Pulsating flow and boundary layers in viscous electronic hydrodynamics

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 an experimental realization. The response depends on viscosity $\ensuremath{\nu}$, driving frequency $\ensuremath{\omega}$, and ohmic heating coefficient $\ensuremath{\gamma}$ via the dimensionless complex variable $\frac{{L}^{2}}{\ensuremath{\nu}}(i\ensuremath{\omega}+\ensuremath{\gamma})=i\mathrm{\ensuremath{\Omega}}+\mathrm{\ensuremath{\Sigma}}$. While at small $\mathrm{\ensuremath{\Omega}}$, we recover the static solution, a different regime appears at large $\mathrm{\ensuremath{\Omega}}$ with the emergence of a boundary layer. This includes a splitting of the location of maximal flow velocity from the center towards the edges of the boundary layer, 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.