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Absolute instability of a thin viscoelastic plate in an air flow. (English) Zbl 1064.74096
Summary: Motivated by aerodynamic applications, we study the destabilization of a thin viscoelastic plate by the stress produced by a flow of air on one side or both sides of the plate parallel to the plate. The research is a follow up on our recent discovery [cf. the authors, Cold Reg. Sci. Technol. 33, 77–89 (2001)] that a floating ice layer is destabilized by wind stress, and the unstable wave packets related to the buckling mode propagate against the wind. The viscoelastic Kelvin-Voigt model is used for describing the viscous damping. The quiescent state of the plate is computed as a function of the stress applied. We use the Lifshitz-Landau thin-plate treatment for obtaining a description of the dynamics of small disturbances in a viscoelastic plate under stress in the framework of Kirchhoff-Love model. The dispersion relation of the model is computed; it is a polynomial of wavenumber and frequency. Stability computations are performed for a great variety of values of physical parameters, for the stress produced by both a laminar as well as a turbulent air flow.
Particular attention is given to plates made of aluminum or steel. For vanishing viscosity, it is found that all the non-zero wavenumbers in the model are unstable for any non-zero value of the stress applied. A non-vanishing viscosity of plates made of aluminum or steel produces a short-wave cut-off of the unstable wavenumbers but makes practically no influence on the growth rate of the normal modes close to its maximum value. In all the cases treated, it was found that all the unstable normal modes possess negative phase speeds, and the maximum of the growing wave packet propagates against the direction of the air stress applied. In all the cases, the model is found to be absolutely unstable. The absolute instability characteristics are computed as functions of the Reynolds number based on the distance from the leading edge of the plate.
We argue that the results of the analysis suggest that the destabilization of a variety of flows having plates as solid boundaries, such as boundary layers on the surfaces of flying vehicles, channel flows, duct flows etc., cannot be treated without taking account of the absolute instability of the plates involved.

MSC:
74H55 Stability of dynamical problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74K20 Plates
74D05 Linear constitutive equations for materials with memory
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