# Analytical Model Fragmentation in Creeping Flow Based on Bateman Equations

#### Abstract

The quantification of fragmentation of particles advected by a fluid in a microchannel should be useful for the design and optimization of technology for drug delivery. In this work we use the Shannon entropy to evaluate the level of fragmentation in an analytical model of eroding particles advected by a creeping flow. We assume a particle to be made of primary fragments bound together. The erosion is the breakage of a primary fragment out of a given particle. Particles are advected by laminar flow and they disperse because of the shear stresses imparted by the fluid. The time evolution of the numbers of particles of different sizes is described by a set of coupled differential equations. Those equations are mathematically equivalent to the Bateman equations that govern nuclear radioactivity. We find, by solving these differential equations, the numbers of particles of each possible size as functions of time. Using the particle size distribution we compute the entropic fragmentation index which varies from 0 for a monodisperse system to 1 for an extreme polydisperse system. The time dependence of the index exhibits a maximum at some intermediate time as the system starts monodisperse (large size particle) and evolves through a polydisperse regime at intermediate times to a monodisperse (small size particle) at late times. An analysis of the spatial dependence of the fragmentation index reveals regions of the cross section that are not conductive to fragmentation.

*This paper has been withdrawn.*