Vortex line representation for flows of ideal and viscous fluids

Abstract

It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid coincides with the equations of motion of a charged {\it compressible} fluid moving due to a self-consistent electromagnetic field. Transition to the Lagrangian description in a new hydrodynamics is equivalent for the original Euler equations to the mixed Lagrangian-Eulerian description - the vortex line representation (VLR). Due to compressibility of a "new" fluid the collapse of vortex lines can happen as the result of breaking (or overturning) of vortex lines. It is found that the Navier-Stokes equation in the vortex line representation can be reduced to the equation of the diffusive type for the Cauchy invariant with the diffusion tensor given by the metric of the VLR.