Multiple Equilibrium Solutions in Bleck-type Models of Drop Coalescence and Breakup

Abstract

One of the best ways to understand the workings of a dynamical system is to construct a phase portrait that shows the equilibrium points of the system and a sufficient number of solution paths to indicate whether or not the equilibria attract or repel the solutions. Such a phase portrait has been constructed for a three-bin model of the coalescence/breakup equation. The portrait confirms the currently held belief that there is a unique, stable equilibrium drop size distribution. The portrait also shows that the three-bin model has three nonphysical equilibrium solutions that are unstable saddle points. Evidence points to the fact that these three equilibria represent the discrete model's attempt to reproduce the infinitely many monodisperse drop distributions that are possible in the continuous case. Each monodisperse distribution is itself in an equilibrium state that is unstable with respect to small perturbations that can introduce drops of other sizes. Fluctuations are found to occur in solutions as they approach the stable equilibrium or pass near a saddle point.