Log-stable laws as asymptotic solutions to a fragmentation equation: application to the distribution of droplets in a high Weber-number spray.

Abstract

In this paper, it will be shown that "totally skewed to the left" log-stable distributions are suitable asymptotic solutions to a fragmentation equation. This result generalizes Kolmogorov's work on log-normal distribution for the drops' size number distribution of particles under pulverization. Indeed, Kolmogorov's discrete process is extended to a continuous time Markov process for the volume distribution instead of the number distribution. New hypotheses are then introduced which lead to log-stable distributions as asymptotic solutions of the fragmentation equation. Log-stable laws are then used to fit experimental probability distribution function (pdf) of Simmons and Hanratty measuring drop sizes in a horizontal annular gas-liquid flow at high Weber number [Int. J. Multiphase Flow 27, 861 (2001)]]. Log-stable pdf better fits to the experimental pdf than usual empirical spray pdf and especially, because of the heavy tail of the associated stable distribution, in the small drops part of the distribution.