# Polymer stress growth in viscoelastic fluids in oscillating extensional flows with applications to micro-organism locomotion

## TL;DR

This paper investigates how polymer stresses grow in oscillating extensional flows relevant to microorganism swimming, revealing a transition from linear to exponential stress growth depending on Deborah and Weissenberg numbers, with implications for biological locomotion.

## Contribution

The study combines analytical and numerical methods to identify stress growth regimes and explains stress concentration phenomena in viscoelastic fluids during oscillatory flows.

## Key findings

- Stress transitions from linear to exponential growth at a Deborah number-dependent Weissenberg number.
- Numerical simulations confirm theoretical stress behavior in oscillating flow geometries.
- High amplitude and Deborah number flows lead to asymmetric trajectories and significant stress growth.

## Abstract

Simulations of undulatory swimming in viscoelastic fluids with large amplitude gaits show concentration of polymer elastic stress at the tips of the swimmers.We use a series of related theoretical investigations to probe the origin of these concentrated stresses. First the polymer stress is computed analytically at a given oscillating extensional stagnation point in a viscoelastic fluid. The analysis identifies a Deborah number (De) dependent Weissenberg number (Wi) transition below which the stress is linear in Wi, and above which the stress grows exponentially in Wi. Next, stress and velocity are found from numerical simulations in an oscillating 4-roll mill geometry. The stress from these simulations is compared with the theoretical calculation of stress in the decoupled (given flow) case, and similar stress behavior is observed. The flow around tips of a time-reversible flexing filament in a viscoelastic fluid is shown to exhibit an oscillating extension along particle trajectories, and the stress response exhibits similar transitions. However in the high amplitude, high De regime the stress feedback on the flow leads to non time-reversible particle trajectories that experience asymmetric stretching and compression, and the stress grows more significantly in this regime. These results help explain past observations of large stress concentration for large amplitude swimmers and non-monotonic dependence on De of swimming speeds.