Duke Physics

Senior Thesis

Berk Sensoy

April 25, 1999

Thesis Committee: Henry Greenside, Robert Behringer, Ronen Plesser

Welcome to my research page. This page is dedicated to the work I did while a senior at Duke University majoring in Physics. I worked with Professor Henry Greenside investigating pattern formation in a convecting fluid. Geometrical patterns are found in many natural phenomena; among them are clouds and crystals. Much important research involves investigating the reasons why some types of patterns are preferred over others in certain systems. This is important because the geometrical configuration of patterns in systems relates strongly to many physical properties. The nature and mechanisms of these relations are not well understood, because of the high degree of complexity of pattern-forming systems.

My research involved the pattern formation of a convecting fluid in an annular domain. A convecting fluid is arranged in rolls of rising and falling fluid. Annular domains are interesting because the geometry of the domain conflicts with the natural configuration of the rolls. My research was one of the first studies of a convecting fluid in an annular domain. Our predictions are based on a numerical study of a model equation that captures many of the important dynamics of convection. Below is an image of a pattern we predict, borrowed from Professor Greenside's home page.

Snapshot from a simulation investigating how the dynamics of a sustained nonequilibrium system depends on the boundaries of the system. This color density plot shows a field u(t,x,y) that satisfies a Generalized Swift-Hohenberg model in an annular domain of inner radius r₁=10, outer radius r₂=80, fluid depth d=3.14, and reduced Rayleigh number r=0.5. The blue and red stripes correspond to convection rolls of falling and rising fluid respectively. One sees spiral, target, and dislocation defects as well as a perpendicular orientation of the convection rolls with respect to the lateral boundaries. From Berk Sensoy's Duke undergraduate physics thesis "Pattern Formation of a Convecting Fluid in an Annular Domain (1998)".

ABSTRACT: Nonequilibrium physics is concerned with the dynamics of systems that have been driven out of equilibrium. This thesis summarizes some theoretical research concerning how the lateral boundaries of a nonequilibrium sys- tem a ect the types of patterns that can form in that system. Using the Swift-Hohenberg equation, a widely studied two-dimensional model of three- dimensional Rayleigh-B enard convection, we study pattern formation and the transition to chaos in a convection cell with an annular geometry. This analysis was accomplished with a newly-developed computer code, speci - cally designed for the case of an annular geometry, that is capable of inte- grating the Swift-Hohenberg equation e ciently over long periods of time and for large spatial domains. We rst investigate the dependence of ob- served steady-state patterns on several parameters: the initial conditions, the aspect ratio, and the reduced Rayleigh number . We then discuss the relationships between possible steady-state patterns analytically predicted by Cross[Phys. Rev. A, 25, 1065 (1982)] and the steady-state patterns predicted by our numerical model. We nd that the preferred steady-state pattern is locally periodic rolls separated by topological defects and discuss the depen- 1 dence of roll wavenumber on linear stability as analyzed by Greenside and Coughran[Phys. Rev. A, 30, 398 (1983)]. A Lyapunov functional density and the average heat transport per unit area are calculated for each steady- state pattern to aid in the analysis of their relative stabilities. Finally, we investigate the Swift-Hohenberg equation coupled to a vertical vorticity eld, the Generalized Swift-Hohenberg Model. We nd that for a regime corre- sponding to a moderate-Prandtl-number uid in a cell of su ciently large aspect ratio, no steady-state pattern is attained: starting from random initial conditions, the uid motion exhibits characteristics similar to spiral-defect chaos. Our predictions can be tested by laboratory experiments on uids with moderate to large Prandtl numbers near onset.

Here's the thesis in PDF format (about 3 MBytes). Here's the thesis in postscript (about 11 MBytes). Enjoy!

If you'd like to talk about this work, send me email at bas7@phy.duke.edu.