Vibrational modes of a thin sheet in a pressurized chamber

We investigate the intricate dynamics arising from the vibrations of a thin sheet in a compressible fluid medium. The system comprises a closed chamber that is filled with an ideal gas and is divided into two parts by a thin sheet. We develop an analytical model that combines the elasticity of thin sheets with the hydrodynamics of compressible fluids to analyze how the sheet's material properties and compression affect the system's lowest mode of vibration. We employ a linear stability analysis around three distinct solutions of the static equations and show that the lowest mode consists of either a vortexlike patterned flow that is localized around the area of the sheet or an extended, oscillating net flow. While these eigenmodes are influenced mainly by the ratio between the thermal energy of the fluid and the bending rigidity of the sheet, the frequency of the vibrations also depends on the mass ratio between the sheet and the fluid. Notably, our findings can provide guidelines for the design of technological applications that require precise control over elastohydrodynamic interactions.