Fluid Deformation in Random Unsteady Flow

Fluid deformation controls myriad processes in random flows including mixing and dispersion, stress development in complex fluids, colloid transport and deposition, droplet breakup and emulsification, fluid-structure interaction, chemical reactions and biological activity. Despite this, fundamental aspects are not well understood, including the link between the Lagrangian velocity gradient tensor $\boldsymbol\epsilon$ and deformation measures such as Lyapunov exponents ($\lambda_{\infty,i}$), their finite-time counterparts (FTLEs) and the right Cauchy-Green tensor $\mathbf{C}$. We address these knowledge gaps by developing an \emph{ab initio} stochastic model of fluid deformation in ergodic and stationary random unsteady flows. We show that although the Lagrangian velocity process is non-Markovian and non-Fickian, temporal decorrelation in unsteady random flows results in Fickian evolution of $\boldsymbol\epsilon$. Application of an objective coordinate transform renders $\boldsymbol\epsilon^\prime$ upper triangular, the basis vectors of which exponentially converge to Lyapunov vectors. As such, the diagonal components of $\boldsymbol\epsilon^\prime$ correspond to increments of the Lyapunov spectra, while the off-diagonal components objectively quantify shear and vorticity. This leads to a stochastic model of Lagrangian fluid deformation as a simple Brownian process that provides a direct link between $\boldsymbol\epsilon^\prime$ and fluid deformation. We develop closed-form expressions for the evolution of $\mathbf{C}$ and the FTLEs, and apply the stochastic model to numerical results for a model 2D unsteady flow and 3D forced homogeneous isotropic turbulence, returning excellent agreement with direct calculations of deformation measures.