Viscous flow through a finite-width slit: Boundary conditions and dissipation

TL;DR

This paper investigates viscous electronic flow through a finite slit, revealing that the velocity profile adheres to a partial-slip boundary condition regardless of slip length, and explores temperature distributions under different heat conductivities.

Contribution

It demonstrates that the unique viscous flow solution satisfies both no-stress and no-slip boundary conditions, leading to a universal partial-slip boundary behavior in this geometry.

Findings

Velocity profile obeys partial-slip boundary condition for all slip lengths.

Temperature distribution becomes asymmetric with low heat conductivity.

Dissipation and resistance are purely viscosity-induced in this setup.

Abstract

We study the hydrodynamic viscous electronic transport in a two-dimensional sample separated into two semi-infinite planes by a one-dimensional infinite barrier. The semi-infinite planes are electrically connected via the finite-size slit in the barrier. We calculate the current through the slit assuming finite voltage drop between the planes and neglecting disorder-induced Ohmic resistance, so that dissipation and resistance are purely viscosity-induced. We find that the only solution to the Stokes equation in this geometry, which yields a finite dissipation at finite resistance (and, hence, is not self-contradictory), is the one that fulfills both the no-stress and no-slip boundary conditions simultaneously. As a remarkable consequence, the obtained velocity profile satisfies the so-called "partial-slip" (Maxwell) boundary condition for any value of the slip length, which drops out from all final results. We also calculate the electronic temperature profile for the small and large heat conductivity, and find asymmetric (with respect to the barrier) temperature patterns in the former case.