Fluid Deformation in Random Unsteady Flow

Fluid deformation controls myriad processes including mixing and dispersion, development of stress in complex fluids, droplet breakup and emulsification, fluid-structure interaction, chemical reactions and biological activity. We develop a simple stochastic model for fluid deformation in random unsteady flows such as homogeneous isotropic turbulence. We show that although the Lagrangian velocity process is non-Markovian and non-Fickian, temporal decorrelation due to the unsteady nature of the flow renders the evolution of the Lagrangian velocity gradient tensor to be Fickian. Application of a coordinate transform renders the velocity gradient tensor upper triangular, eliminating vortical rotation and decoupling principal stretches from shear deformations, leading to a stochastic model of fluid deformation as a simple Brownian process. We develop closed-form expressions for the evolution of the Cauchy-Green tensor and show that the finite-time Lyapunov exponents are Gaussian distribution. Application of this model to DNS calculations of forced isotropic turbulence at Taylor-scale Reynolds number Re_λ≈ 433, confirms the underlying model assumptions and provides excellent agreement with theoretical results.