Stretching distributions in chaotic mixing of droplet dispersions with unequal viscosities

Abstract

The stretching behavior of droplet dispersions with viscosity different from the matrix fluid is examined in chaotic and regular flows, in the limit of zero interfacial tension. Computations use a Lagrangian particle method, with the microstructure for each particle based on an exact solution for ellipsoidal droplets in the dilute limit. Two closed, two-dimensional time-periodic flows are considered: flow between eccentric cylinders and the sine flow. In regular flows with viscosity ratio of five or greater, many droplets display oscillatory motion and never experience large stretching. The global average stretch grows linearly in a regular flow at a rate that decreases as viscosity ratio increases. In contrast, chaotic flows gradually stretch and orient high-viscosity droplets, such that the droplets asymptotically follow the stretching of the underlying flow. Consequently, for long times, droplet stretching statistics display the universal features shown by passive fluid elements in a chaotic flow: the geometric mean stretch grows exponentially at the rate of the Lyapunov exponent, and the log of the principal stretch ratio, scaled by its mean and standard deviation, settles to an invariant global probability distribution and an invariant spatial distribution. These results demonstrate that chaotic flows are highly effective at stretching microstructures that do not stretch readily in regular flows, and show that the stretching ability of a chaotic flow can be concisely described, independent of the viscosity ratio of the dispersion that is being mixed.