988 research outputs found
Wilsonian Approach to Fluid/Gravity Duality
The problem of gravitational fluctuations confined inside a finite cutoff at
radius outside the horizon in a general class of black hole geometries
is considered. Consistent boundary conditions at both the cutoff surface and
the horizon are found and the resulting modes analyzed. For general cutoff
the dispersion relation is shown at long wavelengths to be that of a
linearized Navier-Stokes fluid living on the cutoff surface. A cutoff-dependent
line-integral formula for the diffusion constant is derived. The
dependence on is interpreted as renormalization group (RG) flow in the
fluid. Taking the cutoff to infinity in an asymptotically AdS context, the
formula for reproduces as a special case well-known results derived
using AdS/CFT. Taking the cutoff to the horizon, the effective speed of sound
goes to infinity, the fluid becomes incompressible and the Navier-Stokes
dispersion relation becomes exact. The resulting universal formula for the
diffusion constant reproduces old results from the membrane
paradigm. Hence the old membrane paradigm results and new AdS/CFT results are
related by RG flow. RG flow-invariance of the viscosity to entropy ratio is shown to follow from the first law of thermodynamics together with
isentropy of radial evolution in classical gravity. The ratio is expected to
run when quantum gravitational corrections are included.Comment: 34 pages, harvmac, clarified boundary conditio
Particles and fields in fluid turbulence
The understanding of fluid turbulence has considerably progressed in recent
years. The application of the methods of statistical mechanics to the
description of the motion of fluid particles, i.e. to the Lagrangian dynamics,
has led to a new quantitative theory of intermittency in turbulent transport.
The first analytical description of anomalous scaling laws in turbulence has
been obtained. The underlying physical mechanism reveals the role of
statistical integrals of motion in non-equilibrium systems. For turbulent
transport, the statistical conservation laws are hidden in the evolution of
groups of fluid particles and arise from the competition between the expansion
of a group and the change of its geometry. By breaking the scale-invariance
symmetry, the statistically conserved quantities lead to the observed anomalous
scaling of transported fields. Lagrangian methods also shed new light on some
practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy
NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID AROUND A ROTATING AND TRANSLATING RIGID BODY
We consider the linearized and nonlinear systems describing the motion of incompressible flow around a rotating and translating rigid body Ɗ in the exterior domain Ω = ℝ3 \ D , where Ɗ ⊂ ℝ3 is open and bounded, with Lipschitz boundary. We derive the L∞-estimates for the pressure and investigate the leading term for the velocity and its gradient. Moreover, we show that the velocity essentially behaves near the infinity as a constant times the first column of the fundamental solution of the Oseen system. Finally, we consider the Oseen problem in a bounded domain ΩR := BR ∩ Ω under certain artificial boundary conditions on the truncating boundary ∂BR, and then we compare this solution with the solution in the exterior domain Ω to get the truncation error estimate
Global stability of vortex solutions of the two-dimensional Navier-Stokes equation
Both experimental and numerical studies of fluid motion indicate that
initially localized regions of vorticity tend to evolve into isolated vortices
and that these vortices then serve as organizing centers for the flow. In this
paper we prove that in two dimensions localized regions of vorticity do evolve
toward a vortex. More precisely we prove that any solution of the
two-dimensional Navier-Stokes equation whose initial vorticity distribution is
integrable converges to an explicit self-similar solution called ``Oseen's
vortex''. This implies that the Oseen vortices are dynamically stable for all
values of the circulation Reynolds number, and our approach also shows that
these vortices are the only solutions of the two-dimensional Navier-Stokes
equation with a Dirac mass as initial vorticity. Finally, under slightly
stronger assumptions on the vorticity distribution, we also give precise
estimates on the rate of convergence toward the vortex.Comment: 35 pages, no figur
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