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Concept# Collision

Summary

In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word collision refers to incidents in which two or more objects collide with great force, the scientific use of the term implies nothing about the magnitude of the force.
In physics, collisions can be classified by the change in the total kinetic energy of the system before and after the collision:

- If most or all of the total kinetic energy is lost (dissipated as heat, sound, etc. or absorbed by the objects themselves), the collision is said to be inelastic; such collisions involve objects coming to a full stop. An example of such a collision is a car crash, as cars crumple inward when crashing, rather than bouncing off of each other. This is by design, for the safety of the occupants and bystanders should a crash occur - the frame of the car absorbs the energy of the crash instead.
- If most of the kinetic energy is conserved

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We study the evolution of a system composed of N non-interacting particles of mass m distributed in a cylinder of length L. The cylinder is separated into two parts by an adiabatic piston of a mass M ≫ m. The length of the cylinder is a fix parameter and can be finite or infinite (in this case N is infinite). For the infinite case we carry out a perturbative analysis using Boltzmann's equation based on a development of the velocity distribution of the piston in function of a small dimensionless parameter ε = √(m/M). The non-stationary case is solved up to the order ε ;; our analysis shows that the system tends exponentially fast towards a stationary state where the piston has an average velocity V. The characteristic time scale for this relaxation is proportional to the mass of the piston (τ0 = M/A where A is the cross-section of the piston). We show that for equal pressures the collisions of the particles induce asymmetric fluctuations of the velocity of the piston which leads to a macroscopic movement of the piston in the direction of the higher temperature. In the case of the finite model a perturbative approach based on Liouville's equation (using the parameter α = 2m/(M + m)) shows that the evolution towards thermal equilibrium happens on two well separated time scales. The first relaxation step is a fast, deterministic and adiabatic evolution towards a state of mechanical equilibrium with approximately equal pressures but different temperatures. The movement of the piston is more or less damped. This damping qualitatively depends on whether the ratio R = Mgas/M between the total mass of the gas and the mass of the piston is small (R < 2) or large (R > 4). The second part of the evolution is much slower ; the typical time scales are proportional to the mass of the piston. There is a stochastic evolution including heat transfer leading to thermal equilibrium. A microscopic analysis yields the relation XM(t) = L(1/2 - ξ(at)) where the function ξ is independent of M. Using the hypothesis of homogeneity (i.e. the values of the densities, pressures and temperatures at the surface of the piston can be replaced by their respective average values) introduced in the previous analysis the observed damping does not show up. This can be explained by shock waves propagating between the piston and the walls at the extremities of the cylinder. In order to study the behaviour of the system there is hence a need to adequately describe the non-equilibrium fluids around the piston. We carry out an analysis of the infinite case, based on the perturbative approach introduced earlier. In this case the initial conditions are chosen in such a manner that the piston on average stays at the origin. It is shown that it is possible to describe the evolution of the fluids in such a way that it is coherent with the two laws of thermodynamics and the phenomenological relationships. Finally we study the case of a constant velocity of the piston in a finite cylinder. Such a condition and elastic collisions allow us to derive an explicit expression for the distribution of the fluids and hence for the hydrodynamics fields. This expression reveals the presence of shock waves between the piston and the extremities of the cylinder.

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A space-time adaptive algorithm to solve the motion of a rigid disk in an incompressible Newtonian fluid is presented, which allows collision or quasi-collision processes to be computed with high accuracy. In particular, we recover the theoretical result proven in [M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1345-1371], that the disk will never touch the boundary of the domain in finite time. Anisotropic, continuous piecewise linear finite elements are used for the space discretization, the Euler scheme for the time discretization. The adaptive criteria are based on a posteriori error estimates for simpler problems.

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