This simulation compares the efficiency of rocket nozzles. Each rocket is made of a circular container, and a nozzle, which is straight for the left rocket, and bell-shaped for the right one. The particles in the containers are heated up and then released, producing a vertical thrust that is able to counteract gravity, as long as there are enough particles in the container.
The bell-shaped nozzle increases the efficiency by taking advantage of collisions with escaping particles. For its shape, I used a deformed parabola of equation x = sqrt(a² - 0.5*y) where -y is the distance below the aperture, and a is a constant defining the width of the aperture.
There are 1337 particles in total, which interact via a Lennard-Jones potential. Particles inside the cavities are coupled to a thermostat, the temperature of which is shown at the top. Particles outside the cavity are not coupled to the thermostat, but experience some viscous friction to avoid numerical instabilities. The particles and walls of the containers interact through a harmonic potential with large spring constant. The total mass of each container has been chosen 40 times the total mass of all particles. The ground is also modeled by a repelling harmonic potential.
The simulation has two parts, showing the same evolution with two different playback speeds:
Normal speed: 0:00
Time lapse: 1:17
The particles' color indicates their kinetic energy: red particles move faster than blue ones.
The temperature is controlled by a thermostat, 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 [ Ссылка ]
Render time: 1 hour 50 minutes
Color scheme: Turbo, by Anton Mikhailov
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Music: "Tick Tock" by Jimmy Fontanez
Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). [ Ссылка ]
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Current version of the C code used to make these animations:
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Some outreach articles on mathematics:
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(in French, some with a Spanish translation)
