Lagrangian description of three-dimensional viscous flows at large Reynolds numbers

Boundary layer theory is used to show that, at large Reynolds numbers, the three-dimensional Navier–Stokes equations can be rewritten in a form with diffusion velocity that was previously known for the cases of two-dimensional and axisymmetric flows. Relying on this hypothesis, a closed system of equations that is a development of a similar model for the indicated special cases is derived to describe fluid flows in the Lagrangian approach. Simultaneously, a number of mathematical issues are investigated. The existence of an integral representation for the velocity field with integrals with respect to Lagrangian coordinates is proved by analyzing the equations of motion of selected Lagrangian particles and applying the theory of ordinary differential equations with parameters. An equation describing the vorticity flux from the body surface is derived.