A note on the modelling of the bouncing of spherical drops or solid spheres on a wall in viscous fluid

Abstract

A generalized description of the rebound of spherical drops or solid spheres over a wall is proposed using two parameters: a coefficient of restitution that compares the velocity of restitution to the velocity before impact and the contact time with the wall. During the bouncing, the incident kinetic energy is transferred into deformation energy (stored on the surface for the case of liquid drops or in the bulk for the case of solid particles) and then restored into kinetic energy allowing the particle to leave or not the wall. The corresponding criteria is given by the Stokes number that compares the inertia of the particle (added mass included) and the viscous force exerted on the particle during the drainage of the film formed between the particle and the wall. The general behavior of the coefficient of restitution observed in many experiments can be modelled for solid spheres as well as spherical drops by the use of a unique simple correlation depending on this Stokes number. For solid particles, the contact time with the wall in viscous flows is found to be of the same order as that predicted by the Hertzian theory; hence, the contact with the wall can be described as a discontinuity in the particle motion. On the other hand, for liquid drops, the contact time is significant and of the same order as other characteristic time scales of the particle motion. Therefore, to properly describe the rebound process, both a restitution coefficient and a contact time must be considered. Finally, a simple model is proposed and its predictions are compared with experiments performed for millimetric toluene drops in water.