Nonlinear dynamics of a thin liquid film on an axially oscillating cylindrical surface subjected to double-frequency forcing

摘要

The nonlinear dynamics of a thin axisymmetric liquid film on a horizontal cylindrical substrate subjected tonan axial double-frequency forcing that consists of two components of different amplitudes and frequencies andna possible phase shift is considered in this paper. A nonlinear evolution equation governing the spatiotemporalndynamics of the film interface has been derived in the long-wave limit. Similar to the case of a single-frequencynforcing considered in our earlier work, there exists a critical forcing amplitude below which the film undergoesna long-time capillary rupture typical for a static cylinder, whereas above it the film remains continuous. We findnthat it is possible to arrest the rupture even if the forcing parameters of each of the two components correspondnseparately to the domain where rupture takes place. It is shown that the critical forcing amplitude is easilyndetermined via a single-frequency case when the two forcing frequencies are equal. In the case of differentnforcing amplitudes and frequencies, the variation of the critical forcing amplitude as a function of the frequencynratio exhibits a unique behavior displaying the emergence of spikes. A related case of an amplitude-modulatednsingle-frequency forcing is also addressed here. For a sufficiently small frequency of the amplitude modulation,na significant increase of the pattern amplitude is observed. In the case of commensurate forcing frequencies, thenflow is found to be quasiperiodic.