INSTABILITY OF A LIQUID-FILM FLOW OVER A VIBRATING INCLINED PLANE

The problem of the onset of instability in a liquid layer flowing down a vibrating inclined plane is formulated. For the solution of the pro blem, the Fourier components of the disturbance are expanded in Chebyc hev polynomials with time-dependent coefficients. The reduced system o f ordinary differential equations is analysed with the aid of Floquet theory. The interaction of the long gravity waves, the relatively shor t shear waves and the parametrically resonated Faraday waves occurring in the film flow is studied. Numerical results show that the long gra vity waves can be significantly suppressed, but cannot be completely e liminated by use of the externally imposed oscillation on the incline. At small angles of inclination, the short shear waves may be exploite d to enhance the Faraday waves. For a given set of relevant how parame ters, there exists a critical amplitude of the plane vibration below w hich the Faraday wave cannot be generated. At a given amplitude above this critical one, there also exists a cutoff wavenumber above which t he Faraday wave cannot be excited. In general the critical amplitude i ncreases, but the cutoff wavenumber decreases, with increasing viscosi ty. The cutoff wavenumber also decreases with increasing surface tensi on. The application of the theory to a novel method of film atomizatio n is discussed.