Applied Interdisciplinary Mathematics (AIM) Seminar

I will discuss two problems featuring the interaction between fluids and structures. The first involves propulsion through a fluid by flexible appendages, as in swimming or flying. A small-amplitude fluid model combined with an efficient Chebyshev PDE solver and conformal mapping techniques enables efficient solutions and generalizes to multiple flapping bodies through the so-called prime function. New results may shed light on collective behavior, such as schooling and flocking. The second problem involves the formation of sharp karst pinnacles found in stone forests around the globe. Recent experiments demonstrate how the interaction between dissolution and gravitationally-induced natural convective flows can create ultra-sharp features that resemble these karst structures. Boundary-layer analysis reduces shape evolution to a single integro-PDE on the surface. While previous numerical evidence suggested the formation of geometric shocks, here we find a class of exact equilibrium solutions with large, but finite, curvature. We show the laboratory-generated profiles converge to this solution, which may offer clues about karst pinnacles.
Speaker(s): M. Nick Moore (US Naval Academy)