UC Davis

Dissolving stretchable polymers in a Newtonian fluid can give the fluid a new property calledviscoelasticity. Viscoelastic fluids exhibit fascinating behaviors due to the feedback force created by stretched polymers trying to return to their original length. These fluids come with many mathematical challenges, including open questions about the dynamics of flows as well as numerical difficulties. The two parts of this dissertation each relate to one of those overarching categories. The first part of this dissertation is a broad exploration of a doubly periodic parallel shear flow known as Kolmogorov flow. This flow was chosen in part because it is one of the simplest flows knowntoexhibitcoherentstructuresknownasboth‘narwhals’and‘arrowheads,’ aswellasachaotic phenomenon known as elastic turbulence, both of which are active areas of study. Elastic turbulence sharesmanyfeaturesofNewtonianturbulence, butisadistinctandpoorlyunderstoodphenomenon. In particular, finding a dynamical origin or clear “route to chaos” has not been previously achieved for viscoelastic fluids. We catalog a large variety of flow states, including elastic turbulence, that can be reached with varying levels of elasticity in the fluid and various domains. These chapters contain many intriguing results, the most significant of which is the first evidence of a route to chaos in viscoelastic fluids. An initial instability in Kolmogorov flow generates traveling wave solutions in the form of coherent structures, which in turn lose stability as elasticity is increased. Oscillations in the traveling waves emerge, which proceed through a period doubling cascade and eventually become chaotic. The second part of this dissertation relates to numerical methods and will introduce what we call the Double Immersed Boundary (DIB) method. This is a modification of the existing Immersed Boundary (IB) method, which is very useful for simulating fluid-structure interactions, but cannot achieve convergence of the velocity gradients near boundaries, which are of particular importance to viscoelastic fluid simulations. The DIB method remedies this problem for the special case where solutions are only required on one side of the boundary. Naturally, there are tradeoffs, and the DIB method leads to challenging conditioning and stability issues that can be addressed in a variety of ways.