Vortex methods in two-dimensional fluid dynamics
type: Item
instance of: scholarly article
title: Vortex methods in two-dimensional fluid dynamics
zbMATH Open document ID: 0545.76027
published in: Lecture Notes in Physics
publication date: 1984-01-01T00:00:00Z
review text: The authors discuss some recent results on vortex methods for the time evolution of planar incompressible fluids. Their analysis is macroscopic and is mostly based on a well-known and old idea: a two-dimensional incompressible ideal fluid behaves like a Hamiltonian system. The distribution of vorticity can be envisaged as formed by ''particles'', infinitesimal vortices, whose motion is described by Hamiltonian ordinary differential equations. Such dynamical systems are studied which yield solutions of the Euler equations. When the vorticity is concentrated on points, the evolution of the fluid reduces to a Hamiltonian system with a finite number of degrees of freedom, called the vortex model. Its relation to Euler flows under suitable limits is investigated here (sections 1-5). In the presence of viscosity a vortex dynamics can still be considered, the vortex motion being described by stochastic ordinary differential equations. This entails diffusion and delocalization of the vortices. The present point of view provides a good understanding of Navier-Stokes evolution in the absence of physical boundaries (sections 6-8). The physically most interesting case, in which obstacles occur in the fluid, is discussed in section 9, via the Chorin method. This case is more involved and has not been completely understood as yet.
Mathematics Subject Classification ID: 76B47
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76-01
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 3868839
zbMATH Keywords: vortex methods
zbMATH Keywords: time evolution of planar incompressible fluids
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: distribution of vorticity
zbMATH Keywords: Euler equations
zbMATH Keywords: finite number of degrees of freedom
zbMATH Keywords: diffusion and delocalization
zbMATH Keywords: obstacles
zbMATH Keywords: Chorin method
author: Carlo Marchioro
author: Mario Pulvirenti
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3868839