# Stokes flow analogous to viscous electron current in graphene

## TL;DR

This paper demonstrates the analogy between viscous electron flow in graphene and viscous fluid flow in microfluidic devices, showing vortex formation and validating a 2D theoretical model for 3D Stokes flow.

## Contribution

It provides experimental evidence and numerical validation of vortex behavior in viscous flows, extending the analogy to electronic systems in graphene.

## Key findings

- Vortices observed in viscous fluid flow analogous to electronic systems.
- A threshold for vortex formation is identified.
- A 2D model accurately captures 3D Stokes flow features.

## Abstract

Electron transport in two-dimensional conducting materials such as graphene, with dominant electron-electron interaction, exhibits unusual vortex flow that leads to a nonlocal current-field relation (negative resistance), distinct from the classical Ohm's law. The transport behavior of these materials is best described by low Reynolds number hydrodynamics, where the constitutive pressure-speed relation is Stoke's law. Here we report evidence of such vortices observed in a viscous flow of Newtonian fluid in a microfluidic device consisting of a rectangular cavity$-$analogous to the electronic system. We extend our experimental observations to elliptic cavities of different eccentricities, and validate them by numerically solving bi-harmonic equation obtained for the viscous flow with no-slip boundary conditions. We verify the existence of a predicted threshold at which vortices appear. Strikingly, we find that a two-dimensional theoretical model captures the essential features of three-dimensional Stokes flow in experiments.