Transient and asymptotic behaviour of the binary breakage problem

Abstract

The general binary breakage problem with power-law breakage functions and two families of symmetric and asymmetric breakage kernels is studied in this work. A useful transformation leads to an equation that predicts self-similar solutions in its asymptotic limit and offers explicit knowledge of the mean size and particle density at each point in dimensionless time. A novel moving boundary algorithm in the transformed coordinate system is developed, allowing the accurate prediction of the full transient behaviour of the system from the initial condition up to the point where self-similarity is achieved, and beyond if necessary. The numerical algorithm is very rapid and its results are in excellent agreement with known analytical solutions. In the case of the symmetric breakage kernels only unimodal, self-similar number density functions are obtained asymptotically for all parameter values and independent of the initial conditions, while in the case of asymmetric breakage kernels, bimodality appears for high degrees of asymmetry and sharp breakage functions. For symmetric and discrete breakage kernels, self-similarity is not achieved. The solution exhibits sustained oscillations with amplitude that depends on the initial condition and the sharpness of the breakage mechanism, while the period is always fixed and equal to ln 2 with respect to dimensionless time.