Jeffery—Hamel Asymptotics for Steady State Navier—Stokes Flow in Domains with Sector-like Outlets to Infinity

Abstract. We consider a stationary problem for the Navier—Stokes equations in a domain $ \Omega\subset R^2 $ with a finite number of "outlets" to infinity in the form of infinite sectors. In addition to the standard adherence boundary conditions, we prescribe total fluxes of velocity vector field in each "outlet", subject to the necessary condition that the sum of all fluxes equals zero. Under certain restrictions on the aperture angles of the "outlets", which seem close to being necessary, we prove that for small fluxes this problem has a solution which behaves at infinity like the Jeffery—Hamel flow with the same flux, and we prove that this solution is unique in the class of solutions satisfying the energy inequality (4.4). We also study the problem with another type of additional condition at infinity, that which involves limiting values of the pressure at infinity in the outlets. Finally, we present a simplified construction of a small Jeffery—Hamel solution with a given flux based on the contraction mapping principle.