Michael C Sostarecz
One of my projects at Penn State dealt with the Motion and shape of a viscoelastic drop falling through a viscous fluid (published in the Journal of Fluid Mechanics). For small drops, surface tension dominates and the steady shape is a sphere. As volume is increased, elastic stresses cause the drop to deform into an oblate spheroid. Progressing further, the drop loses vertical symmetry and develops a dimple at the trailing edge. This can be seen as the inverse problem to the well-known cusped air bubble you can see if you flip over a bottle of shampoo. However, in our case the cusp pointed inside:
Modeling this problem gave us a system of PDEs (Partial Differential Equations) for the flow fields of the two fluids and the free boundary in between. Using a double perturbation analysis, we solved these equations to obtain a prediction for the shape. The predicted shapes below use the same measured volumes and velocities as in the top picture:
For volumes around 1ml, the dimple becomes unstable and we see a filament dragged downward by a falling pendent drop. Beyond the validity of a perturbation analysis, our prediction has the same instability as the experiment. The filament then undergoes a Rayleigh instability and breaks into many smaller oil droplets. For volumes larger than around 5ml, the filament coalesces with the leading edge of the drop before the filament would break up. This then stabilizes the filament and the resulting steady state shape of the free surface is a torus.
Some of these experimental images on drops were presented in the 20th Annual Gallery of Fluid Motion at the 2002 APS Division of Fluid Dynamics 55th Annual Meeting. Our winning entry, Dynamics Inside Polymer Drops: From Dimple to Rayleigh Instability to Torus was selected to be published in a special section of the September 2003 issue of Physics of Fluids.