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Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures

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Abstract

We give a survey of recent results on flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are considered. The focus of the discussion here is on the interesting mathematical aspects of physical phenomena occurring in aeroelasticity, such as flutter and divergence. This leads to a partial differential equation treatment of issues such as well-posedness of finite energy solutions, and long-time (asymptotic) behavior. The latter includes theory of asymptotic stability, convergence to equilibria, and to global attracting sets. We complete the discussion with several well known observations and conjectures based on experimental/numerical studies.

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Notes

  1. Note that, as mentioned above, U is normalized by the speed of sound and thus is what engineers refer to as the Mach number.

  2. For definiteness and some simplification we concentrate on the clamped boundary conditions for the displacement u.

  3. Though, there are also calculations performed at the physical level which demonstrate that the effect of the flow can be destabilizing in certain regimes—see Remark 4.1 below.

  4. For the corresponding definitions and basic facts we refer to [22, 68].

  5. See also Remark 4.1 for other (physically motivated) expressions the “piston” term).

  6. The reference [14] contains a result for the behavior of solutions to the piston theoretic plate (with RHS given by \(p^*\) above) as \(U\rightarrow \infty \). The result, however, is only valid for arbitrarily small time intervals, and hence does not provide information about the behavior of solutions for arbitrary t.

  7. More complete analysis and details are given in [53].

  8. This can be seen from the results in Sect. 6.3.6 in [22] in the subsonic case.

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Acknowledgments

The authors would like to dedicate this work to Professor A.V. Balakrishnan, whose pioneering and insightful work on flutter brought together engineers and mathematicians alike. E.H. Dowell was partially supported by the National Science Foundation with grant NSF-ECCS-1307778. I. Lasiecka was partially supported by the National Science Foundation with grant NSF-DMS-0606682 and the United States Air Force Office of Scientific Research with grant AFOSR-FA99550-9-1-0459. J.T. Webster was partially supported by National Science Foundation with Grant NSF-DMS-1504697.

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Authors and Affiliations

  1. Kharkov National University, Kharkov, Ukraine

    Igor Chueshov

  2. Duke University, Durham, NC, USA

    Earl H. Dowell

  3. University of Memphis, Memphis, TN, USA

    Irena Lasiecka

  4. IBS, Polish Academy of Sciences, Warsaw, Poland

    Irena Lasiecka

  5. College of Charleston, Charleston, SC, USA

    Justin T. Webster

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  1. Igor Chueshov
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  2. Earl H. Dowell
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  3. Irena Lasiecka
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  4. Justin T. Webster
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Correspondence to Justin T. Webster.

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In memory of A.V. Balakrishnan.

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Chueshov, I., Dowell, E.H., Lasiecka, I. et al. Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures. Appl Math Optim 73, 475–500 (2016). https://doi.org/10.1007/s00245-016-9349-1

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