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Turbulent Flows in Electron Hydrodynamics: Conductivity and Vorticity

Kanad Bhattacharya

TL;DR

This work addresses turbulence in electron hydrodynamics by analyzing two geometries: a rectangular channel under an electric field and a Corbino disk under a magnetic field. It combines weak-turbulence perturbation theory and Kraichnan's turbulence spectra with Boltzmann and Navier–Stokes frameworks to derive a non-trivial $W^4$ correction to conductivity in the rectangular geometry and to characterize vorticity dynamics and edge effects in the Corbino geometry using vorticity-based formulations and Bessel-function solutions. The key contributions include a concrete $W^4$ scaling correction from both weak and strong turbulence analyses, an Orr–Sommerfeld-based perturbation spectrum, and detailed vorticity-velocity structures near boundaries under varying magnetic fields, supported by numerical illustrations. The results highlight pronounced boundary-driven behavior and suggest avenues for more rigorous turbulence modeling and extension to additional geometries and spin- and damping-informed physics.

Abstract

In this article, we attempt to understand various aspects of turbulent flows in electron hydrodynamics. We analyze a rectangular channel geometry in the presence of an electric field and a Corbino geometry in the presence of a magnetic field. In the former geometry, we analyze the conductivity of the fluid as well as the frequency spectrum of perturbations about the Poiseuille flow. While the normal Poiseuille flow has an associated conductivity which scales as $W^2$, we find a correction which scales as $W^4$ in the case of non-linear flows, where $W$ is the characteristic length of the system. In the Corbino geometry, we analyze the velocity, vorticity and magnetic fields. We find that the vorticity can span across a wide range near the edge of the geometry, a behavior that can be reflected in the velocity and magnetic fields.

Turbulent Flows in Electron Hydrodynamics: Conductivity and Vorticity

TL;DR

This work addresses turbulence in electron hydrodynamics by analyzing two geometries: a rectangular channel under an electric field and a Corbino disk under a magnetic field. It combines weak-turbulence perturbation theory and Kraichnan's turbulence spectra with Boltzmann and Navier–Stokes frameworks to derive a non-trivial correction to conductivity in the rectangular geometry and to characterize vorticity dynamics and edge effects in the Corbino geometry using vorticity-based formulations and Bessel-function solutions. The key contributions include a concrete scaling correction from both weak and strong turbulence analyses, an Orr–Sommerfeld-based perturbation spectrum, and detailed vorticity-velocity structures near boundaries under varying magnetic fields, supported by numerical illustrations. The results highlight pronounced boundary-driven behavior and suggest avenues for more rigorous turbulence modeling and extension to additional geometries and spin- and damping-informed physics.

Abstract

In this article, we attempt to understand various aspects of turbulent flows in electron hydrodynamics. We analyze a rectangular channel geometry in the presence of an electric field and a Corbino geometry in the presence of a magnetic field. In the former geometry, we analyze the conductivity of the fluid as well as the frequency spectrum of perturbations about the Poiseuille flow. While the normal Poiseuille flow has an associated conductivity which scales as , we find a correction which scales as in the case of non-linear flows, where is the characteristic length of the system. In the Corbino geometry, we analyze the velocity, vorticity and magnetic fields. We find that the vorticity can span across a wide range near the edge of the geometry, a behavior that can be reflected in the velocity and magnetic fields.

Paper Structure

This paper contains 12 sections, 29 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conductivity and Perturbation Frequencies in a Rectangular Geometry
  3. Conductivity in Weak Turbulence
  4. Conductivity in Strong Turbulence
  5. Frequency of perturbations
  6. Vorticity in Corbino Geometry
  7. In the absence of a Magnetic Field
  8. In the presence of a Magnetic Field
  9. Qualitative Behavior and Asymptotics
  10. Vorticity and Velocity
  11. Conclusions and Future Work
  12. Acknowledgments

Figures (2)

  • Figure 1: Vorticity field in a Corbino geometry with radii $r_{\mathrm{min}}=2\times 10^{-6}\ \mathrm{m}$ and $r_{\mathrm{max}}=6\times 10^{-6} \mathrm{m}$
  • Figure 2: Velocity field in a Corbino geometry with radii $r_{\mathrm{min}}=2\times 10^{-6}\ \mathrm{m}$ and $r_{\mathrm{max}}=6\times 10^{-6} \mathrm{m}$. The length of arrows represents the strength of velocity field. (The color serves as an aid for our eyes, where yellow indicates a stronger velocity while blue indicates a weaker velocity.)