Nonextensive statistics in viscous fingering

TL;DR

This paper demonstrates that viscous fingering in Hele-Shaw flow exhibits nonextensive statistical behavior, with velocity fields following $q$-distributions, linking turbulence, nonextensive mechanics, and fingering phenomena.

Contribution

It provides the first analysis of viscous fingering using nonextensive statistics, revealing $q$-Gaussian velocity profiles and power law distributions in simulated flows.

Findings

Velocity fields follow $q$-distributions.

Spatial $q$-Gaussian profiles observed.

Power law velocity distributions across measurements.

Abstract

Measurements in turbulent flows have revealed that the velocity field in nonequilibrium systems exhibits $q$-exponential or power law distributions in agreement with theoretical arguments based on nonextensive statistical mechanics. Here we consider Hele-Shaw flow as simulated by the Lattice Boltzmann method and find similar behavior from the analysis of velocity field measurements. For the transverse velocity, we obtain a spatial $q$-Gaussian profile and a power law velocity distribution over all measured decades. To explain these results, we suggest theoretical arguments based on Darcy's law combined with the non-linear advection-diffusion equation for the concentration field. Power law and $q$-exponential distributions are the signature of nonequilibrium systems with long-range interactions and/or long-time correlations, and therefore provide insight to the mechanism of the onset of fingering processes.