Nonlinear stability of a fluid-loaded elastic plate with mean flow

It has been known for some time that the unsteady interaction between a sim ple elastic plate and a mean flow has a number of interesting features, whi ch include, but are not limited to, the existence of negative-energy waves (NEWs) which are destabilized by the introduction of dashpot dissipation, a nd convective instabilities associated with the how-surface interaction. In this paper we consider the nonlinear evolution of these two types of waves in uniform mean flow. It is shown that the NEW can become saturated at wea kly nonlinear amplitude. For general parameter values this saturation can b e achieved for wavenumber k corresponding to low-frequency oscillations, bu t in the realistic case in which the coefficient of the nonlinear tension t erm tin our normalization proportional to the square of the solid-fluid den sity ratio) is large, saturation is achieved for all k in the NEW range. In both cases the nonlinearities act so as increase the restorative stiffness in the plate, the oscillation frequency of the dashpots driving the NEW in stability decreases, and the system approaches a state of static deflection tin agreement with the results of the numerical simulations of Lucey tt al . 1997). With regard to the marginal convective instability, we show that t he wave-train evolution is described by the defocusing form of the nonlinea r Schrodinger (NLS) equation, suggesting (at least for wave trains with com pact support) that in the long-time limit the marginal convective instabili ty decays to zero. In contrast, expansion about a range of other points on the neutral curve yields the focusing form of the NLS equation, allowing th e existence of isolated soliton solutions, whose amplitude is shown to be p otentially significant for realistic parameter values. Moreover, when slow spanwise modulation is included, it turns out that even the marginal convec tive instability can exhibit solitary-wave behaviour for modulation directi ons lying outside broad wedges about the flow direction.