperature of all particles, which are seen to heat up due to the interaction with the obstacle, though the temperature oscillates more here than in the previous simulation. Since the particles arriving on the obstacle are kept at constant temperature, the speed of sound should hopefully not change too much, resulting in a cleaner shock wave. 
One thing that is not quite satisfactory here is the behavior of particles hitting the sharp angle of the wedge. I had to add a repulsive potential at the tip, in addition to the repulsive potential from the sides of the wedge, to avoid numerical instabilities, that could occur if a particle enters the repulsive region sideways. However it seems I have not gotten the computation of the repulsion quite right, because some particles accelerate a lot when approaching the tip. 
The Mach number is the ratio between the obstacle's speed and the speed of sound in the gas, which however has only been estimated from when the shock wave forms. It may be possible to figure out the speed of sound in a more accurate way. 
This simulation has two parts, showing the same evolution with two different color schemes:
Kinetic energy: 0:00
Number of neighbors: 2:09
In the first part, colors depend on the kinetic energy, and thus on the speed of particles. Red particles move faster than blue ones. Note that the speed is measured with respect to the rest reference frame of the particles, which is why blue particles tend to move to the left. In the second part of the video, the color depends on the number of particles within a given distance, allowing to estimate the density of the gas. Blue regions have a higher density then red ones.
The boundary conditions on the circular object are given by a stiff harmonic potential. The thermostat is implemented here with the "Nosé-Hoover-Langevin" algorithm introduced by Ben Leimkuhler,  Emad Noorizadeh and Florian Theil, see reference below. The idea of the algorithm is to couple the momenta of the system to a single random process, which fluctuates around a temperature-dependent mean value. Lower temperatures lead to lower mean values. 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: 1 hour 49 minutes
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Turbo" by Gunnar Olsen

Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). https://doi.org/10.1007/s10955-009-9734-0
http://www.maths.warwick.ac.uk/~theil/HL12-3-2009.pdf

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)