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A numerical simulation is developed to study the time-dependent shapes of droplets in chaotic mixing, as a function of interfacial tension and droplet-to-matrix viscosity ratio. The two-dimensional, time-periodic Newtonian flow between eccentric cylinders is used as a prototype mixing flow. The microstructure is modeled as three-dimensional ellipsoidal droplets, ignoring breakup and coalescence. A Lagrangian particle method is used to follow the microstructure. When interfacial tension is small (global capillary number is large), the major axes of the droplets exhibit the same stretching statistics as passive fluid elements and droplets with zero interfacial tension in chaotic flows: the geometric average of the stretch ratio grows exponentially with time, at a rate equal to the Lyapunov exponent of the flow, while the log of the major-axis stretch of the droplets, when scaled by its instantaneous mean and standard deviation, has a time-invariant, Gaussian global distribution and a non-uniform, fractal, and time-invariant spatial distribution. In this regime the stretch of the longest droplet axis is insensitive to interfacial tension, but the shape of the cross section is very sensitive: initially spherical droplets deform first into ribbons or sheets, but eventually transform into axisymmetric threads. The larger the global capillary number, the longer the sheet morphology persists, but sheet-like structures are always transient, and the sheets relax to threads if mixing goes on too long.

Stretch and Shape Distributions of Droplets with Interfacial Tension in Chaotic Mixing