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The paper deals with the Stokes problem, associated with a flow of a viscousincompressible fluid through a spatially periodic profile cascade. We useresults from [32] (the maximum regularity property in the $L^2$-framework) and[33] (the weak solvability in $W^{1,r}$), and extend the findings on themaximum regularity property to the general $L^r$-framework (for $1<r<\infty$).Using the reduction to one spatial period $\Omega$, the problem is formulatedby means of boundary conditions of three types: the conditions of periodicityon curves $\Gamma_0$ and $\Gamma_1$, the Dirichlet boundary conditions on$\Gamma_{\rm in}$ and $\Gamma_P$ and an artificial "do nothing"-type boundarycondition on $\Gamma_{\rm out}$ (see Fig. 1). We show that, although domain$\Omega$ is not smooth and different types of boundary conditions "meet" in thevertices of $\partial\Omega$, the considered problem has a strong solution withthe maximum regularity property for "smooth" data. We explain the sense inwhich the "do nothing" boundary condition is satisfied for both weak and strongsolutions.

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework