Vortices in Ideal Fluids

Abstract

The observation that vorticity is of great importance in nature can be traced back hundreds of years, beginning at least to Leonardo da Vinci and Descartes. For any three-dimensional vector field v we know the fundamental fact that that provided the vector field v vanishes outside a compact set in R³, then the vector field v decomposes into a solenoidal part and a gradient part. This fact is sometimes called “the fundamental theorem of advanced calculus” The solenoidal part of v, sometimes denoted curl v, is called vorticity when the vector field in question is the velocity vector of a fluid. This means, in usual discussions, assuming the velocity is zero at infinity, that the vorticity can be distinguished from the irrotational part of the fluid by a rotational movement. In symbols we have

• v = velocity vector of a fluid in ℜ³, of compact support

• curl v = vorticity of the fluid in ℜ³

• grad v = irrotational part of v

• v = curl v + grad v