## Sarah Elizabeth Ritche Patterson Research Assistant, Ph D Student

I am currently a fifth year graduate student in mathematics at Duke University. My thesis advisor is Anita Layton.

### Current Research Interests

My current research interests include computational fluid dynamics, numerical analysis, and mathematical modeling. I work under the direction of Anita Layton.

Immersed Interface Method
The immersed interface method is a technique for solving differential equations that contain a singularly supported force which causes solutions to be non-smooth or even discontinuous. These forces are could be the result of an elastic interface that is immersed in a fluid. For this method, the fluid equations are solved on an Eulerian grid while the interface is represented using Lagrangian variables that are not restricted to the underlying fluid grid. The immersed interface method corrects discontinuities in the fluid solution by using the force implicitly to compute the magnitudes of jumps in the solutions and their derivatives.

Modeling Blood Flow
I am interested in modeling the myogenic response of the afferent arterial to changes in pressure. In this application, the blood vessel walls are represented as an infinitely thin elastic tube that is immersed in fluid. This tube can actively contract and dilate in response to environmental changes. This interface exerts a singularly supported force into the internal and external fluid. Therefore, I use the Immersed Interface Method to accurately solve the fluid equations and capture the vessel wall movement.

Extentions of  Immersed Interface Method to an Open Tube
The immersed interface method traditionally applies only to interfaces that are closed. Previous researchers have overcome this challenge by modeling blood vessels as a tube with half-spherical caps on each end. Flow is driven by adding a source and sink to opposite ends of the vessel. This, however, creates unatural flow around the source and sinks. Additionally, there is no canonical way to specify these source and sink terms.   I am currently working to extend the Immersed Interface Method to work in an open tube.  By adding a fictitious interface outside of the computational fluid domain, similar correction terms can be computed for this open tube model. This model will allow me to better capture how blood flows in the micro-circulation, since I will not need to add in an unnatural source and sink term.

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• Teaching & Mentoring

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