Non-linear modal interactions in the oscillations of a liquid drop in a gravitational field

Non-linear modal interactions in the dynamics of a vibrating drop are exami ned. The partial differential equations governing the drop vibrations are f ormulated assuming potential flow and incompressibility. The solution is ex pressed in terms of the eigenfunctions of the (linearized) Laplace operator in spherical coordinates. A small parameter epsilon is introduced to scale the (small) deformation of the drop surface from its position of equilibri um. A 2: 1 internal resonance is then imposed between the second and third modes of the resulting discretized system, and the ensuing non-linear modal interactions are studied using the method of multiple scales. A bifurcatio n in the slow dynamics of the system is detected that leads to amplitude mo dulations of the drop oscillations. The method employed in this work is gen eral and can be used to study other types of non-linear interactions involv ing two or more drop modes. (C) 2001 Elsevier Science Ltd. All rights reser ved.