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Small droplets in turbulent flows can undergo highly variable deformationsand orientational dynamics. For neutrally buoyant droplets smaller than theKolmogorov scale, the dominant effects from the surrounding turbulent flowarise through Lagrangian time histories of the velocity gradient tensor. Herewe study the evolution of representative droplets using a model that includesrotation and stretching effects from the surrounding fluid, and restorationeffects from surface tension including a constant droplet volume constraint,while assuming that the droplets maintain an ellipsoidal shape. The model iscombined with Lagrangian time histories of the velocity gradient tensorextracted from DNS of turbulence to obtain simulated droplet evolutions. Theseare used to characterize the size, shape and orientation statistics of smalldroplets in turbulence. A critical capillary number, $Ca_c$ is identifiedassociated with unbounded growth of one or two of the droplet's semi-axes.Exploiting analogies with dynamics of polymers in turbulence, the $Ca_c$ numbercan be predicted based on the large deviation theory for the largest FiniteTime Lyapunov exponent. Also, for sub-critical $Ca$ the theory enablespredictions of the slope of the power-law tails of droplet size distributionsin turbulence. For cases when the viscosities of droplet and outer fluid differin a way that enables vorticity to decorrelate the shape from the strainingdirections, the large deviation formalism based on the stretching properties ofthe velocity gradient tensor loses validity and its predictions fail. Evenconsidering the limitations of the assumed ellipsoidal droplet shape, theresults highlight the complex coupling between droplet deformation, orientationand the local fluid velocity gradient tensor to be expected when small viscousdrops interact with turbulent flows.

Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence