The Maximum Regularity Property of the Steady Stokes Problem Associated with a Flow Through a Profile Cascade

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 \(\varOmega \). 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 \(\varGamma _{\hspace {-1.1pt} 0}\) and \(\varGamma _{\hspace {-1.1pt} 1}\), the Dirichlet boundary conditions on \(\varGamma _{\hspace {-1.1pt} \mathrm {in}}\) and \(\varGamma _{\hspace {-1.1pt} p}\) and an artificial “do nothing”–type boundary condition on \(\varGamma _{\hspace {-1.1pt} \mathrm {out}}\). (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 \(\varOmega \) is not smooth and different types of boundary conditions meet in the corners of \(\varOmega \), the considered problem has a strong solution with the so called maximum regularity property.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

    Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand Comp, New York (1965)

    Google Scholar 

  2. 2.

    Al Baba, H., Amrouche, Ch., Escobedo, M.: Semi-group theory for the Stokes operator with Navier–type boundary condition in \(L^{p}\)–spaces. Arch. Ration. Mech. Anal. 223(2), 881–940 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Amrouche, Ch., Escobedo, M., Ghosh, A.: Semigroup theory for the Stokes operator with Navier boundary condition in \(L^{p}\) spaces. arXiv:1808.02001v1 [math.AP] (6 Aug 2018)

  4. 4.

    Bruneau, C.H., Fabrie, P.: New efficient boundary conditions for incompressible Navier–Stokes equations: a well–posedness result. Math. Model. Numer. Anal. 30(7), 815–840 (1996)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chen, G.Q., Qian, Z.: A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions. Indiana Univ. Math. J. 59(2), 721–760 (2010)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chen, G.Q., Osborne, D., Qian, Z.: The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non–flat boundaries. Acta Math. Sci. 29B(4), 919–948 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dauge, M.: Stationary Stokes and Navier–Stokes systems on two– or three-dimensional domains with corners. Part I: linearized equations. SIAM J. Math. Anal. 20, 74–97 (1989)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dolejší, V., Feistauer, M., Felcman, J.: Numerical simulation of compressible viscous flow through cascades of profiles. Z. Angew. Math. Mech. 76, 301–304 (1996)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Feistauer, M., Neustupa, T.: On some aspects of analysis of incompressible flow through cascades of profiles. In: Operator Theory, Advances and Applications, vol. 147, pp. 257–276. Birkhäuser, Basel (2004)

    Google Scholar 

  10. 10.

    Feistauer, M., Neustupa, T.: On non-stationary viscous incompressible flow through a cascade of profiles. Math. Methods Appl. Sci. 29(16), 1907–1941 (2006)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Feistauer, M., Neustupa, T.: On the existence of a weak solution of viscous incompressible flow past a cascade of profiles with an arbitrarily large inflow. J. Math. Fluid Mech. 15, 701–715 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady State Problems. Springer, New York (2011)

    Google Scholar 

  13. 13.

    Geissert, M., Heck, H., Hieber, M.: On the equation \(\mathrm{div}\,u = g\) and Bogovskii’s operator in Sobolev spaces of negative order. In: Partial Differential Equations and Functional Analysis. Oper. Theory Adv. Appl., vol. 168, pp. 113–121. Birkhüser, Basel (2006)

    Google Scholar 

  14. 14.

    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Google Scholar 

  15. 15.

    Grisvard, P.: Singularités des solutions du probléme de Stokes dans un polygone. Université de Nice (1979)

  16. 16.

    Grisvard, P.: Elliptic Problems in Non-smooth Domains. Pitman Advanced Publishing Program, Boston (1985)

    Google Scholar 

  17. 17.

    Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 22, 325–352 (1996)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Google Scholar 

  19. 19.

    Kato, T., Mitrea, M., Ponce, G., Taylor, M.: Extension and representation of divergence–free vector fields on bounded domains. Math. Res. Lett. 7, 643–650 (2000)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kellog, R.B., Osborn, J.E.: A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21, 397–431 (1976)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Kozel, K., Louda, P., Příhoda, J.: Numerical solution of turbulent flow in a turbine cascade. Proc. Appl. Math. Mech. 6, 743–744 (2006)

    Article  Google Scholar 

  22. 22.

    Kračmar, S., Neustupa, J.: Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities. Z. Angew. Math. Mech. 74(6), 637–639 (1994)

    MATH  Google Scholar 

  23. 23.

    Kračmar, S., Neustupa, J.: A weak solvability of a steady variational inequality of the Navier–Stokes type with mixed boundary conditions. Nonlinear Anal. 47(6), 4169–4180 (2001)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Kračmar, S., Neustupa, J.: Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality. Math. Nachr. 291(11–12), 1–14 (2018)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Kučera, P.: Basic properties of the non-steady Navier–Stokes equations with mixed boundary conditions ina bounded domain. Ann. Univ. Ferrara 55, 289–308 (2009)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kučera, P., Beneš, M.: Solution to the Navier–Stokes equatons with mixed boundary conditions in two-dimensional bounded domains. Math. Nachr. 289(2–3), 194–212 (2016)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Kučera, P., Skalák, Z.: Solutions to the Navier–Stokes equations with mixed boundary conditions. Acta Appl. Math. 54(3), 275–288 (1998)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompresible Flow. Gordon and Breach Science Publishers, New York (1969)

    Google Scholar 

  29. 29.

    Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)

    Google Scholar 

  30. 30.

    Medková, D.: The Neumann problem for the planar Stokes system. Ann. Univ. Ferrara 58, 307–329 (2012)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Medková, D.: One problem of the Navier type for the Stokes system inplanar domains. J. Differ. Equ. 261, 5670–5689 (2016)

    Article  Google Scholar 

  32. 32.

    Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Paris (1967)

    Google Scholar 

  33. 33.

    Neustupa, T.: Question of existence and uniqueness of solution for Navier–Stokes equation with linear “do-nothing” type boundary condition on the outflow. In: Lecture Notes in Computer Science, vol. 5434, pp. 431–438 (2009)

    Google Scholar 

  34. 34.

    Neustupa, T.: The analysis of stationary viscous incompressible flow through a rotating radial blade machine, existence of a weak solution. Appl. Math. Comput. 219, 3316–3322 (2012)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Neustupa, T.: A steady flow through a plane cascade of profiles with an arbitrarily large inflow: the mathematical model, existence of a weak solution. Appl. Math. Comput. 272, 687–691 (2016)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Sohr, H.: The Navier–Stokes Equations. The Eelementary Functional Analytic Approach. Birkhäuser, Basel (2001)

    Google Scholar 

  37. 37.

    Straka, P., Příhoda, J., Kožíšek, M., Fürst, J.: Simulation of transitional flows through a turbine blade cascade with heat transfer for various flow conditions. EPJ Web Conf. 143, 02118 (2017). https://doi.org/10.1051/epjconf/201714302118

    Article  Google Scholar 

  38. 38.

    Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1977)

    Google Scholar 

Download references

Acknowledgements

The author acknowledges the support of the European Regional Development Fund-Project “Center for Advanced Applied Science” No. CZ.02.1.01/0.0/0.0/16_019/0000778.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tomáš Neustupa.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Neustupa, T. The Maximum Regularity Property of the Steady Stokes Problem Associated with a Flow Through a Profile Cascade. Acta Appl Math 172, 3 (2021). https://doi.org/10.1007/s10440-021-00396-4

Download citation

Keywords

  • The Stokes problem
  • Artificial boundary condition
  • Maximum regularity property

Mathematics Subject Classification (2010)

  • 35Q30
  • 76D03
  • 76D07