Research

This was some work that I did in order to have the tools to look at composed multivariate Taylor series. I have submitted it to the journal of Applicable Analysis and Discrete Mathematics.

Abstract: How do we take repeated derivatives of composed multivariate functions? for one-dimensional functions, the common tools consist of the Faá di Bruno formula with Bell polynomials; while there are extensions of the Faá di Bruno formula, there are no corresponding Bell polynomials. In this paper, we generalize the single-variable Bell polynomials to take vector-valued arguments indexed by multi-indices which we use to rewrite the Faá di Bruno formula to find derivatives of f(g(x)).

The pdf for a presentation I gave on research I did during the spring of 2018. I presented this work at the 71st Annual Meeting of the APS Division of Fluid Dynamics.

Abstract: Two-dimensional point vortices in Euler fluids are a common tool to model airfoils and geophysical flows. For instance, ocean vortices can interact with coastlines and other geometries that affect their motion. Such interactions have been studied around circular islands and in bays. While the flow external to an elliptical island is known and has applications to airfoil theory, the corresponding flow inside an elliptical boundary has not been studied. Here, we show an analytic solution for the flow field due to an ideal two-dimensional point vortex in an elliptical boundary, as well as the motion of the vortex. We use conformal mapping to find an image system that satisfies the boundary conditions of no normal flow through the ellipse walls. This results in flow fields similar to those within a circular boundary but with the streamlines stretched to fill the ellipse. Similar to the circular case, the point vortex traces out concentric ellipses as it moves around the model bay.

The pdf to a presentation I gave on Summer research I did in Leipzig on the stability of Barchan sand dune fields

Abstract: Barchan sand dunes are crescent-shaped dunes that are are found when there is limited sand and unidirectional wind. A single isolated dune is unstable — it will grow to infinity or shrink to nothing with no stable equilibrium. However, we observe large fields of these dunes with all the dunes around the same size and similar distances apart. Clearly there is some stability mechanism. My project was to write a simulation to further explore barchan dune fields.

Here's a write-up for a quick problem that my professor, Greg Elliott, handed me. The goal was to find the shape of the equipotential surfaces on earth due to the moon and sun, or for any combination of celestial bodies.

Abstract: In this paper we demonstrate a derivation for some simple equations describing the equipotential surfaces of tidal forces. We find three equations with varying degrees of simplifying assumptions. Using these equations we map these surfaces for the earth-moon and the earth-moon-sun systems.