Forced Fluid Dynamics from Gravity

TL;DR

This paper extends the fluid/gravity correspondence to include boundary dilaton fields and curved metrics, deriving dual solutions that obey covariant Navier-Stokes equations with forcing, and constructing the dual horizon and entropy current.

Contribution

It generalizes previous fluid/gravity duality to incorporate boundary dilaton fields and curvature, providing explicit second-order solutions and a framework for studying forced fluid flows and turbulence.

Findings

Derived dual solutions with boundary dilaton and curvature effects.

Explicit second-order stress tensor and entropy current expressions.

Confirmed results with known rotating black hole solutions.

Abstract

We generalise the computations of arXiv:0712.2456 to generate long wavelength, asymptotically locally AdS_5 solutions to the Einstein-dilaton system with a slowly varying boundary dilaton field and a weakly curved boundary metric. Upon demanding regularity, our solutions are dual, under the AdS/CFT correspondence, to arbitrary fluid flows in the boundary theory formulated on a weakly curved manifold with a prescribed slowly varying coupling constant. These solutions turn out to be parametrised by four-velocity and temperature fields that are constrained to obey the boundary covariant Navier Stokes equations with a dilaton dependent forcing term. We explicitly evaluate the stress tensor and Lagrangian as a function of the velocity, temperature, coupling constant and curvature fields, to second order in the derivative expansion and demonstrate the Weyl covariance of these expressions. We also construct the event horizon of the dual solutions to second order in the derivative expansion, and use the area form on this event horizon to construct an entropy current for the dual fluid. As a check of our constructions we expand the exactly known solutions for rotating black holes in global AdS_5 in a boundary derivative expansion and find perfect agreement with all our results upto second order. We also find other simple solutions of the forced fluid mechanics equations and discuss their bulk interpretation. Our results may aid in determining a bulk dual to forced flows exhibiting steady state turbulence.