We consider a steady flow of an incompressible viscous fluid in a two-dimensional unbounded domain with unbounded boundaries. The domain has two outlets. The part of the domain upstream is a cylinder and the part of the domain downstream is a wedge. In the part of the domain upstream, the velocity is required to approach the Poiseuille flow. In the part of the domain downstream, the velocity is required to approach Jeffery-Hamel's flow. This problem has been treated by C.J. Amick and L.E. Frankel, V.A. Solonnikov and many others. Recently, T. Kobayashi obtained the unique solution of Jeffery-Hamel's flows in a wedge. Therefore we reconsider this problem. In this paper, we succeed in proving the existence of such a steady flow under the restricted flux condition which depends only on the part of the domain upstream and downstream.
Teppei Kobayashi. "Steady Navier-Stokes equations with Poiseuille and Jeffery-Hamel flows in $\mathbb R^2$." Rocky Mountain J. Math. 49 (6) 1909 - 1929, 2019. https://doi.org/10.1216/RMJ-2019-49-6-1909