On the Cauchy problem for axi-symmetric vortex rings
نویسنده
چکیده
We consider the classical Cauchy problem for the 3d Navier-Stokes equation with the initial vorticity ω0 concentrated on a circle, or more generally, a linear combination of such data for circles with common axis of symmetry. We show that natural approximations of the problem obtained by smoothing the initial data satisfy good a-priori estimates which enable us to conclude that the original problem with the singular initial distribution of vorticity has a solution. We impose no restriction on the size of the initial data.
منابع مشابه
Axi-Symmetric Deformation Due to Various Sources in Saturated Porous Media with Incompressible Fluid
The general solution of equations of saturated porous media with incompressible fluid for two dimensional axi-symmetric problem is obtained in the transformed domain. The Laplace and Hankel transforms have been used to investigate the problem. As an application of the approach concentrated source and source over circular region have been taken to show the utility of the approach. The transforme...
متن کاملThe Collapse of an Axi-symmetric, Swirling Vortex Sheet
An axi-symmetric and swirling vortex sheet is investigated as the simplest ow in which there is non-trivial vortex stretching and as a possible setting for studying vortex cancellation and singularity formation. Rayleigh's criterion indicates linear stability of a single sheet but instability for other conngurations of sheets. Due to the simplicity of vortex sheet problems, the linear modes and...
متن کاملAnalytical Modeling of Axi-Symmetric Sheet Metal Forming
The cup drawing is a basic deep drawing process. Thus, understanding the mechanics ofthe cup drawing process helps in determining the general parameters that affect the deep drawingprocess. There are mainly two methods of analysis; experimental and analytical/numerical.Experimental analysis can be useful in analyzing the process to determine the process parameters thatproduce a defect free prod...
متن کاملZero Modes of Rotationally Symmetric Generalized Vor- tices and Vortex Scattering
Zero modes of rotationally symmetric vortices in a hierarchy of generalized Abelian Higgs models are studied. Under the finite-energy and the smoothness condition, it is shown, that in all models, n self-dual vortices superimposed at the origin have 2n modes. The relevance of these modes for vortex scattering is discussed, first in the context of the slow-motion approximation. Then a correspond...
متن کاملApplication of Adaptive Quadrature to Axi-symmetric Vortex Sheet Motion
Studies of the formation of ne structures on free-surfaces in liquids, such as curvature singularities or interface pinching, demand that the motion of the interface must be computed very accurately. Boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one. On the other hand, the boundary integrals are singular, and their accurate...
متن کامل