Dr. David Ambrose, one of 13 new professors within the College of Arts and Sciences, held his first Dean's Seminar on October 15th, 2008. The first-year professor spoke to a group of approximately 20 students and faculty members in Disque 109 about his research as a mathematician specializing in analysis and scientific computing.

Through the use of concrete examples, Ambrose highlighted the relevance of his work with free surface flows. Many real-life fluid flows involve a free surface with no boundary containing or shaping the fluid's movement, and through the study of these flows, practical uses can be developed.

Beginning with geophysical examples, Ambrose explained that water waves are the most frequently explored free surface flows. By studying water waves, scientists can potentially predict the occurrence of destructive natural phenomenon like rogue waves, and design warning systems to mitigate their impact. Keeping with this idea, studying the free flow of water waves can lead to the harnessing of electrical energy, which could be used to generate power. Examining the free flow of groundwater could illuminate for scientists the way pollutants mix with water, which leads to a better understanding of how pollution spreads. Similarly, the mixing of various free-flowing fluids occurs in the process of oil drilling.

Continuing to emphasize the practicality behind studying free surface flows, Ambrose gave examples of fluid-structure interactions relative to scientific advancement. By observing animals and insects swimming and using the water to control their movements, new things could be discovered about the various means of locomotion. Studying the flow of blood and its relationship to veins and arteries could potentially save lives if blockage prevention is further dissected. For defense specialists, exploring supercavitation, or the process used in launching torpedoes in the water, certainly seems worthwhile.

Although he spoke knowledgably about the physical reality of free surface flows, Ambrose's work does not directly correlate with these concrete examples.

"I don't really study these practical examples. I study equations," he told the audience. "The hope is that the sets of differential equations I study have an impact in a practical way."

Some of the areas Ambrose explores when working with partial-differential equations are the well-posedness (or reliability and uniqueness) of the equation, its stability and instability, further modeling for similar equations, and the existence of special solutions.

The explicit and implicit approaches are two methods employed to solve the partial-differential equations implicated in free surface flows. In the explicit approach, you assume there is a free surface and track it, while in the implicit approach, the equation must be coupled with another equation where a free surface is implied.

In his research, Ambrose has used these approaches to understand curvature mathematically, as it affects free surface flow. Instead of using the Cartesian coordinates to describe the position of the curve, he describes the position using the curve itself. Ambrose has used this idea while performing computational simulations of free surface problems. He has proved two theorems using this theory in 2D and 3D free flowing fluids, related to short-time well-posedness.