Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity

Sayantani Bhattacharyya, Shiraz Minwalla, Spenta R. Wadia

TL;DR

The paper shows that incompressible non-relativistic Navier–Stokes equations arise as a universal scaling limit of relativistic hydrodynamics, with metric fluctuations acting as a forcing akin to a background EM field. It demonstrates that spatial special conformal transformations descend to accelerated-boost NS symmetries, and it derives a gravity dual for any NS solution via a scaling of asymptotically AdS black-brane geometries. It also presents a simple forced steady-state shear flow whose nonlinearities lead to instabilities at high Reynolds numbers, suggesting a dual route to turbulence and offering a holographic perspective on turbulent dynamics. The results collectively illuminate how gravity holography encodes nonrelativistic fluid dynamics and turbulent phenomena.

Abstract

We note that the equations of relativistic hydrodynamics reduce to the incompressible Navier-Stokes equations in a particular scaling limit. In this limit boundary metric fluctuations of the underlying relativistic system turn into a forcing function identical to the action of a background electromagnetic field on the effectively charged fluid. We demonstrate that special conformal symmetries of the parent relativistic theory descend to `accelerated boost' symmetries of the Navier-Stokes equations, uncovering a possibly new conformal symmetry structure of these equations. Applying our scaling limit to holographically induced fluid dynamics, we find gravity dual descriptions of an arbitrary solution of the forced non-relativistic incompressible Navier-Stokes equations. In the holographic context we also find a simple forced steady state shear solution to the Navier-Stokes equations, and demonstrate that this solution turns unstable at high enough Reynolds numbers, indicating a possible eventual transition to turbulence.

The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity

TL;DR

The paper shows that incompressible non-relativistic Navier–Stokes equations arise as a universal scaling limit of relativistic hydrodynamics, with metric fluctuations acting as a forcing akin to a background EM field. It demonstrates that spatial special conformal transformations descend to accelerated-boost NS symmetries, and it derives a gravity dual for any NS solution via a scaling of asymptotically AdS black-brane geometries. It also presents a simple forced steady-state shear flow whose nonlinearities lead to instabilities at high Reynolds numbers, suggesting a dual route to turbulence and offering a holographic perspective on turbulent dynamics. The results collectively illuminate how gravity holography encodes nonrelativistic fluid dynamics and turbulent phenomena.

Abstract

We note that the equations of relativistic hydrodynamics reduce to the incompressible Navier-Stokes equations in a particular scaling limit. In this limit boundary metric fluctuations of the underlying relativistic system turn into a forcing function identical to the action of a background electromagnetic field on the effectively charged fluid. We demonstrate that special conformal symmetries of the parent relativistic theory descend to `accelerated boost' symmetries of the Navier-Stokes equations, uncovering a possibly new conformal symmetry structure of these equations. Applying our scaling limit to holographically induced fluid dynamics, we find gravity dual descriptions of an arbitrary solution of the forced non-relativistic incompressible Navier-Stokes equations. In the holographic context we also find a simple forced steady state shear solution to the Navier-Stokes equations, and demonstrate that this solution turns unstable at high enough Reynolds numbers, indicating a possible eventual transition to turbulence.

Paper Structure

This paper contains 20 sections, 55 equations, 3 figures.

Table of Contents

  1. Introduction and Discussion
  2. Scaling the Navier-Stokes Equation
  3. The Navier-Stoke Equation as a universal limit of fluid dynamics
  4. A scaling limit
  5. Charged Fluids
  6. Reduction of Cauchy Data
  7. Symmetries
  8. Dilatations
  9. Special Conformal Transformations
  10. The Full Symmetry Group
  11. The gravity solution dual to fluid dynamics in the scaling regime
  12. The dual bulk metric
  13. Comments on the bulk metric
  14. Simple Steady State Shear flows at arbitrary Reynolds Number
  15. Stability Analysis
  16. ...and 5 more sections

Figures (3)

  • Figure 1: The imaginary part of the frequency $\gamma$ plotted against $a$, for $\kappa=0$, $c=1$ and the root with $K=1$. Note that $Im (\gamma)=2$ at $b=0$ and decreases monotonically as $|b|$ is increased.
  • Figure 2: The real part of the frequency $\gamma$ plotted against $a$, for $\kappa=0$, $c=1$ and the root with $K=1$. Note that $Re (\gamma)=-1$ at $b=0$ and decreases monotonically as $|b|$ is increased.
  • Figure 3: The curve that separates the region of stability (below) from instability (above) of the eigenvalue at $K=1$, plotted as a function of the Reynolds number on the $y$ axis versus $b$ on the $x$ axis. We have used $\kappa=0$, $c=1$ in the calculations that generated this plot.