The incompressible non-relativistic Navier-Stokes equation from gravity

By Sayantani Bhattacharyya, Shiraz Minwalla, S. Wadia. 2008

183 citations recorded by the source corpus.

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 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a1b001ed5e3c3a9d988ede6d5884b9639afc3f9c

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Citations in the corpus

Wilsonian approach to fluid/gravity duality (2010, background): “13) in which velocities scale as v → ǫv → 0 (see [23] for a nice discussion).”
A Holographic View on Physics out of Equilibrium (2010, background): “We argue that the correspondence is indeed relevant and worth vigorously pursuing further (see [148, 158, 223, 225] for thoughts on these issues).”
Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes (2013, methodology): “incompressible Navier-Stokes equation at low velocities [65], this connects directly to classic results on non-relativistic two dimensional turbulence.”
Weak Field Black Hole Formation in Asymptotically AdS Spacetimes (2009, methodology): “responding spacetime is locally (tube wise) well described by a black brane metric with a value of the temprature that varies with ~x(see (5.1) and the discusson arounf it in section 5). According to [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], the subsequent evolution of”
Gravity and hydrodynamics: lectures on the fluid-gravity correspondence (2009, methodology): “In fact, this limit was discussed recently in the context of the fluid-gravity correspondence in [81] (see also [115, 116] for closely related results), where the authors showed that starting from a parent relativistic conformal fluid dynamical system one can recover incompressible Navier–Stokes equ”
The holographic fluid dual to vacuum Einstein gravity (2011, background): “In a recent paper [6] (see [7–14] for earlier relevant work), a remarkable relation between incompressible non-relativistic fluids in (d+1) dimensions satisfying the Navier-Stokes equation and (d + 2)-dimensional Ricci flat metrics was found.”
From Navier-Stokes to Einstein (2011, methodology): “4) in [1] in which the cosmological constant is taken to zero and the boundary can be pushed to any radius - in particular to the interesting near-horizon region.”
Galilean Conformal Algebras and AdS/CFT (2009, methodology): “The symmetries of this equation were then studied by [41] (see also [42]).”
DE SITTER MUSINGS (2012, methodology): “The problem mimics similar analyses in the context of the near horizon regions of black hole geometries [45, 46, 47, 48, 49, 50] beginning with the seminal work of Damour [51].”