Stochastic and Deterministic Processes in Fragmentation and Sedimentation
Open Access
- Author:
- Higley, Michael
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- November 17, 2009
- Committee Members:
- Andrew Leonard Belmonte, Dissertation Advisor/Co-Advisor
Andrew Leonard Belmonte, Committee Chair/Co-Chair
Qiang Du, Committee Member
Diane Marie Henderson, Committee Member
Francesco Costanzo, Committee Member - Keywords:
- Galton board
fragment size
settling speed - Abstract:
- In this work we present results of analysis, experiment and simulation of two phenomena involving stochastic and deterministic aspects. In the first case we present a modeling framework for 1D fragmentation in brittle rods, in which the distribution of fragments is written explicitly in terms of the probability of breaks along the length of the rod. This work is motivated by the experimental observation of several preferred lengths in the fragment distribution of shattered brittle rods after dynamic buckling. Our approach allows for non-constant spatial breaking probabilities, which can lead to preferred fragment sizes. The resulting relation is shown to qualitatively match experimentally observed fragment distributions, as well as some other commonly reported distributions such as a power law with a cutoff. We also present experimental observations of the trajectories and average velocities of solid spheres falling through a curtain of rising bubbles in water. For the quiescent case (no bubbles), the Reynolds numbers are on the order of 1,000, and the average terminal velocity is determined by the form (inertial) drag. The main effect of the introduction of bubbles is to slow down the spheres. In some regimes (smaller or lighter spheres), there is an added random lateral motion to the sphere paths. In this way, a solid sphere sinking in a bubbly fluid and a solid sphere falling through a crowded bed of rigid obstacles (in air) share two common traits: the settling speed is slowed by the obstructions, and the sphere exhibits random lateral motion. We present a mathematical model which begins as an adaptation of Galton's board to the sedimenting sphere. This allows us to introduce various physical effects of the bubbly fluid, and test their importance, particularly that of bubble collisions. Comparison is made with experimental results.