# Lyapunov dimension of elastic turbulence

## TL;DR

This paper estimates the Lyapunov dimension of the attractor in the 2D Oldroyd-B model to understand elastic turbulence in polymer solutions at low Reynolds numbers, combining mathematical analysis with numerical simulations.

## Contribution

It provides a novel estimate of the Lyapunov dimension for the Oldroyd-B model's attractor, linking elastic turbulence to dynamical systems theory.

## Key findings

- Lyapunov dimension increases with Weissenberg number
- Bounded quantities support the existence of an attractor
- Numerical simulations validate the analytical estimates

## Abstract

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as 'elastic turbulence' when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics requires an assumption that an attractor of the Oldroyd-B model exists : numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations.

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Source: https://tomesphere.com/paper/1701.02366