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NORTHWESTERN UNIVERSITY

Geometric Aspects of Mixing and Segregation in Granular Tumblers

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

for the degree

DOCTOR OF PHILOSOPHY

Field of Chemical Engineering

By

James F. Gilchrist

EVANSTON, ILLINOIS

June 2003

ii

© Copyright by James F. Gilchrist 2003

All Rights Reserved

iii

ABSTRACT

Geometric Aspects of Mixing and Segregation in Granular Tumblers

James F. Gilchrist

Recently, much attention has been given to granular systems due to their

widespread use in industrial processes and their often baffling physical behavior. In the

midst of the emerging field of complex systems, granular materials are quickly becoming

a prototypical system exhibiting spontaneous organization. Flowing or vibrated mixtures

of granular materials, differing in physical properties such as size or density,

spontaneously segregate. Constitutive equations describing the behavior of granular

materials, analogous to the Navier-Stokes equations, do not exist. This field, as

compared to fluids, is poorly understood.

Mixing in fluids may be enhanced by the addition of time-periodic perturbations

to the flow, generating chaos. This same concept is applied to granular flow in a tumbler.

A simple two-dimensional continuum model developed by Khakhar et al. 1997b

describing continuous flow in a circular cross section of a tumbler is modified to describe

flow in noncircular geometries. Time-periodic flow in a square is chaotic and sensitive to

fill level. When bi-disperse materials are tumbled in a square, there is direct competition

between mixing and segregation, and pattern formation is nontrivial – segregation

resembles invariant structures of the underlying flow. This model is extended to describe

iv

flow in a sphere and a cube to shed light on the interplay between segregation and chaotic

flow in three-dimensions.

The competition between granular mixing and segregation is also investigated in a

novel spherical tumbler capable of undergoing independent programmed motions in two

orthogonal directions. The primary mode of operation is a combination of rotating and

rocking of the axes. Space-time plots are used to compare experimental results with

surface Poincaré sections obtained using a continuum model of the flow. A phase plot

showing modes of segregation – band formation/no axial bands – in the

frequency/amplitude domain is used to organize the experimental results; segregated

bands are remarkably robust and survive rocking amplitudes of as much as 60 degrees.

Details differ, but the phenomenon occurs both in dry materials and under slurry

conditions.

v

ACKNOWLEDGEMENTS

The following people greatly supported me in my endeavors:

First, I am thankful for the guidance, support and inspiration of my advisor, Professor

Julio M. Ottino. His work in chaotic mixing drew me to graduate school at Northwestern

University, and his ongoing creative approach to research inspired me to pursue an

academic career.

I greatly appreciate the advice and opportunities given to me by my undergraduate

advisors, Professors P. A. Ramachandran and Curt Thies, and that of my co-opportunity

mentor, David Sextro. I also greatly value the conversations with and much valued

advice I received from Professors Devang Khakhar, Rich Lueptow, Seth Lichter, Mary

Silber, Paul Umbanhowar, Bill Cohen, Vassily Hatzimanikatis, and Luis Amaral, each

challenging me to approach my work from different perspectives.

I also appreciate the friendship and support of the members of the Ottino Lab throughout

the years, including Gerald Fountain, Joseph McCarthy, Paul DeRoussel, Marc Horner,

Kurt Smith, (Kats Miyake), Kimberly Hill, Nitin Jain, Peter Andresén, Stanley Fiedor,

Stephen Cisar, Steve Meier, and Ashley Smart.

I thank my family for their love and support, as always. Most of all, I thank my wife, the

love of my life, who believes I can achieve more than I could ever dream.

vi

For Lisa

vii

TABLE OF CONTENTS

Abstract............................................................................................................................. iii

List of Figures................................................................................................................... ix

1 Introduction.................................................................................................................1

2 Kinematics of Mixing in 2D and 3D containers .......................................................5

2.1 Introduction.......................................................................................................5

2.2 2D Tumblers .....................................................................................................8

2.2.1 Model of a Circular Tumbler ..............................................................8

2.2.2 Model of a Noncircular Tumbler ......................................................14

2.2.3 Mixing in a Square............................................................................19

2.3 Extension to Three-dimensional Flow ............................................................33

2.3.1 Geometry and Bi-axial Protocol ....................................................33

2.3.2 Kinematics of a Bi-axially Rotated Sphere and Cube ...................37

3 Competition Between Mixing and Segregation in 2D and 3D containers............45

3.1 Modeling Inter-particle Interactions ...............................................................45

3.1.1 Collisional Diffusion......................................................................45

3.1.2 Segregation ....................................................................................47

3.2 Experimental Details.......................................................................................51

3.2.1 Quasi-two Dimensional Experiments ............................................51

3.2.2 Three-dimensional Bi-axial Experiments ......................................54

3.3 Mixing and Segregation in a Quasi-two Dimensional Tumbler .....................59

viii

3.3.1 Chaotic Mixing with Diffusion......................................................59

3.3.2 Competition between Segregation and Chaotic Advection ...........67

3.4 Segregation in a Sphere and Cube: Bi-axial Rotation ....................................73

4 Flow in a Rotating-Rocking Sphere ........................................................................87

4.1 Model ..............................................................................................................87

4.2 Quasi-Periodic and Chaotic Path-lines ...........................................................93

4.3 Poincaré Maps.................................................................................................97

4.4 Flow Bifurcations..........................................................................................101

5 Rocking Spherical Tumbler ...................................................................................109

5.1 Axial Segregation in Various Geometries ....................................................109

5.2 Experimental Details.....................................................................................116

5.3 Formation of a Band and Spots.....................................................................121

5.4 Space-time Plots............................................................................................123

5.5 Phase Diagram ..............................................................................................127

5.6 Comparison to Computations .......................................................................135

5.7 Granular Slurries ...........................................................................................143

5.8 Rocking Cubes ..............................................................................................148

6 Summary and Outlook ...........................................................................................153

References.......................................................................................................................156

Appendix A ....................................................................................................................161

ix

LIST OF FIGURES

1.1 Examples of organization in granular systems ........................................................3

2.1 Schematic view of flow regimes in a rotating cylinder ...........................................6

2.2 Flow in a circle.........................................................................................................9

2.3 Stretching of a passive blob in a circle ..................................................................11

2.4 Dependence of circulation time on a, 0δ , and h in a circle ...................................13

2.5 Symmetric rotation of a circle at h = 0 and h = 0.5 ...............................................15

2.6 Random-shaped tumbler ........................................................................................16

2.7 Computed streamlines for a half-full circle and square .........................................20

2.8 Two trajectories within a square ............................................................................21

2.9 Poincaré map of a half-full circle and square ........................................................22

2.10 Blob advection and perimeter growth in a circle and square.................................24

2.11 Poincaré maps of the square with filling near half-full..........................................26

2.12 Sensitivity of filling with respect to 0δ .................................................................27

2.13 Mixing in a square, 22.5%-50% full ( 0.055.0 <<− h ) ........................................29

2.14 Mixing in a square, 50%-72.5% full ( 55.00.0 << h )............................................30

2.15 Poincaré maps and mixing of a blob in ¼ and ¾ full square.................................32

2.16 Half-full rotating sphere.........................................................................................35

2.17 Streamline crossing, coordinates and flow in a bi-axially rotated sphere..............36

2.18 Coordinates relative to a bi-axially rotated cube ...................................................38

x

2.19 Schematic diagram of periodic points in a bi-axially rotated sphere.....................39

2.20 Regular regions in a bi-axially rotate sphere .........................................................41

2.21 Schematic diagram of periodic points in a bi-axially rotated cube........................43

2.22 Regular regions in a bi-axially rotated cube ..........................................................44

3.1 Effect of diffusion on a passive blob .....................................................................48

3.2 Quasi-two dimensional apparatus ..........................................................................52

3.3 Three-dimensional tumbler and rotation protocol .................................................55

3.4 Blob experiments and computations......................................................................60

3.5 Mixing of tracer particles in a square ....................................................................61

3.6 Intensity of segregation vs. mixer rotation of tracer particles in a square .............62

3.7 Computations of intensity of segregation vs. mixer rotation for mixers of different

shapes and sizes: top-bottom initial condition.......................................................64

3.8 Computations of intensity of segregation vs. mixer rotation for mixers of different

shapes and sizes: left-right initial condition...........................................................65

3.9 Segregation in half-full circle and square ..............................................................68

3.10 Segregation pattern in a square with different fillings...........................................70

3.11 Bottom view of a cube after 6 and 50 rotations .....................................................74

3.12 Bottom view of an long tumbler with square cross section ...................................75

3.13 Computation of segregation in bi-axially rotated sphere.......................................78

3.14 Experiments in bi-axially rotated sphere ...............................................................79

xi

3.15 Depiction of predicted possible outcomes in the bi-axially rotated cube ..............81

3.16 Computations of segregation in bi-axially rotated cube ........................................83

3.17 Comparison of an bi-axially rotated S-system and D-system with computations .84

4.1 Rocking protocol and coordinates .........................................................................88

4.2 Surface velocities at the mid section of the free surface........................................91

4.3 Relationship between rocking angle and instantaneous flow direction .................92

4.4 Sphere rotating on angle without rocking..............................................................94

4.5 Typical chaotic path-line........................................................................................95

4.6 Typical quasi-periodic path-line ............................................................................96

4.7 Time evolution of the radius of a quasi-periodic and a chaotic trajectory.............98

4.8 Variety of different Poincaré maps in the A-f plane ............................................100

4.9 Concentric Poincaré maps....................................................................................102

4.10 Bifurcation of regular regions (�45=A , 6175.05775.0 << f ). ......................103

4.11 Bifurcation of regular regions (�45=A , 795.07875.0 << f ) ...........................105

4.12 Evolution of the system at �45=A and 968.0953.0 ≤≤ f ................................106

4.13 Evolution of the system in region of �60=A and 960.0930.0 ≤≤ f ................108

5.1 Tumblers of different geometry ...........................................................................111

5.2 Self-organization of axial bands in a slurry .........................................................112

5.3 Axial segregation observed in a tumbler with square cross section ....................114

xii

5.4 Formation of an axial band in a sphere................................................................117

5.5 Experimental setup and rocking protocol ............................................................118

5.6 Segregation patterns in a rotating-rocking sphere ..............................................122

5.7 Spatio-temporal plots of classic axial segregation...............................................124

5.8 Construction of spatio-temporal composite images.............................................126

5.9 Evolution of the Froude number as a function of frequency and amplitude .......128

5.10 Series of experiments at A = 45º ..........................................................................129

5.11 Experimental phase diagram................................................................................131

5.12 Qualitative comparison of experimental and frequency-locking phase

diagrams...............................................................................................................134

5.13 Comparison between a band and computed quasi-periodic trajectories ..............137

5.14 Comparison between regular regions in model and spots in experiments...........138

5.15 Transition between a band and spots in experiment compared to the model ......140

5.16 Experiments under slurry conditions ...................................................................144

5.17 Odd patterns under slurry conditions...................................................................146

5.18 Model and slurry experiment displaying period-6 periodic points......................147

5.19 Rocking cube with f = 1 .......................................................................................150

5.20 Rocking cube with f ~ 0.8 ....................................................................................151

A.1 Computations of a rocking sphere for A = 5º and .5 � f < 1.5 .............................162

A.2 Computations of a rocking sphere for A = 10º and .5 � f < 1.5 ...........................163

xiii







A.9 Computations of a rocking sphere for A = 42.5º and .5 � f < 1.5 ........................170








1

CHAPTER 1

INTRODUCTION

Recently, much attention has been given to granular systems due to their

widespread use in industrial processes and their often baffling physical properties.

Granular processes are found widely in nature in processes such as dune formation,

sediment deposits in river deltas, and also in catastrophic events such as landslides.

Granular-flow studies have recently received substantial attention within the physics and

engineering communities (Jaeger et al. 1996a, b, Jaeger & Nagel 1992, Bridgewater

1995). Despite this recent attention, the dynamics of granular materials is still poorly

understood when compared to fluids. There are no constitutive equations analogous to

the Navier-Stokes equations in fluids capable of describing general behavior of granular

flow. Most industrial mixing applications are approached on an ad hoc basis, often with

significant complications. Scale-up of laboratory-scale processes is often impossible.

The behavior of most systems cannot be predicted outside their operating parameters.

At the same time, granular materials are quickly becoming a prototypical system

in the emerging field of complex systems. A complex system may be loosely defined as a

large number of interactive constituents or agents capable of exhibiting spontaneous

organization. For instance, vibrated granular beds exhibit complex pattern formation and

can spontaneously organize into coherent structures known as oscillons (figure 1.1 top)

and other lattice-like structures (Umbanhowar et al. 1996). Also, flowing mixtures of

2

granular materials, differing by physical properties such as size or density, spontaneously

segregate (figure 1.1 bottom) (Makse et al. 1997). Segregation is detrimental to mixing.

The processes of interest in this dissertation are mixing and segregation and, in

particular, how they interact with each other. This issue is examined in a partially filled

rotating drum. In a circular container, segregation is relatively well understood. Smaller

and denser particles migrate towards the core of the circle, organizing material in the

radial direction. Granular tumblers are used widely in industry and are found in a large

variety of shapes and sizes. The flow in these systems is poorly understood. The general

expectation seems to be that the more convoluted the geometry, the better the obtainable

mixing. However, prediction of mixing in a complicated geometry is well beyond the

descriptive abilities of current models of granular flow.

The first attempts to describe granular mixing were based on analogies with fluid

mixing. For example, Lacey 1954 speaks of “convective mixing”, “dispersive mixing”

and “shear mixing”. In the past two decades, the study of dynamical systems has greatly

enhanced understanding of mixing in fluid systems (Ottino 1989). This knowledge is

now being extended to build descriptions of granular mixing. Recently, continuum

models have been developed to describe steady flow of granular materials in the cross

section of a rotating drum (Khakhar et al. 1997b). In chapter 2, this model is extended to

cylinders with noncircular cross sections where flow is time-periodic. This model is also

used to describe discontinuous time-periodic flow that generates mixing in the axial

direction of three-dimensional tumblers. As will be shown in chapter 3, extending this

3

Figure 1.1: Examples of self-organization in granular systems. Top – vibrated

materials spontaneously coordinate into coherent structures such as “oscillons”

(Umbanhowar et al. 1996). Bottom – mixtures of flowing granular materials, differing

by physical properties such as size, spontaneously segregate into bands as they are

poured to form a heap (Makse et al. 1997).

4

model further with the addition of collisional diffusion and segregation due to buoyancy

allows description of systems consisting of mono-disperse and bi-disperse materials.

The result of the mixing-segregation interaction is nontrivial. The segregation

patterns predicted by this model are compared to patterns produced in experiments in

both quasi-two dimensional tumblers and in a sphere and a cube rotated to produce three-

dimensional flow. This is examined in experiments of both S-systems (size segregation)

and D-systems (density segregation). Experiments in a sphere and a cube are conducted

under dry and slurry conditions (where air is completely replaced by a more viscous

fluid). Non-invasive techniques are used to visualize the shape of segregated structures.

Experimental studies in rocking cylinders suggest that mixing is enhanced by the

incorporation of axial advection (McCarthy et al. 1996, Wightman et al. 1998a). Chapter

4 considers mixing in a rotating-rocking sphere where mixing is found to be sensitive to

both the amplitude and frequency of rocking. Chapter 5 examines an experimental

realization of this system, presenting direct comparisons between segregation patterns in

the sphere experiments under both dry and slurry conditions.

5

CHAPTER 2

KINEMATICS OF MIXING IN 2D AND 3D CONTAINERS

This chapter extends the 2D model derived by Khakhar et al. 1997b for

mixing in a circle to convex geometries and 3D containers. The main

focus is to understand, from a continuum perspective, how chaotic

advection affects mixing. Chaos may be produced by changing the tubler

geometry, changing the degree of filling, and introducing flow in three-

dimensions.

2.1 Introduction

Flow in a rotating cylinder is well defined and can be classified into different

regimes (Henein et al. 1983, Rajchenbach 1990). These regimes are depicted in figure

2.1. At low rotational speeds (quantified in terms of the Froude number, gLFr /2ω= ,

where g is the acceleration due to gravity, L is the length scale of the system, and ω is

the rotational speed), the flow occurs as discrete avalanches; one avalanche stops before

the next one begins (this is the so-called avalanching or slumping regime). A simple

geometrical model describes overall flow (Metcalfe et al. 1995). Material rotates as a

solid body until the free surface reaches a critical angle, at which the material flows and

then comes to rest. Each avalanche comprises of a wedge of particles near the free

6

Fig

ure

2.1

: S

chem

atic

vie

w o

f fl

ow

reg

imes

in a

rota

ting c

yli

nder

wit

h i

ncr

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ota

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ω.

In t

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rom

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.

7

surface, transported from the upper to the lower part of the container. Mixing occurs

within the wedge and across intersections of consecutive wedges. Geometry and fill level

have significant impact, especially when geometrical features of the tumbler are

asymmetric (such as baffles considered in McCarthy et al. 1996).

At higher speeds a steady flow is obtained with a thin cascading layer at the free

surface of the rotating bed (this is the continuous flow regime, also known as the rolling

or cascading regime). The free surface is nearly flat. At still higher speeds, particle

inertia effects become important, causing the free surface to form an “S” shape and

particles become airborne (cataracting regime). There are many different models that

describe flow in both the rolling and cataracting regime based on assumptions derived

from experimental observations. 1=Fr roughly corresponds to the critical speed for

centrifuging where material only rotates solid body and no mixing occurs. Flow is more

complicated when tumbling cohesive particles or mixtures of particles differing in

physical properties (for instance, size). From here on, only the continuous flow or rolling

regime is considered. The particles will range in size from 0.2 mm to 3 mm diameters.

Cohesion is not an issue.

Some of the important features of flow in the rolling regime are revealed using

NMR (Nakagawa et al. 1993). The flowing layer is fluidlike and the velocity profile is

nearly linear. Rajchenbach et al. 1995 show similar results near the midpoint of the

flowing layer for a two-dimensional system. More recently, a more detailed study

analyzes the velocity profiles across the entire layer for different rotation rates and bead

sizes (Jain et al. 2002). They find that the streamwise velocity profile scales along the

8

free surface as ( )( )[ ]22 /1/ LxLu −ωδ where ω is the rotation rate, u is the average

velocity, δ is the local depth of the flowing layer, and x is the position along the free

surface ( 0=x is the midpoint and the free surface is 2L long – see figure 2.2). Jain et al.

2003 also include variation of the interstitial fluid properties (replacing air with water),

finding similar results.

The next section builds on the model developed by Khakhar et al. 1997b

describing flow in two-dimensional circle in the rolling regime. This model uses the

following assumptions (based on experimental results):

1) The shear profile along the layer is linear.

2) The free surface is flat.

3) Material rotates as a solid body beneath the flowing layer.

Additions to this model describing collisional diffusion and segregation are incorporated

in chapter 3.

2.2 2D Tumblers

2.2.1 Model of a circular tumbler

Consider a circular tumbler, as shown in figure 2.2. The circle is rotated about its

center, denoted C. The free surface is flat and the filling is measured by h, the distance of

the free surface from C. There are two regions of flow. Most material rotates as a solid

body without slip. Near the free surface, material is convected by a rapid shear flow.

Coordinates associated with this flow are at the center of the free surface, O. A thin

9

Figure 2.2: Flow in a circle partially full of material (light gray region, here shown as

both half-full and more than half-full with height h) and rotated clockwise about the

center of the circle at constant rate, ω . All material is in solid body rotation except a thin

region near the free surface. The boundary between solid body rotation and shear flow is

denoted as ( )xδ . Variables in the model are scaled by ω and L, the half-length of the

free surface, and the coordinate system is centered on the free surface. Trajectories

follow along closed streamlines, shown as dotted lines. When more than half-full, the

circle forms an unmixed core at the center (dark gray circular region shown on right).

10

boundary layer of thickness ( )2

0 1 x−= δδ (spatial variables are made dimensionless

with L, the half-length of the free surface) separates solid body rotation from shear flow

(Khakhar et al. 1997b). The maximum depth of the flowing layer, 0δ , is estimated from

experiments, and ranges roughly from 0.03 to 0.10.

A general form for the velocity field of the rapid shear flow that conserves mass

in the flowing layer is

��

��

�−+=

a

xy

aauv

δ1

1 (2.1)

1+

��

��

�−=a

yyxvδ

(2.2)

Time is scaled with respect to ω/1 . The average velocity is ( )02/1 δ=u . In this

generalized form, 1=a corresponds to a linear profile, 2/3=a is refered to as the

Bagnold profile (Bagnold, 1954), and ∞=a is plug flow. For any value of a, this model

produces closed streamlines (shown as dotted lines in figure 2.2). Particle motion along

these streamlines is calculated by integrating the above velocity equations in the layer

and rotating solid body in the bed. Particles spend much more time in solid body rotation

than in the flowing layer.

Consider a passive blob of material passing through the flowing layer (figure 2.3).

This blob consists of about 104 closely spaced particles. As it enters the layer, it is

stretched by the shear flow and then exits into the solid-body region. The net effect is

11

Figure 2.3: Stretching of a passive blob of 104 particles (colored gray and black). Top –

time evolution of a blob of tracer particles as it moves through the flowing layer. As

material passes through the layer, it rotates 180� and stretches in the θ -direction. Bottom

– the same blob after 20 rotations stretches and connects through the flowing layer (initial

condition in gray). The angle of repose is arbitrary. For convenience, the free surface is

shown as horizontal.

12

stretching in the θ -direction and 180° rotation of the blob. At the bottom of figure 2.3,

the mixing pattern is shown in a half-full circle after 20 rotations.

The distribution of circulation times for the closed streamlines (time of one

revolution about each streamline) is given by ( )rTC . Plots in figure 2.4 show the effect

of a, 0δ , and h on ( )rTC . The velocity profiles with a ranging from 2/3 to 3/2 are

depicted in figure 2.4a. To calculate CT , a straight line of initial conditions in the radial

direction is advected, and the time for a particle at distance r to return to its initial

position is recorded. The circulation times are almost independent of the parameter a

(figure 2.4b). Because of this, mixing occurs in the same manner and at nearly the same

rate for all velocity profiles considered. Therefore, a simple linear shear, 1=a , is used

from here on.

Figure 2.4c shows that the deviation in circulation times becomes more

pronounced along r for increasing 0δ . For 00 =δ , all trajectories have the same

circulation time and there is no mixing. Also, changing h significantly changes the

circulation times in the tumbler (figure 2.4d). Both the thickness of the layer and the

filling of the tumbler affect the mixing rates significantly more than the form of the

velocity in the layer. Most simulations in this thesis are calculated using 05.00 =δ

unless specified otherwise.

Increasing h does not change the dynamics of the flowing layer, however it causes

particles to spend more time in solid body rotation. This is the reason why h has such a

large effect on the circulation rate. It should also be noted that h is the only parameter in

13

Fig

ure

2.4

: C

ircu

lati

on r

ate,

T(r

).

The

thre

e dif

fere

nt

vel

oci

ty p

rofi

les

are

show

n i

n (

a),

and h

ave

alm

ost

no e

ffec

t on c

ircu

lati

on r

ate

(b).

H

ow

ever

, ci

rcula

tion r

ate

dep

ends

slig

htl

y o

n t

he

dep

th o

f th

e fl

ow

ing

layer

(c)

and g

reat

ly o

n t

he

fill

ing (

d).

14

a physical system that is user-defined. Instead of calculating the circulation rate at each

r, the average circulation time (T*) can roughly be calculated by dividing the time for a

full rotation (in dimensionless variables, π2 ) by the percent filling. For instance, in a

half-full mixer all of the material passes through the layer in ½ rotation (T* = ½). For

more than half-full circles (specifically 0δ>h ), a core of material that never reaches the

flowing layer resides in the center. Because circulation rate depends on r, there is only

one streamline on which all points return after *Tt = . This is shown in figure 2.5 at

various fillings. An initial line of points is advected for T*. In the half-full case, the

points near the center rotate past their initial locations, whereas points near the boundary

do not pass the original line of points. There is one point that returns exactly to its

original location (corresponding to the crossing of the initial line and the final line of

points). This point and all points on the same streamline are invariant with respect to T*

rotations (so called periodic points). This has little relevance in a circle; however, in

chaotic systems it is the periodic points and regions near these points that define the local

mixing.

2.2.2 Model of a noncircular tumbler

Consider a general, two-dimensional, non-circular tumbler filled to some extent

(figure 2.6). Only convex tumblers are considered to ensure that the free surface only

intersects with the boundaries at only two points. The system is rotated about its

15

Figure 2.5: Symmetric rotation of a circle with h = 0 and h = 0.5. The black line is the

location of the gray line after rotating T*. A periodic point exists where the gray and

black lines cross (except for the center of the circle when more than half-full) and the

dotted line is a streamline on which all points return after T*.

16

Figure 2.6: Irregularly-shaped tumbler. The center of the free surface, O, moves relative

to the centroid, C. This picture is limited to convex tumblers such that the flowing layer

never intersects the container boundaries except at x = -p1 and x = p2. The half-length of

the flowing layer (L), the height (h) and the lateral position of the center of the free

surface (s) are now functions of time.

17

centroid, C, at a constant rate ω ; coordinates are associated with the flowing layer are at

the center of the free surface, O. One can think of a circle as a special case of this system

where the shape of the region of flow is constant. Without this restriction, the flowing

layer changes periodically with rotation. O moves relative to C (except in the case of

half-full and a 180� rotationally symmetric geometry, considered in Khakhar et al.

1999a). Also, the length of the free surface changes with time. Elperin and Vikhansky

1998a also present a description of mixing in various geometries and perform a detailed

analysis of the symmetries for different geometries at various fillings. However, their

billiards-like model does not consider the dynamics within the flowing layer. For

instance, no mixing is predicted in a half-full circle or square. The basic assumptions and

the model described below are also presented in Khakhar et al. 2001a,c as applied to heap

formation, Orpe et al. 2001 as applied to tumbling, and in Khakhar et al. 2001a where the

model is applied to both cases. More recently, the model has been used to scale

experimentally measured velocities in slurries (Jain et al. 2003). Surprisingly, the model

works well even under slurry conditions.

Experiments by Khakhar et al. 1999a show that in different sized circular

containers, the maximum depth of the flowing layer scales with the length of the free

surface. Assume that the layer geometry changes slowly with time compared to the

dynamics of the flow. The flow is then quasi-steady and similar to that for a circle with

the same layer geometry. As discussed in section 2.1, inertial effects of the solid body

rotating particles in the rolling regime are negligible. With these added assumptions, the

velocities in the flowing layer in a noncircular container are

18

( ) ( )( ) ��

�

��

� −+=θδθθ

,12

xhyuLvx (2.3)

( )( ) ( )( )

2

, ��

��

� −−−=θδθθ

xhysxv y (2.4)

As before, these equations are dimensionless with L, now denoted as the half-length of

the free surface at a specified angle, and ω/1 . ( )θh is the dimensionless height of the

free surface at any angle of rotation relative to C (as shown in figure 2.6). The area of the

region of flow under ( )θh remains constant and depends on the initial fill fraction of the

tumbler. ( )θs , the horizontal position of O, is determined by calculating the x-

coordinates p1 and p2 (relative to C) where the free surface intersects the container

boundaries. From this, the lateral location of the center and the half-length of the free

surface are ( ) ( ) 2/12 pps +=θ and ( ) ( ) 2/12 ppL −=θ respectively. The shape of the

flowing layer changes in time as

( ) ( ) ( ) ��

�

�

��

�

��

��

�−=2

0 1,θ

θδθδL

xLx (2.5)

These equations in the layer, with solid body rotation underneath it, describe flow in any

partially full convex container. It should be noted that different functionalities for layer

profiles (e.g., ( )2

0 /1 Lx−= δδ ), give essentially the behavior as described in the next

section, indicating that global aspects (i.e., the shape of the container) control the

important details of the physics.

19

All information about geometry enters into the velocity equations through the

functions ( )θL , ( )θh , and ( )θs , which are time-periodic with each rotation. As

suggested by several studies on fluid mixing, time modulation of a two-dimensional flow,

where superimposed streamlines at different times intersect, is sufficient to produce

chaotic advection. This is demonstrated in figure 2.7 for a half-full square. Closed

streamlines in the circle can be thought of as barriers against transport in the radial

direction. In the square, streamlines associated with two different orientations cross in

the flowing layer. This picture becomes more complex when the square is not half-full.

2.2.3 Mixing in a square

As already suggested in figure 2.3, mixing in a circle is simple. Deformation of

material elements is linear with time. It is also easy to calculate the locations of periodic

points. This is not the case in a square. First consider a half-full square. L in this

container is taken as half the width of the square. Figure 2.8 shows two particle

trajectories (each for 50 rotations). The large circle represents the maximum width of the

flowing layer (particles never leave the square). The top trajectory appears to be quasi-

periodic, wrapping around on a closed surface. The bottom trajectory appears to

randomly enter and exit the flowing layer.

A Poincaré map is a way to show long-time behavior of initial conditions in

different parts of the flow (figure 2.9). To produce this map, particles are advected and

their locations are plotted at the end of each ½ rotation for 200 rotations (as noted in the

20

Figure 2.7: Computed streamlines for a half-full circle and square. The circle has closed

streamlines that act as barriers to transport in the radial direction. The square is shown at

two different orientations. The shaded region in the square shows where intersections

occur between streamlines associated with each orientation.

21

Figure 2.8: Two trajectories within a square. Neither the plot of the quasi-periodic

trajectory (top) nor that of the chaotic trajectory (bottom) gives much insight into mixing.

Particles are always within the boundaries of the square.

22

Figure 2.9: Poincaré map of a half-full circle and square. In a circle, points are always

located on closed streamlines. In the square, different regions of mixing exist. Regular

regions have quasi-periodic orbits. A chaotic sea exists in the middle where mixing is

enhanced.

23

circle, all half-full containers have an average circulation time of a ½ rotation). Each

color represents a different initial condition. In a circle, the Poincaré map shows only

regular motion (“regular” may be roughly interpreted as the opposite of chaotic), and the

invariant curves obtained coincide with the streamlines. However, in a square, regions of

chaotic motion are interspersed with regions of regular motion. Regular regions with

quasi-periodic orbits exist around elliptical points. In these regions, the mixing that

occurs is analogous to that found in a circle. The “chaotic sea” that surrounds hyperbolic

points is where mixing is enhanced (there are two hyperbolic points, one of which is

located in the flowing layer). KAM surfaces, chains of high periodicity periodic points

that surround the regular regions, act as a barrier to transport.

The differences between mixing in the chaotic region and the regular regions

become apparent when two passive blobs are advected and their positions are plotted

after 20 rotations (figure 2.10 top). Material in the regular region is trapped, only mixing

with material in its vicinity. Material in the chaotic regions is spread throughout a larger

region of the granular bed. The rate of stretching in the chaotic region may be quantified

by tracking the perimeter length of the blob as a function of time (figure 2.10 bottom).

The perimeter of the blue blob in the chaotic region grows exponentially. The blob is

stretched along unstable manifolds of the hyperbolic point. This stretching deforms the

blob into a thin filament which folds back upon itself repeatedly as it follows a

heteroclinic orbit along the stable manifold of the other hyperbolic point. As the blob

approaches the other hyperbolic point, it begins to stretch along an unstable manifold and

the process repeats.

24

Figure 2.10: Blob advection in a square (top) and perimeter growth in a square and circle

(bottom). Material initially located in the regular region is trapped, only mixing locally.

A blob that starts in the chaotic region gets stretched and folded, enhancing mixing. The

length of the perimeter of the blob in the chaotic region increases exponentially with

number of rotations. The same blob in a circle only has linear growth of its perimeter.

25

Consider the symmetries of a square. The square has symmetries such that the

boundaries return to their original location after a ¼ rotation. This means that the

functions ( )θL , ( )θh , and ( )θs are periodic with every ¼ rotation. This suggests that

drastic changes may occur in the square for ¼, ½ and ¾ fillings (corresponding to

average circulation times of 2/π , π , and 2/3π ). The regular regions in the half-full

square are associated with periodic points that return every half-rotation. Consider the

system when it is slightly more or less than half-full. The average circulation time

increases with increasing h. Therefore, when the container is not half-full, no points can

return to their original location after just ½ rotation. The Poincaré maps for

1.01.0 <<− h are shown in figure 2.11. Beginning at half-full, with increasing h the

regular regions migrate toward the center of the tumbler and shrink until they disappear.

All that remains is chains of high period periodic points (KAM surfaces) that inhibit

transport in the radial direction. With decreasing h, the regular regions also shrink as

they move toward the outer corners of the square. Once again, eventually only KAM

surfaces remain. When KAM surfaces exist, the topology is very similar to the circle.

Clearly, the topology of the flow is extremely sensitive to fill level.

Recall that in a circle the distribution of circulation times is broader for larger

values of 0δ . In the square, the effects are as follows (figure 2.12). When 0δ is small,

small changes in h cause the regular regions to disappear. When 0δ is deeper, the

26

Figure 2.11: Poincaré maps of the square with filling near half-full. From half-full,

increasing h causes the regular regions to migrate toward the center and shrink. With

decreasing h, the regions shrink as they migrate toward the boundaries. The topology of

the underlying flow at 45% and 55% full are similar, both consisting mainly of KAM

surfaces that inhibit transport in the radial direction.

27

Figure 2.12: Sensitivity of topology to filling for different 0δ . Top – the radial position

of the regular regions seen at half-full is dependent on h. Here, positions are plotted for

different layer depths ( 2 is the length of the diagonal from center of the square to the

corner). Bottom – the rate at which these regular regions disappear is inversely

proportional to 0δ .

28

topology of the flow is insensitive to fill level. The position of the regular region, r*, is

plotted as a function of changing h for different values of 0δ . Plotting the slopes of these

lines shows that the rate of this transition is inversely proportional to 0δ . This suggests

that if the depth of the flowing layer is very thin, mixing is very sensitive to the fill level.

Therefore, any error in measuring fill level translates as a large effect on mixing. Small

deviations in fill level within a noncircular tumbler can result in significantly different

mixing.

Figure 2.13 shows Poincaré maps from 0.055.0 <<− h . As one decreases h, first

the topology primarily consists of high period chains of islands. The periodicity of the

chains in each map decreases with smaller fillings. At less than 35% full, most of these

chains are destroyed, apparently leaving only period 2, 3 and 4 islands. The rest of this

map appears chaotic. 5.0−=h corresponds to a symmetric filling with respect to the

square (25% full). Here, a single island passing through the layer exists. The shape of

this island is different than previous regular regions. Instead of an elliptical point, the

periodic point in this island is hyperbolic. Analogous to less than half-full, multiple

bifurcations occur in the range 55.00 << h (figure 2.14). With increasing filling, the

regular regions disappear, and chains of high period periodic points form and migrate

outward. A diamond shaped core appears in the center. The same type of regular regions

seen at 5.0−=h also forms at 5.0=h . Only one of three regions is stretched through the

layer in this map, and it is apparent that these regions are not centered about elliptical

points.

29

Figure 2.13: Mixing in a square, 22.5%-50% full ( 0.055.0 <<− h ). The flow goes

through multiple bifurcations with changing h. Regions of poor mixing form near 25%

and 50% full (symmetric fillings).

30

Figure 2.14: Mixing in a square, 50%-72.5% full ( 55.00.0 << h ). Regions of poor

mixing form around 50% and 75% full (symmetric fillings). A core region exists for all

fillings where 05.0>h .

31

At half-full, a hyperbolic point is centered in the chaotic region. At ¼ and ¾ full,

it appears as if there are hyperbolic points; however mixing in these regions is poor

(figure 2.15). A blob that starts in the chaotic region stretches and folds to eventually

spread throughout the chaotic region. However, material from this blob never infiltrates

into the regions near the hyperbolic points. Flow in these regions is as follows. In the ¼

full case, as material near a hyperbolic point passes through the flowing layer, it is

stretched along unstable manifolds and rotated 180�. This material then solid body

rotates ¼ rotation until it begins to enter the layer once again. The net rotation of this

material as it re-enters the flowing layer is 270�, unlike the half-full case where a blob in

the regular regions rotates 360� (180� from solid body rotation and 180� while passing

through the flowing layer). Once again, material stretches along the unstable manifolds

as it passes through the flowing layer. However, this acts to deform this material to its

original shape (90� rotation is the same as reversing the stability of each eigenvector).

Every two passes through the layer results in 180� rotation of material without stretching.

The same occurs in the ¾ full square; however, material rotates 450� with each ¾ rotation

of the square. In this region, no mixing occurs.

It is clear that variations in geometries give rise to drastically different mixing

topologies. However in each system, the combined symmetries of both the container and

the degree of filling dictate how mixing occurs. For instance, a triangle (and any odd

number regular polygon) is not symmetric to ½ rotations. However, it is expected that

when a triangle is 1/3 or 2/3 full, regular regions will form in the corners as seen in the

square. Thus far, only mixing in the cross section of an elongated drum is considered.

32

Figure 2.15: Poincaré maps and mixing of a blob in ¼ and ¾ full square. Typically,

hyperbolic points locally enhance mixing. However, this blob stretches throughout the

chaotic region, avoiding regions around the hyperbolic points (labeled as A).

Poincaré section Blob advection

A A

A

33

Another way to introduce time-periodicity into the system is to change the direction of

flow, producing flow along the axis of rotation. As described in the next section, the

above models can also be used to describe discontinuous flow in certain three-

dimensional geometries.

2.3 Extension to Three-Dimensional Flow

2.3.1 Geometry and bi-axial protocol

The above model considers only 2D flow in a cross section. However, if applied

to a longer container, it does not imply that different cross sections along the axis of

rotation must be identical. For instance, a circle can either be a cross section of a

cylinder or a sphere. This can be extrapolated for many different shapes as long as the

geometry does not induce flow across each plane of flow. For this to be true, each cross

section along the axis must have constant area under the free surface. Assume that the

flow scales by the local length of the free surface in each cross section in the direction of

flow and neglects any effect of shear in the axial direction. In the actual system, this

assumption may not be accurate, especially considering end-effects that arise in

experiments in almost all rotating cylinders.

With this additional assumption, the model described above may be used to model

flow in noncylindrical geometries. The length of the free surface now depends on axial

position ( ( )zLL = , where z points orthogonal to x and y-directions). In the rolling

regime, the free surface is a flat plane. However, based on experimental results, the free

34

surface is not necessarily flat (Zik et al. 1994). In a container where the radius varies

sinusoidally in the z-direction, they report that geometry slightly changes the angle of

repose. Also, Elperin and Vikhansky 2000 model the shape of the free surface in an

ellipsoid with flow in the axial direction.

Consider flow in a half-full sphere (figure 2.16). The free surface is a circular

plane and steady flow in each cross section scales as ( ) 21 zzL −= (dimensionless with

radius). We assume that each cross section behaves as described above where particles

move along closed streamlines. As is seen in noncircular tumblers, periodic modulation

of the flow can enhance mixing by inducing chaos. This is straightforward in a sphere.

Imagine rotating for a given amount of time, stopping and then rotating on a different

axis. Cycling between the two axes causes streamlines associated with the each flow

direction to cross each other in the flowing layer (figure 2.17 top). Changing directions

of the flow is simply modeled by translating particles to the new coordinate system

associated with rotation on an alternate horizontal axis. The flow is assumed to start and

stop instantaneously and rotation that does not produce a flowing layer is neglected.

The parameters in this system are the number of axes of rotation, the angle

between axes, the amount of rotation about each axis, and the order of rotations about

different axes. One can imagine a scheme to describe this parameter space and the

symmetries (similar to the symmetry analysis by Franjione and Ottino 1992). Consider a

simple protocol of two-axis rotation as shown at the bottom right of figure 2.17. The

axes are orthogonal with a rotation of π=T about each axis; this is referred to as the

35

Figure 2.16: Half-full rotating sphere. The velocity on the surface scales with the length

of the free surface in the direction of flow. The horizontal axis of rotation is arbitrary,

flow in any direction is identical.

36

Figure 2.17: Top – streamline crossing at the surface, coordinates and flow in a bi-

axially rotated sphere. Flow associated with rotation about two orthogonal axes is shown

in red and blue. Bottom – the coordinate system for each flow and the protocol depicting

consecutive rotations.

37

“bi-axial” protocol. Obviously, mixing will be different with more involved schemes, but

this can be used as a reference case. A bi-axially rotated sphere may be the simplest

system exhibiting axial transport and chaos.

The same model applies to a bi-axially rotated cube (figure 2.18). The cube is

invariant to rotations of 90� on each axis of rotation face-center on different sides of the

cube. Flow in each cross section is described as flow in a square. Like the sphere, bi-

axial rotation causes advection in the axial direction. Understanding of dynamics in a

circle and a square facilitates the analysis of these two systems.

2.3.2 Kinematics of a bi-axially rotated sphere and cube

As seen in the square, invariant regions describing the underlying flow can be

located in terms of the periodic points of the flow. The location of periodic points in both

the bi-axially rotated sphere and cube can be located by knowing the dynamics of a circle

and square and the symmetries of the bi-axial flow. A qualitative analysis yields

considerable insight. First, consider the sphere. In a half-full circle rotated π=*T , there

exists a single streamline such that all points return to their original location (as depicted

in figure 2.5). This streamline of periodic points is located at r* in the region of solid

body rotation (figure 2.19). Computations indicate that 2/1* ≅r (note: a closed integral

equation can be written by calculating the Hamiltonian for the velocity field; the exact

position of r* is slightly dependent on the depth of the flowing layer, 0δ ). Such a

streamline exists in each axial cross section of flow on each axis, therefore tracing a

38

Figure 2.18: Left – coordinates in a bi-axially rotated cube. Each cross section is

modeled as a square, the dynamics of which are depicted in the Poincaré map on the

right.

39

Figure 2.19: Schematic diagram of periodic points in a bi-axially rotated sphere. In each

cross section, a streamline exists on which points return to their original location after ½

rotation. In the sphere, this creates an invariant surface of periodic points in any direction

of flow, shown on the right for two orthogonal directions of flow.

40

surface of periodic points within the sphere. In each cross section along the z-axis, this

streamline is located at 2/1* 2zr −≅ within that cross section (here, r is relative to the

circle, not spherical coordinates). In Cartesian coordinates, the equation describing this

ellipsoidal surface within the region of solid body rotation is ( ) 4/1 222 zyx −≅+ . This

surface of periodic points exists for rotation in any direction. Therefore, after a ½

rotation on one axis and then a ½ rotation on the orthogonal axis, periodic points exist at

the intersection of this surface and its projection in the orthogonal direction. The

intersection of these two surfaces form two curves in the zx = and zx −= planes at

( ) 221¼ xzy −−≅ .

Seeding initial conditions near these periodic points and advecting them with the

bi-axial flow indicates the local stability of these points. Periodic points located in the

zx −= plane are elliptic. Particles starting near these points are trapped in a regular

region and circulate about a given elliptical point, as shown in figure 2.20. In three-

dimensional flow, this elliptic point has two imaginary eigenvalues and one null

eigenvalue. The points near the elliptic points rotate in a plane orthogonal to the radial

direction. The volume of the regular regions is small compared to the volume of the

sphere. All trajectories outside of the regular regions appear to be chaotic. Periodic

points in the zx = plane are hyperbolic. Each of these points also has a null eigenvalue

associated with the radial direction, and stretching occurs in the plane normal to the radial

direction for each point.

41

Figure 2.20: Regular regions in a bi-axially rotated sphere. Green particles trapped in

the regular regions circulate about the periodic points in the zx −= plane. Particles

outside the regular regions (not shown) reside in a chaotic sea. The yellow curve marks

the locations of the hyperbolic points.

42

The periodic points in the bi-axially rotated cube can be found using a similar

analysis (figure 2.21). In the half-full square, there are four periodic points. Two

elliptical points are centered within the regular regions in the corners (their coordinates

are roughly (-0.5, -0.5) and (0.5, -0.5)). Two hyperbolic points are located in the chaotic

sea, one in the center of the granular bed (near (0, -0.5)) and the other is located in the

flowing layer at this orientation of the square. In the cube, each cross section is identical

and these points extend as lines in the axial direction for rotation in either direction. The

ten remaining periodic points after bi-axial rotation are located at the crossing of these

lines, nine points in the bed and one in the flowing layer. Advecting initial conditions

close to these points reveals that only two remain elliptical, trapping particles in

surrounding regular regions (figure 2.22). Other particles near the other periodic points

are mixed in a chaotic sea. The stretching near a given hyperbolic point is difficult to

quantify due to the effect of neighboring hyperbolic points.

Time-periodic flow enhances mixing in both spheres and cubes. Another

parameter that can be explored is the degree of filling. In a sphere, mixing will be similar

to the half-full case, where periodic points are easily calculated and mixing is enhanced.

At fill levels higher than 0δ>h , an unmixed core exists in the center, but the volume of

this core is smaller in a bi-axially rotated sphere than steady rotation on a single axis. In

a cube, changing the fill level has a drastic effect on the topology of the underlying flow.

Mixing is very different in cubes that are slightly more or slightly less than half-full.

43

Figure 2.21: Schematic diagram of periodic points in a bi-axially rotated cube. Four

periodic points exist in the cross section, two of which are elliptic centered within regular

regions. In the cube, this creates lines of periodic points associated with two orthogonal

directions of flow, shown on the right.

44

Figure 2.22: Regular regions in a bi-axially rotated cube. Top and side view show green

points trapped in regular regions and red points in the chaotic region (all but a few points

are removed to assist visualization of the regular regions). Analogous to the sphere,

regular regions exist along the zx −= plane in only two corners. The other periodic

points are located in a chaotic sea throughout the rest of the granular bed.

45

CHAPTER 3

COMPETITION BETWEEN MIXING AND SEGREGATION

IN 2D AND 3D CONTAINERS

This chapter discusses examples where transverse segregation and chaotic

advection co-exist. A quasi-two dimensional circular tumbler leads to

classic radial segregation, which is well understood. In a square,

segregation interplays with the underlying chaotic advection resulting in

dynamic equilibrium patterns. This interplay is also investigated in three-

dimensional systems such as bi-axially rotated spheres and cubes.

3.1 Modeling Inter-particle Interactions

3.1.1 Collisional Diffusion

Consider the model describing advection in a circular tumbler given by Eqns. 2.3-

2.5. The effect of collisional diffusion is incorporated in terms of the model developed

by Savage (1993). The collisional diffusion coefficient, Dcoll, is given by

( )dydvdqD x

coll2η= (3.1)

where dydv x / is the velocity gradient across the layer, and d is the particle diameter.

The prefactor ( )ηq , obtained by Savage via particle dynamics simulations, is a function

46

of the solids volume fraction, η ; in simulations, η is assumed to be a constant and

025.0=q is obtained by fitting to experimental data for mixing of identical particles in a

rotating cylinder. In terms of the model in section 2.2, the dimensionless form of

diffusion becomes

2

0

2

0

2

025.02

025.0δδdduDcoll == , (3.2)

where 12 0 =δu . Here, the dimensionless variables u, d , and 0δ are the average

velocity and the depth of the flowing layer at 0=x , respectively.

Collisional diffusion enters as a Langevin term in the particle advection equations.

Denote S as a white-noise term such that, upon integration over a time interval (Δt) it

gives a Gaussian random number with variance 2DcollΔt. The effects of diffusion may be

masked by convection. In our experiments the Péclet number along the layer, collDu / (a

measure of the relative importance of convection to diffusion) is approximately 102. The

Péclet number in the direction normal to the flow is factor 2

0δ smaller ( 1.003.0 0 << δ in

most cases). Thus, diffusion is important only in the y-direction.

The term S is added to vy (Eq. 2.4) such that

( ) ( )( ) ��

�

��

� −+=txthytuLv x

,12

δ, (3.3)

( )( ) ( )( ) S

txthytsxv y +��

�

��

� −−−=2

,δ, (3.4)

47

now making {vx, vy} a set of stochastic differential equations. The physical picture is as

follows: a passive blob advected in the flowing layer is deformed into a filament by the

shear flow and blurred by collisional diffusion until particles exit the layer (figure 3.1).

Diffusion enhances mixing by generating transport in the r-direction. This is the only

mechanism of transport across confining streamlines.

3.1.2 Segregation

Consider D-systems. The effects of segregation are incorporated in terms of drift

velocities with respect to the mean mass velocity (Khakhar et al. 1997b). Segregation,

like the effects of collisional diffusivity, is significant only in the direction normal to the

flow (apparent, again, when the Péclet number in each direction is considered). The

segregation velocity for the more dense particles (labeled 1) can be written as

( )( )d

Dv colls

yφργ −−−

=112

1 (3.5)

and for the less dense particles (labeled 2) as

( )

dD

v collsy

φργ −=

122 (3.6)

Here, sγ is the so-called dimensionless segregation velocity, ρ is the density ratio, d is

the particle diameter, and ( )tyx ,,φ is the number fraction of the more dense particles.

For elastic particles, sγ appears to be inversely proportional to the granular temperature;

48

Figure 3.1: Effect of diffusion on a passive blob. As the blob passes through the flowing

layer, diffusion blurs the edges of the blob, spreading across streamlines, thus enhancing

mixing.

Without diffusion With diffusion

49

however, for real particles a simple expression for sγ is not available, and it is treated as

a fitting parameter to match the rate of segregation. This model has been tested in

circular (nonchaotic) containers (Khakhar et al. 1999a). However, the main interest here

is how the final segregation pattern depends on container geometry and filling.

To add segregation into the advection model, assume first that the mean flow is

still the same as if all particles were identical, so that Eqs. 3.3 and 3.4 still apply. There

are two sets of advection equations; each corresponds to particle 1 or 2. The y-

component of the dynamical system representing the motion of the more dense particles

(labeled 1) is:

( ) ( )( )d

DS

hysx

dtdy colls φργ

δ−−

−+��

�� −

−−=112

2

11 , (3.7)

Whereas for the less dense particles (labeled 2), Eq. 3.7 becomes:

( ) ( )d

DS

hysx

dtdy colls φργ

δ−

++��

�� −

−−=12

2

22 , (3.8)

The two equations given above, combined with the corresponding equations for the x-

coordinates for each of the species, describe the evolution of the interpenetrating continua

from a Lagrangian viewpoint. Computations using this formulation are straightforward.

The number of particle trajectories simulated is roughly the same as the number of beads

in the corresponding experiment. Computation time increases by an order of magnitude

as compared to calculating Poincaré maps in chapter 2, computation time increases by an

order of magnitude. However, these calculations are significantly faster than discrete

element methods (DEM) for the same number of particles. Particles are labeled less or

50

more dense and are randomly distributed in the domain. They are advected according to

the equations of motion for each type of particle. The number fraction field, ( )tyx ,,φ , is

determined by defining a grid over the flowing layer and calculating the fraction of the

more dense particles in each bin of the grid.

The above model is a reasonable representation for the case of equal-sized

particles with different density. However, for mixtures of different-sized particles the

flow may be significantly affected by the composition of the layer. The local density

strongly depends on the local particle concentration. Consequently, the velocity field and

concentration field can be coupled with the proper volume fractions and size ratio of

particles. This issue is considered by Khakhar et al. 2001b. A model attempting to

capture this coupling alters the shape and thickness of the flowing layer based on local

concentration. As the authors point out, the resulting model is quite complicated.

This model (Eqns. 3.7 and 3.8) is also applied to 3D containers. Flow in three-

dimensional tumblers is also considered. The model represents flow in containers when

the flow only occurs in the transverse direction. The geometries of the tumblers and

symmetries of the protocol described in this chapter are chosen to ensure this. One

correction is necessary: a three-dimensional grid is defined over the flowing layer to

calculate the number fraction field (now ( )tzyx ,,,φ ). In the physical system, diffusion in

the z-direction is small. In a tumbler with consecutive rotations on two orthogonal axes,

diffusion in the z-direction is much smaller than the y-direction. Simulations with and

without a diffusive component in the z-direction result in roughly the same final pattern.

51

3.2 Experimental Details

3.2.1 Quasi-two Dimensional Experiments

Experiments in a quasi-two dimensional tumbler (figure 3.2) roughly approximate

a two-dimensional cross section of a three-dimensional tumbler. The analogy is only to

approximate since the packing density is greatly affected by the walls. The container in

this apparatus consists of a clear front-plate (Plexiglas), a center-plate made of either

foam core or aluminum, and an aluminum back-plate. The shape of the container is cut

into the center-plate, in this case either a circle or a square. The separation between the

front and back plates is only a few particle diameters (~6 mm). The center of the back-

plate is affixed to a long shaft that supports the weight of the tumbler and is grounded to

reduce static electricity on the beads. The container is rotated at a fixed angular velocity

using a computer controlled stepper motor (Compumotor) such that flow is in the rolling

regime (typically between 1 and 5 rpm). The actual value of the speed of rotation is not

significant; the mixing and segregation dynamics are roughly the same as rotation

produces flow that is continuous and produces a flat free surface (the so-called rolling

regime). As mentioned earlier, rotation rate only slightly affects the depth of the flowing

layer.

Two kinds of experiments are conducted:

(1) Evolution of a tagged blob – same particles, different colors

(2) Final shape of a phase boundary – particles of differing physical properties

52

Figure 3.2: Quasi-two dimensional apparatus. The container is a template of any shape

with a front and back plate. The front is Plexiglas and the back is aluminum that is

grounded to reduce static charge. The computer-controlled motor rotates the container at

a constant rate.

53

In the first case, experimental studies of blob deformation are carried out to

provide insight into the mixing process. Experiments using a mono-disperse material and

a tagged initial condition (material of a different color) are much more difficult than those

exhibiting segregation. Only experiments in half-full containers are considered. One

note of caution: the flowing layer expands. Therefore the container is filled slightly less

than half-full with spherical, noncohesive beads ( 2.0=d mm, Quackenbush). The goal

is to have the system precisely half-filled under dynamic conditions. A circular blob of

colored beads is positioned at the desired location in the bed. This is accomplished by

inserting a plastic template in the chaotic region, enclosing a circular region of beads.

Beads inside the template are replaced with tagged beads and the template is removed.

Great care is used to avoid agitating the material while positioning the blob, replacing the

front plate, and attaching the container to the motor. Digital photographs are taken at

different times to record the progress of mixing.

Quantitative comparisons between theory and experiment can be made in terms of

the intensity of segregation (Danckwerts 1952). The intensity of segregation, I, is

calculated as

( )2/1

1

2

1 ��

�

�

��

�

�

−

−= � =

NI

N

i mi φφ

, (3.9)

which is essentially the normalized standard deviation of the concentration of tracer

particles from the value for perfect mixing. The local concentration, iφ at the grid point

i, is measured at N uniformly distributed points within the granular bed and compared to

54

the average concentration, mφ . The location of the tagged particles is determined by

thresholding the image. The local concentration is calculated by measuring the area

coverage in each square of a template (in computations, iφ is the local number fraction

calculated using x, y-coordinates of each particle).

Experiments studying segregation are significantly easier to perform. D- and S-

systems are examined using a variety of noncohesive spherical beads with sizes ranging

from 0.8 to 2 mm and densities of 2.5 and 7.8 g/cm3 (glass and steel, respectively). In

this case, the tumbler is filled to various levels. All systems are initially well mixed.

More accurately, the tumbler is manually agitated until the resulting segregation pattern

has no apparent structure. Segregation experiments in each container for each fill-level

were repeated several times.

3.2.2 Three-dimensional Bi-axial Experiments

A sketch of the apparatus used for three-dimensional experiments is shown at the

top of figure 3.3. The apparatus is designed to perform a variety of protocols in different

tumblers. The same frame is used for all three-dimensional flow studies. The container

in the center, here shown as a sphere, is rotated by independent programmed motions in

two orthogonal directions (denoted as axis A and axis B). This is achieved by the

container first being secured into the apparatus on axis A. This is mounted into a rigid

platform that rotates about an orthogonal axis B. Adjustments on axis A allow the

55

Figure 3.3: Three-dimensional tumbler and rotation protocol. This apparatus allows

independent motion on two orthogonal axes. The rotation protocol shown at the bottom

produces alternating time-periodic flows on each axis. In this example, each axis rotates

β+�360 and then counter rotates β− so flow occurs for a full rotation and the free

surface is re-leveled at the beginning of each half cycle.

56

mounting of different size/shape containers (the maximum width allowed is ~ 18 cm) and

centering of the container. A stepper motor (Compumotor), mounted on the inner

platform produces rotation on axis A. This inner platform is mounted on and rotates

about axis B. Two uprights support axis B. A second stepper motor mounted on one of

the uprights drives the rotation of the inner platform. These two uprights are mounted on

a main platform that has legs that can be adjusted to level the entire apparatus.

Any amount of rotation on each axis prescribed in any order can be investigated.

For instance, if clockwise rotation of a specified duration on axis A and B is denoted U

and V respectively and counter clockwise rotation U’ and V’, one could produce any

consecutive repetition of rotations such as UV, UVVU, or even UVU’V’ which would

return any container to its original orientation. One can vary the amount of rotation on

each axis, producing even more complicated protocols (e.g U1V1V2U2U1’V1’V2’U2’,

where subscripts pertain to different amounts of rotation). Detailed analysis of the

symmetries of the rotation protocol is straightforward (Franjione and Ottino 1992). Even

the speed can be prescribed, but as mentioned in the quasi-two dimensional system, the

dynamics are not sensitive to rotation rate as long as the flow is within the rolling regime.

Furthermore, as will be described in chapter 5, simultaneous rotation of both axes is

possible.

While any set of rotations is possible, the motion of the container is physically

limited. This apparatus does not allow control of any arbitrary set of rotations of the

container. For instance, the angle of axis A to horizontal depends on the amount of

rotation on axis B. Considering this, logical decisions regarding rotation protocol are

57

necessary to produce a desired flow. Flow does not start until the free surface reaches the

angle of critical stability ( iβ ), and the flow remains at the angle of repose β after rotation

ends (the difference between these two angles for the spherical beads used in these

studies typically �10<− ββ i ). Also, it is obvious that when axis B is near vertical no

flow results from rotation on axis A.

Figure 3.3 also shows an example of a rotation protocol designed to produce the

bi-axial flow protocol as will be modeled in section 3.3. Motions on each axis are

discrete, where the flow is fully developed after just a few degrees after start up (roughly

ββ −i ) and flow comes to rest quickly after stopping. To directly compare the results of

the experiment to the simple model, both the container and the free surface should start

level. This protocol, using above terminology, is U1U2’V1V2’. Rotation U1 is β+�360 ,

where flow occurs for approximately 360º of rotation from iβ until β+�360 . U2’ is

counter clockwise rotation of β− . This returns the container and free surface to level

(without flow) before starting rotations on the orthogonal axis, V1 and V2’.

Visualization of the segregation pattern resulting from different flows is difficult

in three-dimensional systems. Until recently, the only method was to sample material by

the use of thieves – long tubes used to extract particles from the middle of the bed.

However, this technique is invasive – disturbing the final state of the granular bed – and

it is shown to poorly represent concentrations throughout the system (Wightman et al.

1996). Another invasive technique is to solidify and slice the granular bed to reveal the

internal structure. This is accomplished by pouring liquid agar into the bed and allowing

58

it to cool. This technique has been shown to be rather successful (but difficult and time

consuming). Scanning the granular bed using NMR is a rather successful non-invasive

way to visualize the interior. However, results using NMR are limited by speed and

precision (especially in rapid flows), but have been used successfully by Nakagawa et al.

1993 to characterize granular flow and by Nakagawa et al. 1997 to study radially

segregated structures. This technique is only possible in some systems. Bead choice is

critical; for instance, glass beads are invisible. Also, the apparatus cannot have any metal

parts near the magnet. Accessibility to this equipment is limited.

Another possibility, but one that changes the physics of the system, is to run

experiments under slurry conditions. Jain et al. 2001 performed experiments

investigating axial segregation – producing alternating bands of different types of

particles along the axis of rotation – under slurry conditions. Using a liquid brings

substantial benefits. Using a suitable fluid (for instance, ~66% NaI solution) that roughly

matches the refractive index of the larger/lighter clear beads making them translucent.

This same methodology is used in experiments described in this chapter using 0.2 mm

black and 0.8 mm clear glass beads.

In dry systems, index matching is not possible. However, we use a technique that

works reasonably well using large clear beads (2.0 mm glass) and small fluorescent beads

(0.8 mm glass). Using UV light, the fluorescent beads glow brightly and the beads

closest to the container walls become visible (fluorescent beads within ~ 10 particle

diameters from the boundary). In experiments described here, the size of the sphere is R

= 70 mm for the dry systems, R = 40 mm for the slurry experiments. The cube used for

59

dry experiments is 100 mm and 60 mm on a side for slurry conditions. The smaller

containers are used in slurry experiments to minimize the overall weight of the container,

beads, and fluid.

3.3 Mixing and Segregation in a Quasi-Two Dimensional Tumbler

3.3.1 Chaotic Mixing with Diffusion

Consider half-full two-dimensional tumblers. Figure 3.4 shows computations and

experiments conducted in the circle and square. The initial condition is inset in the top

left corner of each picture – in the square, the blob is placed in the chaotic region. After

one pass through the flowing layer, the blob of darker particles is spread into a wedge. In

a circle, diffusion appears to mix better than stretching of the underlying flow – the blob

is mostly blurred across closed streamlines.

A comparison between theory and experiment is shown in figure 3.5 for blob

deformation in a square. There is good qualitative agreement between the two.

Comparison to the Poincaré map shows that several features of the underlying flow

structure are apparent even when diffusion is included: blobs are stretched along unstable

manifolds, and the density of particles in the regular regions is small. This is analogous

to passive dyes aligning with unstable manifolds of the flow as observed in the mixing of

fluids (Swanson and Ottino 1990). A quantitative comparison between theory and

experiment is shown in figure 3.6. There is again good agreement between the two for

the variation of intensity of segregation with time. Note that there are no adjustable

60

Figure 3.4: Blob experiments and computations. All particles are the same (0.2 mm

glass spheres) except for color. After just two rotations, diffusion spreads the initial

condition (upper left corner of experimental pictures) into a wedge.

Experiment Computation

61

Figure 3.5: Mixing of tracer particles in a square. This is a comparison of an experiment

using colored glass beads (left) and a simulation using the model (right). The number of

rotations for each image is listed in the lower right corner.

62

Figure 3.6: Variation of the intensity of segregation as a function of rotation

corresponding to the experimental and theoretical results in figure 3.5. The filled circles

represent experimental values and the open circles denote values obtained from the

model. Error bars result from uncertainties in image processing.

63

– the depth of the flowing layer, 0δ , is measured in the experiment using streakline

images.

While diffusion blurs the invariant structures of the underlying flow in laboratory

scale tumblers, in larger systems this is not so. Since the depth of the layer 0δ is

proportional to the flow distance L, the Péclet number scales as 2)/(~ dLPe . Therefore,

increasing the size of the tumbler decreases the rate of collisional diffusion. This can be

illustrated by means of computations comparing mixing in the square and circle, for large

and small sized containers. Figures 3.7 and 3.8 show the variation of the intensity of

segregation as a function of time for four different mixers and two different initial

conditions. In figure 3.7, the initial distribution corresponds to one material layered on

top of the other with colored particles on the bottom; the other initial distribution (figure

3.8) corresponds to the left-half of the bed containing colored particles and the right-half

of the bed white particles. In practice, the top-bottom placement is more practical. The

intensity of segregation for all cases is initially 5.0=I , which corresponds to the

theoretical value for a completely segregated mixture containing 50% of the colored

particles. In each figure, the distribution of particles in a large circle and square are

shown after 20 rotations.

The decay in intensity of segregation with time for the different tumblers and

initial conditions is different. In the case of the circular mixer, increase in the mixer size

results in a significant reduction in the rate of mixing. This is also the case in the square

mixer. However, comparing at equal sizes (i.e., equal Péclet numbers) we see that the

64

Figure 3.7: Variation of the intensity of segregation vs. mixer rotation obtained for

mixers of different shapes and sizes predicted by computations. The initial condition is

initially unmixed top-bottom. The large systems correspond to a Péclet number 410=LPe , and the small systems to 210=SPe , which indicates a mixer size ratio of 10.

The final distributions of particles in the large tumblers after 20 rotations are shown at the

bottom.

65

Figure 3.8: Variation of the intensity of segregation vs. mixer rotation obtained for

mixers of different shapes and sizes predicted by computations. The difference between

this and results in figure 3.7 is the initial condition (left-right here). Clearly the initial

distribution has an effect on mixing in the square. At the bottom, after 20 rotations the

distribution of particles in the large square shows black and white particles trapped in the

two separate regular regions.

66

square container mixes faster than the circular one for both the sizes considered here.

Furthermore when compared at equal sizes (i.e., equal Péclet numbers) it is apparent that

the square container mixes faster than the circle. These results indicate that chaotic

advection becomes more important with increasing mixer size. Chaotic mixing is thus

essential for practical applications, given that large mixers are the norm in industrial

processes.

One could argue, however, that a good tumbler design should mix well regardless

of the initial condition. Mixing in a large square is clearly biased by the initial

distribution. Material within the regular regions mix isolated from the rest of the flow.

Therefore, mixing is better when the regular regions contain both black and white

particles, as is the case in the top-bottom orientation. This is clear in the picture at the

bottom of figure 3.7. In the left-right initial condition, each regular region only has one

type of particle initially isolated from both the chaotic regions and the other type of

particle. In fact, looking closely at the graph in figure 3.8, mixing in both a circle and

square is initially very similar (as seen in figure 3.4). Only after a given number of

rotations is it clear that the material in the square container mixes faster. In the large

container it is clear that there is little mixing between black and white particles in the

regular regions, even after 20 rotations. In fact, although mixing is occurring, the square

looks similar to the initial condition where most black particles are on the left side.

67

3.3.2 Competition Between Segregation and Chaotic Advection

Consider the same systems as above but now with a bi-disperse granular mixture

(as considered in Hill et al. 1999a,b). Under flow bi-disperse granular materials

segregate. The focus here is on the final segregation pattern without regard to dynamics

that may depend on the initial conditions. The easiest case to understand is the circle.

The classic radially segregated structure is shown on the left in figure 3.9. In this case,

the dynamics in the flowing layer cause the smaller (more dense) particles to move down

in the layer. These particles have a low probability of reaching the end of the layer,

leading to a segregated core –segregation is complete after just 1 or 2 revolutions. The

model clearly captures this behavior. The closed streamlines represented by the Poincaré

map are shown below these segregation patterns and the boundary between the two

“phases” lies along the streamlines.

The case of the square is more complicated. First consider the half-full square.

Materials begin segregating radially in 1 or 2 revolutions. After this, smaller/denser

material moves toward the corners, deforming the interface between the two phases and

forming two lobes. Comparing the pattern in the experiment and predicted by the

computations to the Poincaré map, the segregated core clearly tags the regular regions of

the flow. This is significant and explained in more detail below. The two lobes are not

symmetric to reflections across 0=x , however they are rotationally symmetric (the

pattern seems to “pinwheel” clockwise). The Poincaré map appears symmetric across

68

Figure 3.9: Segregation in half-full circle and square - from top to bottom, experimental

D-system, experimental S-system, computation, and Poincaré map. The phase boundary

roughly conforms to the close streamlines (circle) or the regular regions (square).

69

0=x . However, there is a hyperbolic point at (0, -0.5) near which the underlying flow is

asymmetric across 0=x . Its unstable manifolds initially follow along the vectors (-1, 1)

and (1, -1), pulling material away from the hyperbolic point. The phase boundary

between the two materials seems to roughly follow along a heteroclinic orbit around the

regular region into the other hyperbolic point (located in the flowing layer).

As shown in chapter 2, the underlying advection patterns are very sensitive to the

degree of filling. Likewise, the segregation patterns are also sensitive to filling (figure

3.10). Parallel S-system and D-system experiments, computations, and Poincaré maps

are shown for 25%, 45%, 50%, 55%, and 75% full squares. Clearly the Poincaré map

predicts the shape of segregation (with the exception of the 55% full S-system). In the

Poincaré maps for the 45% and the 55% cases, the KAM surfaces inhibit mixing in the

radial direction. In the D-system and computations, the phase boundary is roughly traces

these KAM surfaces. The stripes formed in the 55% full S-system clearly show another

process at work. In S-systems at this filling, the layer may alternate between nearly

100% small particles and 100% large particles. Thus, depending on the composition of

the layer, the flow may become time-periodic under steady rotation. Khakhar et al.

(2001b) demonstrates that this also happens in a circle with the correct filling and

material parameters. A model describing this phenomenon alters the shape of a

piecewise linear flowing layer based on the local concentration. This is a clear example

of sensitivity to material properties.

70

Figure 3.10: Segregation patterns in a square with different fillings. From left to right –

experimental results in a D-system and S-system, Poincaré map, and computation. The

model captures the behavior except for the 55% full S-system where the layer velocities

are coupled to the concentration. In all other cases, the topology of the underlying flow

may assist in predicting the shape of segregation.

71

Symmetric fillings with regard to the square are 25%, 50% and 75% full (noted in

section 2.2, the boundaries are ¼ rotationally symmetric). For instance all material in a

25% full square passes through the flowing layer every ¼ rotation. Regions of poor

mixing in the underlying flow attract the smaller/denser material. In both the 25% and

75% full square it appears that the phase boundary roughly follows along the unstable

manifolds of the hyperbolic points in the chaotic region, as is seen in the 50% full square.

The segregation patterns for 25% and 75% full appear to “pinwheel” in the opposite

direction as the half-full case. In the middle bottom of each map, there is a hyperbolic

point. The unstable manifolds are rotated 90� as compared to those at half-full.

On first observation, it is surprising that the underlying flow, without considering

segregation, predicts the segregation patterns for most of the systems described above.

The Lagrangian formulation is useful for interpreting the behavior of the system. The

simplest case is when there is no diffusion ( 0=S ). Note first that the introduction of the

segregation drift velocity results in a qualitative change in the dynamical behavior of the

system (Hill et al. 1999b). In the case of a binary system (see Eqs. 3.7 and 3.8) the

divergence of the flow gives

( )

��

��

�∂∂−

=⋅∇=⋅∇yd

vv s φργ 1221 (3.9)

where v1 and v2 are velocities in phase space of species 1 and 2. Thus, the space for each

species contracts in the layer as the larger particles sink to lower y values (i.e.,

0/ <∂∂ yφ ). In pure regions, corresponding to 1,0=φ , the segregation velocities vanish

(see Eqs. 3.5 and 3.6) and volume is conserved. The case of circular mixers is easily

72

visualized. The denser particles sink to lower positions in the layer while lighter particles

rise, which translates into a radial motion with the denser particles moving to the central

core and the less dense moving to the periphery. At equilibrium there is complete

segregation with all of the denser particles occupying the central core and all of the

lighter particles in the periphery, with the boundary between the two being given by the

streamline bounding an area fraction equal to the volume fraction of the denser beads. In

noncircular mixers there is also an initial tendency for the denser particles to initially

move to form a central core and for the lighter particles to migrate to the periphery.

However, in this case streamlines are not invariant curves, and thus the core is distorted

and mixed because of advection.

The key to the structure formation lies in the invariant curves of the mapping. An

invariant curve composed entirely of one of the species is preserved by the mapping

(which now coincides with that for a single component because the segregation velocity

vanishes). Thus, the structure formed is determined by the surviving KAM curves of the

Poincaré section for the single-component system. The region within the outermost

KAM curve enveloped by the core formed by the dense particles is invariant as is the

peripheral pure region of light particles outside the innermost KAM curve. The effect of

diffusion primarily blurs the structures. When the core size exceeds that of the invariant

region, chaotic advection and diffusion act in concert to mix the particles outside the

invariant region (Hill et al. 1999b).

This analysis, as described by Hill et al. (1999b), is relatively successful in

explaining the phenomena seen in two-dimensional tumblers. In fact, Hill et al. (1999b)

73

reports that if three different sizes of beads are tumbled, with only a small amount of the

middle size, the middle sized beads reside near the phase boundary that is seen with only

two materials (they use this technique only to make the phase boundary more clear).

Beads of multiple sizes segregating and producing flow in the rolling regime could

possibly give “experimental contour plots” of the underlying flow. One also might ask to

what extent do these results apply to the full three-dimensional case. This is considered

in the next section where chaotic advection in bi-axially rotated containers interacts with

segregation.

3.4 Segregation in a Sphere and Cube: Bi-Axial Rotation

Consider now the interplay of chaotic advection and segregation in three-

dimensional granular systems. In a cylindrical tumbler rotating on its primary axis, the

flow is essentially two-dimensional with most flow occurring in the cross section.

Because of this, the dynamics in the transverse direction are similar to that of a 2D flow.

First, smaller/denser particles segregate to the middle of the granular bed – classic radial

segregation is seen in containers that have a circular cross section, and lobe formation is

seen in containers with a square cross section. This is evident in a half-full cube (figure

3.11) – after just a few rotations (the bottom of the cube rotates from left to right) the two

lobes of the segregated core are seen from the bottom as two horizontal lines. From this

perspective, smaller beads are glowing from behind the larger clear beads. Already it is

clear that there are some slight end effects preventing the lobes from extending, as seen

here, completely from the top of the picture to the bottom. After many rotations, these

74

Figure 3.11: Bottom view of a cube after 6 and 50 rotations (rotation moves image from

left to right). Two lobes form as seen in the two-dimensional segregation. They

eventually contract along the axis of rotation to roughly form a band. The beads are

brighter in the bottom picture as they reach toward the boundaries of the cube.

Axis of

Rotation

6 rotations

50 rotations

75

Figure 3.12: Bottom view of a long tumbler with square cross section (Hill et al. 1999b).

Even in the presence of axial segregation, chaotic advection clearly plays a role in the

two-dimensional cross section.

Axis of

Rotation

Axis of

Rotation

76

lobes contract toward the middle due to segregation along the axis of rotation. In an

elongated tumbler, multiple bands form after many rotations, but this underlying

segregated structure is still apparent (figure 3.12 from Hill et al. 1999b). The

combination of axial banding and underlying chaotic advection is discussed in more

detail in chapter 5.

The bi-axial protocol is designed to generate a simple example of a chaotic three-

dimensional flow. The aim is also to eliminate axial banding in order to study how

segregation in the transverse plane interacts with three-dimensional flow. Consecutive

rotations on orthogonal axes produce flow in one direction that is intended to cancel the

effects of axial segregation formed by flow in the previous direction – this may or may

not be the case, but axial bands do not exist. The location of periodic points of this flow

are given analytically in chapter 2 – there are both regular regions and chaos in both the

sphere and the cube. In the sphere, the periodic points are located in the '' zx =

(hyperbolic) and the '' zx −= (elliptic) planes at ( ) 221¼ xzy −−≅ . Two regular

regions exist in the '' zx −= corners. Hyperbolic points scattered throughout the bed

enhance mixing, the strongest of which is located in the center of the granular bed

(mixing near this point enhanced relative to flow in each direction).

The primary difference between these two geometries is that flow in a cube is

chaotic at any instant, whereas in the sphere, it is the time periodicity of the protocol that

generates chaos. As described above, the underlying flow has a large effect on the shape

of the phase boundary between the segregated regions in two-dimensional flows. It is

77

unclear if this same behavior will exist in three dimensions. If so, smaller/denser beads

will be pulled towards the regular regions such that the segregation will form

“experimental” Poincaré maps. This is computationally investigated in a half-full sphere

and cube.

Consider a bi-disperse system in a bi-axially rotated sphere. Computations using

the model given by Eqs. 3.3, 3.7 and 3.8 show that the denser material migrates to the

center after just one or two cycles. Segregation is almost purely radial (figure 3.13 – red

particles are more dense, green are less dense). No mechanism driving axial segregation

is built into the model. There is little evidence that the segregation is affected by the

underlying chaotic advection. Slices in the x-y and z-y planes at z or 5.0,0,5.0−=x

respectively show that each slice looks roughly the same as segregation seen in a circle.

Denser material does not appear to favor the regular regions of the flow, nor does it avoid

the regions of chaos. Different volume ratios of beads exhibit the same pattern.

Two different types of experiments are used to further investigate these results.

First, large clear beads and small fluorescent beads (2.0mm and 0.8mm respectively) are

tumbled bi-axially in the sphere. Under UV light, the smaller beads all seem to reside in

the center (figure 3.14). After as many as 30 cycles, this pattern remains the same. There

are no details captured from this bottom view picture that indicate anything but almost

purely radial segregation (as predicted by the model). To check that this non-invasive

technique is accurate, another experiment is performed under the same conditions using

larger red and smaller blue beads and solidified with agar after 30 cycles. A center cut in

78

Figure 3.13: Computation of segregation in a bi-axially rotated sphere. Segregation

appears to occur mostly in the radial direction. Slices are in the x’y’ and z’y’ planes at –

0.5, 0.0 and 0.5 along the orthogonal axis.

79

Figure 3.14: Experiments in bi-axially rotated sphere. Top picture is the bottom view of

the sphere partially full with fluorescing smaller beads. In order to verify that the

segregation is mostly radial, the granular bed is solidified with agar and sliced. The slice

is in the x’y’ plane at 0' =z .

80

the x-y plane along 0' =z reveals roughly the same pattern. The smaller blue beads form

an almost purely radially segregated core.

There is good qualitative agreement between the experiment and computations.

Both predict radial segregation. It is somewhat surprising that the underlying chaotic

advection seems to have little effect on the segregation pattern. This fact can be

explained by taking a closer look at the Poincaré map of this flow (figure 2.20). The

volume of the regular regions is small. Also, the points trapped in the regular region

appear as if they are all located roughly at the same radius. If regular regions “attract”

smaller beads, they could do so without distorting the phase boundary in the radial

direction. In the same way, the hyperbolic points in this flow have unstable manifolds

that seem to point orthogonal to the radial direction. If the phase boundary follows along

the unstable manifolds, as it does in the two dimensional experiments, there would be no

evidence of distortion in the radial direction. Aside from this, it is also surprising that

segregation is mostly radial considering that flow in S-systems can be more complicated

than represented by this simple model (as seen in the 55% full square in figure 3.10).

The bi-axially rotated cube presents even more intriguing behavior. Prior to

performing computations, imagine two scenarios (figure 3.15). One scenario assumes

that the regular regions of the three-dimensional map will attract the smaller/denser

material. This would result in two lobes forming two spots covering the regular regions

in the Poincaré map. The other scenario assumes that dynamics in the two-dimensional

flow are dominant such that invariant structures formed by three-dimensional flow

81

Figure 3.15: Depiction of possible outcomes. The final segregation pattern may

resemble picture A at the end of each rocking cycle if regular regions in the three-

dimensional flow captures smaller/denser particles analogous to the two-dimensional

system (the regular regions are shown to the right of A from figure 2.22). However, if

lobe formation is only an artifact of two-dimensional flow, the picture may have either

four lobes (B) or two elongated lobes (C) at the end of each cycle (like those shown to the

right of C from figure 3.10).

82

(different than those in two-dimensions) have little consequence on the shape of the

phase boundary.

The result of this could be four lobes, one in each corner (the regions where

regular regions exist for flow in both directions). Moreover, the biaxial protocol could

simply produce a time-periodic pattern, where elongated lobes form from flow in each

direction associated with the direction of flow in each half of the cycle.

In computations, a three-lobed segregation pattern that forms after 6 cycles (figure

3.16) does not correspond to either of the above scenarios considered in figure 3.15.

Computations in a bi-axially rotated cube show that red (more dense) particles form

segregated lobes. This is clear in vertical slices (figure 3.16, left), as was done in the

sphere (figure 3.13), but the lobes are easier to see when the bottommost particles

(particles below 7.0−=y ) are removed (figure 3.16, right). Three lobes are formed, all

of which avoid the center of the cube. Two of these lobes, those located in the upper and

lower right corners, are very clear. The other lobe on the left side resembles one of the

two elongated lobes formed when rotating on a single axis (figure 3.11) but less coherent.

Clearly this is unlike the radial segregation seen in the sphere – the shape of the

associated three-dimensional phase boundary is complicated.

While this computational result is quite surprising, it correctly predicts the

segregation pattern seen in experiments (figure 3.17). In dry experiments, the three-lobe

structure that arises after just a few cycles is very clear when using fluorescent particles

and UV light. The lobes avoid the center of the granular bed, two in the upper and lower

83

Figure 3.16: Computations after 6 cycles of bi-axial rotation in a cube (left – vertical

slices, right – horizontal slice). The segregation pattern avoids the center and migrates

into three different lobes of denser (red) material. In the “bottom view”, the bottommost

material is removed to show an interior slice at y = -0.7

84

Figure 3.17: Comparison of an S-system, D-system and computations of pattern

formation in a bi-axially rotated cube. Similarities between the different experiments are

remarkable. This complex pattern is captured by the model.

85

right corners and the third on the left side. This pattern is very reproducible and stable

for at least 30 cycles. In fact, this pattern is seen twice in each rocking cycle. This

picture shows the pattern at the end of the rocking cycle. Because of the symmetry of the

protocol (flow could start in either direction), this pattern appears again, rotated 90�

counter-clockwise half way through the cycle.

The experiment and computation are in good qualitative agreement. The relative

locations, sizes and shapes of the lobes in the experiments are roughly matched by the

computations. This is surprising in light of the following:

a) The model essentially only considers dynamics in the transverse direction.

b) Computations are of a D-system (simple), whereas the experiments are an S-

system (complicated – flow may couple with local concentration).

Consider now a similar, perhaps more complicated experiment run under slurry

conditions. The interstitial fluid properties are not considered as if the particles are in a

vacuum. Air likely has a slight effect on the flow, but replacing air with a more viscous

fluid makes this system even more complicated. Lubrication forces between particles in

the flowing layer change the depth of the flowing layer and the rate of segregation (Jain

et al. 2002). For this reason, it is remarkable that the same three-lobe pattern forms in a

bi-axially rotated cube under slurry conditions (figure 3.17 also shows the dry case and

computations and the Poincaré map predicting this pattern). Substituting air with a NaI

solution allows index matching of the larger glass beads, making the smaller black beads

visible. Furthermore, this system is in a smaller cube with different sized beads (0.2 mm

and 0.8 mm). Comparing the resolution of each of the two experimental pictures

86

suggests that the new technique using fluorescent beads and UV lighting in the dry

condition may reveal the internal structure as well, if not better than index matching. The

relative brightness of fluorescent beads gives depth to the contours of the internal

segregated structure.

Comparing these experiments and the computation, it is clear that the material

parameters appear to have little influence on the final segregation patterns in almost all of

the two and three-dimensional systems considered in this chapter. In these systems, the

shape of segregation seems to follow along with the invariant structures of the underlying

flow. If this is universally true, evaluating the shape of the phase boundaries of

segregated patterns could be used as a tool that suggests the underlying flow. As

mentioned in the quasi-two dimensional system, mixtures of three or more different sizes

of beads segregate so that the middle phase resides between the small and large sized

beads. Multiple phase boundaries would show more of the underlying flow behavior in a

single experiment.

87

CHAPTER 4

KIMEMATICS OF A ROTATING-ROCKING SPHERE

This chapter considers chaotic advection within a rocking-rotating sphere.

Modeling is straightforward, but the underlying dynamics are not; for a

fixed amplitude, small changes in rocking frequencies give rise to

radically different segregated structures. The topology of the underlying

flow has implications on the stability of segregation in the experimental

system.

4.1 Model

Consider a half-full sphere. The sphere rotates on one main or primary axis

denoted A which in turn is moved in a prescribed manner about the axis labeled B (figure

4.1). The rotation on axis A induces a flow that mixes primarily in the x-y plane.

Rocking the main axis of rotation induces flow in the axial direction z. Conventional

thinking indicates that adding axial flow should enhance mixing beyond diffusion-limited

axial transport, as occurs in three-dimensional tumblers rotated on a single axis.

Moreover, time-periodic flow induces chaos, which also enhances mixing.

The spherical geometry allows modeling as follows. The amount of rotation

around axis A is denoted θ . The angle of axis A to the horizontal is ( )θγ fAsin= .

88

Figure 4.1: Rocking protocol and coordinates. In this protocol, the sphere has constant

rotation on axis A and rocking motion on axis B. Note that the flow is in the x-direction

at an angle β to x’-direction relative to the lab.

89

Thus, for a frequency 1=f , one rocking cycle occurs every full revolution on axis A.

f/2πθ = corresponds to one cycle. The instantaneous angular velocity about axis B is

( )θθγ fAfdd cos/ = .

The free surface is assumed flat – therefore it is always circular. Coordinates

associated with this model are as follows.

• The flowing layer frame (x, y, z) – chosen such that the x-axis always

points in the flow direction, the y-axis points opposite gravity, and the

z-axis is orthogonal to the xy plane.

• The laboratory frame (x’, y’, z’) – chosen such that the x’-axis points

along axis B, y’-axis points opposite gravity, and the z’-axis is

orthogonal to the x’y’ plane.

The origin for both of these coordinate systems is stationary at the center of the circle that

is also the center of the sphere. The instantaneous angle between x and x’ is denoted as

β .

Assume that within the flowing layer each cross section in the flow direction is

independent of its neighbors. Moreover, assume that the flow readjusts instantaneously

to changes in orientation of the sphere. Surface velocities at the air-granular bed

interface scale with the local free surface half-length 21 zL −= (figure 4.2). The

flowing layer is comprised of a thin lens-like region of form ( )( )2

0 /1 LxL −= δδ - the

same as described in section 2.2.

90

The flow direction relative to lab coordinates is given by β . This angle is

determined by the ratio of angular velocity about axis B and axis A, where

( ) ��

��

�=

γθ

cos

/tan 1- dd��

. (4.1)

Rocking amplitudes of �90≥A are not considered, therefore �� 9090 <<− β for all f.

The equations for velocity in the flowing layer are

��

�� +=

δyuLMvx 12 (4.2)

2

��

��−=

δyMxvy . (4.3)

The function M in (Eq. 4.2) and (Eq. 4.3) is given by

( )2

2cos ��

��+=

θγγ

ddM . (4.4)

Particle path-lines are calculated by integrating the above equations in the flowing layer

with solid body rotation in the bed. A pictorial representation of the relationship of

rocking angle and direction of flow is depicted in figure 4.3. The flow only points in the

x’ direction when 0/ =θγ dd . This model applies to any prescribed motion about axis

B. Consider sinusoidal rocking as a perturbation to a base case where axis A is always

horizontal and each cross section is a simple circle (as described in section 3.1). When A

and f are non-zero, flow occurs in the axial direction. Large values of A and f produce

flow that is almost perpendicular to the x’-direction. The goal is to investigate the

behavior of the system in the A-f phase space.

91

Figure 4.2: Surface velocities at the midsection of the free surface. Velocity scales with

the length of the free surface 21 zL −= . The y-axis points out of the page.

92

Figure 4.3: Relationship between rocking angle and instantaneous flow direction.

93

4.2 Quasi-periodic and Chaotic Path-lines

Consider now the flow generated within the sphere. A few general observations

are in order. In general, most streamlines produced by solid body rotation intersect the

flowing layer. However, a cone-shaped stream-surface exists where material within it

circulates without intersecting the surface flow. This region is illustrated for a rotating

sphere that is rotated at a constant angle of inclination in figure 4.4 – without rocking

streamlines close onto themselves. The size of this region depends on the value of γ ,

and all streamlines intersect the flowing layer when 0=γ . This cone is analogous to the

core in the center of a more than half-full tumbler. With rocking, this cone-shaped core

grows and shrinks periodically. Following a path-line (figure 4.5) shows that some

particles are temporarily trapped within this region. These trajectories occasionally get a

“kick”, temporarily rotating without entering the flowing layer. This happens to most of

the particles that are located near the projection of axis A throughout the granular bed.

Other initial conditions give rise to quasi-periodic trajectories (figure 4.6). These quasi-

periodic trajectories may be confined to a small region or spread seemingly throughout

the sphere. In either case, mixing is poor.

In the lab frame (x’, y’, z’), particles flow on hemispherical surfaces when rotating

as a solid body in the bed. Consider tracking the radial position of a particle with a quasi-

periodic trajectory. After passing through the flowing layer, it returns close to the same

hemispherical surface when exiting the flowing layer (figure 4.7 - r is calculated each

time the trajectory leaves the flowing layer). The figure shows the dynamics after ~200

94

Figure 4.4: Sphere rotating on angle. Without rocking, flow produces closed

streamlines. Flow in the cone-shaped region does not intersect the flowing layer.

95

Figure 4.5: Typical chaotic path-line. Occasionally, this trajectory is temporarily trapped

in the cone-shaped region depicted in figure 4.4.

Top

Front

Side

96

Figure 4.6: Typical quasi-periodic path-line. Although the particle explores much of the

container in the axial direction (top picture), mixing is poor in the region neighboring this

particle at any instant.

Top

Front

Side

97

rotations. It is clear that the distance traveled in the r-direction is small and the behavior

is nearly periodic. In this graph, the aggregate numerical error during these 200 rotations

is comparable to the width of the line plotted. On the other hand, a chaotic trajectory also

moves slightly in the r-direction undergoing small, seemingly random, jumps. Mixing in

the chaotic regions is very poor in the radial direction. All of these trajectories, chaotic or

quasi-periodic, are trapped within a small range of r over many rotations of the tumbler,

effectively flowing on a quasi-two dimensional hemispherical surface.

4.3 Poincaré Maps

Poincaré maps are computed by plotting the particle position at the end of each

rocking cycle. In these computations, locations are plotted for 200 cycles. By definition,

Poincaré plots yield more information about the long-time dynamics than path-lines.

Because all particles roughly return to the same surface, the initial conditions for each

plot have the same initial radial position. Most of the maps described in this chapter have

initial conditions starting at 95.0=r , giving information only about the dynamics near

the container walls. However, even though this technique is limited, this method will

allow comparison between the model and experiments. Most experiments (unlike those

mentioned in section 3.2) reveal only what is in immediate contact with the containers’

walls.

98

Figure 4.7: Time evolution of the radius of a quasi-periodic trajectory (black) and a

chaotic trajectory (red). The radius is calculated each time the particle leaves the flowing

layer over ~200 rotations. The quasi-periodic trajectory has small periodic motion in the

r-direction. The chaotic trajectory is irregular, but also has relatively small radial motion.

This suggests that most dynamics occur within hemispherical “shells” with little radial

transport.

99

As the frequency is decreased, the tumbler rotates multiple times in each rocking

cycle. As 0→f , each computation takes an excessively long time. Therefore, most

computations are limited to the range 5.15.0 << f . Many Poincaré plots were computed

within this range in order to extract the most important aspects in the flow. These maps

are organized in Appendix A with only a few highlighted in this chapter.

Figure 4.8 shows a sample of the various dynamics observed. View is from the

bottom, where the main direction of rotation moves material from top to bottom – and

rocking is side to side (i.e. left to right and right to left). Quasi-periodic trajectories are

closed orbits; chaotic regions appear as a seemingly random scatter of particles. All of

these maps are 180º rotationally symmetric. This symmetry is exploited to reduce

computational time by only calculating half of the particle paths represented by each

map. Most maps at low amplitudes or low frequencies look similar to figure 4.8a

(�30=A , 925.0=f ), where almost all trajectories are quasi-periodic and chaotic regions

are small, located near the ends of the sphere, almost out of view.

The map in figure 4.8b (�5.42=A , 6.0=f ) shows that large regular regions

where particles circulate about a periodic point can form. Higher order periodic points

exist at different frequencies forming chains of islands (figure 4.8c, �5.47=A , 65.0=f ).

Some periodic points are hyperbolic (figure 4.8d, �60=A , 95.0=f ), where stretching is

enhanced only locally. In this figure, all trajectories shown appear to be chaotic. At

higher values of f, regular regions are smaller or non-existent (figure 4.8e,

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Figure 4.8: Variety of different Poincaré maps in the A-f plane. The model displays rich

dynamics including clearly visible quasi-periodic trajectories (a), regular regions (b, c, e),

hyperbolic points (d), and widespread chaos (f). g and h show large chaotic regions on

either side isolated by quasi-periodic trajectories acting a barrier to mixing. The lower

graph gives the parameters for each map.

101

�40=A , 475.1=f ), and at the highest values of amplitude and frequency computed,

apparently all trajectories are chaotic (figure 4.8f, �70=A , 475.1=f ). In this case, there

may be periodic points with regular regions that are not resolved. Figures 4.8g,

h(�70=A , 5.0=f and

�40=A , 325.1=f respectively) show that these chaotic regions

may reach near the center, but as long as quasi-periodic trajectories exist, there is no

transport from one hemisphere to the other.

Concentric Poincaré maps shown in figure 4.9 - the interior revealed as if peeling

an onion - represent the dynamics at each radial position (�45=A , 6.0=f ,

9358.03258.0 << r ). For ease of comparison, each map is plotted with same average

density of points – less initial conditions in the innermost maps. The relative size of the

chaotic regions compared to the “surface” shown is relatively the same at each radial

position. However, it seems that structures existing near the boundaries (near 1=r )

move from right to left, then shrink and disappear as 0→r . In fact, the way the maps

transform as r decreases is similar to the way the maps transform with decreasing

frequency. This will become clearer in the next section where the sensitivity of regular

regions to both f and A is investigated.

4.4 Flow Bifurcations

Consider the regular regions that are shown in figure 4.8b. The existence of the

periodic points at the center of these regular regions is sensitive to f. The way in which

these regular regions that appear at a given frequency grow and shrink is shown in

102

Figure 4.9: Concentric Poincaré maps. Map A corresponds to 9358.0=r and the last

map (B) to 3258.0=r . It is apparent that the dynamics near the boundary ( 1=r ) vary

slightly from the inner dynamics. The graph at the bottom shows the location of this

computation in the A-f phase space. This phase plot is important in summarizing results

in chapter 5.

103

Figure 4.10: Bifurcation of regular regions. Poincaré maps over a very narrow range of

frequencies show creation and extinction of regular regions associated with periodic

points.

104

consecutive maps at increasing frequencies (figure 4.10, �45=A , 6175.05775.0 << f ).

With increasing frequency, these points originate at the sides of the map near the chaotic

region with quasi-periodic trajectories in the middle of the map between them. These

points move toward the center, and then pass each other as they continue to cross to

opposite sides. This transition also occurs over a shorter range of frequency points of

higher periodicity (figure 4.11, �45=A , 795.07875.0 << f ).

Consider the interaction between rocking and circulation in the sphere. The

average circulation time in the direction of rotation for a half-full container is a ½

rotation. Periodic points in this system result from period-doubling – when the rocking

time and rotation time are commensurate. Period-doubling is most apparent for low-

order ratios (e.g., 0:1, 1:1, 1:2). Near 1=f , the dynamics should reflect transition when

the rocking-rotation relationship is 1:1.

There is a bifurcation near this point (figure 4.12, �45=A , 968.0953.0 << f ).

At the lowest frequencies shown, the quasi-periodic trajectories seem to dilate in the

center of the map and neck down as they pass through the layer. At slightly higher

frequencies, two hyperbolic points appear (highlighted with “*”). These hyperbolic

points move apart and reconnect in the flowing layer (not shown). In the final maps,

quasi-periodic trajectories once again arise. Trajectories neck down in the center of the

map and they dilate in the flowing layer. This is a phase shift; the point of dilation moves

½ φπ / .

105

Figure 4.11: Bifurcation of regular regions. Same type of bifurcation as demonstrated in

figure 4.10 with period 4 periodic points.

106

Figure 4.12: Evolution of the system in region of �45=A as f is increased from

953.0=f (map A) to 968.0=f (map B). This map changes from mostly quasi-periodic

to two hyperbolic points (*) and back over a narrow range of frequencies. Notice that the

dilation of quasi-periodic trajectories in map A is seen passing through the layer in map

B.

*

107

This same bifurcation is apparent at higher amplitudes over a slightly wider range

of frequencies (figure 4.13, �60=A , 960.0930.0 << f ). Note that the bifurcation has

shifted to a slightly lower frequency. In the graph at the bottom of the figure, a gray

region is sketched as to mark the location and relative range of this bifurcation. Although

the computations do not reflect this bifurcation as 0→A , the periodic points must exist

until 0≡A . At the highest values of A, this period-doubling eventually leads to chaos.

The rocking-rotating sphere demonstrates a diverse range of dynamics as

compared to flow in two-dimensional and even the bi-axially rotated sphere and cube. It

is clear that there are likely other phenomena occurring in this system or systems such as

the rich variety seen here that warrant closer investigation. However, the novelty of the

behavior is not as surprising as the ability of this model to mimic various phenomena

seen in the represented experiment, as will be seen in the next chapter.

108

Figure 4.13: Evolution of the system in region of �60=A increasing f from 930.0=f

(map A) to 960.0=f (map B). This bifurcation occurs near the same frequency range

as figure 4.12. The “Bifurcation Region” denotes that the range of frequencies increases

with increasing values of A. This figure will be compared with experimental results in

figure 5.10.

109

CHAPTER 5

ROCKING SPHERICAL TUMBLER

This chapter addresses the stability of axial segregation in a spherical

tumbler in the presence of perturbations leading to axial flow. Prior to

these experiments, axial segregation was thought to only occur primarily

in long cylinders rotating on a single axis. It does however occur in

perturbed spheres. Axial banding in a rocking spherical tumbler is

surprisingly robust. A model of underlying advection without segregation

reflects stability of these bands.

5.1 Axial Segregation in Various Geometries

Oyama (1939) is the earliest study to report materials segregating into alternating

bands of different sizes in the now classic system of a horizontal cylindrical drum

rotating on its main axis. Since then, there have been few variations of the experiments

yielding similar phenomena. However, surprisingly, little is known about the basic

mechanisms that lead to axial segregation. More generally, axial segregation can occur

in S-systems, D-systems, or when particles differ by other physical properties such as

shape. These systems first exhibit standard radial segregation after 1-5 rotations as

supported by evidence from MRI (Nakagawa 1994; Hill et al. 1997; Metcalfe and

Shattuck 1996). It has been suggested that radial segregation is a necessary precursor to

110

axial segregation. After 10-100 rotations the particles separate further into bands along

the axis of rotation, leading to relatively pure mono-disperse regions. The dynamic angle

of repose has been shown to depend on concentration, and one theory explains band

formation as being driven by a positive feedback mechanism magnifying initial

concentration fluctuations (Das Gupta et al. 1991; Hill and Kakalios 1994). Another

observation frequently reported is that axial bands commonly form first near the ends.

The end-plates affect the angle of repose, and it is typically noted that few smaller or

more dense particles are found at the ends after just a few rotations. The effects of the

end-plates can eventually migrate inward with consecutive band formation toward the

center.

There are no scaling laws to predict the initial number of bands. After bands

form, they may undergo a coarsening process where neighboring bands spontaneously

merge together. Experiments (Choo et al. 1998, Fiedor and Ottino 2003) and theory

(Aranson et al. 1999) suggest that the rate of coarsening that reduces the number of bands

is logarithmic. However, recent experiments indicate that at certain parameters bands can

travel in the axial direction (Choo et al. 2002) and form and merge continuously in time

(Fiedor and Ottino 2003).

There are several other works that are relevant to the material presented in this

chapter. The following experiments are noted in order from simple to more complicated

(shown in figure 5.1). Axial banding is seen under slurry conditions, where a viscous

liquid replaces air (Jain et al. 2001). As mentioned earlier, an advantage of operating

systems under slurry conditions is that a suitable viscous liquid can roughly match the

111

Figure 5.1: Tumblers of different geometry ordered from generating simple to

complicated flow. a) and f) are widely used throughout industry.

112

Figure 5.2: (From Jain et al. 2001) Self-organization in a slurry consisting of equal

parts per volume of 0.2 and 1 mm glass beads. The liquid, used to match the refractive

index of the clear beads, is a 66% by weight solution of NaI in water. Pictures are

taken from the primed position, and the direction of lighting is indicated by the light

bulbs. Top image (AA’) and side image (BB’) show the internal structure consisting of

a core of small beads within clear large beads. Side image (CC’) shows only the

surface view.

113

refractive index of transparent glass beads, revealing the internal structure of the system

(figure 5.2 – the bi-axial rotation experiment in section 3.4 uses similar particles and

fluid). Top view shows the internal structure consisting of a core of small beads within

large clear beads. Surprisingly, in the systems studied by Jain et al. (2001), axial

segregation occurs faster under slurry conditions than in the corresponding dry systems.

Axial segregation also occurs in cylinders with non-circular cross sections (Hill et

al. 1999b). As seen in chapter 2, flow in a non-circular cross section is chaotic.

Surprisingly, axial bands also form in cylinders with square cross sections, even though

segregation in the cross section is considerably more complex than in the classic cylinder

(figure 5.3). The standard axial segregation pattern is clearly observed in the top of the

flowing layer (figure 5.3a), while two lobes are evident on the bottom view of the mixer

(figure 5.3b), as is seen in the two-dimensional 50% full square (described in chapter 3).

It appears that classical radial segregation is not a necessary condition for axial

segregation. In closer analysis, patterns formed at other fill levels of the square also co-

exist with axial banding (Hill et al. 2001).

The experiments of Zik et al. (1994) are quite relevant to segregation in a sphere.

They examine axial segregation in containers with modifications in geometry along the

axial direction. Two different geometries were considered, one with a periodic variation

in radius alternating thick and thin regions, and the other a helical tube. A difference

should be noted. In both of these contours the rotation is slow, producing discrete

avalanches. In the thick-thin tube, axial segregation is induced in materials that did not

form bands in a straight tube of comparable radius. Smaller beads form bands in the

114

Figure 5.3: Axial segregation observed in a tumbler with square cross section. Views

correspond to orientation shown to the right. (a) - The standard axial segregation pattern

is clearly observed in the top of the flowing layer. (b) - In the bottom view, two lobes are

evident, as is seen in the two-dimensional square mixer.

115

“bellies” and bands of large beads in the “necks”. They report that the location of bands

is reversed at lower fill levels. In the helical tube, material completely segregates to

opposite ends of the container. Another geometry similar to this is a double-cone blender

studied by Alexander et al. (2001), also forming axial bands. Bands of smaller beads

form in the middle of the container at lower rotation rates. This pattern reverses at higher

rotation rates with larger particles forming bands in the middle. These bands coarsen in

time. In both of these studies, the authors suggest that geometry produces a variation in

the angle of repose in the axial direction, aiding in the initiation of axial segregation.

The majority of tumblers used in industry produce some degree of flow in the

axial direction, attempting to destroy segregated structures and enhance mixing. The

geometry and motion of these tumblers is typically complicated (e.g., V-blenders,

double-cones, and cubes – some of these rotated on an oblique axis as shown in figure

5.1). Various mixing studies have been performed in many specific (often patented)

tumblers. Bands formed from axial segregation have not been reported in any tumblers

with periodic axial flow. It is a common assumption that axial mixing inhibits axial

segregation. In attempting to quantify this, Wightman and Muzzio (1998b) perturb flow

in a cylinder with rocking motions, however they only consider a few different

frequencies at low amplitudes. The initial condition is unmixed with each material on

different ends of the container in the axial direction. Their system, without rocking,

produces axial bands. Concentration of material is evaluated after numerous rotations

either by solidifying the granular bed and imaging or by inserting dividers, sieving the

particles, and weighing the amounts as a function of axial distance. With symmetric

116

rocking with amplitude of 5º, the segregation pattern investigated at three different

rocking frequencies is complex. They report segregation in the axial direction, however

there is no evidence of distinct alternating bands. At one frequency, they see progression

towards homogeneous mixing. At another frequency, they see poor axial mixing –

deviating very little from the initial condition. They note that there are higher

concentrations of larger beads near the ends after 1000 rotations. It is unclear whether

these patterns are steady after 1000 rotations or even if distinct bands rise to top of the

flowing layer. In Shinbrot et al. (2001), rocking angles are varied from 0º to 90º.

Concentrations differed along the axis, but once again there is no evidence of axial bands

(except, of course, at 0º).

We investigate 3D mixing and segregation within a sphere. This is a simpler

system than those systems producing 3D flow listed above. The main reason for this is

that in a sphere the free surface remains constant regardless of the orientation. Modeling

of continuous flow within the sphere is also possible. Without rocking, band formation in

a sphere is expected considering the bands that arise in a double-cone blender (Alexander

et al. 2001). In an S-system, smaller beads do form a band in the center of the sphere

(figure 5.4), agreeing with behavior seen by Zik et al. (1994).

5.2 Experimental Details

The experimental apparatus is the same as described in section 3.2. Figure 5.5

shows the experimental setup and rocking protocol. The only modification to this system

from the bi-axial protocol (chapter 3) is to produce simultaneous rotation about two axes.

117

Figure 5.4: Formation of an axial band in a sphere (bottom view). Top – a band

without rocking captured without using a UV filter. Bottom – a coherent band can also

form with rocking. Each band connects through the flowing layer (not shown). From

this perspective, rotation on axis A moves the particles shown from left to right.

118

Figure 5.5: Experimental setup and rocking protocol. Continuous rotation on axis A

produces flow. Rocking on axis B causes time-periodic flow that changes direction

relative to flow without rocking.

119

The motor control indexer (Compumotor) allows simultaneous control of up to four

motors. It is designed with its own programming language to ensure synchronization of

any combination of these motors. Our system has two motors where a motor produces

motion on each axis. The stepper motors have 25,000 steps per 360º of rotation

providing smooth motion. The first motor rotating axis A has constant angular velocity.

The second produces sinusoidal motion on axis B, where the rocking angle is defined as

( )tA πφγ 2sin= . Rocking is scaled by the constant angular velocity on axis A,

πθω 2// =t and ωφ /=f such that ( )θγ fAsin= . Thus, for 1=f , one rocking cycle

occurs in one full rotation on axis A. The indexer synchronizes all motions with an

internal clock. The coordinates to describe most phenomena in this chapter will be in the

lab frame – the x’, y’, z’ coordinate system described in chapter 4 and shown in figure

4.1; some aspects of the pattern in the flowing layer are described in terms of the frame of

reference relative to the direction of flow (x, y, z).

Patterns arise at the boundary consisting of particles resting against the inside

bottom surface of the tumbler. For clear imaging, most experiments are performed with

larger black beads (2 mm) and smaller fluorescent green beads (0.8 mm) illuminated by

UV lamps. A stationary digital camera captures images of these patterns from below.

Air is the interstitial fluid. The sphere ( mm 140=r ) in most of these experiments is

half-full with granular material and rotated such that the flow is always in the rolling

regime with a flat free surface. The mass ratio between small and large beads is typically

1:2 – there are roughly 500,000 beads in the sphere (estimated from the volume of the

120

sphere, packing fraction and size of individual beads). Most of the results described

below use these materials unless otherwise specified.

A potential difficulty in using S-systems should be noted. It was mentioned in

chapter 3 that size segregation may be more complicated than density segregation. Under

certain conditions, the velocities in the flowing layer are coupled to local concentration.

The flowing layer may become rich in small particles or consist entirely of large

particles. This produces time dependent speeds in the shear layer making S- systems

more complicated than D-systems. While evidence of this sort of coupling is not seen in

our experiments, it is likely to exist at other fill levels or with other materials.

A few experiments are also conducted under slurry conditions, where the

interstitial fluid is water. The radius of the sphere is 40 mm and the particle sizes are 0.8

mm (zirconium silicate) and 0.2 mm (glass), so both size and density are different

( 2/ 21 ≈ρρ ). These particular beads were chosen for clear imaging, however it should

be noted that they segregate quickly in a cylindrical drum forming many axial bands after

just 5 - 10 rotations (Fiedor and Ottino 2003). Size segregation dominates, causing

smaller beads to form a radial core after 1 or 2 rotations before re-emerging at the surface

as bands or other patterns.

Because segregation dominates, initial conditions are never “well-mixed”.

However, the initial distribution of particles in all experiments is set by erratic motions of

the tumbler on its two axes, resulting in each initial condition being unique. Transients

from the initial condition are first destroyed by radial segregation in one or two cycles.

121

In the dry case, it is clear from many repeated experiments that the final segregation

pattern is independent of the initial condition. However, it appears that certain patterns

are transient and dependent on the initial condition in slurry conditions.

5.3 Formation of a Band and Spots

Consider the dry system described above. Rotation on a single axis produces a

stable single band after ~5 minutes of rotation at 5 rpm. In fact, this combination of

materials in this container never produces more than one band. When rocking, the three

most commonly observed patterns are shown in figure 5.6: bands, spots, and no visible

segregation pattern (where all smaller beads segregate away from the walls of the

granular bed). An axial band wraps around the bottom of the granular bed and connects

through the flowing layer. For different frequencies, the band may be very wavy or

relatively straight, however within the flowing layer, it always aligns roughly in the

direction of flow. Also, this band is often observed close to center of the flowing layer at

all times (in the frame of the flowing layer where flow is in the x-direction, it is located at

or near 0=z ). The final pattern begins to emerge after ~20 cycles – roughly the same

amount of time as without rocking. Surprisingly, at certain frequencies, bands are stable

even when rocking angles are greater than 60º. The band either only consists of smaller

beads or consists of both small and large beads (denoted as “diffuse”).

When spots form, the regions may also be well segregated with relatively sharp

boundaries or diffuse. The spots are typically round, but in some cases, they are V-

shaped or four-pointed stars. Typically, they are only seen from the bottom (i.e., only

122

Figure 5.6: Segregation patterns in a rotating-rocking sphere – typical patterns as seen

from the bottom. The bands are either relatively straight or wavy. Spots are located at

various regions at the end of a rocking cycle and sometimes all of the small beads are

segregated to the interior of the granular bed (no pattern).

123

pass through the flowing layer) twice in each rocking cycle at every π /f. It is difficult to

capture images of these spots when they are located at either end of the tumbler and not

at the bottom of the container. All of the patterns produced from sinusoidal rocking are

symmetric; each half cycle has the same pattern as the previous half cycle rotated 180º in

the xz plane. At some frequencies, there is no observable segregation pattern at the

boundaries of the container; all small beads segregate to the center of the granular bed.

5.4 Space-time Plots

Spatio-temporal (space-time) plots are used to display the shape of the pattern at

the boundary throughout an entire rocking cycle. A somewhat similar technique has been

used previously in describing the time evolution of multiple bands in a cylinder. In

cylinders, a digital picture of the bottom of the container is taken after Δt – typically the

time for a single rotation. Each image consists of a thin horizontal part of the picture

(typically 1 line of pixels) simply to capture the number of bands and the width of each

band. These pictures are then stacked as shown in figure 5.7 to represent the evolution of

the pattern in the axial direction in time (time is shown vertically).

The rocking sphere is time-periodic. Therefore, rather than looking at the

temporal evolution of the patterns over multiple cycles, a variation of this technique is

used to capture the shape of the final pattern over an entire rocking cycle. Pictures of the

bottom of the sphere are taken using digital video at 30 frames per second ( s 30/1=Δt ).

At a rotation rate of 5 rpm, 360 pictures are captured – one picture per degree of rotation.

124

Figure 5.7: Spatio-temporal plots of classic axial segregation (courtesy Stan Fiedor).

The long-time dynamics are clear using this technique; bands coarsen over time

(smaller beads are black, larger beads are clear). Little is known about the underlying

dynamics driving the merging of bands.

125

One degree of rotation translates into a distance along the equator of the sphere of

approximately ( ) mm 9.0360/mm 1402360/2 == �� ππr . In each individual digital

picture, this corresponds, roughly, to two pixels. By extracting the two center columns of

pixels from each picture and aligning them sequentially (as shown in figure 5.8), the

segregation pattern is shown in one picture for an entire cycle. An axial band appears as

a straight line (it is almost always located at the bottom of the container) and spots are

periodic regions of smaller beads (bottom). The thin wavy line in this picture is the seam

between two hemispheres of the acrylic container. The smoothness of this line

demonstrates both the continuity of the picture as well as the smoothness of the rocking

motion.

A spatio-temporal image of the container without rocking is analogous to a

cylindrical projection in cartography, mapping the spherical surface of Earth onto a flat

surface (Bugayevskiy and Snyder 1995). The data at the top and bottom of the picture is

skewed such that beads near the ends of the sphere are represented wider and thinner than

reality and are out of focus relative to the center of the picture. With rocking, for

instance, the line representing the seam undergoes almost perfect sinusoidal motion,

however in the image, it is represented by a curve that is roughly sinusoidal, but flattened

at the maxima and minima. Another issue is that parts of the frame of the apparatus

obscure parts of the picture at high values of the amplitude, A. First axis A is in view,

concealing beads behind it. At the highest amplitudes, the platform holding the motor for

axis A significantly blocks the view of the container. Since the larger beads are black

126

Fig

ure

5.8

: C

onst

ruct

ion o

f sp

atio

-tem

pora

l co

mposi

te i

mag

es.

The

cente

r pix

els

of

each

fra

me

from

dig

ital

vid

eo o

f th

e bott

om

vie

w o

f th

e tu

mble

r ar

e cr

opped

. T

hes

e sl

ices

of

the

pic

ture

s ar

e hori

zonta

lly a

ligned

in

tim

e.

Th

e to

p a

nd

bo

tto

m o

f th

e pic

ture

are

z’

= ±

1.

The

dif

fere

nce

bet

wee

n b

ands

and s

pots

is

clea

rly s

een

at t

he

bo

tto

m.

127

and typically are found in these obscured regions, this effect is minimal. Later in this

chapter, in pictures depicting experiments under slurry conditions, the region of the

picture where the apparatus is in view is masked in black (see figures 5.16 and 5.18).

These issues should be taken into account when comparing the experimental results to the

model.

5.5 Phase Diagram

When changing A and f, it is essential to maintain operation within the rolling

regime. However, as the system starts to rock, the Froude number changes in a time-

periodic manner. The Froude number at any instant is defined as

( )( )

gdtdR

Fr

2

22 cos ��

��+

=

γγω. (5.1)

Fr is a function of both rotation rate, ω , rocking angle, γ , and the rate of rocking on axis

B, dtd /γ . Figure 5.9 shows the evolution of Fr during one typical rocking cycle. At

high rocking frequencies and moderate amplitudes, the flow enters the cataracting regime

when axis A is nearly horizontal ( 0=γ at f/,0 πθ = , 221~ fAFr + ) due to the high

accelerations produced by rocking. At the highest amplitudes considered, the flow enters

the discrete avalanching regime at f/90=θ and f/270=θ in the cycle, and when A is

greater than 75º (roughly 90º minus the dynamic angle of repose) the flow temporarily

comes to a stop. At this angle, not enough material advects solid body into the flowing

128

Figure 5.9: Evolution of the Froude number as a function of frequency and amplitude

( mm 70=R ). At low A and f, acceleration due to rocking and rotation is relatively

constant. The regimes shown on the right are approximate. At high values of A, the

minima of this curve are low enough such that, at those times in the cycle, the flow

becomes unsteady (~1 rpm). When large A is combined with higher values of f, the

maxima correspond to times when the flow enters the so-called “cataracting” regime (~8

rpm).

129

Fig

ure

5.1

0:

Ser

ies

of

pic

ture

s at

A =

45º.

P

atte

rns

chan

ge

from

no a

ppar

ent

pat

tern

to a

ban

d t

o s

pots

to a

ban

d a

gai

n a

nd o

ther

pat

tern

s at

hig

her

fre

quen

cies

. C

lose

inves

tigat

ion s

how

s th

at t

he

tran

siti

on f

rom

a b

and

to s

pots

is

extr

emel

y s

ensi

tive

to f

requen

cy.

130

layer to sustain continuous flow. Experiments at low frequencies take much longer;

therefore experiments are arbitrarily cut off at 2.0=f .

We wish to explore how amplitude and frequency affect axial banding. Consider

a set of experiments at �45=A (figure 5.10). At 45.0<f , there is no observable

pattern. However, with increasing f, the system displays different patterns (band, no

band, or no pattern). Besides this, as 0→A , banding must be stable for all frequencies

(the no rocking case). These results are reproducible – each experiment is conducted as

few as twice and as many as 20 times. At this point, it is unclear how amplitude and

frequency affects axial banding.

To fully display the effect of rocking amplitude and frequency, a phase diagram is

constructed to map the different patterns (figure 5.11). These results are quite surprising.

Axial banding, the “phase” denoted in gray, is remarkably stable across the region

investigated. The right and top boundaries of this diagram roughly outline where the

flowing layer remains in the rolling regime. Approximately 227 different sets of

parameters (roughly 1000 individual experiments including repetition) were sampled to

construct this diagram. Rocking is relatively insignificant up to �5~A for all

frequencies. This is unlike the rocking cylinder considered by Wightman and Muzzio

1998b) where the highest amplitude considered is �5=A . For increasing amplitude,

there is still a large range of frequencies where an axial band is stable. This region

steadily shrinks with increasing amplitude. At �30≈A , spots are stable for a narrow

range of frequencies near 85.0=f and 2.1=f . The region of stable bands is split at

131

Figure 5.11: Experimental phase diagram with “Arnold tongues”. The pictures in figure

5.10 are performed along the dashed line at �45=A (the bottom line of pictures

corresponds to the focus region in the top right). Stability of a band is surprisingly

sensitive to the frequency and amplitude. Each of 227 data points represents 2-20

experiments and the resolution is smaller than the size of each point.

132

higher amplitudes. An alternate way to characterize the landscape of this map is to

describe the regions where axial banding is not stable. Regions of axial banding

instability grow from ),85.,30( =°= fA ),2.1,30( =°= fA and ).7.1,30( =°= fA

Now reconsider the set of experiments at �45=A described in figure 5.10. This

phase diagram now explains and clarifies what appeared as inconsistent behavior seen in

figure 5.10. If experiments were performed at amplitudes �30<A , the region of spots

would not have been seen. The lack of a band at higher frequencies could have been

construed as simply high enough Froude number such that the flow is unstable.

However, this bifurcation occurs at even higher frequencies at low amplitudes, and

therefore low Froude numbers. From an engineering viewpoint, the above-described

sensitivity to operating parameters is undesirable. It suggests that current geometries

may mix or unmix depending on the precise value of operating conditions. It also

suggests that tumbler designs that cannot be adapted or discarded when tested at only a

few sets of parameters. These transitions may continue at even higher frequencies. As

shown in the phase diagram, spots form at �5=A , 6.1=f , but bands are once again

stable at �5=A , 2=f . It is unclear if more experiments are performed whether these

separate but similar transitions would reveal information about this process.

Consider the overall shape of the phase regions. The shape of this A-f phase

diagram shows structures reminiscent of “Arnold tongues” (figure 5.12 from Hilborn

1994). Arnold (1983) introduced a simple model – the “sine circle map” – to study the

phenomenon of frequency-locking between coupled oscillators. Frequency-locking

133

occurs when two oscillators with nearly commensurate frequencies “lock” when coupled

by weak nonlinear interaction. Two frequencies are considered commensurate when their

ratio, Ω, is rational. Real world examples of this exist in planetary motion (e.g., the same

side of the moon always faces earth) and in the synchronization of mechanical clocks

residing on the same table, first noted by Huygens in 1665 (vibrations in the table provide

the weak nonlinear interaction). In the simple case of two oscillators studied by Arnold,

these nonlinear interactions cause frequency-locking near all rational frequency ratios,

but are strongest near low-order frequency ratios – such as 0:1, 1:1, and 1:2. Denote the

magnitude of the nonlinear interaction as K. When the frequencies are nearly in phase,

the nonlinear interactions are large even for low values of K compared to the difference

in frequencies. This acts to form a “sink” or attractor. The regions in a K- Ω phase map

where oscillators do not frequency-lock are called denoted as “tongues”.

In the experimental phase diagram, f is scaled by full rotations of the tumbler.

Consider a rescaling of our variables in order to draw similarities between the

experimental phase diagram and classic frequency-locking phase diagram (both shown in

figure 5.12). The rocking frequency is scaled by the average circulation time

2//* fT ==Ω ωφ . ½* =T is the dimensionless average circulation time for any half-

full tumbler (defined in section 2.2). A value of 1=Ω corresponds to one rocking cycle

per one circulation of a particle (on average). The rocking is the source of the nonlinear

behavior – a perturbation to simple flow on one axis. Therefore, A is the magnitude of

the nonlinearity – analogous to K in the sine circle map. In the rocking sphere, note the

134

Figure 5.12: Qualitative comparison of experimental phase diagram and frequency-

locking (bottom diagram from Hilborn 1994). Rescaling to compare cycles to material

circulations, 2/f=Ω . There are many similarities between the stability of bands and

the regions that do not frequency-lock. In the rocking sphere, note the regions of spots

growing upward from (f, A) of (0.85, 30�), (1.2, 30�), and (1.7, 5�) corresponding

roughly to ratios of 2:1, 3:1, and 4:1.

135

regions of spots growing upward from (f, A) of (0.85, 30�), (1.2, 30�), and (1.7, 5�).

These correspond roughly to ratios of 2:1, 3:1, and 4:1. It should be noted that further

investigation is likely to reveal another region of frequency-locking in the neighborhood

of (0.43, 30�) corresponding to a 1:1 relationship. While there are many similarities, this

analogy to the sine-circle map is only qualitative.

5.6 Comparison to Computations

Consider now the relationship of the experimental results to the computer

simulations of chapter 4 and appendix A. The model considers flow in a rocking-rotating

sphere. It is important to note that this model only describes advection without

segregation. This is a simplification of the experimental system with a binary material

represented as a single component.

In spite of these caveats, by constructing spatio-temporal pictures from the

computations, direct comparisons are made between model and the experiments. Points

in surface Poincaré maps in chapter 3 are convected and their locations are recorded 180

times throughout one rocking cycle. At each time t during the cycle, the centermost

points ( )180/180/ ππ <<− x are re-plotted z vs. t ( 11 ≤≤− z and φπ /20 ≤≤ t ). As in

the experimental spatio-temporal composites, when the hemispherical surface is projected

onto a flat surface, the chaotic regions near the top and bottom are distorted, typically

making them seem larger in relation to the dynamics in the center.

136

There is qualitative agreement between the shapes of bands seen in experiment

and corresponding Poincaré maps (figure 5.13). This set of spatio-temporal composites

of the experiment (top) and computation (below) at �45=A and 0.1=f show how a

wavy axial band is mimicked by the corresponding quasi-periodic trajectories. The

centermost quasi-periodic trajectories in the Poincaré map are highlighted in green to

assist visualization. For aesthetic reasons and ease of comparison, all of the spatio-

temporal pictures are formatted to the same dimensions. Otherwise, the length of each

picture would be proportional to φπ /2 . The qualitative similarity is striking (see figure

5.13 and 5.14)

There are many frequencies where the model shows periodic points with

surrounding regular regions (figure 5.14), possibly corresponding to spots formed in the

experiment. However, the periodicity of these regular regions does not match the

periodicity of the spots that remain when the band becomes unstable. On the other hand,

at roughly the same frequencies as the experiment, the model does predict a dilation of

the quasi-periodic trajectories where flow stagnates. The periodicity, relative size, and

orientation of these stagnation regions are roughly the same as spots. Experiments in the

dry system at conditions exhibiting the same periodicity as the other Poincaré maps

shown here were not found. It is possible that they exist in a very narrow range of

frequencies or they do not exist at all – the magnitude of the rocking may not produce

large enough perturbations to the flow.

137

Fig

ure

5.1

3:

Qual

itat

ive

com

par

ison b

etw

een a

ban

d i

n t

he

exper

imen

t an

d t

he

shap

e of

the

quas

i-

per

iodic

tra

ject

ori

es i

n t

he

mo

del

A =

45�

and f

= 1

.0.

On t

he

left

, th

e re

d l

ine

dep

icts

the

cente

r

pix

els

sam

ple

d t

hro

ughout

the

cycl

e to

com

pose

the

spat

io-t

empora

l co

mposi

te o

n t

he

right.

T

he

cente

rmost

quas

i-per

iodic

tra

ject

ori

es i

n t

he

model

are

shad

ed i

n g

reen

.

138

Figure 5.14: Comparison between regular regions in model and spots in experiments.

From top to bottom: (A = 40�, f = 1.5), (A = 45�, f = 0.6), (A = 45�, f = 0.957), and the

experiment is at (A = 45�, f = 0.842). In the model, regular regions of various

periodicity exist, however in this experimental system only spots with the bottom

periodicity are seen.

139

Consider now the narrow range of frequencies where the band bifurcates into

spots and back into bands (shown in figure 5.10). Closer investigation of this bifurcation

is possible by creating successive spatio-temporal plots of the patterns at frequencies

between 824.0=f and 856.0=f (figure 5.15). The band appears to “pinch-off” at

distinct times in the rocking cycle. After pinch-off, these spots contract. At slightly

higher frequencies, these spots stretch out back into bands. Interestingly, these spots are

at the same location within the cycle where the bands originally pinched-off at lower

frequencies. The bands initially break in the cycle when 0=γ . The spots, originally

located at 0/ =θγ dd , migrate to this part of the cycle when 0=γ . It should be noted

that this pinch-off is not a dynamic process, as each of these pictures is the final

segregation pattern after many rocking cycles in separate experiments.

Alongside the experimental pictures in figure 5.15, the computations from figure

4.12 are shown as spatio-temporal plots. The transition over this narrow range of

frequencies is similar. The phase-shift seen in the experiments is mimicked in these

consecutive plots. The quasi-periodic trajectories become wavy, dilating at one point in

the cycle and necking at another. At a specific frequency, two hyperbolic points exist in

the region where trajectories are dilating. These points drift apart as the frequency

increases. Because of the periodicity of the flow, as a pair of hyperbolic points drifts

apart, one approaches the hyperbolic point associated with the previous cycle and the

other approaches a hyperbolic point associated with the next cycle. The hyperbolic

points in each new pairing converge. They eventually annihilate each other, creating a

140

Figure 5.15: Transition between a band and spots in experiment compared to the model.

Bifurcation from a band to spots and back to a band occurs over a narrow range of

frequencies (A = 45º). The model shows a similar transition between quasi-periodic

trajectories and hyperbolic points and back.

141

part of the flow where quasi-periodic trajectories once again exist in the center of the

map. These quasi-periodic trajectories once again have regions where they diverge and

contract as seen at frequencies lower than the bifurcation, however the regions have an

apparent phase-shift of 90º compared to pictures at lower frequencies.

This bifurcation is remarkably similar to the transition from a band to spots and

back to a band. The model, without modeling segregation, may be used to investigate the

range over which the band is unstable and the phase-shift of spots throughout the

transition. The most remarkable aspect of this correspondence is that this happens even

though the dry S-systems considered in this particular example are, in general,

significantly more taxing from a modeling standpoint than the corresponding D-systems

(as mentioned in section 3.3.2). However, if no significant time effects arise from the

flow composition in the layer, as is the case in the experiments reported here, the system,

once in steady state, behaves essentially as a single fluid and can be modeled as such.

This model does not directly predict frequency-locking, as suggested by the

experimental phase diagram. The hyperbolic points in the surface Poincaré maps in

figure 5.15 are created via period-doubling. Like frequency-locking, periodic points exist

in the map when the frequencies associated with rocking and circulation are

commensurate. A represents the magnitude of the nonlinear terms, and increasing A

increases the range where periodic points exist. Although this mechanism is very similar,

there are distinct differences between period-doubling and frequency-locking. For

instance, sources and sinks required to produce frequency-locking cannot exist in a

volume-preserving model.

142

However, constitutive equations describing axial segregation are not available.

However, consider the relationship between radial segregation and the underlying flow,

described in chapter 3. It is noted that the regular regions behave as “attractors” for one

type of material when the system is still well mixed – materials preferentially segregate to

opposite regions of the flow. The system could be viewed as two co-existing maps, each

associated with one material. These two maps would have matching sets of sinks and

sources driving the two materials into different regions of the flow. The strength of these

sinks and sources would be relative to local concentration and proportional to differences

in material properties driving segregation. When materials reach a final segregated state,

segregation only exists at the phase boundaries. Inside each phase, the materials are

mono-disperse and flow as described by the model. The phase boundaries have no force

analogous to surface tension in liquids, therefore they must follow along the underlying

invariant structures produced by flow of each mono-disperse material – as suggested in

chapter 3 in two-dimensional systems. If sources and sinks do exist, this allows the

possibility of frequency-locking suggested in the experimental phase diagram. Once

again, this suggests that final segregation patterns are independent of material properties

causing segregation. If this is true, then the same phenomena should also exist under

slurry conditions (analogous to the dry and slurry conditions tested in bi-axially rotated

cubes shown in figure 3.17).

143

5.7 Granular Slurries

Consider now experiments under slurry conditions. The number of experiments

considered in this analysis is much less than performed in the dry case. Only a

qualitative description will be given, paying particular attention to differences between

dry and slurry conditions. This system primarily behaves in a similar way as the dry

materials in the rotating-rocking sphere described above. Bands are common, and at

certain frequencies spots exist. The sensitivity of these patterns is roughly the same as in

the dry system, and spots are located in the same part of the cycle.

Figure 5.16 (top) shows that when rotated on a single axis, these particles initially

form multiple bands after ~10 rotations. These bands coarsen over time into a single

central band of smaller beads, similar to those seen in the S-system under dry conditions

in a double-cone blender (Alexander et al. 2001). The rate of this coarsening has not yet

been investigated because of the few number of bands (the rate of coarsening has not

been explored in any geometry other than rotating cylinders). In rotating-rocking

experiments, multiple bands exist at frequencies where a single band is stable in the dry

system, (figure 5.16 middle - regions in the picture where the frame of the apparatus

comes into view has been blacked out so as to focus on the granular material). Often,

these wavy bands also coarsen over time. After hundreds of rotations (and rocking

cycles), the smaller beads eventually form only one or two centrally located bands. At

other frequencies, spots form when bands are not stable (figure 5.16 bottom). These

patterns seem most similar to those described in the dry system. There are other

parameters where the pattern is different than that seen in the corresponding dry system.

144

Fig

ure

5.1

6:

Exper

imen

ts u

nder

slu

rry c

ondit

ions.

T

op i

s an

exper

imen

t w

ithout

rock

ing w

her

e

mult

iple

ban

ds

are

form

ed.

Mid

dle

show

s m

ult

iple

ban

ds

wit

h r

ota

ting a

nd r

ock

ing (

A =

45º,

f =

0.9

67

).

Bo

tto

m s

ho

ws

spo

ts (

A =

10º,

f =

1.6

7).

In

the

mid

dle

pic

ture

, a

bla

ck m

ask i

s use

d t

o

cover

the

area

wher

e th

e fr

ame

of

the

appar

atus

com

es i

nto

vie

w.

145

Some of these experiments produce bands that apparently fork into multiple bands

(figure 5.17 top). The stability of these bands is unclear. While a central band always

exists, after splitting, one of the branches ends abruptly. Band formation and

disappearance on the order of a single rotation has not been previously reported. It is

unclear if this is the final pattern.

At the same amplitude but higher frequency, wavy bands are stable, but the

smaller beads now migrate to the ends of the sphere and larger beads form the bands in

the middle (figure 5.17b). This phenomenon (referred to as “big-out”) is observed in a

double-cone blender (Alexander et al. 2001) by changing the rotation rate of the tumbler.

Preliminary experiments suggest that this “big-out” pattern may depend on the initial

condition. The pattern is reproducible when starting with a mixed initial condition (in

which the tumbler is rotated quickly and a mixture of small and large particles exist at the

bottom of the container) and is stable for many hundreds of cycles. However, when the

initial condition is a leftover pattern from another experiment, bands form, but smaller

beads may once again form the bands in the center of the tumbler (also stable for

hundreds of cycles). The pattern itself does not seem to be a transient effect – bands are

consistently formed in both cases.

Another qualitative difference between the dry and slurry conditions is the

number of spots. In the dry case, no more than two spots per rocking cycle have been

observed. However, six spots can form under the slurry condition at �40=A and

33.1=f (figure 5.18). The model captures this pattern at �40=A and 5.1=f . The

146

Fig

ure

5.1

7:

Odd p

atte

rns

under

slu

rry c

ondit

ions.

T

he

top (

A =

10º,

f =

1.1

7)

corr

esponds

to a

pat

tern

that

, in

spit

e of

appea

ring p

erio

dic

, m

ay n

ot

hav

e re

ached

a f

inal

sta

te y

et.

The

bott

om

pic

ture

(A

= 1

0º,

f =

1.3

33)

dep

icts

rev

erse

axia

l se

gre

gat

ion,

wher

e th

e la

rger

bea

ds

form

ban

ds

in

the

cente

r of

the

spher

e.

147

Fig

ure

5.1

8:

Model

and s

lurr

y e

xper

imen

t (t

he

par

amet

ers

are

A =

40�,

f =

1.5

and A

= 4

0�,

f_=

_1.3

33 r

espec

tivel

y),

both

dis

pla

yin

g p

erio

d-6

per

iodic

poin

ts.

This

is

not

seen

in t

he

dry

syst

em.

This

sugges

ts t

hat

the

slurr

y m

ay b

e m

ore

sen

siti

ve

to t

he

forc

ing.

In t

he

exper

imen

t, a

bla

ck m

ask c

over

s th

e ar

ea w

her

e th

e fr

ame

of

the

appar

atus

cover

s par

t of

the

spher

e.

148

spots in the slurry are very small and diffuse, but clearly segregate to the same region as

the regular regions in the model. This suggests that using beads with different material

properties possibly could reflect more details of the computed invariant structures. The

spatio-temporal picture of this is less clear than the still images because these spots are

small. At this rocking amplitude, axis A is visible, and covers part of the bottom view of

the granular bed. Again, the area where the frame of the apparatus comes into view has

been blacked out so as to focus on the granular material.

5.8 Rocking Cubes

Replacing the sphere with a cube makes the flow significantly more complicated.

Only the basic elements of this system are described here. In all protocols investigated

thus far, the forcing of the system is either periodic with rotation or periodic with rocking

frequency. This system is a combination of both. Consider holding the cube face-

centered by an axis of rotation. A cube is invariant to 90� rotations and multiples thereof.

In a rocking cube, this means the flow can only be periodic at rocking frequencies of

4/* nf = where n is a positive integer.

At all other frequencies, the forcing is quasi-periodic. This is analogous to

rocking the sphere according to a quasi-periodic function, however the shape of the free

surface in a cube is also quasi-periodic (the free surface is an ever-changing polygon). A

rocking cylinder (considered in experiments by Wightman et al. 1998a,b) also has a

periodic free surface shape (the shape free surface in a half-full cylinder ranges from a

149

rectangle when 0=γ to an ellipse roughly when ( )WR /tan 1−>γ where W is the half

width of the main axis of rotation), but it is always periodic with the rocking frequency.

One other consideration: the flow in a cube depends on the initial orientation of the cube

at the beginning of the experiment. A cycle that starts with the free surface parallel to a

side of the cube should give different results than one that starts with the free surface

crossing a diagonal of the cube and so on.

Consecutive pictures of a rotating-rocking cube show segregation patterns (figure

5.19). Because of the geometry, spatio-temporal pictures are impractical. In a cube

rocking with f = 1, a steady pattern forms, and this pattern has similar qualities as both

axial banding and spots. The smaller fluorescent beads do not reach the boundaries.

Some analogies may be made between what is seen in the bi-axial experiments, where

smaller and denser material is located away from the chaotic region in center of the cube.

When the ratio of rocking frequency to circulation rate is quasi-periodic, no

“final” segregation pattern can form. Pictures show that the smaller beads can even

migrate towards the edges of the container, producing reverse axial segregation (figure

5.20). However, this is temporary. Later, the smaller material congregates in the center

of the tumbler. Because of this, each type of material is able to explore most of the

granular bed. This may enhance practical processes such as spray coating, where it

would be beneficial for all material to eventually reside on the free surface. Also,

because the pattern is dynamic, the material properties may also interplay with the

segregation pattern. Altering the magnitude of radial or axial segregation, for instance by

150

Figure 5.19: Rotating-rocking cube with f = 1. This series of pictures show a final

pattern through one rocking cycle. The resulting pattern from changing the flow

direction and the shape of the free surface is complex and depends on the initial

orientation of the cube.

151

Figure 5.20: Rotating-rocking cube with 8.0≈f . This series of pictures shows the

segregation pattern over one rocking cycle. Momentarily, the smaller beads are segregating to the edges of the cube. After a few cycles, the pattern will look different

– the free surface shape is quasi-periodic.

152

performing the same experiments under slurry conditions, may alter a delay in the system

to evolve from one pattern to another.

As container shapes, materials, and flow become more complicated, it is unclear

how well models of the underlying flow will predict segregation structures. As in all

continuum models, the assumptions break down at extreme limits. For example, a

breakdown occurs when the flow velocities become coupled with concentration, as seen

with the streaks formed in the 55% full S- system square in figure 3.10. When this is not

the case, the continuum models of flow of mono-disperse beads used throughout this

dissertation are remarkably accurate in predicting the qualitative segregation patterns.

153

CHAPTER 6

SUMMARY AND OUTLOOK

This dissertation explores granular mixing and segregation in various geometries.

Segregation in a circle is simple. Changing the geometry from simple flow in a circle to

chaotic flow in other shapes has a large effect on the resulting segregation. Invariant

regions in the underlying flow dictate the shape of the phase boundary between materials

differing by physical properties. Both the flow and resulting pattern formation are found

to be sensitive to fill level. In a bi-axially rotated sphere and cube, chaos is generated,

but it is unclear how the resulting segregation patterns relate to the underlying flow.

However, incorporating collisional diffusion and segregation into the model correctly

predicts both simple radial segregation in a sphere and a three-lobe pattern that forms in a

cube.

A simple model of underlying flow in a rotating-rocking sphere demonstrates how

regular regions and chaos are sensitive to the rocking frequency and amplitude.

Experiments within a spherical tumbler produce an axial segregated band when rotated

on a single axis and this band is surprisingly stable when the flow is perturbed with the

addition of rocking. The underlying principles of axial segregation are poorly

understood, but by predicting the sensitivity of the underlying flow to the amplitude and

frequency of rocking, the shape and stability of this band are predicted qualitatively. In

both the bi-axial sphere and cube and rotating-rocking sphere, experiments under both

154

dry and slurry conditions mostly exhibit the same patterns, suggesting that the final

segregation patterns are independent of material properties. It is unclear whether this is

true in a rocking cube, where flow can be quasi-periodic and shape of the segregated

region would likely continuously evolve with the underlying flow.

It is evident that throughout this dissertation there is a rough correspondence

between the model and the experimental results. The most remarkable aspect of the

correspondence is the similarity that is seen even in the dry S-systems which are often

significantly more taxing from a modeling standpoint than the corresponding D-systems

(as mentioned in section 3.3.2). However, if no significant time effects arise from the

composition in the flowing layer, as is the case in the experiments reported here, the

system, once in steady state, behaves essentially as a single fluid, and can be modeled as

such.

It is clear that our model works well under these conditions. This is unexpected,

as the model does not contain segregation. How can we then rationalize the agreement

between experiments and computations? The answer becomes apparent once one realizes

that segregated regions are invariant regions – segregated domains are mapped onto

themselves. Segregation, once completed, has to follow the template provided by the

Poincaré map. It is also obvious that the only place where segregation takes place is in

the layer itself, since is it the only place with non-zero shear rates; in fact the layer is the

only point of entry of materials into segregated domains. Testaments to this are seen

throughout this dissertation.

155

Short of developing constitutive equations, there are many routes to further

understanding of granular flow and segregation. Currently, most research focuses on

existing prototypical systems. However, it is likely that many simple experiments,

analogous to those described in this dissertation, have yet to be discovered. Granular

research is still in a “discovery” phase. New phenomena emerge within novel

experiments that stretch the understanding of granular dynamics past its current

boundaries, and these new phenomena will likely give opportunity to observe and

quantify poorly understood fundamental processes such as axial segregation.

Aside from studying systems exhibiting different behavior, in general it is

necessary to further develop tools of investigation. With more sophisticated scanning

and imaging techniques, it may be possible to automate analysis of granular systems.

One can imagine a feedback system where a segregation pattern is interpreted through

image analysis and this information is then used to alter parameters of the system, such as

rotation rate or rocking frequency. Most of the results presented here are largely

independent of material properties, however systems are difficult to explain when local

concentration couples with the flow. If material parameters could be tuned during an

experiment, the flow and the resulting segregation pattern could be controlled. With the

emergence of nanotechnology, it is even possible that intelligent tools of analysis will

eventually be embedded within the material itself – perhaps tumbling spherical

microchips and analyzing the data of the network of interactions when segregation is

present.

156

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APPENDIX A

The following figures are Poincaré maps produced by the model detailed in

chapter 4. Each individual figure consists of 40 maps, each of which represent

computations at the same amplitude, A, with frequency ranging from 475.15.0 ≤≤ f .

Overall, these took roughly 500 hours of computational time (Pentium IV, 2.4GHz) plus

additional time to plot maps. Many more computations not shown investigate specific

regions (as seen in figures 4.10, 4.11, and 4.12). It is useful to find a specific pattern,

note its amplitude and frequency, and then inspect figures of neighboring amplitudes

(figures) to see if that pattern arises again at the same or nearby frequency. Many

features exist that do not have a specific trend because their range of stable frequencies

are narrower than 0.025.

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A.1: Computations of a rocking sphere for A = 5º and 0.5 � f < 1.5.

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A.9: Computations of a rocking sphere for A = 42.5º and 0.5 � f < 1.5.

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