# A minimal model for a hydrodynamic fingering instability in microroller suspensions

## TL;DR

This paper develops a minimal continuum model to explain the hydrodynamic fingering instability in microroller suspensions, accurately capturing experimental and simulation results through a simple two-line rotlet system.

## Contribution

The paper introduces a minimal two-line rotlet continuum model that reproduces the fingering instability and its lengthscale selection driven solely by hydrodynamics.

## Key findings

- Model reproduces observed lengthscale selection
- Dispersion relation matches simulations and experiments
- Instability driven by combined advective and transverse flows

## Abstract

We derive a minimal continuum model to investigate the hydrodynamic mechanism behind the fingering instability recently discovered in a suspension of microrollers near a floor [Driscoll et al. Nature Physics, 2016]. Our model, consisting of two continuous lines of rotlets, exhibits a linear instability driven only by hydrodynamics interactions, and reproduces the lengthscale selection observed in large scale particle simulations and in experiments. By adjusting only one parameter, the distance between the two lines, our dispersion relation exhibits quantitative agreement with the simulations and qualitative agreement with experimental measurements. Our linear stability analysis indicate that this instability is caused by the combination of the advective and transverse flows generated by the microrollers near a no-slip surface. Our simple model offers an interesting formalism to characterize other hydrodynamic instabilities that have not been yet well understood, such as size scale selection in suspensions of particles sedimenting adjacent to a wall, or the recently observed formations of traveling phonons in systems of confined driven particles.