A sufficient condition for the absence of two-dimensional instabilities of an elastic plate in a duct with compressible flow ^∗

We study the time-harmonic resonance of a finite-length elastic plate in a fluid in uniform flow confined in a duct. Although the resonance frequencies are usually real, the combined effects of plate elasticity and of a flow can create complex frequencies, different from the usual so-called scattering frequencies, corresponding to instabilities. We study theoretically the existence of instabilities versus several problem parameters, notably the flow velocity and the ratio of densities and of sound speeds between the plate and the fluid. A three dimensional volume in the parameters space is defined, in which no instability can develop. In particular it corresponds to a low enough velocity or a light enough plate. The theoretical estimates are validated numerically.