Characterization of Chaotic Motion in a Rotating Drum Open Access

Davidheiser, James E (2009)

Permanent URL: https://etd.library.emory.edu/concern/etds/xg94hq20s?locale=en
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Abstract

Abstract
Characterization of Chaotic Motion in a Rotating
Drum
By
James E. Davidheiser
Numerous studies in the past have demonstrated the potential for geometrically sim-
ple fluid systems to produce complicated dynamical behavior. In particular, small
collections of non-Brownian particles moving within viscous fluids can follow chaotic
trajectories. In this work, we study a rotating drum filled with pure glycerol and
three large, heavy particles.
In studying these rotating drum systems, we have found a rich and varied phase
space, made up of several previously unseen behaviors. With varying rotation rate,
the particles can undergo straightforward cascading periodic behavior, and grouped
periodic cascades we have labeled as doublet and triplet states. Furthermore, we find
two regimes of qualitatively distinct chaotic behavior, with one type biased to either
side of the drum, and the other lacking bias.
The rotating drum experiment serves as a simple model system to demonstrate
chaotic behavior in fluid dynamical systems. The existence of such model systems
gives a baseline to which other systems can be compared and better understood,
and our use of robust, easily implemented measurements serves as a straightforward
comparison point which can be applied to various other chaotic fluid systems.

Table of Contents

1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Stokeslet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Chaotic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Chaos in Pure Sedimentation . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Orbits and Bounded Behavior in Sedimentation . . . . . . . . . . . . 10
1.6 The Rotating Drum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Theory 15
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Drum Spin-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Ekman Pumping . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Fluid Effects Due to Beads Moving Within the Drum . . . . . . . . . 18
2.5 Particle-Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Instrumentation and Methods 24
3.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1 The Drum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Temperature Control . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.3 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Particle Tracking Software . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Notes and limitations . . . . . . . . . . . . . . . . . . . . . . . 314

Results and Analysis 34
4.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.1 Description of the Five Phases . . . . . . . . . . . . . . . . . . 35
4.2.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.3 Transients and Motor Stability . . . . . . . . . . . . . . . . . 44
4.3 Qualitative Fluid Behavior . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Periodic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.2 Doublet Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.3 Triplet Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.4 Biased Chaotic Regime . . . . . . . . . . . . . . . . . . . . . . 48
4.3.5 Fully Chaotic Regime . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Reduced Dimensionality Analysis . . . . . . . . . . . . . . . . . . . . 52
4.5.1 Reduced Dimensionality . . . . . . . . . . . . . . . . . . . . . 52
4.5.2 2-D Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.4 Bin Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Summary and Outlook 64
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Bibliography 68
A Video Capture Software 72
A.1 Software Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.1.1 Pixelink Software . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.1.2 Installing Stream-Pixelink . . . . . . . . . . . . . . . . . . . . 73
A.2 Setting up the PixeLINK IEEE1394 Camera . . . . . . . . . . . . . . 74
A.2.1 Basic Controls Tab . . . . . . . . . . . . . . . . . . . . . . . . 75
A.2.2 Region of Interest Tab . . . . . . . . . . . . . . . . . . . . . . 76
A.2.3 External Control Tab . . . . . . . . . . . . . . . . . . . . . . . 77
A.3 Running Stream-Pixelink . . . . . . . . . . . . . . . . . . . . . . . . . 78
B Video Tracking Software 80
B.1 Software Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
B.2 TrackFramebeta.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
B.3 TrackMovieSeq.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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