# The relationship between viscoelasticity and elasticity

## TL;DR

This paper explores the connection between viscoelastic and elastic behaviors in polymer solutions, clarifying model relationships, energy dynamics, and implications for elastic thread stability during capillary instability.

## Contribution

It introduces a detailed analysis of the kinematics of elastic liquids, linking Lagrangian and Eulerian descriptions, and provides new insights into energy splitting and stability analysis.

## Key findings

- Relationship between viscoelasticity and elasticity clarified
- Energy decomposition into reversible and dissipative parts established
- Numerical schemes demonstrate elastic thread behavior during capillary instability

## Abstract

We consider models for elastic liquids, such as solutions of flexible polymers. They introduce a relaxation time $\lambda$ into the system, over which stresses relax. We study the kinematics of the problem, and clarify the relationship between Lagrangian and Eulerian descriptions, thereby showing which polymer models correspond to a nonlinear elastic deformation in the limit $\lambda\rightarrow\infty$. This allows us to split the change in elastic energy into reversible and dissipative parts, and thus to write an equation for the total energy, the sum of kinetic and elastic energies. As an illustration, we show how the presence or absence of an elastic limit determines the fate of an elastic thread during capillary instability, using novel numerical schemes based on our insights into the flow kinematics.