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Initial-boundary value problems for solid–fluid composite structures

  • N. Chinchaladze
  • R. P. Gilbert
  • G. Jaiani
  • S. Kharibegashvili
  • D. Natroshvili
Article
  • 116 Downloads

Abstract

We investigate a three-dimensional mixed initial-boundary value problem arising in the dynamical solid–fluid interaction theory. A 3D domain occupied by an incompressible and viscous Stokes fluid may be bounded or unbounded, while a domain occupied by an elastic body immersed in the fluid is assumed to be bounded. On the basis of the results obtained for an elastic inclusion of an arbitrary geometrical shape, we derive a special model and analyze in detail the case when an elastic inclusion is a thin prismatic shell, in particular a plate of variable thickness. Here, we apply I. Vekua’s dimension reduction method in the elastic part which reduces 3D solid–3D fluid interaction problems to the 2D solid–3D fluid interaction problems and which is important from the practical point of view since it takes into account intrinsic differences of the dimensions of solid and fluids part. The main goal of the paper was to study the strain–stress state of the elastic part under the action of the Stokes flow. The corresponding mechanical model is described mathematically as a transmission problem for the linear Stokes system and the dynamical Lamé equations in the corresponding domains with appropriate initial conditions along with the boundary and interface conditions. For 3D solid–3D fluid dynamical interaction problems, we prove the uniqueness and existence theorem. Further, considering the case when the elastic inclusion is a thin prismatic shell of variable thickness, we apply the N = 0 approximation of Vekua’s hierarchical model for the elastic field in the solid part. In contrast to the usual classical streamline conditions, in the case under consideration, on the cut surface, there appear non-local boundary conditions. We prove unique solvability of the non-classical boundary value problem that leads to the existence results for the solid–fluid interaction problem with a thin elastic inclusion.

Mathematics Subject Classification (2010)

Primary 74F10 Secondary 74K25 

Keywords

Solid–fluid interaction dynamical problems Vekua’s hierarchical models Elastic prismatic shell Non-local boundary conditions Initial-boundary value problem 

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References

  1. 1.
    Avalishvili M., Gordeziani D.: Investigation of two-dimensional models of elastic prismatic shell. Georgian Math. J. 10(1), 17–36 (2003)MathSciNetMATHGoogle Scholar
  2. 2.
    Bielak J., MacCamy R.: Symmetric finite element and boundary integral coupling methods for fluid-solid interaction. Q. Appl. Math. 49, 107–119 (1991)MathSciNetMATHGoogle Scholar
  3. 3.
    Chinchaladze N., Gilbert R.: Cylindrical vibration of an elastic cusped plate under action of an incompressible fluid in case of N = 0 approximation of I. Vekua’s hierarchical models. Complex Var. Theory Appl. 50(7–11), 479–496 (2005)MathSciNetMATHGoogle Scholar
  4. 4.
    Chinchaladze N., Gilbert R., Jaiani G., Kharibegashvili S., Natroshvili D.: Existence and uniqueness theorems for cusped prismatic shells in the N-th hierarchical model. Math. Method Appl. Sci. 31, 1345–1367 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology: Integral Equations and Numerical Methods. Vol. 4, Springer, Berlin (1990)Google Scholar
  6. 6.
    Everstine G., Henderson F.: Coupled finite element/boundary element approach for fluid-structure interaction. J. Acoust. Soc. Am. 87, 1938–1947 (1990)CrossRefGoogle Scholar
  7. 7.
    Fichera, G.: Existence Theorems in Elasticity. Handbuch der Physik, Bd. 6/2. Springer, Heidelberg. (1973)Google Scholar
  8. 8.
    Goldenveizer, A.L.: Theory of Elastic Thin Shells. Nauka, Moscow. (1976, in Russian)Google Scholar
  9. 9.
    Goldenveizer A.L., Kaplunov J.D., Nolde E.V.: Asymptotic analysis and refinement of Timoshenko-Reisner-type theories of plates and shells. Trans. Acad. Sci. USSR. Mekhanika Tverd. Tela 25(6), 126–139 (1990)Google Scholar
  10. 10.
    Gordeziani, D.: To the exactness of one variant of the theory of thin shells. Dokl. Acad. Nauk. SSSR 216(4), 751–754 (1974, Russian)Google Scholar
  11. 11.
    Hardy G.H., Littlwood J.E., Pólya G.: Inequalities. The University press, Cambridge (1934)Google Scholar
  12. 12.
    Hsiao, G.C.: On the boundary field equation methods for fluid-structure interactions. In: Proceedings of the 10.TMP, Teubner Texte zur Mathematik, Bd., vol. 134, pp. 79–88. Stuttgart-Leipzig. (1994)Google Scholar
  13. 13.
    Hsiao G.C., Kleinman R.E., Roach G.F.: Weak solutions of fluid-solid interaction problems. Math. Nachr. 218, 139–163 (2000)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jaiani G.: Cusped Shell-Like Structures, SpringerBriefs in Applied Science and Technology. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Jaiani G., Kharibegashvili S., Natroshvili D., Wendland W.L.: Two-dimensional hierarchical models for prismatic shells with thickness vanishing at the boundary. J. Elast. 77(2), 95–122 (2004)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Jentsch L., Natroshvili D.: Non-local approach in mathematical problems of fluid-structure interaction. Math. Methods Appl. Sci. 22, 13–42 (1990)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jentsch L., Natroshvili D.: Interaction between thermoelastic and scalar oscillation fields. Integr. Equ. Oper. Theory 28, 261–288 (1997)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Jones D.: Low-frequency scattering by a body in lubricated contact. Q. J. Mech. Appl. Math. 36, 111–137 (1983)MATHCrossRefGoogle Scholar
  19. 19.
    Kaplunov J., Nolde E., Rogerson G.A.: An asymptotic analysis of initial-value problems for thin elastic plates. Proc. R. Soc. A Math. Phys. Eng. Sci. 462(2073), 2541–2561 (2006)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kaplunov J.D., Nolde E.V., Veksler N.D.: Asymptotic description of peripheral waves in scattering of a plane acoustic wave by a spherical shell. Acustica 76, 10–19 (1992)MATHGoogle Scholar
  21. 21.
    Kaplunov J.D., Nolde E.V., Veksler N.D.: Asymptotic formulae for the modal resonance of peripheral waves in the scattering of an oblique incident plane acoustic wave by a cylindrical shell. Acustica 80, 280–293 (1994)MATHGoogle Scholar
  22. 22.
    Kupradze V.D., Gegelia T.G., Basheleishvili M.O., Burchuladze T.V.: Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland Publishing Company, Amsterdam (1979)MATHGoogle Scholar
  23. 23.
    Ladizhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1963)Google Scholar
  24. 24.
    Landau L.D., Lifshitz E.M.: Continuum Mechanics. Physmatgiz, Moscow (1954)Google Scholar
  25. 25.
    Lawrie J.B., Guled I.M.M.: On tuning a reactive silencer by varying the position of an internal membrane. J. Acoust. Soc. Am. 120(2), 780–790 (2006)CrossRefGoogle Scholar
  26. 26.
    Lawrie J.B.: Orthogonality relations for fluid-structural waves in a three-dimensional, rectangular duct with flexible walls. Proc. R. Soc. A 465, 2347–2367 (2009)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Vol. 1, Springer, New York (1972)Google Scholar
  28. 28.
    McLean W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, UK (2000)MATHGoogle Scholar
  29. 29.
    Muskhelishvili N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Nauka, Moscow (1949)Google Scholar
  30. 30.
    Natroshvili, D., Kharibegashvili, S., Tediashvili, Z.: Direct and inverse fluid-structure interaction problems. Rend. Mat., Serie VII, 20 Roma (2000), 57–92Google Scholar
  31. 31.
    Nolde E.: Qualitative analysis of initial-value problems for a thin elastic strip. IMA J. Appl. Math. 72(3), 348–375 (2007)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Natroshvili D., Sadunishvili G.: Interaction of elastic and scalar fields. Math. Methods Appl. Sci. 19(18), 1445–1469 (1996)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Nečas J.: Méthodes Directes en Théorie des Équations Élliptiques. Masson Éditeur, Paris (1967)MATHGoogle Scholar
  34. 34.
    Rudin, W.: Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424 pp. ISBN: 0-07-054236-8, 46-01; McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, xiii+397 pp. 46–01. (1973)Google Scholar
  35. 35.
    Sanchez-Palencia E.: Non-homogeneous Media and Vibration Theory. Springer, Berlin (1980)MATHGoogle Scholar
  36. 36.
    Schwab Ch.: A-posteriori modeling error estimation for hierarchical plate models. Numer. Math. 74(2), 221–259 (1996)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Temam R.: Navier-Stokes Equations Theory and Numerical Analysis. Revised edition. With an appendix by F. Thomasset. North-Holland Publishing Co., Amsterdam (1979)Google Scholar
  38. 38.
    Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North Holand, Amsterdam (1978)Google Scholar
  39. 39.
    Vekua I.N.: Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston (1985)MATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • N. Chinchaladze
    • 1
  • R. P. Gilbert
    • 2
  • G. Jaiani
    • 1
  • S. Kharibegashvili
    • 1
  • D. Natroshvili
    • 1
  1. 1.Iv. Javakhishvili Tbilisi State University, I. Vekua Institute of Applied MathematicsTbilisiGeorgia
  2. 2.University of DelawareNewarkUSA

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