The weak Stokes problem associated with a flow through a profile cascade in Lr-framework

TL;DR

This paper investigates the weak steady Stokes problem for incompressible fluid flow through a periodic profile cascade within the Lr-framework, establishing existence, uniqueness, and boundary condition interpretation of solutions.

Contribution

It introduces a mathematical model for the Stokes problem with mixed boundary conditions in a periodic domain and proves fundamental properties of the weak solutions in the Lr setting.

Findings

Existence and uniqueness of weak solutions in W^{1,r}(Omega)

Continuous dependence of solutions on data

Clarification of the 'do nothing' boundary condition interpretation

Abstract

We study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the Lr-framework. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain Omega. The corresponding Stokes problem is formulated by means of three types of boundary conditions: the conditions of periodicity on the "lower" and "upper" parts of the boundary, the Dirichlet boundary conditions on the "inflow" and on the profile and an artificial "do nothing"--type boundary condition on the "outflow". Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in W^{1,r}(Omega) and its continuous dependence on the data. We explain the sense in which the "do nothing" boundary condition on the "outflow" is satisfied.