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The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

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  • Tomáš Neustupa

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$L^r$‐setup. The mathematical model used here is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated using 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 W1,r(Ω)$\mathbf {W}^{1,r}(\Omega )$ and its continuous dependence on the data. We explain the sense in which the “do nothing” boundary condition on the “outflow” is satisfied.

Suggested Citation

  • Tomáš Neustupa, 2023. "The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework," Mathematische Nachrichten, Wiley Blackwell, vol. 296(2), pages 779-796, February.
    • Handle: RePEc:bla:mathna:v:296:y:2023:i:2:p:779-796
    DOI: 10.1002/mana.202000320
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    References listed on IDEAS

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    1. Stanislav Kračmar & Jiří Neustupa, 2018. "Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality," Mathematische Nachrichten, Wiley Blackwell, vol. 291(11-12), pages 1801-1814, August.
    2. Neustupa, Tomáš, 2016. "A steady flow through a plane cascade of profiles with an arbitrarily large inflow_The mathematical model, existence of a weak solution," Applied Mathematics and Computation, Elsevier, vol. 272(P3), pages 687-691.
    3. Michal Beneš & Petr Kučera, 2016. "Solutions to the Navier–Stokes equations with mixed boundary conditions in two-dimensional bounded domains," Mathematische Nachrichten, Wiley Blackwell, vol. 289(2-3), pages 194-212, February.
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