The oscillations of a drop laying on a substrate (sessile drop) is a subject critical to a number of emerging technologies, such as 3D printing (rapid prototyping), aerosol drug delivery, and the fabrication of semiconductor chips using immersion lithography. This grant will support research to develop a mathematical framework to predict and control the droplet dynamics on solid substrates in order to advance the aforementioned technologies, among others. Results from this research will, for example, provide an enhanced understanding of how drops change in shape during the 3D printing process which could lead to the ability to pinch-off multiple satellite droplets from a single inkjet for high resolution printing that would transform the industry through enhanced capabilities and significant improvement in printing speeds. Since rapid prototyping is widely used in the manufacturing industry, this research will benefit the U.S. economy as a whole. The research project is integrated with undergraduate education through the Creative Inquiry program at Clemson University where students will develop an experimental drop dynamics demonstration unit to be used as a module for undergraduate dynamics and vibrations courses to enhance their understanding of drop oscillations.
More than a century ago, Lord Rayleigh (1879) predicted that a free liquid drop will oscillate with a degenerate spectrum, reflecting the underlying rotational symmetry of the sphere, consistent with Noether?s theorem. Sessile drops differ from free drops through their interaction with the solid substrate and the symmetry can be broken spatially through dynamic wetting effects and temporally through mechanical excitation. Systems with broken symmetry need a mathematical framework around which to organize the associated spectra, because classical theories of spectral ordering are often not applicable. This research plans a mathematical framework using a bifurcation theoretic approach. The fluid dynamical field equations are mapped to the free-surface boundary using a boundary integral approach which results in an integrodifferential equation for the interface shape. This equation is recast as an operator equation from which one can systematically predict how the spectrum varies with the symmetry-breaking parameters. Simple spectral rules-of-thumb are immediate from this framework and can be used to develop a classification system. Mathematical homotopy can be established through deformation of the solid boundary which allows the sessile drop to be related to the liquid bridge and bead-on-fiber. Parametric oscillations in sessile drops will be analyzed using Floquet theory and a frozen interface approximation or dual-spectral method.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
