  Small droplets in turbulent flows can undergo highly variable deformations
and orientational dynamics. For neutrally buoyant droplets smaller than the
Kolmogorov scale, the dominant effects from the surrounding turbulent flow
arise through Lagrangian time histories of the velocity gradient tensor. Here
we study the evolution of representative droplets using a model that includes
rotation and stretching effects from the surrounding fluid, and restoration
effects from surface tension including a constant droplet volume constraint,
while assuming that the droplets maintain an ellipsoidal shape. The model is
combined with Lagrangian time histories of the velocity gradient tensor
extracted from DNS of turbulence to obtain simulated droplet evolutions. These
are used to characterize the size, shape and orientation statistics of small
droplets in turbulence. A critical capillary number, $Ca_c$ is identified
associated with unbounded growth of one or two of the droplet's semi-axes.
Exploiting analogies with dynamics of polymers in turbulence, the $Ca_c$ number
can be predicted based on the large deviation theory for the largest Finite
Time Lyapunov exponent. Also, for sub-critical $Ca$ the theory enables
predictions of the slope of the power-law tails of droplet size distributions
in turbulence. For cases when the viscosities of droplet and outer fluid differ
in a way that enables vorticity to decorrelate the shape from the straining
directions, the large deviation formalism based on the stretching properties of
the velocity gradient tensor loses validity and its predictions fail. Even
considering the limitations of the assumed ellipsoidal droplet shape, the
results highlight the complex coupling between droplet deformation, orientation
and the local fluid velocity gradient tensor to be expected when small viscous
drops interact with turbulent flows.
