# Interaction of Vortices in Weakly Viscous Planar Flows

@article{Gallay2011InteractionOV, title={Interaction of Vortices in Weakly Viscous Planar Flows}, author={Thierry Gallay}, journal={Archive for Rational Mechanics and Analysis}, year={2011}, volume={200}, pages={445-490} }

We consider the inviscid limit for the two-dimensional incompressible Navier–Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter ν, and we choose a time T > 0 such that the Helmholtz–Kirchhoff point vortex system is well-posed on the interval [0, T]. Under these assumptions, we prove that the solution of the Navier–Stokes…

## 42 Citations

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## References

SHOWING 1-10 OF 60 REFERENCES

Interacting vortex pairs in inviscid and viscous planar flows

- Mathematics, Physics
- 2012

The aim of this contribution is to make a connection between two recent results concerning the dynamics of vortices in incompressible planar flows. The first one is an asymptotic expansion, in the…

Generalized Helmholtz-Kirchhoff Model for Two-Dimensional Distributed Vortex Motion

- PhysicsSIAM J. Appl. Dyn. Syst.
- 2009

The two-dimensional Navier–Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations and it is proved that the convergence of this expansion is convergence and the Helmholtz–Kirchhoff model for the evolution of point vortices is recovered.

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

- Physics, Mathematics
- 2005

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…

Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit

- Mathematics
- 2011

We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier–Stokes equations with an axisymmetric background…

Stretched vortices – the sinews of turbulence; large-Reynolds-number asymptotics

- PhysicsJournal of Fluid Mechanics
- 1994

A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an…

Viscous interactions of two co-rotating vortices before merging

- PhysicsJournal of Fluid Mechanics
- 2002

The viscous evolution of two co-rotating vortices is analysed using direct two-dimensional numerical simulations of the Navier–Stokes equations. The article focuses on vortex interaction regimes…

Motion and Decay of a Vortex in a Nonuniform Stream

- Physics
- 1965

The motion of a vortex in a two‐dimensional incompressible nonuniform stream is studied by including the viscous effects in the inner core of the vortex. A systematic procedure is presented by the…

Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity

- Mathematics
- 2004

Abstract.We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L1(R2) for positive times is entirely determined by the trace of…

Uniqueness Theorem for the Basic Nonstationary Problem in the Dynamics of an Ideal Incompressible Fluid

- Mathematics
- 1995

A bstract . The initial boundary value problem is considered for the Euler equations for an incompressible fluid in a bounded domain D ⊂ Rn. It is known [Y1] that uniqueness holds for those flows…

The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in ₃

- Mathematics
- 1971

It is shown here that a unique solution to the Navier-Stokes equations exists in R3 for a small time interval independent of the viscosity and that the solutions for varying viscosities converge…