University of Utah
Department of Mathematics
Office: LCB 317
Many biological processes, specifically blood clotting, involve multi-phase fluid flow where one of the fluids is Newtonian and the others are viscoelastic. My PhD research has led me to investigate problems involving two phase flow, to derive equations and construct numerical algorithms that are capable of accurately simulating the movement of the fluids over time. My approach so far has been to represent the Newtonian fluid in Eulerian coordinates and the visco elastic fluid in Lagrangian coordinates. This approach seems to make physical sense given the systems we are trying to represent though mixing coordinate systems presents a wide array of mathematical and numerical challenges. There are many open questions that remain for how to best construct this type of model in a way that is consistent with basic physical conservation laws, scientific observation, and previous mathematical models that have been demonstrated to be adequate representations of reality.
I spend the majority of my time constructing elaborate numerical algorithms that are intended to approximate solutions to the complex systems of partial differential equations I am studying. In addition to having a high level of accuracy, these algorithms need to be fast and memory efficient. I have designed several GUI's in MatLab that provide a better user interface for running these simulations.
My research is one of the sub-projects of a much greater scientific investigation that uses mathematical techniques to better understand the nature blood clotting and is lead by my PhD adviser, Aaron Fogelson. My hope is that one day my own research will contribute in some way to our understanding of human blood flow and the formation of blood clots, which would increase the capacity of our medical industry and improve the quality of life for millions of people around the world. If I am lucky, the results of my research may prove to be useful in other fields of science and engineering as well.
V. Camacho, A. Fogelson, and J. Keener. Eulerian--Lagrangian Treatment of Nondilute Two-Phase Gels. SIAM Journal of Applied Mathematics 2016 February 18; 76(1): 341 - 367
V. Camacho, R. D. Guy, and J. Jacobsen. Traveling Waves and Shocks in a Viscoelastic Generalization of Bergers' Equation. SIAM Journal of Applied Mathematics 2008 April 11; 68(5):1316-1332
Documentation for Important Numerical Packages
IBAMR - Immersed Boundary Adaptive Mesh Refinement
SAMRAI - Structured Adaptive Mesh Refinement Applications Infrastructure
PETSc - Portable, Extensible Toolkit for Scientific Computation
HYPRE - A library of high performance preconditioners that features parallel multigrid methods for both structured and unstructured grid problems
Silo - A general purpose library and file format for visualizing scientific data