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.

Evaluation of Mixing Performance in Baffled Screw Channel Using Lagrangian Particle Calculations

“…Except for case F, tracer s 1 is folded and stretched when wandering around whole screw channel. From top and front views, we could draw the conclusion that the mixing patterns were distinct from the previous research results because of the very large geometric ratio we used . With the increase of the geometric ratio, tracer covered less z distance.…”

“…After that, numerous studies focused on investigating the mechanism and quantification of chaotic mixing in different models and on finding ways to achieve more efficient chaotic mixing in engineering application . To understand chaotic mixing, various techniques from mathematics and physics, such as Poincaré maps, periodic point analysis, tracer stretching distributions, and so on, are used. Along with the significant advances in chaotic mixing in two‐dimensional periodic flows, the perturbative theory of mixing for three‐dimensional steady flows was initiated by Feingold et al and Mezić and Wiggins and some applications were found .…”

A numerical model for the development of the morphology of disperse blends in complex flow

“…Other models by Muzzio et al (1992) and Florek and Tucker (2005) are excellent for describing the stretching of the droplet morphology in globally chaotic flows. However, these models neglect interfacial tension and thereby omit breakup and coalescence events.…”