Skip to main content
Log in

On the interaction of an elasticwall with a poiseuille-type flow

  • Published:
Ukrainian Mathematical Journal Aims and scope

We study the dynamics of a coupled system formed by the 3D Navier–Stokes equations linearized near a certain Poiseuille-type flow in an (unbounded) domain and a classical (possibly nonlinear) equation for transverse displacements of an elastic plate in a flexible flat part of the boundary. We first show that this problem generates an evolution semigroup S t in an appropriate phase space. Then, under some conditions imposed on the underlying (Poiseuille-type) flow, we prove the existence of a compact finite-dimensional global attractor for this semigroup and also show that S t is an exponentially stable C 0 -semigroup of linear operators in the completely linear case. Since we do not assume any kind of mechanical damping in the plate component, this means that the dissipation of energy in the flow of fluid caused by viscosity is sufficient to stabilize the system.

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

Access this article

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. G. Avalos, “The strong stability and instability of a fluid-structure semigroup,” Appl. Math. Optimiz., 55, 163–184 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Avalos and R. Triggiani, “Semigroup well-posedness in the energy space of a parabolic hyperbolic coupled Stokes–Lamé PDE system of fluid-structure interaction,” Discrete Contin. Dynam. Syst., Ser. S, 2, 417–447 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  3. A.V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam (1992).

    MATH  Google Scholar 

  4. V. Barbu, Z. Grujić, I. Lasiecka, and A. Tuffaha, “Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,” Fluids and Waves. Contemp. Math., 440, 55–82 (2007).

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd ed, Springer, New York (1993).

    MATH  Google Scholar 

  7. I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov (2002); http://www.emis.de/monographs/Chueshov.

  8. I. Chueshov, “A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,” Math. Meth. Appl. Sci., 34, 1801–1812 (2011).

    MathSciNet  MATH  Google Scholar 

  9. I. Chueshov and S. Kolbasin, “Long-time dynamics in plate models with strong nonlinear damping,” Commun. Pure Appl. Anal., 11, 659–674 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  10. I. Chueshov and I. Lasiecka, “Long-time behavior of second-order evolution equations with nonlinear damping,” Mem. AMS, 195, No. 912 (2008).

    Google Scholar 

  11. I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York (2010).

    Book  MATH  Google Scholar 

  12. I. Chueshov and I. Lasiecka, Well-Posedness and Long-Time Behavior in Nonlinear Dissipative Hyperbolic-Like Evolutions with Critical Exponents, Preprint ArXiv, 1204.5864v1.

  13. I. Chueshov, I. Lasiecka, and J. Webster, “Evolution semigroups in supersonic flow-plate interactions,” J. Different. Equat., 254, 1741–1773 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Chueshov and I. Ryzhkova, “A global attractor for a fluid–plate interaction model,” Commun. Pure Appl. Anal., 12, 1635–1656 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. I. Chueshov and I. Ryzhkova, “Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,” J. Different. Equat., 254, 1833–1862 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Chueshov and I. Ryzhkova, “Well-posedness and long time behavior for a class of fluid-plate interaction models,” IFIP Adv. Inform. Commun. Technology, Springer, Berlin (2013), Vol. 391, pp. 328–337.

  17. D. Coutand and S. Shkoller, “Motion of an elastic solid inside an incompressible viscous fluid,” Arch. Ration. Mech. Anal., 176, 25–102 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd ed., Springer, New York (2011).

  19. Q. Du, M. D. Gunzburger, L. S. Hou, and J. Lee, “Analysis of a linear fluid–structure interaction problem,” Discrete Contin. Dynam. Syst., 9, 633–650 (2003).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Grobbelaar-Van Dalsen, “A new approach to the stabilization of a fluid–structure interaction model,” Appl. Anal., 88, 1053–1065 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Grobbelaar-Van Dalsen, “Strong stability for a fluid–structure model,” Math. Meth. Appl. Sci., 32, 1452–1466 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Guidorzi, M. Padula, and P. I. Plotnikov, “Hopf solutions to a fluid–elastic interaction model,” M3AS, 18, 215–269 (2008).

    MathSciNet  Google Scholar 

  24. N. Kopachevskii and Yu. Pashkova, “Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane,” Rus. J. Math. Phys., 5, No. 4, 459–472 (1998).

    MathSciNet  Google Scholar 

  25. O. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York (1969).

    MATH  Google Scholar 

  26. O. Ladyzhenskaya and V. Solonnikov, “Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equations,” Zap. LOMI, 59, 81–116 (1976); English translation: J. Soviet Math., 10, 257–286 (1978).

    MATH  Google Scholar 

  27. J.-L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Dunod, Paris (1968), Vol. 1.

  28. J.-L. Lion, Quelques Methodes de Resolution des Problémes aux Limites Non Lineaire, Dunod, Paris (1969).

    Google Scholar 

  29. B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed., Taylor & Francis, New York (2006).

    Google Scholar 

  30. A. Osses and J. Puel, “Approximate controllability for a linear model of fluid structure interaction,” ESIAM: Contr., Optimiz. and Calc. Variat., 4, 497–513 (1999).

  31. G. Raugel, “Global attractors in partial differential equations,” Handbook Dynam. Syst., Elsevier Sci., Amsterdam (2002), Vol. 2, pp. 885–992.

  32. J. Simon, “Compact sets in the space L p(0, T; B),” Ann. Mat. Pura Appl. Ser. 4, 148, 65–96 (1987).

    Google Scholar 

  33. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1988).

    Book  MATH  Google Scholar 

  34. R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 ed., AMS Chelsea Publ., Providence, RI (2001).

  35. H. Triebel, Interpolation Theory, Functional Spaces, and Differential Operators, North-Holland, Amsterdam (1978).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 1, pp. 143–160, January, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chueshov, I., Ryzhkova, I. On the interaction of an elasticwall with a poiseuille-type flow. Ukr Math J 65, 158–177 (2013). https://doi.org/10.1007/s11253-013-0771-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-013-0771-0

Keywords

Navigation