The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade

TL;DR

This paper proves that the steady Stokes problem for flow through a periodic profile cascade has strong solutions with maximum regularity, despite domain irregularities and mixed boundary conditions.

Contribution

It establishes the maximum regularity property for weak solutions of the steady Stokes problem in a non-smooth domain with mixed boundary conditions.

Findings

Existence of strong solutions with maximum regularity.

Handling of boundary condition complexities at domain corners.

Applicability to flow in periodic profile cascades.

Abstract

We deal with a steady Stokes-type problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain Omega. The corresponding Stokes-type problem is formulated by means of the Stokes equation, equation of continuity and three types of boundary conditions: the conditions of periodicity on the curves Gamma0 and Gamma1, the Dirichlet boundary conditions Gamma-in and Gamma-p and an artificial "do nothing"-type boundary condition on Gamma-out. (See Fig. 1.) We explain on the level of weak solutions the sense in which the last condition is satisfied. We show that, although domain Omega is not smooth and different types of boundary conditions meet in the corners of Omega, the considered problem has a strong solution with the so called maximum regularity property.