We perform a dimension analysis for colliding viscoelastic spheres to show that the coefficient of normal restitution epsilon depends on the impact velocity g as epsilon=1-gamma(1)g(1/5)+gamma(2)g(2/5)-/+..., in accordance with recent findings. We develop a simple theory to find explicit expressions for coefficients gamma(1) and gamma(2). Using these and few next expansion coefficients for epsilon(g) we construct a Pade approximation for this function which may be used for a wide range of impact velocities where the concept of the viscoelastic collision is valid. The obtained expression reproduces quite accurately the existing experimental dependence epsilon(g) for ice particles.

Coefficient of restitution of colliding viscoelastic spheres.

http://cc.oulu.fi/~hsalo/salo2001_icarus153.pdf

Viscous Overstability in Saturn's B Ring I. Direct Simulations and Measurement of Transport Coefficients

Numerical simulations of dense collisional systems

The collisional dynamics of particulate disks

Numerical simulations of collisions between rotating particles

The numerical simulations of 200 mutually colliding, non-identical particles indicate that if elasticity depends on the impact velocity, an equipartition of the random kinetic energy is possible if either the particle masses are close to each other, or the number of small particles significantly exceeds that of large particles. On the other hand, if the large particles dominate, the velocities of the smaller particles are at most a few times greater than those of the large ones. In the case of a constant coefficient of restitution no equipartition is observed.

Numerical simulations of collisions in systems of non-identical particles

The influence of partially elastic collisions, gravitational encounters, and different gravitational potentials is studied in terms of computer simulations involving a ring of 200 identical particles. If the masses of the particles are small, their mutual gravitational attraction is found to produce a decrease in the dispersion of velocities below the collisional equilibrium value, but for heavier particles the dispersion again begins to increase. The vertical component of the force which is produced by the self-gravitation of the ring reduces its thickness. The results of the simulations are in good agreement with the predictions of the collisional theory.

Numerical simulations of collisions in self-gravitating systems

Numerical simulations of 250 mutually colliding particles, revolving in the gravitational field of a central body, indicate that velocity dependent coefficient of restitution, α = α(Ν), can lead to a many particle thick equilibrium state, independent of the initial conditions. The essential requirement for this to take place is that dα dΝ 0, the system either disperses through growing random velocities, or flattens to a near monolayer state, depending on the initial conditions.

Velocity dependent coefficient of restitution and the evolution of collisional systems

Quasi-equilibrium solutions for the pre-planetary disk are studied in terms of Hameen-Anttila's theory (1984) of collisional, self-gravitating systems. The distribution of particle sizes is assumed to follow simple power-law distributions, with a power index in the range of 1.5–5.0. The treatment includes mutual impacts with a velocity dependent coefficient of restitution, as well as gravitational encounters with dynamical friction. The mean gravitational field of the disk is also taken into account. The results indicate that the energy(equi)-partition depends mainly on the index of size distribution, but is also affected by the optical thickness of the system, as well as on the vertical thickness as compared to the particle size. The vertical component of the gravitational field is found to be important, especially when the mass of the system is concentrated on the large particles.

J. Lukkari

Papers

Numerical simulations of collisions in systems of non-identical particles

Numerical simulations of collisions in self-gravitating systems

Velocity dependent coefficient of restitution and the evolution of collisional systems

Velocity dispersion in particulate disks with size distribution