Existence of a steady flow through a rotating radial turbine with an arbitrarily large inflow and an artificial boundary condition on the outflow

Abstract

AbstractWe prove the existence of a steady strong solution to the Navier–Stokes‐type problem in a 2D multiply connected domain, modeling a flow of an incompressible viscous fluid through a rotating radial turbine. We consider the inhomogeneous Dirichlet boundary condition on the inflow Γin and an artificial boundary condition of the “do nothing” type on the outflow Γout. The conditions admit an arbitrarily large flux through the turbine. This is, however, compensated by the requirement that the distance between Γout and the profile family is “sufficiently small”. (Theorem 1.1.) The solution is steady in the rotating frame, that is, in the frame, attached to the rotating turbine. As an auxiliary result, we prove that a function, that takes just two different values in a 1D interval on two complementary measurable sets M and of positive 1D measure, is not in . (Lemma 3.4.)