Math at the Boundary in Sweden

Professor of Mathematical Sciences Michael Siegel

Professor of Mathematical Sciences Michael Siegel will be spending part of a forthcoming sabbatical year in Sweden taking a close look at some important problematic boundaries — specifically moving-boundary problems presented by emulsions that involve the dispersion of microscopic droplets, such as droplets of oil in water. Siegel’s research in Sweden as a visiting professor of numerical analysis at the KTH Royal Institute of Technology is funded by a grant from the Knut and Alice Wallenberg Foundation, one of the largest private supporters of scientific research in Europe.

A member of NJIT’s Department of Mathematical Sciences since 1995, Siegel is an expert in the analysis and numerical computation of moving-boundary problems that arise in fluid mechanics, materials science and biology. A moving boundary is the dynamic interface between diverse phases, like that between oil and vinegar in salad dressing. Emulsions — widely important in foods, personal care products, pharmaceuticals, and many other applications — are mixtures of two or more liquids that are normally unmixable or unblendable.

The challenge is to create more stable forms of such “immiscible” mixtures. Mayonnaise is a classic example. It is an oil-in-water emulsion stabilized with egg-yolk lecithin or other types of food additives. In pharmaceuticals, minute amounts of a medically active agent are often dispersed in an inert medium. The nanoemulsions used to deliver vaccines can consist of active particles 400-600 nanometers in diameter dispersed in soybean oil. A nanometer is one-billionth of a meter.

Model Solutions

Modeling and numerically simulating moving-boundary behavior is an essential aspect of creating new, commercially valuable emulsions and improving many already in use. But as Siegel emphasizes, developing accurate and efficient numerical methods to study a fluid that can have lots of bubbles or drops of a different substance is not an easy task.

The difficulty is further compounded if the fluid includes a surfactant, a substance intended to stabilize an emulsion by preventing constituent drops from coalescing into larger ones or breaking up into smaller droplets. A surfactant works by coating the drops with a molecular barrier that preserves their form. The research that Siegel will pursue in Sweden has significant real-world implications in many fields, including delivery of therapeutic agents either as discrete components or on the surface of different constituent droplets.

Over the years, Siegel’s work with NJIT colleagues, among them Professor Michael Booty, has yielded progress with respect to relatively fast algorithms for numerically simulating moving-boundary problems. In 2015, he will take additional steps toward computing even faster and more accurate solutions with Anna Karin Tornberg, KTH professor of numerical analysis.

Siegel and Tornberg began a collaborative relationship while he was on a previous sabbatical, at New York University’s Courant Institute of Mathematical Sciences. Finding that they had approached the same problems from different perspectives, the two researchers decided to work together to create a synthesis of their efforts.

The Wallenberg grant is an opportunity for Siegel and Tornberg to focus exclusively on this computationally difficult challenge in Sweden. A key goal is to build on the utility of numerical algorithms that they and others have found to work well for problems in two dimensions, and to extend them to more difficult problems in three dimensions.       

“We’re combining forces and approaches,” Siegel says. “By working together and combining our knowledge, and using some promising new approaches to simulating the motion of surfaces, we hope to come up with better and faster methods for solving three-dimensional problems.”

CAMS on Campus

Along with being an educator and researcher, Siegel is director of the Center for Applied Mathematics and Statistics (CAMS) based at NJIT. The center’s activities include bringing national and international experts in the application of mathematics in numerous scientific and technical areas to campus for Frontiers in Applied and Computational Mathematics, an annual conference that CAMS has hosted for more than a decade.

CAMS is also at the forefront of facilitating the use of mathematical modeling and numerical computation across disciplines at NJIT, which is very much in keeping with the collaborative spirit of the university’s current strategic vision. Mathematics is more popular and important than ever before, Siegel says, as a tool for research and discovery in virtually every field. “You see advanced math used to design new materials, analyze social and economic trends, and make sense of all the data being generated in the health sciences and other areas. It’s really a great time to be a math person.”

By Dean Maskevich