Flow-Induced Wave Generated on a Thin Film in a Narrow Gap

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This paper deals with a theoretical stability analysis of an unstable wave generated on a thin film subjected to a fluid flow in a narrow gap. In stability analysis, the basic equations of fluid flow around the thin film are based on the Navier-Stokes equations integrated over a gap width between the thin film surface and side wall, assuming that the gap width is enough small compared with a length of the passage. The structural equation of the thin film in the transverse motion is based on the Kirchhoff-Loves thin-plate model. From these basic equations, the governing equations of the thin film coupled with the fluid flow are obtained, employing the moving boundary conditions of the surface of the thin film. These equations are linearized around the equilibrium position, and the dispersion relation of the wave motion is derived as a function of the flow velocity. As a result, the analytical results show that traveling-wave type unstable wave occurs to the thin film due to the fluid flow in the narrow gap and clarify the dispersion relation of the wave motion, phase velocity (traveling-wave velocity) and growth rate of the wave. Moreover, the analytical results are verified by experiments on the most unstable wave number and critical flow velocity.