On nonsymmetric two-dimensional viscous flow through an aperture (Q2706895)

The paper studies a free boundary problem which describes the flow of a viscous incompressible fluid in a given semi-infinite smooth strip-like domain (with no-slip boundary conditions) through an aperture. The aim is to determine, by using the Navier-Stokes equations, the velocity and the pressure in the fluid after the aperture. Under the assumptions of a small jet opening angle and a small fluid velocity through the aperture, the authors show the existence of a solution to the free boundary problem. The proof is based on the theory of Stokes and Navier-Stokes equations in unbounded domains. The formulation of the problem uses special weighted Hölder spaces that take into account the behaviour of the solution on the boundary and at infinity. The nonlinear problem is firstly solved with a prescribed boundary, by means of fixed point argument, if the flow velocity through the aperture is small. Again, a fixed point argument is used to solve the free boundary problem if the opening angle is small.