Existence of steady flows of a viscous incompressible fluid through a profile cascade and their Lr‐regularity

Abstract

The paper deals with the steady Navier–Stokes problem, describing a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to one spatial period Ω, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves Γ0 and Γ1, the Dirichlet boundary conditions on and ΓP, and an artificial boundary condition on (see Figure 1). For “small data,” we consider the so‐called “do nothing” boundary condition on and prove the existence and uniqueness of a weak and strong solution in the Lr–framework. For “large data,” we consider an appropriately modified “do nothing” condition on and prove the existence of a weak and strong solution.