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

Non-linear modal interactions in the dynamics of a vibrating drop are examined. The partial di!erential equations governing the drop vibrations are formulated assuming potential #ow and incompressibility. The solution is expressed in terms of the eigenfunctions of the (linearized) Laplace operator in spherical coordinates. A small parameter e is introduced to scale the (small) deformation of the drop surface from its position of equilibrium. 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 bifurcation in the slow dynamics of the system is detected that leads to amplitude modulations of the drop oscillations. The method employed in this work is general and can be used to study other types of non-linear interactions involving two or more drop modes. ( 2001 Elsevier Science Ltd. All rights reserved.