Mixing and dispersion of immiscible fluids: Stretching and breakup in chaotic flows

We investigate the stretching and breakup of a drop freely suspended in a viscous fluid undergoing chaotic advection. Droplets stretch into filaments acted on by a complex flow history leading to exponential length increase, folding, and eventual breakup; following breakup, chaotic stirring disperses the fragments throughout the flow. These events are studied by experiments conducted in a time-periodic two-dimensional low Reynolds number chaotic flow. Studies are restricted to viscosity ratios p such that 0.01 $\u3c$ p $\u3c$ 2.8. The experimental results are highly reproducible and illustrate new qualitative aspects with respect to the case of stretching and breakup in linear flows. For example, breakup near folds is associated with a change of sign in stretching rate; this mode of breakup leads to the formation of rather large drops. The dominant breakup mechanism, however, is capillary wave instabilities in highly stretched filaments. Other modes of breakup, such as \u27necking\u27, \u27end-pinching\u27, and \u27fold-pinching\u27 occur as well. We find that drops in low-viscosity-ratio systems, p $\u3c$ 1, extend relatively little, O(10$\sp1$-10$\sp2$), before they break, producing an array of uniformly spaced drops (also known as mother drops) with as many as 19 satellites and sub-satellites in between. Large drop fragments, typically the product of end-pinching and fold-pinching, may again undergo a succession of breakup events, usually about 2-4 times. Drops in systems with p $\u3e$ 1, on the other hand, stretch substantially, O(10$\sp1$-10$\sp4$), before they break producing small fragments that rarely break again. The number of satellites in between two mother drops in these systems never exceeds 5. The experimental results are interpreted in terms of a simple model assuming that moderately extended filaments behave passively; this is an excellent approximation especially for low-viscosity-ratio drops, p $\u3c$ 1. The dynamics of the disturbances on the surface of the extending filament, and consequently the sizes of the resulting mother drops, are computed by means of linear stability theory. The linear stability theory, however, fails to capture the evolution of the undulated filament once the large curvature around the neck region develops. Thus, we resort to a different technique, i.e., boundary integral method, to obtain the exact pinch-off location and simulate the subsequent fluid motions which lead to the formation of satellite drops. Data from 2000-3000 thousands of droplets measured in each experiment after a long time indicate that the mean drop size decreases as the viscosity ratio increases. The repetitive nature of stretching and folding, as well as the self-repeating nature of the breakup process, suggests self-similarity. We find that, indeed, upon scaling, the drop size distributions corresponding to different viscosity ratios can be collapsed into a master curve