FINITE-WAVELENGTH INSTABILITY IN A HORIZONTAL LIQUID LAYER ON AN OSCILLATING PLANE

The linear stability of a thin liquid layer bounded from above by a fr ee surface and from below by an oscillating plate is investigated for disturbances of arbitrary wavenumbers, a range of imposed frequencies and selective physical parameters. The imposed motion of the lower wal l occurs in its own plane and is unidirectional and time-periodic. Lon g-wave instabilities occur only over certain bandwidths of the imposed frequency, as determined by a long-wavelength expansion. A fully nume rical method based on Floquet theory is used to investigate solutions with arbitrary wavenumbers, and a new free-surface instability is foun d that has a finite preferred wavelength. This instability occurs cont inuously once the imposed frequency exceeds a certain threshold. The n eutral curves of this new finite-wavelength instability appear signifi cantly more complex than those for long waves. In a certain parameter regime, folds occur in the finite-wavelength stability limit, giving r ise to isolated unstable regions. Only synchronous solutions are found , i.e. subharmonic solutions have not been detected. In Appendix A, we provide an argument for the non-existence of subharmonic solutions.