Triangular shapes can be used to model fluid droplets

To ease the study of the dynamics of large populations of droplets, there is a need for a minimal liquid droplet model. In a recent paper, Elizabeth Wesson and Paul Steen aim to find the simplest model for droplet motions by replicating parameters, such as the droplet’s surface tension and internal pressure.

The Steiner triangle is a triangular geometry that deforms with a constant area. Its one-to-one mapping to ellipses makes it a useful model for elliptical droplets. The triangle shares a center with the ellipse, and changing its base length corresponds to a change in the drop’s contact line.

Wesson and Steen found such a triangle can predict the small amplitude bouncing and rocking motions of a fluid droplet’s center of mass, and the model exhibits the long-term dynamics and symmetry-breaking motions of real droplets. The results of the model are based on a single relevant parameter, the contact angle between the triangle’s base and one of its sides when the system is at rest.

“We were surprised by the dynamical richness that came out of such a simple model, and that such a simple geometric model can have anything to say about actual drops,” Wesson said.

Though the authors’ goal was to develop a model with minimal degrees of freedom, they note the method is extendable in a variety of directions.

“A natural next step is an investigation of tetrahedral drops similarly governed by surface tension and internal pressure. Then one could model populations of interacting minimal drops,” said Wesson. “The minimal nature would make computations much easier than with spherical cap drops.”

Source: “Steiner triangular drop dynamics,” by Elizabeth Wesson and Paul Steen, Chaos (2020). The article can be accessed at https://doi.org/10.1063/1.5113786 .