Mixed Lagrangian–Eulerian description of vortical flows for ideal and viscous fluids

Abstract

It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid is equivalent to the equations of motion of a charged compressible fluid moving due to a self-consistent electromagnetic field. The velocity of new auxiliary fluid coincides with the velocity component normal to the vorticity line for the primitive equations. Therefore this new hydrodynamics represents hydrodynamics of vortex lines. Their compressibility reveals a new mechanism for three-dimensional incompressible vortical flows connected with breaking (or overturning) of vortex lines which can be considered as one of the variants of collapses. Transition to the Lagrangian description in the new hydrodynamics corresponds, for the original Euler equations, to a mixed Lagrangian–Eulerian description – the vortex line representation (VLR). The Jacobian of this mapping defines the density of vortex lines. It is shown also that application of VLR to the Navier–Stokes equations results in an equation of diffusive type for the Cauchy invariant. The diffusion tensor for this equation is defined by the VLR metric.