The axially symmetric flow-through problem for the Navier-Stokes equations in variables “vorticity-stream function”

Abstract

The axially symmetric steady-state motion of a viscous incompressible fluid in a region of the type of a spherical layer is considered. The boundary problem for the Navier-Stokes equations is formulated in terms of the stream function and vorticity. The solvability of this problem was previously established provided that the fluid flow rate through each layer boundary is sufficiently small. The main result of this study is in proving the solvability of the axially symmetric flow-through problem without limitations on the magnitude of the flow rate. The proof is based on the a priori Dirichlet estimate of the flow velocity field. The asymptotic properties of the solution near the symmetry axis are established conditionally. The proof procedure admits the generalization to the case of axially symmetric problems in multiply-connected domains.