Solvability in a finite pipe of steady-state Navier–Stokes equations with boundary conditions involving Bernoulli pressure

Most methods of numerical simulation require a truncation of an infinite domain to a bounded one, thereby introducing artificial boundaries. We prove existence of weak solutions to the stationary Navier–Stokes equations, simulating the steady flow of a viscous fluid through the pipe. We consider the case when at inflow and outflow boundaries conditions involving the Bernoulli pressure are prescribed and study the problems either with given flow rate of the fluid or the given pressure drop. In both cases we prove existence of a weak solution without any restriction on the data.