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Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems

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Fluid-Structure Interaction and Biomedical Applications

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

Abstract

It is well known that elastic solids, when subjected to a time-periodic load of frequency ω, may respond with a drastic increase of the magnitude of basic kinematic and dynamic quantities, such as displacement, velocity and energy, whenever ω is near to one of the “proper frequencies” of the solid. This phenomenon is briefly described as resonance. Objective of our analysis is to investigate whether the interaction of an elastic solid with a dissipative agent can affect and possibly prevent the occurrence of resonance. We shall study this problem in a broad class of dynamical systems that we call partially dissipative, and whose dynamics is governed by strongly continuous semigroups of contractions. For such systems we will provide sharp necessary and sufficient conditions for the occurrence of resonance. Afterward, we shall furnish a number of applications to physically relevant problems including thermo- and magneto-elasticity, as well as several liquid–structure interaction models.

MSC2010: 76D07, 35M99, 35Q61, 74B05, 74F05, 74F10, 74F15

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Notes

  1. 1.

    For notation, see the end of this introductory section.

  2. 2.

    These conditions amount to say that the Poisson ratio must not be “too large,” a condition that is verified in most common materials; see, e.g., [34].

  3. 3.

    It is worth remarking that, as recently shown in [33], resonance is excluded in nonlinear magnetoelasticity without the above restrictions on the Lamé coefficients, in presence of a suitable nonlinear damping for the elastic material.

  4. 4.

    For the general case when the periodic load has an infinite number of modes, we refer to Remark 3.1.

  5. 5.

    The calculations to follow show lack of resonance also in the case α < 0. However, such an assumption is unacceptable from the physical viewpoint in that it would imply an increase of total energy in absence of external loads.

  6. 6.

    It is worth remarking that there are also “familiar” domains where (3.15) has an infinite number of linearly independent solutions. This happens, for example, when \(\Omega \) is a ball [15, Remark 5.2], and, in fact, in the two-dimensional case, the circle is the only (sufficiently smooth) simply connected domain where (3.15) has an infinite number of linearly independent solutions [8].

  7. 7.

    In this respect, see [31].

  8. 8.

    See, however, also Remark 3.4.

  9. 9.

    Considerations and results reported in this section remain valid also in dimension 2.

  10. 10.

    In this regard, see also Remark 3.3.

  11. 11.

    This means to modify \(\Psi \) by a function of time which, of course, does not affect the load \(\boldsymbol{f}\).

  12. 12.

    Recall that if there exists t 0 > 0 such that \(\|x(t_{0})\| \equiv \| U(t_{0})x(0)\| <\| x(0)\|\) for all x(0) ∈ X, then, by the semigroup property of U(t), all solutions must decay exponentially fast [5, Remark at p. 178].

  13. 13.

    The argument that follows is due to Professor Jan Prüss, to whom we are indebted.

  14. 14.

    Notice that, of course, x k  ∈ D(A), because x(t) ∈ D(A) for all t ≥ 0.

  15. 15.

    See Footnote 3.1.

  16. 16.

    In the case of three-dimensional flow, or else as a “string,” in the two-dimensional case.

  17. 17.

    Notice that the requirement \(u_{t} \in \hat{ L}^{2}(\Gamma )\) follows from (3.97)2 and (3.99)2.

  18. 18.

    \(H_{0}^{2}(\Gamma )\) is endowed with scalar product \((\Delta u_{1},\Delta u_{2})\).

  19. 19.

    Here and in the rest of the proof, all Banach spaces are meant over the field \(\mathbb{C}\) of complex numbers.

  20. 20.

    Recall that \(\textquotedblleft \overline{\,\ \ }\)” = c.c.

  21. 21.

    See the definition of the space \(\mathcal{H}\).

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Acknowledgements

The work of G.P. Galdi and P. Zunino was partially supported by NSF DMS Grant-1311983.

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Authors and Affiliations

  1. Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Giovanni Paolo Galdi, Mahdi Mohebbi, Rana Zakerzadeh & Paolo Zunino

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  1. Giovanni Paolo Galdi
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Correspondence to Giovanni Paolo Galdi .

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Editors and Affiliations

  1. Department of Technical Mathematics, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Bodnár

  2. Swanson School of Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague 1, Czech Republic

    Šárka Nečasová

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Galdi, G.P., Mohebbi, M., Zakerzadeh, R., Zunino, P. (2014). Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems. In: Bodnár, T., Galdi, G., Nečasová, Š. (eds) Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0822-4_3

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