(474a) A Simplified Model for the Evolution of the Drop Size Distribution and Solute Extraction In An Emulsion Under Strong-Flow Conditions

A theoretical study is presented on the evolution of a drop size distribution and solute extraction in an emulsion under strong flow conditions where viscous stresses are comparable to or larger than the capillary stresses that maintain spherical drop shapes.  Under these conditions, drop breakup dominates drop coalescence.  Previous experiments and numerical simulations indicate that the size of daughter drops produced by breakup in a particular fluctuation scale with the critical size drop for the fluctuation; the volume of the parent drop only determines the number of daughter drops produced by a breakup event.  A simplified population balance model predicated on this observation is presented. The essential simplification of our model is the reduction of the usual two-variable distribution function which describes the daughter drops produced by the breakup of a parent drop to a single-variable function that describes the rescaled distribution of daughters produced by a breakup event. 

Numerical and analytical solutions for the evolution of the drop size distribution are obtained.  The results are sensitive to the daughter drop distribution function at short times. At long times, the drop size distribution attains a self-similar form described by an integral equation.  The long-time self-similar drop size distribution is insensitive to the daughter drop distribution function but sensitive to the power-law exponent of the breakup rate.  An extension of the simplified population balance model is described for solute extraction in an emulsion under strong-flow conditions and numerical results are presented.