Nonlocal conservation law in a submerged jet

Landau was the first to obtain the exact solution of Navier-Stokes equations for an axisymmetric submerged jet generated by a point momentum source. The Landau jet is the main term of a coordinate expansion of the flow far field in the case when the flow is generated by a finite size source (for example, a tube with flow). The next term of the expansion was calculated by Rumer. This term has an indefinite coefficient. To determine this coefficient we need a conservation law connecting the jet far field with the source. Well-known conservation laws of mass, momentum, and angular momentum fail to calculate the coefficient. In my talk, I will solve this problem for low viscosity. In this case, the flow satisfies the boundary layer equations that possess a nonlocal conservation law closing the problem. The problem for an arbitrary viscosity remains open.