Nonlinear dynamics of viscous droplets

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Abstract
Nonlinear viscous droplet oscillations are analysed by solving the Navier-Stokes equation for an incompressible fluid. The method is based on mode expansions with modified solutions of the corresponding linear problem. A system of ordinary differential equations, including all nonlinear and viscous terms, is obtained by an extended application of the variational principle of Gauss to the underlying hydrodynamic equations. Results presented are in a very good agreement with experimental data up to oscillation amplitudes of 80% of the unperturbed droplet radius. Large-amplitude oscillations are also in a good agreement with the predictions of Lundgren & Mansour (boundary integral method) and Basaran (Galerkin-finite element method). The results show that viscosity has a large effect on mode coupling phenomena and that, in contradiction to the linear approach, the resonant mode interactions remain for asymptotically diminishing amplitudes of the fundamental mode.