Victor Camacho

Harvey Mudd College Mathematics 2007

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Thesis Proposal: Modeling Wave Propogation in Viscoelastic Fluids
Thesis Advisor: Prof. Darryl Yong
Second Reader: Prof. Jon Jacobsen
Final Thesis Report (PDF): Analysis of Traveling Waves in Viscoelastic Fluids
Final Thesis Report (Latex): Analysis of Traveling Waves in Viscoelastic Fluids
Thesis Poster (PDF): Presentation Days Poster (4/30/07)
Simulation Files (Matlab): visco.m, viscocmd.m

Analyzing Traveling Waves in a Viscoelastic Generalization of Burgers Equation

Viscoelastic fluids are a broad class of fluids that exhibit both viscous and elastic properties. Fluid elasticity is the measure of a fluid's tendency to return to its original shape in the absence of external forces, and fluid viscosity is the measure of its resistance to flow. Common examples of viscoelastic fluids include bio fluids, tempurpedic mattresses, egg white, and corn starch in water. What distinguishes viscoelastic fluids from Newtonian fluids, such as water, is that their viscosity changes when stress is applied. A Newtonian fluid has no elastic property and thus keeps a constant viscosity under shear stress.

In this thesis project I analyzed a pair of nonlinear PDEs describing viscoelastic fluid flow in one dimension. Based on the boundary conditions and parameters, I was able to classify three different categories of traveling wave solutions. I thoroughly describe the numerical schemes and software program, VISCO, that were designed specifically to analyze the model I studied. The simulations lead me to conjecture that the traveling wave solutions are globally stable for perturbatoins with compact support. I give some analytical evidence in support of this hypothesis but am unsuccessful in providing a complete proof.

Graph showing traveling wave

Publications

Camacho, Victor, Bob Guy and Jon Jacobsen. Traveling Waves and Shocks in a Viscoelastic Generalization of Burgers Equation.
[Submitted on 4/10/07]