Skip to main content
SpringerLink
Log in

Existence of a Weak Solution to a Nonlinear Fluid–Structure Interaction Problem Modeling the Flow of an Incompressible, Viscous Fluid in a Cylinder with Deformable Walls

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We study a nonlinear, unsteady, moving boundary, fluid–structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by two-dimensional incompressible Navier–Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the one-dimensional cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (two-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and the balance of contact forces at the fluid–structure interface. We prove the existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme, introduced in Guidoboni et al. (J Comput Phys 228(18):6916–6937, 2009) to numerically solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk–Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York, 1975

  2. Barbu, V., Grujić, Z., Lasiecka, I., Tuffaha, A.: Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model. Fluids and Waves. Contemporary Mathematics, vol. 440. American Mathematical Society, Providence, 55–82, 2007

  3. Barbu V., Grujić Z., Lasiecka I., Tuffaha A.: Smoothness of weak solutions to a nonlinear fluid-structure interaction model. Indiana Univ. Math. J. 57(3), 1173–1207 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beirão da Veiga H.: On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mech. 6(1), 21–52 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Boulakia M.: Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid. C. R. Math. Acad. Sci. Paris 336(12), 985–990 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brezis, H.: Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris. Théorie et applications. [Theory and applications] (1983)

  7. Bukac, M., Canić, S., Glowinski, R., Tambaca, J., Quaini, A.: Fluid-structure interaction in blood flow allowing non-zero longitudinal structure displacement. J. Comput. Phys. (2012, accepted)

  8. Canić, S., Muha, B., Bukac, M.: Stability of the kinematically coupled β-scheme for fluid-structure interaction problems in hemodynamics. arXiv:1205.6887v1 (2012, submitted)

  9. Canić S., Tambaca J., Guidoboni G., Mikelić A., Hartley C.J., Rosenstrauch D.: Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow. SIAM J. Appl. Math. 67(1), 164–193 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Canić S., Hartley C.J., Rosenstrauch D., Tambaca J., Guidoboni G., Mikelić A.: Blood flow in compliant arteries: an effective viscoelastic reduced model, numerics and experimental validation. Ann. Biomed. Eng. 34, 575–592 (2006)

    Article  Google Scholar 

  11. Chambolle A., Desjardins B., Esteban M.J., Grandmont C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7(3), 368–404 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Cheng C.H.A., Coutand D., Shkoller S.: Navier–Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39(3), 742–800 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng C.H.A., Shkoller S.: The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell. SIAM J. Math. Anal. 42(3), 1094–1155 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ciarlet, P.G.: Mathematical Elasticity, vol. III. Theory of Shells. Studies in Mathematics and its Applications, vol. 29. North-Holland, Amsterdam, 2000

  15. Ciarlet P.G., Lods V.: Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations. Arch. Rational Mech. Anal. 136(2), 119–162 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Conca C., Murat F., Pironneau O.: The Stokes and Navier-Stokes equations with boundary conditions involving the pressure. Jpn. J. Math. (N.S.) 20(2), 279–318 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Conca C., San Martín J., Tucsnak M.: Motion of a rigid body in a viscous fluid. C. R. Acad. Sci. Paris Sér. I Math. 328(6), 473–478 (1999)

    Article  ADS  MATH  Google Scholar 

  18. Coutand D., Shkoller S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Rational Mech. Anal. 176(1), 25–102 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Coutand D., Shkoller S.: The interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rational Mech. Anal. 179(3), 303–352 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Cumsille P., Takahashi T.: Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid. Czechoslovak Math. J. 58(133)(4), 961–992 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Desjardins B., Esteban M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Rational Mech. Anal. 146(1), 59–71 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Desjardins B., Esteban M.J., Grandmont C., Le Tallec P.: Weak solutions for a fluid-elastic structure interaction model. Rev. Mat. Complut. 14(2), 523–538 (2001)

    MathSciNet  MATH  Google Scholar 

  23. Donea J.: Arbitrary Lagrangian-Eulerian finite element methods. Computational Methods for Transient Analysis. North-Holland, Amsterdam (1983)

    Google Scholar 

  24. Du Q., Gunzburger M.D., Hou L.S., Lee J.: Analysis of a linear fluid-structure interaction problem. Discrete Contin. Dyn. Syst. 9(3), 633–650 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Feireisl E.: On the motion of rigid bodies in a viscous compressible fluid. Arch. Rational Mech. Anal. 167(4), 281–308 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Fernández, M.A.: Incremental displacement-correction schemes for incompressible fluid-structure interaction: stability and convergence analysis. Numer. Math. doi:10.1007/s00211-012-0481-9 (2012)

  27. Formaggia L., Gerbeau J.F., Nobile F., Quarteroni A.: On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191(6–7), 561–582 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations, vol. I. Linearized steady problems. Springer Tracts in Natural Philosophy, vol. 38. Springer, New York, 1994

  29. Galdi, G.P.: Mathematical problems in classical and non-Newtonian fluid mechanics. Hemodynamical Flows. Oberwolfach Semin., vol. 37. Birkhäuser, Basel, 121–273, 2008

  30. Glowinski, R.: Finite element methods for incompressible viscous flow. Handbook of Numerical Analysis, vol. 9 (Eds. Ciarlet, P.G., Lions, J.-L.). North-Holland, Amsterdam, 2003

  31. Grandmont C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 40(2), 716–737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Guidoboni G., Glowinski R., Cavallini N., Cavallini N., Cavallini N.: Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow. J. Comput. Phys. 228(18), 6916–6937 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Guidoboni G., Guidorzi M., Padula M.: Continuous dependence on initial data in fluid-structure motions. J. Math. Fluid Mech. 14(1), 1–32 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  34. Hundertmark-Zauskova, A., Lukacova-Medvidova, M., Rusnakova G.: Fluid-structure interaction for shear-dependent non-Newtonian fluids. Topics in Mathematical Modeling and Analysis. Necas Center for Mathematical Modeling. Lecture Notes, vol. 7, 109–158, 2012

  35. Kukavica I., Tuffaha A.: Solutions to a fluid-structure interaction free boundary problem. DCDS-A 32(4), 1355–1389 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kukavica I., Tuffaha A., Ziane M.: Strong solutions for a fluid structure interaction system. Adv. Differ. Equ. 15(3–4), 231–254 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Lequeurre J.: Existence of strong solutions to a fluid-structure system. SIAM J. Math. Anal. 43(1), 389–410 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li F.S.: Asymptotic analysis of linearly viscoelastic shells. Asympt. Anal. 36, 21–46 (2003)

    MATH  Google Scholar 

  39. Li F.S.: Formal asymptotic analysis of linearly viscoelastic flexural shell equations. Adv. Math. 35, 289–302 (2006)

    Google Scholar 

  40. Li F.S.: Asymptotic analysis of linearly viscoelastic shells justification of Koiter’s shell equations. Asymptot. Anal. 54(1–2), 51–70 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, New York, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181

  42. Pontrelli, G.: A mathematical model of flow through a viscoelastic tube. Med. Biol. Eng. Comput. (2002)

  43. Quaini A., Quarteroni A.: A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method. Math. Models Methods Appl. Sci. 17, 957–985 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Quarteroni A., Tuveri M., Veneziani A.: Computational vascular fluid dynamics: problems, models and methods. Survey article. Comput. Visual. Sci. 2, 163–197 (2000)

    Article  MATH  Google Scholar 

  45. San Martín, J.A., Starovoitov V., Tucsnak M.: Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rational Mech. Anal. 161(2), 113–147 (2002)

    Article  ADS  MATH  Google Scholar 

  46. Temam R.: Sur la résolution exacte et approchée d’un problème hyperbolique non linéaire de T. Carleman. Arch. Rational Mech. Anal. 35, 351–362 (1969)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and its Applications, vol. 2. North-Holland, Amsterdam, 1977

  48. Velčić I.: Nonlinear weakly curved rod by Γ-convergence. J. Elast. 108, 125–150 (2012)

    Article  MATH  Google Scholar 

  49. Xiao L.M.: Asymptotic analysis of dynamic problems for linearly elastic shells-justification of equations for dynamic Koiter Shells. Chin. Ann. Math. 22B, 267–274 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX, 77204-3476, USA

    Boris Muha & Suncica Canić

Authors
  1. Boris Muha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suncica Canić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suncica Canić.

Additional information

Communicated by A. Bressan

Rights and permissions

Reprints and permissions

About this article

Cite this article

Muha, B., Canić, S. Existence of a Weak Solution to a Nonlinear Fluid–Structure Interaction Problem Modeling the Flow of an Incompressible, Viscous Fluid in a Cylinder with Deformable Walls. Arch Rational Mech Anal 207, 919–968 (2013). https://doi.org/10.1007/s00205-012-0585-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-012-0585-5

Keywords

Search

Navigation