SpringerLink
Log in

Steady-State Stokes and Navier–Stokes Equations in Tube Structures

  • Chapter
  • First Online:
Multiscale Analysis of Viscous Flows in Thin Tube Structures

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

  • 12 Accesses

Abstract

Chapter Steady-State Stokes and Navier–Stokes Equations in Tube Structures revisits the definitions of thin tube structure given in Chap. 1 and formulates the stationary problem for the Stokes and Navier–Stokes equations in thin tube structures. The boundary conditions are homogeneous no-slip on the lateral part of the boundary, and given inflow and outflow velocities. Alternatively, the pressure is given at the inflows and outflows for the Stokes equations and the Bernoulli pressure conditions for the Navier–Stokes equations. The existence and uniqueness of a solution is proved for all settings. The asymptotic expansion of the solution is constructed. The error estimates are proved for the difference of the exact solution and its asymptotic approximations. Method of asymptotic partial decomposition of the domain (MAPDD) is described and justified. Numerical experiments confirm the theoretically established error estimates.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It should be noted that functions in \({W}^{1,2}_{O_N}({\mathcal B})\) are not assumed to be zero at the remaining vertices \(O_l\), where \(l = N_1+1, \ldots , N-1\).

  2. 2.

    A cycle is a finite sequence of distinct edges \(\overline {O_{\alpha } O_{\alpha _1}}, \overline {O_{\alpha _1} O_{\alpha _2}}, \ldots , \overline {O_{\alpha _n} O_{\alpha }}\).

  3. 3.

    It is crucial that \(\mathrm {div}\mathbf {U}=0\) in \(B_\varepsilon \backslash \bigcup \limits _{i=1}^N B_{i}^{\varepsilon ,\delta }\), which means that the integration in the term \(\intop _{B_\varepsilon } p^{(J)} \mathrm {div}\mathbf {U} dx\) is only over the domain \(\bigcup \limits _{i=1}^N B_{i}^{\varepsilon ,\delta }\).

References

  1. R. Becker, D. Capatina, R. Luce, D. Trujillo, Finite elements formulation of general boundary conditions for incompressible flows. Compt. Methods. Appl. Mech. Eng. 295, 240–267 (2015)

    Article  MathSciNet  Google Scholar 

  2. M. Beneš, P. Kučera, Solutions of the Navier-Stokes equations with various types of boundary conditions. Arch. Math. 98, 487–497 (2012)

    Article  MathSciNet  Google Scholar 

  3. M. Beneš, I. Pažanin, M. Radulović, B. Rukavina, Nonzero boundary conditions for the unsteady micropolar pipe flow: well-posedness and asymptotics. Appl. Math. Comput. 427, Article No. 127184 (2022)

    MathSciNet  Google Scholar 

  4. C. Béque, C. Conca, F. Murat, P. Pironneau, A nouveau sur les équations de Stokes et de Navier–Stokes avec des conditions aux limites sur la pression. C.R. Acad. Sci. Paris, Sér I 304(1), 23–28 (1987)

    Google Scholar 

  5. L.M. Bernard, Time-dependent Stokes and Navier-Stokes problems: existence problems with boundary conditions involving pressure, existence and regularity. Nonlinear Anal. Real World Appl. 4(5), 805–839 (2003)

    Article  MathSciNet  Google Scholar 

  6. F. Blanc, O. Gipouloux, G. Panasenko, A.M. Zine, Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure. Math. Models Methods Appl. Sci. 9(9), 1351–1378 (1999)

    Article  MathSciNet  Google Scholar 

  7. S. Blazy, S. Nazarov, M. Specovius-Neugebauer, Artificial boundary conditions of pressure type for viscous flows in a system of pipes. J. Math. Fluid Mech. 9(1), 1–33 (2007)

    Article  MathSciNet  Google Scholar 

  8. D. Bothe, T.K. Kashiwabara, M. Köhne, Strong well-posedness for a class of dynamic outflow boundary conditions for incompressible Newtonian flow. J. Evol. Equ. 17(1), 131–171 (2017)

    Article  MathSciNet  Google Scholar 

  9. M. Braack, P.P. Mucha, Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. C. Conca, Problèmes Mathématiques en Couplage Fluide-Structure (Eyrolles, Paris, 1994)

    Google Scholar 

  12. C. Conca, C. Pares, O. Pironneau, M. Thiriet, Navier-Stokes equations with imposed pressure and velocity fluxes. Int. J. Num. Methods Fluids 20(4), 267–287 (1995)

    Article  MathSciNet  Google Scholar 

  13. A.C. Egloffe, Study of Some Inverse Problems for the Stokes System. Application to the Lungs. Thèse de doctorat, l’Université Pierre et Marie Curie - Paris VI, 2012

    Google Scholar 

  14. J. Fouchet-Incaux, Artificial boundaries and formulations for the incompressible Navier-Stokes equations: applications to air and blood flows. SeMA 64, 1–40 (2014)

    Article  MathSciNet  Google Scholar 

  15. P.M. Gresho, Some current CFD issues relevant to the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 87, 201–252 (1991)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. T. Kim, Regularity of solutions to the Navier-Stokes equations with a nonstandard boundary condition. Acta Math. Appl. Sin. Engl. Ser. 31(3), 707–718 (2015)

    Article  MathSciNet  Google Scholar 

  18. M.V. Korobkov, K. Pileckas, R. Russo, Solvability in a finite pipe of steady-state Navier-Stokes equations with boundary conditions involving Bernoulli pressure. Calc. Var. and Partial Diff. Equ. 59(32), 1–22 (2020)

    MathSciNet  Google Scholar 

  19. S. Kračmar, J. Neustupa, Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier-Stokes variational inequality. Math. Nachrichten 291, 1801–1814 (2018)

    Article  MathSciNet  Google Scholar 

  20. P. Kučera, Basic properties of solutions of the non-steady Navier-Stokes equations with mixed boundary conditions in a bounded domain. Ann. Univ. Ferrara Sez. VII Sci Mat. 55(2), 289–308 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. O.A. Ladyzhenskaia, The Mathematical Theory of Viscous Incompressible Flow (Nauka, Moscow, 1970) (in Russian); (English Edition: Gordon and Breach Science Publishers, New York, 1969)

    Google Scholar 

  23. S.A. Nazarov, K. Pileckas, Asymptotic conditions at infinity for the Stokes and Navier-Stokes problems in domains with cylindrical outlets to infinity. Quad. Mat. Adv. Fluid Dyn. 4, 141–245 (1999)

    MathSciNet  Google Scholar 

  24. S.A. Nazarov, M. Specovius-Neugebauer, Approximation of unbounded domains by bounded domains. Boundary value problems for Lamé operator. St. Petersburg Math. J. 8(5), 879–912 (1997)

    Google Scholar 

  25. B. Nowakowski, G. Stróhmer, In-flow and out-flow problem for the Stokes system. J. Math. Fluid Mech. 22, Article No. 58 (2020)

    Article  MathSciNet  Google Scholar 

  26. G. Panasenko, Method of asymptotic partial decomposition of domain. Math. Models Methods Appl. Sci. 8(1), 139–156 (1998)

    Article  MathSciNet  Google Scholar 

  27. G. Panasenko, Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure. C.R. Acad. Sci. Paris 326, Série IIb, 867–872 (1998)

    Google Scholar 

  28. G. Panasenko, Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure. C.R. Acad. Sci. Paris 326, Série IIb, 893–898 (1998)

    Google Scholar 

  29. G. Panasenko, Multi-scale Modeling for Structures and Composites (Springer, Dordrecht, 2005)

    Google Scholar 

  30. G. Panasenko, Introduction to Multiscale Mathematical Modeling (World Scientific, New Jersey, 2022)

    Book  Google Scholar 

  31. G. Panasenko, K. Pileckas, Pressure boundary conditions for viscous flows in thin tube structures: Stokes equations with locally distributed Brinkman term. Math. Modell. Nat. Phenomena 18, Article No. 17 (2023)

    Article  MathSciNet  Google Scholar 

  32. O. Pironneau, Conditions aux limites sur la pression pour les équations de Stokes et de Navier–Stokes. C.R. Acad. Sci. Paris Sér. I Math. 303(9), 403–406 (1986)

    Google Scholar 

  33. R.L. Sani, P.M. Gresho, Résumé and remarks on the open boundary condition. Int. J. Num. Methos Fluid 18, 983–1008 (1994)

    Article  Google Scholar 

  34. R.L. Sani, J. Shen, O. Pironneau, P.M. Gresho, Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids 50, 673–682 (2006)

    Article  MathSciNet  Google Scholar 

  35. M. Specovius-Neugebauer, Approximation of the Stokes Dirichlet problem in domains with cylindrical outlets. SIAM J. Math. Anal. 30(3), 645–677 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania

    Grigory Panasenko & Konstantin Pileckas

Authors
  1. Grigory Panasenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Konstantin Pileckas
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Panasenko, G., Pileckas, K. (2024). Steady-State Stokes and Navier–Stokes Equations in Tube Structures. In: Multiscale Analysis of Viscous Flows in Thin Tube Structures. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-54630-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation