Fluid Stretching as a Levy Process

TL;DR

This paper investigates how fluid elements stretch in steady 2D random flows, revealing a spectrum of behaviors influenced by intermittent shear, modeled as a Levy process, linking flow structure to deformation statistics.

Contribution

It introduces a novel model connecting flow-induced shear and velocity fluctuations to fluid stretching, demonstrating a Levy walk behavior for elongation in broad velocity distributions.

Findings

Stretching behaviors range from sublinear to superlinear.

Stretching follows a coupled continuous time random walk.

Flow and deformation statistics are explicitly linked.

Abstract

We study the relation between flow structure and fluid deformation in steady two-dimensional random flows. Beyond the linear (shear flow) and exponential (chaotic flow) elongation paradigms, we find a broad spectrum of stretching behaviors, ranging from sub- to superlinear, which are dominated by intermittent shear events. We analyze these behaviors from first principles, which uncovers stretching as a result of the non-linear coupling between Lagrangian shear deformation and velocity fluctuations along streamlines. We derive explicit expressions for Lagrangian deformation and demonstrate that stretching obeys a coupled continous time random walk, which for broad distributions of flow velocities describes a L\'evy walk for elongation. The derived model provides a direct link between the flow and deformation statistics, and a natural way to quantify the impact of intermittent shear events on the stretching behavior, which can have strong anomalous diffusive character.