Pulsating flow and boundary layers in viscous electronic hydrodynamics

TL;DR

This paper analyzes how oscillating electric fields influence viscous electron flows in a 2D channel, revealing a boundary layer regime with distinct flow and response characteristics, relevant for experimental observation in materials like graphene.

Contribution

It provides an analytical study of oscillating viscous electron flow in a simple geometry, identifying a new boundary layer regime at high frequencies and its impact on flow and conductance.

Findings

Emergence of a boundary layer at high oscillation frequencies

Shift of maximum flow velocity towards the channel edges

Frequency dependence of conductance resembles Drude behavior

Abstract

Motivated by experiments on a hydrodynamic regime in electron transport, we study the effect of an oscillating electric field in such a setting. We consider a long two-dimensional channel of width $L$, whose geometrical simplicity allows an analytical study as well as hopefully permitting experimental realisation. The response depends on viscosity $\nu$, driving frequency, $\omega$ and ohmic heating coefficient $\gamma$ via the dimensionless complex variable $\frac{L^2}{\nu}(i\omega +\gamma)=i\Omega +\Sigma$. While at small $\Omega$, we recover the static solution, a new regime appears at large $\Omega$ with the emergence of a boundary layer. This includes a splitting of the location of maximal flow velocity from the centre towards the edges of the boundary layer, an an increasingly reactive nature of the response, with the phase shift of the response varying across the channel. The scaling of the total optical conductance with $L$ differs between the two regimes, while its frequency dependence resembles a Drude form throughout, even in the complete absence of ohmic heating, against which, at the same time, our results are stable. Current estimates for transport coefficients in graphene and delafossites suggest that the boundary layer regime should be experimentally accessible.