Incompressible Réthy Flows in Two Dimensions

This paper deals with the mathematical theory of a two-dimensional incompressible flow with two free boundaries effusing from a semi-infinite nozzle around a given obstacle, which is named the Réthy flows. The interesting and old problem was suggested and tried by M. Réthy in 1895 for a flow around a symmetric wedge. Here, we are concerned with the well-posedness theory of the symmetric Réthy flow with more general geometric conditions to the nozzle and obstacle. Given a mass flux at the inlet of the nozzle, we established the existence of the incompressible symmetric Réthy flows, containing two free boundaries behind the obstacle. Furthermore, the location estimate, the deflection angle estimate of the Réthy flow in the far field, and the asymptotic behavior of the Réthy flow in the upstream and downstream are also obtained. Finally, some results on the uniqueness of the Réthy flow are established.