On Stationary Viscous Incompressible Flow Through a Cascade of Profiles with the Modified Boundary Condition on the Outflow and Large Inflow

Abstract

The paper is concerned with the analysis of the model of incompressible, viscous, stationary flow through a plane cascade of profiles. The problem is formulated in a bounded domain of the form of one space period with suitable boundary conditions on the boundary. Let us recall that there is usually imposed the condition on smallness of the inflow velocity or the condition on smallness of fluxes between various components of the boundary (Specially that the balance of fluid entering and leaving domain is zero for each component of boundary) in known theorems on existence of a weak solution of the boundary-value problem for the Navier–Stokes equation with the nonzero Dirichlet boundary condition, (see e.g., Mathematical Methods in Fluid Dynamics (1993), An Introduction to the Mathematical Theory of the Navier–Stokes Equations (1994), Finite Element Approximation of the Navier–Stokes Equations (1979), Navier–Stokes Equations (1977)). In this paper the case of a large inflow is considered, however the possibility of the large inflow is compensated by certain modification of the boundary condition on the outflow and by a specification on the shape of the domain.