Document Type


Date of Award


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Anand Uttam Oza

Second Advisor

Lou Kondic

Third Advisor

Linda Jane Cummings

Fourth Advisor

Boris Khusid

Fifth Advisor

Daniel M. Harris


Interacting particles are a common theme across various physical systems, particularly on the atomic and sub-atomic scales. While these particles cannot be seen with the human eye, insight into such systems can be gained by observing macroscopic systems whose physical behavior is similar. This dissertation consists of three different chapters, each presenting a different problem related to interacting particles, as follows:

Chapter 1 explores chaotic trajectories of a droplet bouncing on the surface of a vertically vibrating fluid bath, with a simple harmonic force acting on the droplet. The bouncing droplet system has attracted recent interest because it exhibits behaviors similar to those previously observed only in the microscopic quantum realm. In this investigation, an approximation is presented for an existing model that describes a walking droplet's trajectories in the xy-plane. Simulations of the approximate model show that it captures behaviors of the full model, but is less computationally expensive to simulate.

Chapter 2 presents the results of a theoretical investigation of one-dimensional chains of bouncing droplets, with particular attention to the case in which the drop at one end of the chain is oscillated periodically. Numerical simulation results are also presented, and are found to compare favorably with predictions based on linear theory. Both demonstrate the existence of resonant forcing frequencies, as well as the presence of an oscillatory instability in the chain as the forcing acceleration is increased. For sufficiently high forcing frequencies, dynamic stabilization of the chain into a new bouncing state is observed. The results of this investigation highlight the role of temporally nonlocal interactions in the dynamics of this unique system, and have been published in the journal Comptes Rendus Mecanique.

Chapter 3 investigates the phase behavior of colloid-polymer suspensions in a microgravity environment. Phase transitions in such suspensions give valuable insight into atomic-scale phase transitions. While colloidal suspensions exhibit fluid-solid coexistence, the addition of polymer leads to three-phase coexistence. A numerical exploration of the effects of fluid viscosity and other system parameters on the phase behavior of colloid-polymer suspensions is presented. The first few sections of Chapter 3 describe the process of downloading and processing experimental images from a NASA database and extracting information from them in order to quantitatively characterize the separation of the mixture into distinct phases as time progresses. The remainder of this chapter is dedicated to the development and implementation of a phase-field model that couples a Cahn-Hilliard equation for the colloid concentration with the incompressible Stokes equations to include the effects of hydrodynamic interactions between the particles and the surrounding fluid in the low-Reynolds number limit. The model also incorporates the dependence of viscosity on the local colloid concentration, which significantly influences the hydrodynamic behavior. Simulation results for varied system parameters are presented and discussed.



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