Coupled continuous-time random walks for fluid stretching in two-dimensional heterogeneous media

Area of Science:

• Fluid dynamics
• Geophysics
• Statistical mechanics

Background:

• Understanding fluid flow and deformation in heterogeneous media is crucial for various scientific and engineering applications.
• These media exhibit complex stretching behaviors, ranging from sub- to superlinear, influenced by flow structure.
• Identifying the underlying mechanisms driving these behaviors is an ongoing challenge.

Purpose of the Study:

• To investigate the relationship between flow structure and fluid deformation in two-dimensional heterogeneous media.
• To elucidate the origin of subexponential stretching observed in such flows.
• To develop a model that links flow characteristics to deformation statistics.

Main Methods:

• Analysis of steady flows through two-dimensional heterogeneous media.
• First-principles analysis to uncover the origins of stretching behaviors.
• Derivation of explicit expressions for Lagrangian deformation.
• Modeling stretching as a coupled continuous-time random walk, reducing to a Lévy walk for broad velocity distributions.

Main Results:

• Intermittent shear events are identified as the cause of subexponential stretching.
• Explicit expressions for Lagrangian deformation were derived.
• The derived model demonstrates that stretching follows a coupled continuous-time random walk.
• This model simplifies to a Lévy walk for broad distributions of flow velocities.

Conclusions:

• A direct link between flow and deformation statistics in heterogeneous media has been established.
• The impact of intermittent shear events on stretching behavior can be quantified using the derived model.
• The study provides a theoretical framework for understanding complex fluid deformation in heterogeneous environments.