# Sur le problème de la ségrégation des milieux granulaires vibrés: Étude théorique et phénoménologique

Leonardo Trujillo

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Leonardo Trujillo. Sur le problème de la ségrégation des milieux granulaires vibrés: Étude théorique et phénoménologique. Physique mathématique [math-ph]. Université Pierre & Marie Curie; Ecole Supérieure de Physique et Chimie Industrielles, 2002. Français. NNT: 2003PA066325 . tel-03892900

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Thèse de doctorat

# Sur le problème de la ségrégation des milieux granulaires vibrés

## Étude théorique et phénoménologique

Leonardo Trujillo

Laboratoire de Physique et Mécanique des Milieux Hétérogènes

École Supérieure de Physique et Chimie Industrielles

<figure><img src="image_2.png" /></figure>

Université Pierre et Marie Curie

École doctorale SMAE

4. place Jussieu

75252 Paris cedex 05

France
<figure><img src="image_3.png" /></figure>
UNIVERSITÉ DE PARIS VI - PIERRE ET MARIE CURIE

THÈSE

pour obtenir le grade de

DOCTEUR EN SCIENCES

de l'Université de Paris VI

présentée et soutenue publiquement
par

Leonardo TRUJILLO LUGO

le 7 novembre 2003

# Sur le problème de la ségrégation des milieux granulaires vibrés

## ÉTUDE THÉORIQUE ET PHÉNOMÉNOLOGIQUE

Directeur de thèse:

M. Hans J. HERRMANN

Membre du Laboratoire PMMH-ESPCI,
Professeur a l' Université de Stuttgart

## JURY

Président: M. E. CLÉMENT Université de Paris VI

Rapporteur: M. D. BIDEAU Université de Rennes I

Rapporteur: M. R. JULLIEN Université de Montpellier II

Examinateur: M. A. BARRAT Université de Paris XI

Examinateur: M. J. KURCHAN ESPCI

Examinateur: M. S. LUDING Technische Universiteit Delft

préparée au Laboratoire de Physique et Mécanique des Milieux Hétérogènes
École Supérieure de Physique et Chimie Industrielles
**Abstract:** When a granular mixture is subject to strong perturbations the system becomes spatially non-uniform by sorting the grains in terms of their size and/or mass. For example, larger particles rise to the top when subjected to vertical shaking. This is the well known “Brazil nut phenomenon” (BNP) which became a common physical realization to explore non-equilibrium dynamical processes. This thesis is devoted to the study of segregation in granular materials from a theoretical perspective. We show that a *granular hydrodynamic* theory does indeed give a good qualitative description of the Brazil nut phenomenon. In the first part we study the size dependence of the intruder upward velocity and the upward to downward intruder's movement transition. After introducing the kinetic theory for binary granular mixtures, we show a novel mechanism for segregation based on the competition between buoyancy and geometric forces. The main conclusion is that the effect of dissipation have important consequences on the segregation dynamics. On a qualitative level our model satisfactorily reproduces the observed phenomenology, for example, the size dependence on the intruder's upward velocity and the upward to downward intruder's movement transition.

**Keywords:** Granular Materials; Segregation; Brazil Nut Problem; Buoyancy; Dissipation; Granular Mixtures; Kinetic Theory; Inelastic Collisions.
Acknowledgments

I am grateful to Professors Alain Barrat, Eric Clément, Jorge Kurchan and Stefan Luding, for agreeing to be on my jury committee and especially to Professors Daniel Bideau and Rémi Jullien for having accepted to be *rapporteurs* of this thesis.

I thank my advisor, Prof. Hans J. Herrmann, for his guidance, deep criti-
cisms and encouragement. It was indeed a honor to be his student in Paris.
His enthusiasm, leading force and infinite energy have been invaluable pro-
fessional lessons for me. These three years under his supervision have been
truly enlightening.

The last part of this project was carried out with the cooperation of Dr. Meheboob Alam from the Jawaharlal Nehru Centre in Bangalore, India. I am deeply indebted to him for numerous valuable improvements, instructive discussions and communications.

I am grateful for having been taught by Prof. Jorge A. González who infused the mathematician's lifestyle in me. The phone calls at dawn, e-mails and visits have been of invaluable professional and personal support. Our working sessions are always very stimulating.

The little entrance door at the 10 rue Vauquelin will stay in my memory. This thesis was developed at the Laboratoire de Physique et Mécanique des Milieux Hétérogènes, ESPCI. I feel a deep gratitude to the lab for providing excellent working conditions, support and freedom. I am in debt with the director of the PMMH, Prof. José Eduardo Wesfreid, for his unconditional support and hospitality. I thank Prof. Etienne Guyon for informative discussions. Were it not for Ivan, I would be buried somewhere under piles of paper, with a French dictionary in my hand. He and Sorin are *mes copains du labo* and I do not know what I would have done without their friendship and our fascinating conversations about physics and anything else! The PMMH gave me the opportunity to know many people and it is a pleasure to thank F. Auger, P. Brunet, B. Cabane, J. Fernández, S. Firmin, M. Hoyos, K. Kroy, J. Kurchan, H. Nakagawa, G. Rousseaux, F. Virieux, and the visitors J.S. Andrade Jr., A. Garriga, P. Lind, S. Moss de Oliveira, P.M.C. de Oliveira, S. Ruffo and C. Toninelli. I thank Prof. Jerry P. Gollub for the stimulating discussions we had when he was at the PMMH during the summer of 2001. My thanks are also due to Prof. Sidney R. Nagel for discussions concerning the experimental aspects of the convection effect.

During my PhD I spent some time abroad. I am grateful for the kind hospitality of the Institute for Computer Applications (ICA-1) of the University of Stuttgart, Germany, as well as for providing excellent working conditions
and for supporting several visits to Stuttgart. In particular, I wish to thank Raul Cruz-Hidalgo for his superb hospitality and support at the ICA-1. Many thanks to the IAEA and the UNESCO for their hospitality at the International Centre for Theoretical Physics (ICTP), Trieste, Italy during the summer of 2002. I thank C. E. Alves, M. Asfaw, P. Braz e Silva, A. Cabo, D. Cabra, J. Chela-Flores, S. P. González, A. Papa and J. Stephany for their extensive support and for stimulating discussions on various aspects of practicing science while considering the needs of developing countries. Thanks are also due to the Condensed Matter Group at the Universidad Simón Bolívar, Caracas, Venezuela, for their hospitality during our visit there in December 2001, particularly to Professors G. Gutiérrez, L. I. Reyes and R. Paredes for many useful discussions, e-mails and visits in Paris.

Many people helped with *emotional* support and friendship. I am deeply grateful to Adriano Roberto de Lima who encouraged and prodded me during the early stages of my work, *obrigado meu amigo*. It is amazing what friends can pull you through. Over the course of this PhD, many potential therapy sessions were averted by phone calls, e-mails and visits. Many thanks to *les copains* from the Institut Pasteur: Claudine, Miguel, Monica, Vanesa, Manu, Evelyne, Ralf Spoeler, Cristina Grim. Special thanks to Madame Wendy, who is the Guardian Angel of my *niña linda* at the Pasteur and I do not know what I would have done without her assistance (and the secret recipe of the famous Wendy's lemon cake!) Thanks to my friends Alberto, Harold, Oswaldo, Mathieu, Søren, Marta González, Miguel Martín, Claudio, Dianora, Sigifredo, Sole, Susin, Malusinha, and Ale e Irina. I am indebted to Ada and Gustavo for their hospitality in Vienna, constant support and love. My best regards to Eladio, Rosa and Samuel.

I wish to express my sincere gratitude and love to Maria Elena for her support, encouragement, patience and love. She is my *compañera de aventuras y luchas* and these ten years together have been outstanding. My love and my deepest gratitude are to Ligia, who gave me the roots that secured me and wings to fly. With Maria Elena and Ligia I am sure that no matter where I go or where I am, there is no place like home. This thesis is dedicated to you two and, of course, to *los Abuelitos*.
# Contents

<table><tr><td>1</td><td>Introduction</td><td>1</td></tr><tr><td>1.1</td><td>Why do we study segregation in granular materials</td><td>1</td></tr><tr><td>1.2</td><td>Scope of the thesis and organization of the manuscript</td><td>1</td></tr><tr><td>2</td><td>The Brazil nut problem</td><td>3</td></tr><tr><td>2.1</td><td>Granular matter</td><td>3</td></tr><tr><td>2.2</td><td>Vibration-induced segregation: Principal facts</td><td>4</td></tr><tr><td>2.2.1</td><td>A home experiment</td><td>4</td></tr><tr><td>2.2.2</td><td>Laboratory experiments</td><td>5</td></tr><tr><td>2.2.3</td><td>Computer simulations</td><td>13</td></tr><tr><td>2.3</td><td>Segregation mechanisms</td><td>15</td></tr><tr><td>2.3.1</td><td>Upward size-segregation: Brazil nut phenomenon (BNP)</td><td>15</td></tr><tr><td>2.3.2</td><td>Downward segregation: Reverse Brazil nut phenomenon (RBNP)</td><td>17</td></tr><tr><td>2.4</td><td>Some general comments about segregation in granular materials</td><td>18</td></tr><tr><td>2.4.1</td><td>Other examples of granular segregation</td><td>18</td></tr><tr><td>2.4.2</td><td>Industrial relevance</td><td>19</td></tr><tr><td>2.5</td><td>How the Brazil nut goes up or down: A novel interpretation</td><td>20</td></tr><tr><td>2.5.1</td><td>Theory of granular materials</td><td>20</td></tr><tr><td>2.5.2</td><td>Phenomenological theory for segregation</td><td>21</td></tr><tr><td>3</td><td>Single intruder size segregation dynamics</td><td>29</td></tr><tr><td>3.1</td><td>Continuum formulation</td><td>29</td></tr><tr><td>3.1.1</td><td>Preliminaries and description of the granular fluid</td><td>29</td></tr><tr><td>3.1.2</td><td>Local temperature difference</td><td>33</td></tr><tr><td>3.1.3</td><td>Calculation of the granular temperature difference</td><td>36</td></tr><tr><td>3.1.4</td><td>Phenomenological aspects</td><td>39</td></tr><tr><td>3.2</td><td>Thermal expansion</td><td>40</td></tr><tr><td>3.3</td><td>Segregation forces</td><td>41</td></tr><tr><td>3.4</td><td>Conclusions</td><td>45</td></tr><tr><td>4</td><td>The upward and downward movement</td><td>51</td></tr><tr><td>4.1</td><td>Segregation dynamics: Competition between buoyancy and thermal forces</td><td>51</td></tr><tr><td>4.1.1</td><td>Preliminaries</td><td>51</td></tr><tr><td>4.1.2</td><td>Time evolution and settling velocity</td><td>52</td></tr></table>
<table><tr><td>4.1.3 Remarks</td><td>53</td></tr><tr><td>4.2 Analysis</td><td>53</td></tr><tr><td>4.2.1 Temperature and settling velocity</td><td>53</td></tr><tr><td>4.2.2 Phase diagram</td><td>55</td></tr><tr><td>4.3 Discussion</td><td>55</td></tr></table>

5 Aspects of the kinetic theory for binary granular mixtures: The two-temperatures scenario 59

<table><tr><td>5.1 Definitions and preliminaries</td><td>60</td></tr><tr><td>5.1.1 Hydrodynamic fields</td><td>60</td></tr><tr><td>5.1.2 The Boltzmann-Enskog equation for a mixture</td><td>61</td></tr><tr><td>5.1.3 Analysis based on the Boltzmann-Enskog equation</td><td>62</td></tr><tr><td>5.2 Balance equations and hydrodynamics</td><td>63</td></tr><tr><td>5.2.1 Energy balance</td><td>63</td></tr><tr><td>5.2.2 Mass and momentum balance</td><td>65</td></tr><tr><td>5.3 Inelastic dissipation and temperature ratio</td><td>66</td></tr></table>

6 A novel mechanism: Competition between buoyancy and geometric forces 71

<table><tr><td>6.1 Hydrodynamics of granular mixtures</td><td>71</td></tr><tr><td>6.1.1 Preliminaries</td><td>71</td></tr><tr><td>6.1.2 Momentum balance</td><td>72</td></tr><tr><td>6.2 Evolution equation for the rise velocity of the intruder</td><td>73</td></tr><tr><td>6.3 Driving mechanisms: segregation forces</td><td>74</td></tr><tr><td>6.4 Phase diagram and discussion</td><td>77</td></tr></table>

7 Conclusions and perspectives 81

8 Version Française abrégée 85

References 105

Index 117
# 1 Introduction

## 1.1 Why do we study segregation in granular materials

A real granular system is always characterized by some degree of polydispersity in size, shape and mass, which often leads to unwanted segregation.

Segregation is a process in which a homogeneous mixture of particles of different species becomes spatially non-uniform by sorting themselves in terms of their size and/or mass. This phenomenon is rather ubiquitous in the transport and handling of bulk granular mixtures which occur in many industries. In most cases, it is required to maintain a homogeneous mixture during processing, with segregation being the ‘unwanted’ phenomenon. Despite its unwanted consequences, however, segregation occurs spontaneously in vibrofluidized granular mixtures.

The *Brazil nut phenomenon* (BNP)—where, larger particles rise to the top when subjected to strong vertical shaking— became one of the most puzzling phenomena of granular materials research. Different mechanisms have been proposed to explain this phenomenon, for example, percolation, arching, convection, inertia, condensation, buoyancy, etc.

But *how the Brazil nut goes up* is not the only question to answer. Recently it was discovered that a downward intruder's movement occurs as well: *reverse Brazil nut phenomenon* (RBNP). Therefore, the current question is *how the Brazil nut goes up or down?*

We study segregation in granular materials because these questions still remain open! and a theoretical viewpoint of the problem could be of substantial importance in understanding the mechanism of segregation.

## 1.2 Scope of the thesis and organization of the manuscript

This dissertation contains three projects. All three projects are devoted to the study of segregation in granular material from a theoretical perspective (see the bibliography below). We present an alternative interpretation of the experimental Brazil-nut and reverse Brazil nut observations, based on a hydrodynamic formalism for inelastic particles.

This thesis has been laid out as follows: In Chapter 2 we present an overview of the current state of investigations consecrated to the segregation phenomenon. Also the principal assumptions and statements of this thesis have been introduced in this
chapter. In Chapter 3 we have developed a hydrodynamic model for segregation and we discussed the effect of size in the segregation dynamics. In Chapter 4 we extend the model introduced in Chapter 2 to study the effects of intruder's size and density on the upward to downward segregation transition. In Chapter 5 we have presented an introduction to the kinetic theory for binary granular mixtures. In Chapter 6 we have introduced a novel mechanism for segregation based on the competition between buoyancy and geometric forces. Finally, in Chapter 7 we discuss the main results of the thesis, and some phenomenological and experimental implications.

This thesis is based on the following bibliography:

• Hydrodynamic model for particle size segregation in granular media
Trujillo, L. & Herrmann, H. J.
Physica A (2003) **330** 525–548.

• A note on the upward and downward intruder segregation in granular media
Trujillo, L. & Herrmann, H. J.
Granular Matter **5** (2003) 85–89.

• Segregation in a Fluidized Binary Granular Mixture: Competition between Buoyancy and Geometric Forces
Trujillo, L., Alam, M. & Herrmann, H. J.
Europhysics Letters **64** (2003) 190–196.
## 2 The Brazil nut problem

Anyone who has opened a can of mixed nuts knows that the larger nuts tend to be at the top. This phenomenon of segregation occurs spontaneously in fluidized particulate mixtures and is one of the most puzzling problems in granular materials research. Experimental and numerical results related to segregation of granular mixtures are reviewed in the present chapter. We start with a brief discussion about the physics of granular materials as a general context of the present research. The review covers theoretical models and different driving mechanisms proposed in the last years. Attention is focused on fluidized granular systems subject to vertical vibrations and we touch lightly other situations where segregation arises too. We introduce the motivations for the new approach presented in this thesis. In particular we stress the theoretical framework based on a genuinely hydrodynamic description of fluidized granular systems and its implications in a model for segregation. Some industrial aspects are addressed and we comment the granular segregation literature briefly.

### 2.1 Granular matter

Granular materials consist of a collection of discrete macroscopic solid particles interacting via short-range repulsive contact forces. Classical examples are sand, powders, sugar, salt and gravel. No limitation is imposed on the size of the particles, which may range from nanometer scale, as in pigments or aerosols, to that of mined or quarried materials and rocks.

<figure><img src="image_4.png" /><figcaption>Fig. 2.1. Médanos de Coro National Park, Falcón-Venezuela</figcaption></figure>
What are granular materials and why are they interesting?

*A new state of matter.* – Beyond their practical importance, granular materials interest physicists because they are an unusual form of matter with interesting properties that are still not yet fully understood.

*Interdisciplinary effort.* – Investigations in granular material gather teams from chemical engineers, food technologists, powder metallurgists, materials scientists, pharmaceutical scientists, applied mathematicians and physicists.

*Fascinating experiments.* – They exhibit a wealth of interesting phenomena like heaping under vibration, segregation, convection, fluidization, pattern formation, anomalous sound propagation, compaction, and density waves.

*Computer applications.* – Physics on high performance computers modelling granular flows gives a great insight towards the understanding of the dynamics of particulate systems.

*A theoretical challenge.* – There is no accepted set of universal governing equations describing granular materials. Important contributions have been made to improve our understanding of many new aspects using modern tools from statistical mechanics.

*Complex systems.* – Without doubts granular materials are fascinating nonlinear, dissipative, nonequilibrium systems with striking and beautiful behavior.

To conclude this brief account, let us remark the following two aspects directly related to the project presented in this thesis:

1. A real granular system is always characterized by some degree of polydispersity in size, shape and mass which often leads to unwanted segregation.

2. Strong violation of equipartition of energy is observed in systems that are continuously supplied energy to replace the energy lost due to inelastic collisions.

Finally, the reader is encouraged to consult the books [26, 27, 62, 91, 92, 101, 173] which provide an excellent introduction to the physics of granular materials as well as the Refs. [57, 58, 100, 116, 117, 174].

## 2.2 Vibration–induced segregation: Principal facts

In order to make theoretical progress, it is necessary to collect the experimental facts related to the segregation phenomenon. In this section we give a short recipe to invite the reader to do one of the most simple verification of the problem addressed in this thesis. Then, we introduce experimental and computational studies and discuss different general properties which motivate our theoretical effort.

### 2.2.1 A home experiment

You can check yourself the principal experimental statement of this thesis. It is very easy!
All you need is: 1) a recipient, like a glass jar or a transparent vessel, 2) an assembly of small beads (sugar, salt, rice or sand) and 3) one larger *intruder* particle, for example, a plastic ball or a big nut.

<figure><img src="image_5.png" /><figcaption>**Fig. 2.2.** Home made Brazil nut experiment</figcaption></figure>

To do the experiment put the *intruder* inside the recipient along with the small beads. Push the intruder down to “mix” it with the smaller particles. Then, shake the recipient —more or less— vertically. *What happens after shaking the system for a few seconds?* —The reader can verify that the large particle moves to the top!— This experimental outcome could be *a priori* more or less counterintuitive, and becomes one of the most outstanding questions in contemporary studies on granular materials: *Why the Brazil nuts are on top?*

### 2.2.2 Laboratory experiments

#### Setups

Fundamental studies to determine the mechanisms of segregation have been carried out on idealized systems under closely controlled conditions. A systematic development is very complex because there are a huge number of independent design variables with a large number of combinations, including vibration frequency and amplitude, particle shape, particle physical properties such as density and elastic modulus, particle size distribution, wall effects such as geometry and inclination, effects of interstitial air or the influence of liquid on the segregation behavior. The interplay between experiment, computer and mathematical modelling is crucial to outline promising design parameters.

Experimental methods for analyzing vibrofluidized three dimensional granular beds at the single level are not generally available. The central technical problem is how to follow the intruder particle motion inside the granular bed. Most experiments measure the rise time [2, 72, 90, 109, 141, 149, 179, 188, 204, 225, 236]. An alternative to locate the intruder inside the bed is stopping the perturbations and remove all material above the intruder [141]. However, this procedure disturbs the granular packing and alters the intruder velocity [141]. Despite the differences between two and three dimensional devices, the segregation phenomenon arises in both dimen-
sions, e.g., the experiments [54, 64, 65] where carried out in two dimensional cells and [37, 141] in three dimensional containers.

<figure><img src="image_6.png" /><figcaption>Fig. 2.3. Two dimensional experimental setup for the study of vertically vibrated granular materials.</figcaption></figure>

Figure 2.3 shows a typical two dimensional experimental setup for vertically vibrated beds. This system consists of a Plexiglas® cell of two rectangular plates (10 × 15 cm²) separated by two thin plates (2.7 cm,) seen at the right top in Fig. 2.3. The cell is filled with monodisperse spherical beads where the intruder particle is immersed. Under the cell the vertical vibrations are produced with a (VTS150) electromagnetic shaker. Vertical accelerations are measured with an (ADXL250AQC) accelerometer. At the left of the figure 2.3 we can see an oscilloscope to monitor the acceleration, and a signal generator coupled to the amplifier. Usually the cell is illuminated from the rear with a stabilized halogen light source and a plastic fresnel lens [54], or by a stroboscopic flashing light synchronized to the vibration [65]. The particle motion is filmed with a digital high speed camera and the digital frames are processed by a computer with an image processing software.

Let us remark that the vertical vibration is characterized by the normalized acceleration parameter:

$$ \Gamma = \frac{A\omega^2}{g}, \qquad (2.1) $$

where, A is the amplitude of the oscillation, $\omega = 2\pi f$ is the angular frequency and g is the Earth's acceleration (9.81 m s⁻¹.)

As mentioned above, in three dimensions we cannot follow the intruder at po-
sitions below the surface of the granular bed (Fig. 2.4 shows a
standard three dimensional experimental setup.) Different tech-
niques have recently been developed to probe granular flows in
three dimensional geometrical setups, for example, Diffusing-
Wave Spectroscopy (DWS) [176], Magnetic Resonance Imaging
(MRI) [69, 115], Positron Emission Particle Tracking (PEPT)
[230, 231] and Inductive Devices (ID) [183]. To our knowledge
only the MRI and ID techniques have been used to explore the
segregation problem (see references [115] and [183] respectively.)

The experimentalist has to fight with a great number of variables and difficulties to reproduce well controlled experiments. On the other hand, a laboratory setup is not always easy to be re-scaled to real industrial applications. A complete mathematical analysis of the vibrating system coupled to the granular media represents a theoretical challenge. Let us remark that in the present thesis we avoid the complete analysis of how the system is perturbed, i.e., dissipative properties and scaling relations of a granular material on a vibrating plate [152, 154, 169, 178, 228], and we appeal to rough approximation whenever necessary.

It is clearly impossible to cite here all or even a significant fraction of all experi-
mental aspects on the subject of vibrated granular materials. There have been many
articles that make a serious effort on experimental studies of vibro-fluidized beds.
We hope that this abbreviated discussion will stimulate the reader to consult these
more complete sources, e.g., [48–51, 73, 113, 115, 152, 228, 230]

**Some experimental outcomes**

To answer the question *Why the Brazil nuts are on top?* it is essential to know the experimental investigations developed towards the understanding of the segregation phenomenon in vibrated systems. Here we summarize some of the most outstanding experiments carried out over the past years. Early studies were consecrated to the size effect in different regimes such as convection [141] and *vault*, i.e., the successive formation and destruction of arches at low acceleration [64, 65]. A renewed interest in the Brazil nut phenomenon (BNP) is currently taking place. For example, several “new millennium” experiments have been published since 2001 [33, 39, 40, 42, 86, 90, 149, 179, 183, 184, 236]. This renewed interest was motivated by the observation of new complex and unusual phenomena [108, 149, 179, 217]. For example, the reverse buoyancy effect [90, 217, 236], when larger and heavy grains can rise and equally larger but lighters grains can sink. On the other hand, not always segregation occurs in an upward movement, there exist too a segregation to the bottom of the container,

<figure><img src="image_7.png" /><figcaption>**Fig. 2.4:** Three dimensional experimental setup for the study of vertically vibrated granular materials.</figcaption></figure>
due to the interplay between the size and the mass of the particles (The upward to downward transition) [108].

Many studies and applications can be found in the literature of which only a few are cited here. For a classification and review of the literature up to 1976 we address the reader to reference [53] and for recent reviews see [189, 207, 208]. In the following survey we focus on those experiments that have been discussed among the physics community. This is not intended as a comprehensive review; rather, it should provide the major experimental outcomes to bring together the context in which this thesis is placed. We have made an effort to introduce a *historical classification* to sketch the experimental advances that have been achieved in the last years: the *First* (1939–1973), *Second* (1993–1997) and *Third* (1998–2003) periods.

## *The first period*

The history goes back to the year 1939 when Brown published the article *Fundamental Principles of Segregation* [38]. In this article the author acknowledges the importance of the “deleterious effects of segregation… in preparing, transporting, storing, and burning coal” [38]. Brown seems to have been the first to report that when a particulate mixture is subjected to vibration the larger particles appear at the top and the smaller at the bottom of the container [38]. Among the firsts qualitative studies, may it suffice to mention the work of Williams in 1963 [233] and the extensive studies carried out by Rippie and coworkers (1964–1965) characterizing the influence of particle size and distribution [188], the particle size–density interactions [204], and agitation intensity<fnref n="1" /> [72]. From these experiments the authors conclude that the rate of segregation of binary mixtures is proportional to the size ratio, also they found that for ternary mixtures the presence of intermediate size particles reduces the rate of segregation in the system. In 1973 Ahmad and Samalley [2] confirmed that the greater the size of the particle the greater the tendency to segregate for several conditions of vibration frequency and amplitude. Also they observed that a heavier particles segregate less than a lighter one under the same vibratory conditions and the shapes of the particles (of equal mass) did not have a significant effect on segregation. Recording the rising time, Ahmad and Samalley found that at constant vibration frequency an increase in acceleration diminished the segregation time and the contrary occurred with increasing frequency.

<footnote n="1">¹ Let us quote the following observation pointed out by Faiman and Rippei in this work [72]: “*Segregation and mixing processes observed in idealized systems of spherical particles subjected to sine wave motion show an analogy to molecular systems, and agitation of these systems appears analogous to the thermal motion of molecules*”. We remark that this analogy is the core of the theoretical strategy developed in this thesis.</footnote>
From the above brief historical account we note that most of the pioneers' works on segregation were carried out by engineers (see references.) In these preliminary studies the mechanisms of segregation were not yet completely understood.

### The second period

To give a convincing answer to the question of which mechanism causes the Brazil nuts to rise, in the 1990s experiments were carried out by physicist at the University of Chicago, USA (the Chicago Group); the Université de Paris VI, France (the Jussieu Group and the “Groupement de Recherche Physique des Milieux Hétérogènes of the CNRS”); and the Cavendish Laboratory, University of Cambridge and the Loughborough University of Technology, United Kingdom.

Ten years ago Duran, Rajchenbach and Clément [64] reported experiments in a two dimensional packing (Hele-Shaw cell) of small corrugated aluminum beads (1.5 mm-diam) and an intruder cylinder (ranging from 1.2 to 13 mm-diam) for accelerations larger than $Γ_c$, where $Γ_c ≈ 1.1g$ is the acceleration threshold as defined in Ref. [49]. The cell was made with two parallel plates (10 × 15 cm²) at close distance 1.01 times the bead diameter (more design details are given in Refs. [49, 54, 64] and [62] pp. 193–199.) Fig. 2.5 shows a photograph of a disk surrounded by smaller spheres clamped in a vault [64]. Using topological arguments, they found the set of stable positions when the large disk is raised by infinitesimal steps within the packing; and the critical diameter ratio for a continuous ascent (see Figs. 3 and 4 in Ref. [64].)

<figure><img src="image_8.png" /><figcaption>**Fig. 2.5:** Configuration of a disk in an assembly of smaller spheres (From Ref. [64])</figcaption></figure>

Figure 2.6 reveals another intriguing experimental outcome. Contrary to the experiments described above in 1993 Knight, Jaeger and Nagel [141] showed that different single intruder tracers rise at approximately the same rate due to a convection cycle, rising in the middle and dropping in a thin stream along the walls of a vertical cylinder (35 mm diam) filled with spherical glass beads (2 mm diam.) They used different intruder spheres of the same composition (2 mm (□), 6 mm (○), 19 mm (×) in diameter.) In this regime, the tracer particles are trapped on top when they are larger than the width of the thin downward convection zone.

Always in the same spirit of Refs. [49, 64] for two dimensional experiments, Du-

<figure><img src="image_9.png" /><figcaption>**Fig. 2.6:** Depth, $\Delta$, below the top surface of a single tracer as a function of tap number $\tau$ (From [141].)</figcaption></figure>
ran et al. [65] reported in 1994 that, depend-
ing on the vibration amplitude, two different
regimes act on segregation. At high acceler-
ation (Γ = 2.00) the convection process re-
ported in Ref. [141] associated with surface
trapping is dominant. While at low accelera-
tion (Γ = 1.25) the segregation dynamics de-
pends on the size ratio between the big particle
and the smaller particles. They recorded, us-
ing the computer posed photograph technique (CPP) test beads and the intruder
positions (see Fig. 2.7 in the right.) From thesues records a size dependence of the rise
velocity was observed. Another intriguing observation was a size threshold for seg-
regation (this threshold was anticipated in Ref. [131, 134].) Finally, they evidenced
the presence of arches in the segregation dynamics.

**Fig. 2.7:** Left: snapshot of a 2D cell at the beginning
of an experiment. Right: CPP of the 2D vibrated cell in
the *vault regime* (From Ref.[65])

<figure><img src="image_10.png" /><figcaption>**Fig. 2.7:** Left: snapshot of a 2D cell at the beginning of an experiment. Right: CPP of the 2D vibrated cell in the *vault regime* (From Ref.[65])</figcaption></figure>

In contrast to previous results, in 1996 Cooke et al. [54] claimed that the stability
of the intruder is not relevant in their experiments, and convection rolls drive the
segregation dynamics in both the intermittent and continuous regimes. Their exper-
imental method is very close to those of [64,65]. They noted that the segregation
rate increases as the size ratio and the acceleration parameter Γ increase.

The rise time of a single large sphere within a sinusoidally vibrated bed of uniform
particles was studied by Vanel, Rosato and Dave in 1997 [225]. They did three
dimensional experiments using 3 mm acrylic spheres placed in a cylindrical container
agitated by a floor piston, and identified three distinct regimes: For frequencies below
15 Hz strong convection and surface heaping appeared. For intermediate frequencies
between 15 Hz and 40 Hz, surface heaping was not observed. In the latter regime
they noted a size dependence in the segregation rate.

Up to now, the above experiments were carried out in systems containing a single
intruder. Among the few experiments consecrated to binary mixtures (in this sec-
ond period from 1993 to 1997) let us men-
tion those of Brone and Muzzio [37] and
Hsiau and Yu [109]. Both experiments re-
ported a reverse segregation or re-mixing
process when the vibration frequency is
increased above a certain threshold. Brone
and Muzzio observed the formation of ra-

<figure><img src="image_11.png" /><figcaption>**Fig. 2.8:** Reversible segregation [37]</figcaption></figure>

dial segregation patterns in cylindrical containers which depends on the with of the
downward convective flow stream near the walls. They claim that this convective
flow promotes the re-mixed state [37]. Figure 2.8 shows a mixture of 1500 µm and
600 µm glass beads vibrated at 14 Hz (left) and 22 Hz (right) for 20 min. On the other hand, the re-mixing process observed by Hsiau and Yu in two dimensional experiments results from a dense to loose state transition which depends on the vibrational energy [109].

### The third period

From 1998 a series of discoveries have shown that the Brazil nut phenomenon is more intricate than it might at first seem. The subject took a major step forward with the papers of Shinbrot and Muzzio [217], and Hong, Quinn and Luding [108].

Buoyancy belongs to those physical phenomena that have fascinated people since the time of Archimedes of Syracuse<fnref n="2" /> (213 B.C.) In 1998 Shinbrot and Muzzio [217] observe a reverse buoyancy effect: when large heavy grains rise, but equally large light grains sink in a granular bed (let us address the reader to the household experiment described in Ref. [218].) Three years later Hong, Quinn and Luding [108] introduced a theory for the *reverse Brazil nut phenomenon* (RBNP): when larger particles sink to the bottom of the container. Almost overnight, the subject attracted the attention of new important experiments [33, 39, 40, 42, 86, 90, 149, 179, 183, 184, 236].

In 2001 Liffman et al. [149] studied metal disks of different size and density placed in a bed of monodisperse granular material. They showed that the ascent speed of the disk is proportional to the square root of the disk density, approximately proportional to the disk size, and is a function of the disk's depth in the bed. The same year the controversy started when Möbius et al. [179] presented experimental data of the Brazil nut phenomenon density and size dependence contrary with the monotonous dependence reported in Refs. [108, 149, 217]. The most important lesson from this experiment is that the Brazil nut phenomenon is sensitive to the background air pressure [179]. The air drag effect on segregation dynamics was studied the next year by Nahmad-Molinari, Calnul-Chay and Ruiz-Suárez [183]. They reported a new technique to track the movement of beads based in an inductive device, and concluded that the kinetic energy exchange between particles is strongly affected by the air pressure. In another experiment performed by Calnul-Chay and coworkers, they did not find evidence of the reverse Brazil nut phenomenon [42].

In 2002 Burtally, King and Swift [39] studied a binary granular mixture of bronze and glass spheres of similar sizes (90 to 120 µm) observing a series of pattern transitions which depend on vibration frequency and air pressure. They observed that

<footnote n="2">² The “sand laptop”: He is even said to have carried a small wooden tray filled with sand, which he used to draw his figures and work on his mathematical problems. This tray would have been Archimedes’ version of the modern lap top computer (Junior Engineering, Utah State University 1997)</footnote>
when the air pressure is substantially reduced, the segregation is eventually lost (See also [40]).

The reverse buoyancy effect was reproduced in 2002 by Gutiérrez et al. [90], and they presented an analysis based on the fluidization of the granular media in the active–passive–cycles scheme introduced by de Gennes in 1989 [71]. They reported the vertical displacement and velocity of the intruder particles as a function of the density. Figure 2.9 shows a sequence of pictures of the vibrating granular systems. The first three images show the descent movement of a lighter intruder for 0 s, 0.9 s and 1.43 s. The last three pictures show the ascent movement of a heavier intruder for 0 s, 2.27 s and 4.04 s.

<figure><img src="image_12.png" /><figcaption>Fig. 2.9: Reverse buoyancy effect, from [90].</figcaption></figure>

At the time of this thesis, first observations of reverse Brazil nut phenomenon, predicted by Hong, Quinn and Luding [108], have been reported by Breu et. al. [33]. They used spherical beads of various sizes and materials in order to observe the transition from the upward to downward movement. Figure 2.10 shows the temporal evolution of the system studied by Breu et al. [33]. (a) 8 mm glass spheres on top of 15 mm polypropylene beads showing the classical Brazil nut phenomenon. (b) 10 mm bronze spheres on 4 mm glass beads, which show the reverse Brazil nut phenomenon. Let us remark that, in the reverse Brazil nut phenomenon the downward movement appears when the larger particles are heavier than the smaller particles, and it not necessarily implies the reverse buoyancy.

<figure><img src="image_13.png" /><figcaption>Fig. 2.10: Brazil nut / reverse Brazil nut phenomenon, from [33].</figcaption></figure>

The effect of liquid on the dynamic behavior of a binary granular mixture has been recently successfully explored in an experiment carried out by Geromichalos et al. [86]. Using a mixture of 2.5 mm glass beads and smaller glass beads (ranging from 50 µm to 500 µm,) they observed that as liquid was added, the tendency to segregate was diminished depending on the sizes of the particles. However, the tendency to mixing did not improve monotonically with the amount of liquid added (see the segregation phase diagram presented in Ref. [86], Fig. 3.)
Very recently, in relation to the experiment reported in [217] and as anticipated in [90], Yan et. al. [236] analyzed the effect of air on the reverse buoyancy effect. They found that the intruder always rises for dry systems, and the intricate effect that the intruder can acquire both positive and negative buoyancy in presence of air.

From the above overview, let us address some remarks. We can conclude that convection is predominant in systems with a cylindrical geometry [37, 141, 179, 217, 225]. The size dependence of the intruder's upward movement arises when convection streams are absent or weak [33, 39, 54, 65, 109, 149, 225]. The interplay between particle size and mass are crucial for the reverse segregation and buoyancy [33, 39, 42, 90, 217, 236]. Finally, without a doubt the complexity of the problem increases from dry to wet systems [39, 86, 179, 183, 236].

The comparison between our theoretical approach and experiments will be frequently emphasized in the following chapters. We anticipate that our theoretical models are able to explain several experimental results, for example, the size dependence of the intruder's upward velocity, and the upward to downward intruder's movement transition.

### 2.2.3 Computer simulations

Computer simulations give a great insight towards the understanding of the dynamics of vibrated particulate systems. There are different alternatives to model a granular flow, for example, Molecular Dynamics (MD) [99, 102], Monte Carlo (MC) simulations [105] and Cellular Automata (CA) [16, 23, 78]. In particular MD simulations have been used to investigate several aspects of granular media [82]. However, they are limited to relatively small systems (standard simulations involve $10^4 – 10^5$ particles<fnref n="3" />.)

Beside experiments, computer simulations have widely been applied to explain the mechanisms of segregation. In fact, the current interest within the physics community, in fundamental investigations of the segregation phenomenon was stimulated by the early MC simulations of Rosato and co-workers [210, 211], which clearly demonstrated that the larger particles immersed in a sea of smaller particles rise to the top when subjected to strong vertical shaking. This model includes non-sequential

<footnote n="3">³ One of the factors limiting simulations of real physical situations are computing time and memory storage, even when using some of the most powerful computational resources available today [105]. For example, the most time consuming part of a MD simulation is the calculation of the forces between the particles [99, 102]. Recent important achievements offer a possibility to avoid these problems [105, 151], for example, Event-driven (ED) algorithm [7] and the Direct Simulation Monte Carlo Method (DSMC) [28]</footnote>
dynamics<fnref n="4" /> and is parameterized by the shaking amplitude, which measures the size of the periodic perturbation.

The 1987 work of Rosato et al. [211] motivated subsequent important numerical investigations carried out in 1990's in Jülich and Paris. For example, the work of Devillard [59] derives a scaling law relating the segregation velocity with the vibration amplitude, and the study carried out by Dippel and Luding [61] of the geometrical effects of a vibrated system in the absence of convection, on time scales comparable to experimental observations.

Jullien and co-workers [129-131, 133-135] carried out several simulations in the context of large-amplitude, low-frequency vertical shaking. Their model is based on a random deposition model in which particles are deposited, one at a time, via randomly located vertical trajectories onto a horizontal substrate. They reproduced the Brazil-nut phenomenon and the experimental observations of Ahmad and Smalley [2]. Jullien, Meakin and Pavlovitch [131] predicted a critical size ratio below which segregation does not occur. This provoked some controversy [17, 18, 133, 135]. In subsequent works, this segregation threshold was analyzed in the MC simulation reported in Ref. [61] and observed in experiments [65].

Pöschel and Herrmann [195] investigated via MD simulations the relationship between segregation and convection. In Fig. 2.11 we show two snapshots of the first MD simulation in three dimensions of the Brazil nut phenomenon, carried out by Gallas et al. in 1996 [80]. They discuss carefully the conditions of the simulations which should include static and dynamic interactions, such as, translation and rotation.

<figure><img src="image_14.png" /><figcaption>**Fig. 2.11.** Snapshots showing the cylindrical cell with particle on the bottom (left) and on the top (right). See Ref. [80].</figcaption></figure>

Results reported for cylindrical [80] and rectangular [80, 195] containers showed that at low frequencies the big particle remains essentially on the bottom. For in-

<footnote n="4">⁴ That is, the route of one particle to its stable position cannot be computed without simultaneously computing the routes of many other particles [16].</footnote>
termediate frequencies the big particle goes up and remains there. If the frequency
is further increased the “whale effect” appears, this is when the big particle may
eventually dive deeply back into the granular assembly. In 1998, this whale effect was
observed in the experiments carried out by Shinbrot and Muzzio [217]. For other
MD simulations of the effect of convection on size segregation see Refs. [68] and
[146].

The effect of particle density on segregation in vibrated systems, was studied
by Ohtsuki et al. [187], but their MD simulations did not provide any evidence of
a density dependence. Very recently Shishodia and Wassgren [219] showed how a
buoyant force analysis predicts the equilibrium height of an intruder in a vibro-
fluidized bed.

The interplay between size and mass has been considered by Hong, Quinn and
Luding [108]. Besides their new theoretical model for the Brazil nut phenomenon,
they did extensive MD simulations for binary granular mixtures in two and three
dimensions. As we have already mentioned, they discovered that a downward in-
truder's movement occurs as well: *The reverse Brazil nut phenomenon*.

**2.3 Segregation mechanisms**

In the preceding section we have illustrated different features of the nature of the segregation phenomenon in vibrated systems. There has been some controversy about when certain segregation mechanisms may become dominant while others remain latent in some sense [173, 208]. In this section we enumerate different segregation mechanisms that have been used to explain the above experiments and computer simulations.

**2.3.1 Upward size-segregation: Brazil nut phenomenon (BNP)**

*Percolation.*— The Brazil–nut phenomenon has first been explained using geometri-
cal ideas of percolation theory [210, 211], i.e., in a vibrated–bed the smaller particles
are more likely to find a void through which they can percolate down to the bottom,
leaving the larger intruder at the top. Rosato et al. [211] argue that the percolation
effect is controlled by the ratio of volumes of the two types of particles.

*Arching.*— Duran Rajchenbach and Clément [64] formulated a geometrical inter-
pretation based on the *arching effect*, whereby the larger particle is being supported
by the arches of smaller particles, can help to assist the percolation-driven segrega-
tion [64, 65]. A threshold for segregation can be obtained from the geometrical ideas
introduced by Jullien, Meakin and Pavlovitch [131].

*Convection.*— Other mechanism of segregation is thought to be the convective mean flow in the vibrated bed which leads to the formation of two convective cells such that the particles move to the top through the central–axis and returns back to the
bottom near the side-wall [81, 94, 141, 142]. Since the mean-width of the boundary-layer near the wall is typically *half* of that near the central-axis, the larger particles are unlikely to return back to the bottom once they have moved to the top [141]. This mechanism was proposed by Knight, Jaeger and Nagel [141] and reproduced by Duran et al. [65] in two dimensions. However, there are other authors which consider convection as a mechanism that promotes mixing rather than separation (for instance, see Refs. [37] and [208].)

*Diffusion.*— Henrique, Batrouni and Bideau [97] presented numerical studies on granular mixtures (in absence of the gravity.) The system reaches a well mixed homogeneous stationary state in the bidisperse case under homogeneous agitations. However, if there is a gradient of agitation through the system, the system reaches a segregate state. The effect of an energy gradient has been also studied in Ref. [157] (this effect will be briefly discussed later.)

In Ref. [30] Braun has developed a mathematical formalism for segregation based on diffusion-convection equations. In this model convection describes competition between species in vertical direction whereas noise (shaking) allows particles to exchange positions. He shows that for two species the moving grains converge to a unique segregated distribution along the vertical direction.

The study of the diffusion phenomenon is a task of primary importance in statistical physics. Some works have modeled the Brazil nut phenomenon as the diffusion of a rod in a lattice fluid. However, in this thesis we do not discuss these models in detail. Let us address the reader to the seminal work of Alexander and Lebowitz [8] (see [9, 74, 76] for further references.)

*Compaction.*— Caglioti et al. [43], considered the geometrical properties of granular mixtures in the presence of compaction. Their model is based in the concept of geometrical frustration (the “Tetris” model [44]) and relate this effect to the effective average mobilities of two types of particles. The tapping sequence causes compaction of the system [44]. Then, they showed that the compaction of two types of particles is different and results in a segregation of the big particles on the top of the mixture.

*Depletion.*— When the separation of big spheres is less than diameter of the small ones, there is an unbalanced pressure of the “sea” of small spheres which gives rise to an attractive depletion force between the big spheres. This mechanism was first described by Asakura and Oosawa for colloid-polymer mixtures [13]. The attractive force was investigated in detail by Duran and Jullien in 1998 [63]. Their computer simulation shows that larger spheres, initially apart, tend to flocculate during the segregation process. An alternative description of the same effect (in the absence of gravity) is that a free volume of small spheres is gained due to the overlap of the
exclude volume of clustering large spheres; the resulting gain of entropy of the small spheres drives this clustering, and thus induces an effective attraction between large spheres (for instance, see [60, 137].)

*Inertia.*– Shinbrot and Muzzio [217] proposed the following explanation to the reverse buoyancy effect: once per cycle when an intruder is thrown upward, a heavy intruder tends to continue on a parabolic arc, while a light intruder is rapidly forced back down by collisions with particles above. Thus the heavy intruder would leave the bed, permitting smaller particles to infiltrate beneath, while a light intruder would bounce noisily, preventing this infiltration. A similar analysis was given in Refs. [149] and [90].

*Interstitial effect.*– We have already noted in Section 2.2 that the effects of in-terstitial air and liquids are relevant on the granular motion [39, 86, 179, 236]. The interstitial medium modifies the interchange of energy and momentum between par-ticles, affecting the mobilities associated to each particle [86]. On the other hand, the medium exerts a pressure on the grains acting as viscous damping [39, 193, 236].

Let us remark that in the present work we consider only *dry* systems.

### 2.3.2 Downward segregation: Reverse Brazil nut phenomenon (RBNP)

*Condensation.*– Hong, Quinn and Luding [108] proposed a phase diagram for the upward ⇔ downward transition (BNP/RBNP) based on a competition between percolation and condensation.

Hong’s theory of condensation of hard spheres under gravity can be seen as a particular case of the kinetic theory of dense gases [106]. Originally this work was intended to solve the Enskog equation for hard particles under gravity as a gradient expansion in the density. To this end, Hong explored how the kinetic theory can de-scribe the crossover from Boltzmann to Fermi statistics<fnref n="5" />. He discovered that there exists a critical temperature below which a system of hard spheres undergoes a con-densation transition under gravity. We will call this transition *Hong-condensation*. This transition has been confirmed by MD simulations in [198].

The criterion for the crossover from the BNP to the RBNP is determined setting the control parameter for the percolation effect, i.e., the ratio of volumes, and the Hong-condensation for two species. The crossover condition is given by [108]:

$$ \left(\frac{d_l}{d_s}\right)^{d-1} \approx \frac{m_l}{m_s}, \qquad (2.2) $$

<footnote n="5">⁵ Hong and Hayakawa [107] demonstrated that the configurational statistics of weakly excited two-dimensional granular systems in the presence of gravity does not follow the usual Boltzmann statistics as in molecular gases but a new Fermi statistics of spinless Fermions.</footnote>
where, $d_l$, $d_s$, $m_l$ and $m_s$ are the radius and the mass of the large $l$ and small $s$ particle, respectively.

*Edwards' approach.*— The statistical theory of powders proposed by Edwards and co-workers [66, 67, 175] has been applied to the problem of segregation in granular packings [215]. Most recently, Fierro, Nicodemi and Coniglio [77, 185] have generalized the Edwards' approach introducing a canonical distribution characterized by two *configurational temperatures* to the two types of particles and observed a crossover from the BNP to the RBNP.

## 2.4 Some general comments about segregation in granular materials

To conclude the first part of this thesis, let us give an overview of some applied aspects of the segregation problem inherent to industrial processes. Next, we give a short survey of the literature related to the segregation problem. We list the central references together with a selection of those of general background interest.

### 2.4.1 Other examples of granular segregation

The field of segregation in granular materials has become so large that many areas of development can only be touched. A comprehensive review would be beyond the scope of this thesis. Consequently we quote briefly another realizations of segregation.

*Packing.* Segregation in binary mixtures of hard disk packings has been studied in horizontal air table experiments. For example, at intermediate scale, small discs rearrange into chain-like clusters along large discs [87, 221]. Another effect of segregation was found for ballistic deposition models [132, 172]. It was noticed [132] that for certain ratio of sizes all the small particles are located near the bottom of the packing. Ballistic models were extended to the case of shaken systems [131].

*Free flowing granular materials.* When a mixture of small and large grains is

poured into a Hele-Shaw cell, a pile builds up and the small grains are observed to segregate near the top of the pile and the large ones near the bottom of the pile [161]. This segregation is due to the different grain sizes. Moreover, Fig. 2.12 shows the phenomenon of *granular stratification*, i.e., the formation of alternating layers of small-rounded and large-faceted grains. Spontaneous stratification, arises when the grains composing the mixture differ not only in size but also in shape [162–164].

<figure><img src="image_15.png" /><figcaption>**Fig. 2.12:** Granular stratification, see Refs. [162–164].</figcaption></figure>
In inclined chute flow of granular materials comprising particles of identical ma-
terial but different sizes, the small particles fall to the bottom and the larger to the
top of the sheared layer [104, 214].

A mixture of grains differing either in size, shape, or density when rotated in
a drum is seen to undergo radial segregation [41, 52, 162, 205], in which denser or
smaller particles migrate towards the core of the drum. In a horizontally placed
rotating tube the mixture of grains segregates along the axis into bands arranged
along the axis (axial segregation)[10, 189]. Recent developments in flows in rotating
drums and segregation and related models are reviewed in Refs. [168, 189, 190] and
the book [206].

Size segregation has been also investigated by a variety of methods, including, Couette flow geometry [138], horizontal shaking [182, 203] and swirling excitation [15], sawtooth-shaped base [202] and gas fluidized beds [235].

**2.4.2 Industrial relevance**

There are countless situations throughout all areas of industry where flowing par-
ticulate materials are handled as they pass through various processes. For exam-
ple, cosmetics, powder metallurgy, abrasives, solid state communications, ceramics,
pharmaceuticals, foods, detergents, construction, explosives, glass, and most other
mass-produced consumer goods. While mineral processing technologies exploit the
tendency for granular materials to segregate, industrial mixing technologies must
counteract this tendency. An understanding of the mechanisms underlying segrega-
tion can have an impact on technology and practice. The following two common
examples are only a few of the countless situations where segregation can contribute
to losses in product quality/process efficiency that many industrial situations expe-
rience [62, 101, 173, 189, 191].

*Pharmaceutical:* In pharmaceuticals, a 'filler' material is often mixed with the drug during the preconditioning stage prior to compaction in a tablet. Usually a pharmaceutical pill is formed by superposition of different tablets. If the filler material differs in size and/or density to that of the drug then segregation of the tablets mixture as it passes throughout the process prior to compaction will be inevitable. The worst case scenario would be the formation of tablets in which some contain all filler and no active drug, whilst others contain a high proportion of the active drug and little filler. Therefore, the pill's composition becomes inhomogeneous with an incorrect dosage of the drug. It is clear that the consequences are more than economical, the consumer could become vulnerable to administering an incorrect dosage of the drug.
*Foodstuffs:* During processing, particles of different sizes could be mixed together deliberately to give the material a desirable physical property such as smell, taste, feel and appearance, e.g., detergents, soaps and foodstuffs etc. Again, if care is not taken as to the physical properties of the constituents of the mixture, the amount and proportion of dosage of the mixture may lead to out of specification products.

## 2.5 How the Brazil nut goes up or down: A novel interpretation

Now that we have outlined the general features and different proposed mechanisms, we can conclude that the segregation phenomenon can be revealed in an intricate competition between percolation, condensation and convection. Despite all the work that has since been devoted to this issue, it remains an unresolved question of how provide a general framework to explain the segregation phenomenon (for vibrofluidized systems) into a theoretical ground. Under suitable conditions, a noncohesive granular system can be maintained in a fluid-like state by continuous vibration of the container. Previous studies have established that this fluidized state is well described by hydrodynamic theories for inelastic hard particles. Thus an obvious possibility is to try to combine the mathematical background provided by the kinetic theory of dense gases, to construct heuristic models for the Brazil nut phenomenon.

### 2.5.1 Theory of granular materials

To date, there is no unified mathematical framework for understanding all aspects of granular materials. This absence of a general theory can be attributed to the intrinsic physical complexity of the granular systems.

For example, there is no clear separation of *scales* in a granular material, we can distinguish between three levels of description:

*Microscopic* (single grain): mass, shape and mechanical properties [128]. Collision between grains are irreversible processes which cause dissipation of mechanical energy by fragmentation or heating of the colliders [36, 144, 181, 199, 216]. Measurements of collisional properties are essential for understanding several rheological behaviors of granular materials [144], as well as the scaling laws for the energy input of a single particle undergoing vertical vibrations [152, 154, 169, 178, 227, 228].

*Mesoscopic* (several grains): interfaces, forces, torques, conservation laws, volume constraints [216].

*Macroscopic* (bulk, material as a whole): stress/strain relation bulk moduli [147, 239] boundary conditions [110, 118] and behavior of vibrating beds [145].

On the other hand, there is no clear separation of *regimes* in a granular material. For example, when the system is compact and no possible interparticles motion is allowed, the granular material can be considered as a *granular solid*. When the system is subject to strong perturbations, it can arrive to a fluidized state, in this case, the granular material can be considered as a *granular fluid*. These two regimes can
exist “simultaneously” under vertical vibrations, e.g., a metastable (glassy) regime
[140], or a series of alternate passive and active regimes [71] coexisting between the
solid and fluid regimes.

Now that we are familiar with some elementary ideas about granular materials,
we are ready to turn our attention to the main subject of this thesis, the development
of phenomenological models for segregation in vibrofluidized granular systems.

2.5.2 Phenomenological theory for segregation

Granular hydrodynamics

Granular matter in the fluid-like regime can be described at the ‘macroscopic’ level by continuous hydrodynamic equations as a fluid-mechanical phenomenon [34, 89, 93, 126, 200]. We will call this type of equations *Granular hydrodynamics*. These hydrodynamic models have been provided some insight into behavior produced by vibrations using the correspondence between the motion of the molecules of dense gases and the random movement of granular particles (see e.g., [31, 32, 95, 103, 148]).

In contrast to a molecular gas in equilibrium state, where the mean kinetic energy
(mean-square velocity) of a molecule is proportional to the gas temperature (ther-
modynamic temperature), the natural equilibrium state of a granular material is a
static configuration due to the inelastic nature of particle collisions. Therefore, a sta-
tionary fluidized state needs a constant energy flow of energy into the system. This
steady state driven by the energy flux can be assumed as a condition of “thermal
equilibrium” [98]. On the other hand, in analogy to a molecular gas, we can define
a “granular temperature” [186], which is proportional to the mean kinetic energy
of the particles’ velocity. Let us remark that this generalized notion of temperature
is introduced for theoretical convenience to take advantage of a thermodynamical
analogy for granular materials.

In fact, the definition of thermodynamic variables for nonequilibrium states is
straightforward theoretically. The thermodynamic of nonequilibrium states has al-
ways been a matter of debate [70]. Alternatives for a thermodynamic formulation
for granular materials have been proposed by Edwards and coworkers [66, 67, 175],
Herrmann [98], and Hayakawa and Hong [107]. A revival of the Edwards’ approach
is currently taking place, using the ideas and methods of the theory of glasses (see
Barrat et. al. [21, 22] and Maske and Kurchan [165].) Certainly, the problem of a
thermodynamic formulation for granular materials deserves a deeper analysis (cf.
Ref.[136]), but to keep the scope of this thesis manageable we will limit our theo-
retical analysis into the framework of the kinetic theory of gases (i.e., a Maxwell–
Boltzmann statistics.)
The development of statistical descriptions for granular systems involves aver-
aging over the ‘microscopic’ laws for the particles motion to obtain ‘macroscopic’
balance equations for the hydrodynamic fields. A basis for the derivation of gran-
ular hydrodynamic equations (analogous to the Navier–Stokes equations) and de-
tailed expressions for the constitutive relations (transport coefficients) is provided
by the kinetic theory of gases conveniently modified to account for inelastic binary
collisions. See e.g., [119, 124, 125], and the book [196] which provides an excellent
overview of the current knowledge about granular gases. The reader is encouraged
to consult that book, in particular Ref. [224].

The balance laws for a granular fluid can be obtained on the basis of a mean
field kinetic equation, like the Boltzmann or Enskog-Boltzmann equation. The only
“nonclassical” term is the collision rate of dissipation per unit volume per unit
time due to inelastic collisions. Many calculations are based on the kinetic theory
of nonuniform gases [47] using the Chapman–Enskog procedure around a reference
state. For example, in a (elastic) molecular fluid this reference state is the local
equilibrium state. For a granular fluid two alternatives have been proposed: The
homogeneous cooling state (HCS) for a freely evolving system [35, 83, 155], and the
nonequilibrium steady state (NESS) for a uniformly driven (heated) system [46, 197].

In 1987–1989 Jenkins and Mancini [122, 123] introduced a remarkable extension
of the kinetic theory of nonuniform gases to bidisperse granular mixtures. In the
context of the so–called Revised Enskog Theory (RET) for multicomponent mixtures
[150], they derived balance laws and constitutive relations for a dense binary mixture
of smooth, nearly elastic particles by assuming a Maxwellian velocity distribution.
They investigated in detail the use of the theory to a steady rectilinear shearing
flow induced by the relative motion of parallel boundaries in the absence of external
forces. Important modification in these models were introduced by Arnarson and
Willits [12, 234] and most recently by Alam et al. [5]. It is interesting to recall that
there has been also an attempt to extend the kinetic theory of multicomponent
mixtures to systems consisting of an arbitrary number of inelastic particles [237].

Most of the above mentioned kinetic models are based on the crude assumption
of equal partition of species kinetic energy. Essentially, because granular systems
are dissipative, the energy equipartition does not generally hold [114, 160, 170]. Such
lack of energy equipartition has been recently investigated in several theoretical and
numerical works [3, 4, 19, 20, 25, 55, 84, 85, 111, 112, 167, 180, 192, 212, 213], and also
confirmed in air-table table experiments [114] and vibrofluidized beds [75, 229].

In the present thesis we will show how this breakdown of energy equipartition
could be an important ingredient for the dynamics of the segregation process. In
Chapter 5 we will not repeat these detailed works but do survey the same kinetic
theory methods for completeness and we will take the time to extend the discussion
to include different granular temperatures.

Granular hydrodynamic and the segregation phenomenon

Recently Jenkins, Louge and Arnarson, at the Cornell University, Ithaca, New York,
USA, have developed kinetic theories for particle segregation in collisional flows of
dry granular materials [11, 12, 120, 121, 127, 234].

Their work began as an effort to sketch the theoretical framework for the interpre-
tation of a remarkable series of experiments on particle segregation in microgravity<fnref n="6" />
[121]. In Refs. [120] and [11] the species diffusion velocity has been calculated in the
context of the Chapman–Enskog procedure, and following the procedure introduced
in [122] they obtained the weighted difference of the species momentum balances
and the gradients of partial pressure to outline expressions for the interparticle
force. However, they did not integrate the equations to determine the variations of
the fields and the segregation dynamics.

For binary mixtures of inelastic spheres in the presence of a temperature gradi-
ent, Arnarson and Willits [12], found that larger, denser particles tend to be more
concentrated in cooler regions. This result was confirmed by numerical simulations
[97, 157, 171]. However, this mechanism of segregation is a natural consequence of
the imposed gradient of temperature and its not related to the nature of the grains
[97].

Recently, Jenkins and Yoon [127] investigated the upward ⇔ downward transition employing the hydrodynamic equations for binary mixtures. Their theory agrees with the phase diagram for the BNP/RBNP transition introduced in [108]. The driving mechanism for segregation in the hydrodynamic framework is presumably different from that of the percolation–condensation transition.

These theories form the core of the framework of this thesis.

Principal assumptions of this thesis

Now we introduce the following assumptions:

(A) An uniformly fluidized state can be realized when the granular material is vi-
brated in the vertical direction, typically as z₀(t) = A₀ sin(ω₀t), with the am-
plitude A₀ and the frequency ω₀ = 2πf. The normalized acceleration parameter
Eq.(2.1) satisfies Γ ≫ 1.

<footnote n="6">⁶ Project carried out by the NASA Microgravity Science and Application Division and the Cornell University. Currently the experimental device is undergoing airborne flight tests and refinements of its design and data-collection software in anticipation of its scheduled launch to the International Space Station in 2007 [24].</footnote>
(B) Granular matter in the fluid–like regime can be described at the ‘macroscopic’ level by continuous hydrodynamics equations as a fluid–mechanical phenomenon. We will call this type of equations *Granular Hydrodynamics*.

(C) In non-equilibrium systems, the concept of temperature can be generalized to contain kinetic and configurational contributions. However, we restrict our discussion to the case of systems with strong driving forces and we assume that the *granular* temperature is defined proportional to the mean kinetic energy. Therefore, the role of *configurational* temperatures becomes secondary in our approach.

(D) We assume that the system is in a regime where convection can be neglected. However, the hydrodynamic description could be generalized to consider the coupling of the intruders' movement with the bulk convection stream.

(E) We impose horizontal periodic boundary conditions to make the equations analytically tractable. Therefore, there are variations of the hydrodynamic fields only along the vertical direction.

**Main results of this thesis**

Now that we have introduced the general framework and the principal assumptions, we can state the main results of the thesis.

I. We show that a *granular hydrodynamic* theory does indeed give a correct qualitative description of the size dependence of the upward segregation velocity. The proof of this statement relies on a novel mechanism of segregation based on buoyant forces.

II. The inelastic dissipation determines the regimes of the BNP and RBNP.

III. We have identified alternatives types of segregation forces: apart from the Archimedean buoyancy force we have introduced an analog of the thermal buoyancy force and two *geometrical* forces due to the size-disparity between the intruder and the smaller particles.

**Remark**

It is necessary at this point to place the perspectives of this thesis. The theoretical construction carried out in Chapters 3 and 4, deal with the problem of a single intruder immersed in a fluidized granular bed. The system is modelled by inelastic hard spheres or disks. The intruder's presence affects the distribution of kinetic energy in the granular bed. We will introduce an analytical method to calculate this variation of kinetic energy. On the other hand, the system formed by the small particles will be considered as a *granular fluid*. To describe this granular fluid we
appeal to the kinetic theory of dense gases to calculate the constitutive relations of
the system and to derive the expressions for the energy flux in the system.

In Chapters 5 and 6, we follow the kinetic theory description. However, we will consider the case of a binary mixture of hard spheres or disks. This theoretical construction differs with the single intruder case: In the binary mixture approach we will consider the *single intruder limit*. This (mathematical) limit should be interpreted as the limit when the number of intruder particles by volume unit is small (i.e., diluted) and it not necessarily implies the single intruder case. We will point out some similarities between these two approach. However, it is important to clarify that these comparisons are formal and we must be careful with the notion of lack of kinetic energy equipartition due to the inelastic nature of the system.
<figure><img src="image_16.png" /></figure>
Notation

The following notation is used throughout the text in Chapters 3 and 4.

<table><tr><td>d = 2, 3</td><td>dimension of the system</td></tr><tr><td>I</td><td>index which denotes intruder particle</td></tr><tr><td>F</td><td>index which denotes small particles (granular fluid particles)</td></tr><tr><td>r<sub>I</sub></td><td>radius of the intruder particle</td></tr><tr><td>r<sub>F</sub></td><td>radius of the small particles</td></tr><tr><td>&phi; = r<sub>I</sub>/r<sub>F</sub></td><td>size ratio</td></tr><tr><td>e</td><td>restitution coefficient</td></tr><tr><td>L</td><td>length of the container</td></tr><tr><td>V = L<sup>d</sup></td><td>Volume of the container</td></tr><tr><td>v</td><td>velocity</td></tr><tr><td>E</td><td>kinetic energy</td></tr><tr><td>&Delta;E</td><td>change of the kinetic energy</td></tr><tr><td>N</td><td>number of fluid (smaller) particles</td></tr><tr><td>T<sub>g</sub></td><td>granular temperature</td></tr><tr><td>&Delta;T<sub>g</sub></td><td>granular temperature difference</td></tr><tr><td>q</td><td>flux of kinetic energy</td></tr><tr><td>&gamma;</td><td>rate of dissipated energy</td></tr><tr><td>&kappa;</td><td>thermal conductivity</td></tr><tr><td>p</td><td>pressure</td></tr><tr><td>g</td><td>gravitational acceleration</td></tr><tr><td>g</td><td>pair correlation function</td></tr><tr><td>&nu;</td><td>volume fraction</td></tr><tr><td>&omega;</td><td>collision frequency</td></tr><tr><td>&alpha;</td><td>thermal expansion coefficient</td></tr><tr><td>&mu;</td><td>viscosity</td></tr><tr><td>u<sub>s</sub></td><td>settling velocity</td></tr><tr><td>f<sub>b</sub></td><td>buoyancy force</td></tr><tr><td>f<sub>d</sub></td><td>drag force</td></tr></table>
<figure><img src="image_17.png" /></figure>
# 3 Single intruder size segregation dynamics

We present an exhaustive study of the problem of a single intruder particle immersed in a fluidized granular bed. Effects of intruder's size on the segregation dynamics are analyzed in detail from a continuum perspective. We introduce a mechanism of segregation due to buoyant forces. To demonstrate how buoyancy drives the upward intruder's movement, we develop a hydrodynamic scheme for inelastic hard particles. The proposed new mechanism is characterized by the feature of non-equipartition of the granular kinetic energy.

## Introduction

Now that we are familiar with some elementary ideas about the Brazil nut problem, we are ready to turn our attention to the main subject of this thesis, the development of hydrodynamic models for segregation.

In this chapter, we show that a granular hydrodynamic theory does indeed give a correct qualitative description of the Brazil nut phenomenon (BNP). We propose a segregation mechanism based on the difference of densities between different regions of the system, which give origin to a buoyant force that acts on the intruder. The difference of densities is caused by the difference between the mean kinetic energy in the region around the intruder and the medium without intruder. The dissipative nature of the collisions between the particles is responsible for this mean energy difference, and modifies the mobility of the particles. We restrict our analysis to the single intruder case in two and three dimensions.

## 3.1 Continuum formulation

### 3.1.1 Preliminaries and description of the granular fluid

We consider an intruder particle of mass $m_I$ and radius $r_I$ immersed in a granular bed. The granular bed is formed of $N$ monodisperse particles of mass $m_F$ and radius $r_F$. We assume that the material density of all particles is the same. The particles are modelled by inelastic hard disks ($d = 2$) or spheres ($d = 3$) in a $d$-dimensional volume $V = L^d$ of size $L$. The size ratio is denoted $\phi = r_I/r_F$. The particles interact via binary encounters. The inelasticity is specified by a restitution coefficient $e \le 1$. We assume this restitution coefficient to be a constant, independent on the impact
velocity and the same for the fluid particles and the intruder. The post collisional velocities $\mathbf{v}'$ are given in terms of the pre-collisional velocities $\mathbf{v}$ by

$$ \mathbf{v}'_{1,2} = \mathbf{v}_{1,2} \mp \frac{m_{red}(1+e)}{m_{1,2}} [(\mathbf{v}_1 - \mathbf{v}_2) \cdot \hat{\mathbf{n}}] \hat{\mathbf{n}}, \quad (3.1) $$

where the labels 1 or 2 specify the particle, $\hat{\mathbf{n}}$ is the unit vector normal to the tangential contact plane pointing from 1 to 2 at the contact time, and the reduced mass $m_{red} = m_1 m_2 / (m_1 + m_2)$. To calculate the dissipated energy we consider that energy is dissipated only by collisions between pairs of grains. In a binary collision the energy dissipated is proportional to $\Delta E = -m_{red}(1 - e^2)v^2/2$, where $v$ is the mean velocity of the particles.

The granular temperature $T_g$ is defined proportional to the mean kinetic energy $E$ associated to the velocity of each particle

$$ \frac{d}{2} T_g = \frac{E}{N} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{1}{2} m_i v_i^2 \right). \quad (3.2) $$

We expect a continuum limit to hold for $N \gg 1$, when the small particles may be considered as forming a granular fluid [93]. In order to develop an analytic study, we assume that the uniformly heated granular fluid can be described by the standard hydrodynamic equations [93] derived from kinetic theories for granular systems [83, 119, 124, 125, 143]. In this study, we focus on a steady state with no macroscopic flow.

The balance equation for the energy is

$$ \nabla \cdot \mathbf{q} = -\gamma, \quad (3.3) $$

where $\mathbf{q}$ is the flux of energy and $\gamma$ is the average rate of dissipated energy due to the inelastic nature of the particles collisions. The constitutive relation for the flux of energy,

$$ \mathbf{q} = -\kappa \nabla T_g, \quad (3.4) $$

defines the thermal conductivity $\kappa$. Consequently, we have

$$ \nabla \cdot (\kappa \nabla T_g) = \gamma. \quad (3.5) $$

Let us recall that an uniformly fluidized state can be realized when the granular material is vibrated in the vertical direction, typically as $z_0(t) = A_0 \sin(\omega_0 t)$, with the amplitude $A_0$ and the frequency $\omega_0 = 2\pi f$, so that one can define a reference velocity $u_0 = A_0 \omega_0$. As a first approximation, the effect of the external force due to the gravitational field is neglected in the description of the granular flow. Experimentally
this correspond to the regime where the normalized acceleration parameter satisfies the condition $\Gamma \gg 1$. Therefore, the momentum balance, in the steady state, implies that the pressure $p$ is constant throughout the system.

The hydrodynamic equations close with the state equation, the collisional dissipation $\gamma$ and the transport coefficients for a granular medium. In the limit $N \gg 1$ the constitutive relations are determined as function of the properties of the small grains. The transport coefficients are assumed to be given by the Enskog theory for dense gases [47] in the limit of small inelasticity.

The total pressure should be essentially equal to that of the small particles, the contribution of the intruder being negligible, since $N \gg 1$. For a dense system the pressure is related to the density by the virial equation of state, which in the case of inelastic particles is [83, 125]

$$p = nT_g [1 + (1 + e)(d - 1)G], \quad (3.6)$$

where $n = N/V$ is the number density and $G = \nu g_0$, where $\nu = \Omega_d n r_F^d / d$ is the volume fraction, with $\Omega_d = 2\pi^{d/2}/\Gamma(d/2)$ as the surface area of a $d$-dimensional unit sphere and $g_0$ is the pair correlation function.

For disks $g_0$ is taken to be that proposed by Verlet and Levesque [226]:

$$g_0 = \frac{1}{1 - \nu} + \frac{9}{16} \frac{\nu}{(1 - \nu)^2}, \quad (3.7)$$

with the area fraction $\nu = n\pi r_F^2$, and for spheres that proposed by Carnahan and Starling [45]:

$$g_0 = \frac{1}{1 - \nu} + \frac{\nu}{2(1 - \nu)^2} \left[ 3 + \frac{\nu}{2(1 - \nu)} \right], \quad (3.8)$$

with the volume fraction $\nu = 4\pi n r_F^3 / 3$.

It is important to mention that the Eqs. (3.7) and (3.8) only work for moderate densities. In Ref. [156] Luding and Strauß showed that $g_0$ should diverge at the close packing limit ($\nu = \nu_{max}$), rather than at $\nu \to 1$. Therefore, in the present model we assume that $\nu < \nu_{max}$.

The state-dependent thermal conductivity possesses the general form [119, 125]

$$\kappa = \kappa_0 \sqrt{T_g}, \quad (3.9)$$

where the prefactor $\kappa_0$ is a function of the fluid particle properties, and can be calculated using a Chapman–Enskog procedure through the solution of Boltzmann–Enskog transport equation [47, 83, 119, 125]. The explicit expressions of these prefactors are given in Appendix A.

To estimate the collisional dissipation rate $\gamma$ we consider the loss of average kinetic energy per collision and per unit time. In a binary collision the kinetic energy
dissipated can be expressed in terms of the granular temperature as $\Delta E = -(1 - e^2)T_g/2$. For the fluid particles, the average collision frequency $\omega_F$ is proportional to $\omega_F \sim \sqrt{T_g}$, and we assume that it is given by the Enskog collision frequency [47]

$$ \omega_F = \frac{\Omega_d}{\sqrt{2\pi}} ng_0 (2r_F)^{d-1} \left(\frac{2}{m_F}\right)^{1/2} T_g^{1/2}. \quad (3.10) $$

This form for the frequency of collisions is justified for a granular medium. This is a consequence of the fact that the average spacing between nearest neighbor $s$ is supposed to be less than the grain diameter ($s \ll 2r_F$) [93]. Multiplying $\Delta E$ by the collision rate $\omega_F$ and the number density $n = N/V$, we obtain the collisional dissipation rate $\gamma_F$ for the fluid particles

$$ \gamma_F = \frac{\Omega_d}{2\sqrt{2\pi}} (1 - e^2) n^2 g_0 (2r_F)^{d-1} \left(\frac{2}{m_F}\right)^{1/2} T_g^{3/2}. \quad (3.11) $$

In order to simplify the mathematical notation let us express $\gamma_F$ as

$$ \gamma_F = \xi_F T_g^{3/2}, \quad (3.12) $$

where the dissipation factor $\xi_F$ contains the prefactors which multiply $T_g^{3/2}$ in Eq.(3.11), this is

$$ \xi_F = \frac{\Omega_d}{2\sqrt{2\pi}} (1 - e^2) n^2 g_0 (2r_F)^{d-1} \left(\frac{2}{m_F}\right)^{1/2}. \quad (3.13) $$

To understand the essential features of the intruder's presence in the granular medium, it is adequate to adopt a simplified point of view. If the mean velocity of the fluid particles is $v \propto \sqrt{T_g/m}$, the flux of fluid particles which hits the intruder's surface can be estimated as $nv$. Multiplying this flux by the area of the intruder $\Omega_d r_I^{(d-1)}$, we can calculate the number of fluid particles which hit the surface of the intruder per unit time, and written in terms of the granular temperature we have

$$ \omega_I = \frac{\Omega_d}{\sqrt{2\pi}} n \tilde{g}_0 (r_F + r_I)^{d-1} \left( \frac{m_I + m_F}{m_I m_F} \right)^{1/2} T_g^{1/2}, \quad (3.14) $$

where $\tilde{g}_0$ is the pair correlation function of the granular fluid in presence of the intruder.

The local density of kinetic energy dissipated in the region near the intruder is

$$ \gamma_I = \frac{\Omega_d}{2\sqrt{2\pi}} (1 - e^2) \frac{n}{V} \tilde{g}_0 (r_F + r_I)^{d-1} \left( \frac{m_I + m_F}{m_I m_F} \right)^{1/2} T_g^{3/2}. \quad (3.15) $$

In a simplified form Eq.(3.15) can be expressed as
$$ \gamma_I = \xi_I T_g^{3/2}, \qquad (3.16) $$

where the dissipation factor $\xi_I$ is defined as

$$ \xi_I = \frac{\Omega_d}{2\sqrt{2\pi}}(1-e^2)\frac{n}{V}\tilde{g}_0(r_F + r_I)^{d-1}\left(\frac{m_I + m_F}{m_I m_F}\right)^{1/2}. \qquad (3.17) $$

Finally, in order to estimate $\gamma_I$ an explicit expression for $\tilde{g}_0$ is needed. To our knowledge, no theory exists for the exact calculation of the pair correlation function of a single intruder (tracer particle) immersed in a granular fluid. Rather than going into details concerning the calculation of $\tilde{g}_0$, we propose to use the generalization of Eqs. (3.7) and (3.8) for binary mixtures [122, 166], restricted in our case to the limit where $N \gg 1$ and $r_I > r_F$. For two dimensions we can verify that

$$ \tilde{g}_0 \approx \frac{1}{1 - \nu} + \frac{9 \nu}{8 (1 - \nu)^2}, \qquad (3.18) $$

and for three dimensions

$$ \tilde{g}_0 \approx \frac{1}{1 - \nu} + \frac{\nu}{(1 - \nu)^2} \left[ 3 + \frac{\nu}{1 - \nu} \right]. \qquad (3.19) $$

Let us remark that $g_0$ and $\tilde{g}_0$ are not very different and both quantities tend to one in the diluted limit. Here too, we assume that $\nu < \nu_{max}$.

### 3.1.2 Local temperature difference

The intruder's presence modifies the local temperature of the system due to the collisions that happen at its surface. The number of collisions on the surface increases with the size of the particle, but the local density of dissipated energy diminishes. From Eq.(3.5) we can calculate within a sphere of radius $r_0$ the value of the temperature in the granular fluid in the presence of the intruder and compare it with the temperature in the granular fluid without intruder, we will denote these temperatures $T_1$ and $T_2$ respectively (see Fig. 3.1). These spherical regions are considered to be placed in the reference frame of the intruder particle. This is a simple method to estimate the temperature difference between a region with intruder and a region without intruder $\Delta T_g = T_1 - T_2$.

Let us concentrate on solutions with radial symmetry. The solutions of Eq.(3.5), for an arbitrary dimension $d$, satisfy the equation

$$ \frac{1}{r^{d-1}} \frac{d}{dr} \left( r^{d-1} \kappa_0 T_g^{1/2} \frac{dT_g}{dr} \right) = \xi T_g^{3/2}. \qquad (3.20) $$

This nonlinear differential equation can be simplified by the fact that the pressure is considered constant throughout the system and remembering that $p \sim T_g$. So, linearizing Eq.(3.20) the resulting equation may be written in terms of $w \equiv T_g^{1/2}$,
<figure><img src="image_18.png" /><figcaption>Fig. 3.1. Schematic representation of the regions used to calculate the granular temperature. (a) Region around the intruder within a sphere of radius $r_0$ and (b) region without intruder.</figcaption></figure>

$$ \frac{d^2w}{dr^2} + \frac{d-1}{r} \frac{dw}{dr} = \lambda^2 w, \qquad (3.21) $$

where

$$ \lambda^2 = \frac{\xi}{2\kappa_0}. \qquad (3.22) $$

It is useful at this point to see the implications of the requirement of constant pressure invoked to derive Eq. (3.21). This estimate is suggested by the fact that the effect of the gravitational field can be neglected in the regime of strong perturbations. Then, the momentum balance equation for fluid particles satisfies $\nabla p = 0$. This is a good approximation if the kinetic energy is much larger than the change of the gravitational potential energy experienced over the average spacing between grains. However, we can not forget that, the granular fluid is a system out of “thermal” and mechanical equilibrium. A change in the local granular temperature does change the pressure of the system. To make the model analytically tractable, we have neglected all the gradients of the pressure. This heuristic assumption is based on the fact that a dense granular fluid can be considered as an incompressible system [93]. Qualitatively, this picture is correct if the “isothermal compressibility” $k_p$ (see Eq. (3.58), Section 4) is bigger than the coefficient of “thermal expansion” $\alpha$ (see Eq. (3.57), Section 4). To be more concrete, using the thermodynamic relation $(\partial p/\partial T)_V = \alpha/k_p$, it follows that if $k_p \gg \alpha$, then $\partial p/\partial T \simeq 0$.

Therefore, in the present model, the gradients $\partial p/\partial T$ and $\nabla p$ do not contribute to the description of the granular fluid. The effect of the gravitational field must be included in the description of the system at the level of individual grains. These rough approximations, which should be sufficiently accurate for our purpose, might not always be valid. It is necessary to be aware that in proceeding along this way some part of the dynamics may be lost. For example, a complete analysis of Eq. (3.20) requires much more information, such as the behavior of “compressive waves” arising
from the variations of the pressure, which drives the system toward the mechanical
equilibrium.

The collisional dissipation rate can be decomposed in two parts. We propose this decomposition supposing that the energy dissipation around the intruder is dominated by the collisions between the small grains and the intruder, then the dissipation rate in this region is given by Eq. (3.15). In the rest of the system the dissipation rate is dominated by the collisions between small grains only. In this case the dissipation is given by Eq. (3.11).

We can therefore consider the following two cases:

1. The “inhomogeneous case” when the intruder is localized in the center of the spherical region ($r = 0$), see Fig. 3.1(a). The dissipation factor $\xi$ can be decomposed in two parts: $\xi = \xi_I$ for the region near the intruder ($r = r_I$), and $\xi = \xi_F$ for the region ($r_I < r \le r_0$), where $r_0$ is the radius of the considered region.
For the inhomogeneous case we can write Eq. (3.21) as

$$ \frac{d^2w}{dr^2} + \frac{d-1}{r} \frac{dw}{dr} = \begin{cases} \lambda_I^2 w & \text{for } 0 < r \le r_I, \\ \lambda_F^2 w & \text{for } r_I < r \le r_0, \end{cases} \qquad (3.23) $$

where, $\lambda_I^2 \equiv \xi_I/2\kappa_0$ and $\lambda_F^2 \equiv \xi_F/2\kappa_0$. The solution of Eq. (3.23) is determined by the boundary condition imposed upon the system. As boundary condition we suppose that the system is enclosed by an external surface of radius $r_0$ at temperature $T_g(r_0) = T_0$ (respectively, $w(r_0) = w_0$).

Denote by $T_{1-}(r)$ the granular temperature for the region ($0 < r \le r_I$), and $T_{1+}(r)$ the granular temperature for the region ($r_I < r \le r_0$). Then we have, respectively, $w_-(r) \equiv \sqrt{T_{1-}(r)}$ and $w_+(r) \equiv \sqrt{T_{1+}(r)}$. The intruder's presence imposes internal boundary conditions. On the inner surface, the temperature should satisfy

$$ w_-(r)|_{r=r_I} = w_+(r)|_{r=r_I}. \qquad (3.24) $$

The flux of energy also imposes another internal boundary condition. If we suppose the flux of energy continuous on the inner surface, from Eq. (3.4) the granular temperature should satisfy

$$ \left. \frac{dw_-(r)}{dr} \right|_{r=r_I} = \left. \frac{dw_+(r)}{dr} \right|_{r=r_I}. \qquad (3.25) $$

2. The “homogeneous case” when the spherical region ($0 < r \le r_0$) does not contain any intruder and we have $\lambda_I = 0$, with the boundary condition $T_g(r_0) = T_0$ ($w(r_0) = w_0$), see Fig. 3.1(b).
### 3.1.3 Calculation of the granular temperature difference

Now we are ready to construct the analytical expression for $\Delta T_g = T_1 - T_2$ for two and three dimensions.

#### Solution for 2d

The solutions to Eq. (3.23) for $d = 2$ are a linear combination of the modified Bessel function of order zero $w_1(r) = \{I_0(\lambda r), K_0(\lambda r)\}$ [1]. The general solution is

$$w_{-}(r) = A_{-}I_{0}(\lambda_{I}r) + B_{-}K_{0}(\lambda_{I}r) \quad \text{for } 0 < r \le r_{I}, \qquad (3.26)$$

and

$$w_{+}(r) = A_{+}I_{0}(\lambda_{F}r) + B_{+}K_{0}(\lambda_{F}r) \quad \text{for } r_{I} < r \le r_{0}, \qquad (3.27)$$

where $A_-$, $A_+$, $B_-$ and $B_+$ are constants that must be determined from the boundary conditions.

The function $K_0(\lambda r)$ diverges when $r \to 0$, then

$$B_{-} = 0. \qquad (3.28)$$

When $r = r_0$, Eq. (3.27) should satisfy the boundary condition

$$w_{+}(r)|_{r=r_{0}} = w_{0}, \qquad (3.29)$$

this is,

$$A_{+}I_{0}(\lambda_{F}r_{0}) + B_{+}K_{0}(\lambda_{F}r_{0}) = w_{0}. \qquad (3.30)$$

On the inner surface, boundary condition (3.24) $w_-(r_I) = w_+(r_I)$ leads to

$$A_{-}I_{0}(\lambda_{I}r_{I}) = A_{+}I_{0}(\lambda_{F}r_{I}) + B_{+}K_{0}(\lambda_{F}r_{I}) \\ \implies A_{-} = A_{+}\frac{I_{0}(\lambda_{F}r_{I})}{I_{0}(\lambda_{I}r_{I})} + B_{+}\frac{K_{0}(\lambda_{F}r_{I})}{I_{0}(\lambda_{I}r_{I})}. \qquad (3.31)$$

The inner boundary condition (3.25) leads to

$$A_{-}\lambda_{I}I_{1}(\lambda_{I}r_{I}) = A_{+}\lambda_{F}I_{1}(\lambda_{F}r_{I}) - B_{+}\lambda_{F}K_{1}(\lambda_{F}r_{I}) \\ \implies A_{-} = \left(\frac{\lambda_{F}}{\lambda_{I}}\right) \left[A_{+}\frac{I_{1}(\lambda_{F}r_{I})}{I_{1}(\lambda_{I}r_{I})} - B_{+}\frac{K_{1}(\lambda_{F}r_{I})}{I_{1}(\lambda_{I}r_{I})}\right]. \qquad (3.32)$$

Equating Eqs. (3.31) and (3.32) we find

$$\frac{A_{+}}{B_{+}} = \frac{\lambda_{F}I_{0}(\lambda_{I}r_{I})K_{1}(\lambda_{F}r_{I}) + \lambda_{I}I_{1}(\lambda_{I}r_{I})K_{0}(\lambda_{F}r_{I})}{\lambda_{F}I_{0}(\lambda_{I}r_{I})I_{1}(\lambda_{F}r_{I}) - \lambda_{I}I_{1}(\lambda_{I}r_{I})I_{0}(\lambda_{F}r_{I})}, \\ \equiv \Theta_{AB}. \qquad (3.33)$$
From Eqs. (3.30) and (3.33) the constant $B_+$ should be

$$B_+ = \frac{w_0}{\Theta_{AB}I_0(\lambda_F r_0) + K_0(\lambda_F r_0)} \qquad (3.34)$$

Substituting Eq. (3.34) into (3.33) we have

$$A_+ = \frac{w_0 \Theta_{AB}}{\Theta_{AB} I_0(\lambda_F r_0) + K_0(\lambda_F r_0)} \qquad (3.35)$$

Substituting Eqs. (3.34) and (3.35) into (3.31) we have

$$A_- = \frac{\Theta_{AB} I_0(\lambda_F r_I) + K_0(\lambda_F r_I)}{\Theta_{AB} I_0(\lambda_F r_0) + K_0(\lambda_F r_0)} \left( \frac{w_0}{I_0(\lambda_I r_I)} \right) \qquad (3.36)$$

The granular temperature in the inhomogeneous case is

$$T_1(r) = \begin{cases} (A_- I_0(\lambda_I r))^2 & \text{for } 0 < r \le r_I, \\ (A_+ I_0(\lambda_F r) + B_+ K_0(\lambda_F r))^2 & \text{for } r_I < r \le r_0, \end{cases} \qquad (3.37)$$

where the constants $A_-$, $A_+$ and $B_+$ are given by Eqs. (3.36), (3.35) and (3.34), respectively.

In the homogeneous case the prefactor $\lambda_I = 0$. Then the granular temperature $T_2(r)$ is

$$T_2(r) = \left( \frac{I_0(\lambda_F r)}{I_0(\lambda_F r_0)} \right)^2 T_0. \qquad (3.38)$$

Now we are interested in determining the temperature difference $\Delta T_g$ between cases 1 and 2 in the granular fluid. For this we calculate the granular temperatures at $r = r_I$. So, Eqs. (3.37) and (3.38) lead to

$$T_1(r_I) = \left( \frac{\Theta_{AB} I_0(\lambda_F r_I) + K_0(\lambda_F r_I)}{\Theta_{AB} I_0(\lambda_F r_0) + K_0(\lambda_F r_0)} \right)^2 T_0, \qquad (3.39)$$

$$T_2(r_I) = \left( \frac{I_0(\lambda_F r_I)}{I_0(\lambda_F r_0)} \right)^2 T_0. \qquad (3.40)$$

Then, the temperature difference is

$$\Delta T_g = \left[ \left( \frac{\Theta_{AB} I_0(\lambda_F r_I) + K_0(\lambda_F r_I)}{\Theta_{AB} I_0(\lambda_F r_0) + K_0(\lambda_F r_0)} \right)^2 - \left( \frac{I_0(\lambda_F r_I)}{I_0(\lambda_F r_0)} \right)^2 \right] T_0 \qquad (3.41)$$

in two dimensions.
## Solution for 3D

When $d = 3$, the solution of Eq.(3.23) is given in terms of the spherical modified Bessel functions of zero order $w_1(r) = \{i_0(\lambda r) = \sinh(\lambda r)/\lambda r, k_0(\lambda r) = e^{-\lambda r}/\lambda r\}$ [1]. The general solution in this case is

$$w_{-}(r) = A_{-}i_{0}(\lambda_{I}r) + B_{-}k_{0}(\lambda_{I}r) \quad \text{for } 0 < r \le r_{I}, \qquad (3.42)$$

and

$$w_{+}(r) = A_{+}i_{0}(\lambda_{F}r) + B_{+}k_{0}(\lambda_{F}r) \quad \text{for } r_{I} < r \le r_{0}. \qquad (3.43)$$

The function $k_0(\lambda r)$ diverges when $r \to 0$, then

$$B_{-} = 0. \qquad (3.44)$$

The constants $A_-$, $A_+$ and $B_+$, are calculated from the boundary conditions in a similar way as before:

$$A_{-} = \frac{\Theta_{AB}i_{0}(\lambda_{F}r_{I}) + k_{0}(\lambda_{F}r_{I})}{\Theta_{AB}i_{0}(\lambda_{F}r_{0}) + i_{0}(\lambda_{F}r_{0})} \left( \frac{w_{0}}{i_{0}(\lambda_{I}r_{I})} \right), \qquad (3.45)$$

$$A_{+} = \frac{w_{0}\Theta_{AB}}{\Theta_{AB}i_{0}(\lambda_{F}r_{0}) + k_{0}(\lambda_{F}r_{0})}, \qquad (3.46)$$

$$B_{+} = \frac{w_{0}}{\Theta_{AB}i_{0}(\lambda_{F}r_{0}) + k_{0}(\lambda_{F}r_{0})}, \qquad (3.47)$$

where in this case the factor $\Theta_{AB}$ is

$$\Theta_{AB} = \frac{\lambda_{F}i_{0}(\lambda_{I}r_{I})k_{1}(\lambda_{F}r_{I}) + \lambda_{I}i_{1}(\lambda_{I}r_{I})k_{0}(\lambda_{F}r_{I})}{\lambda_{F}i_{0}(\lambda_{I}r_{I})i_{1}(\lambda_{F}r_{I}) - \lambda_{I}i_{1}(\lambda_{I}r_{I})i_{0}(\lambda_{F}r_{I})}. \qquad (3.48)$$

The granular temperature in the inhomogeneous case in 3D is

$$T_{1}(r) = \begin{cases} (A_{-}i_{0}(\lambda_{I}r))^{2} & \text{for } 0 < r \le r_{I}, \\ (A_{+}i_{0}(\lambda_{F}r) + B_{+}k_{0}(\lambda_{F}r))^{2} & \text{for } r_{I} < r \le r_{0}, \end{cases} \qquad (3.49)$$

where the constants $A_-$, $A_+$ and $B_+$ are given by the Eqs.(3.45), (3.46) and (3.47).

In the homogeneous case the prefactor $\lambda_I = 0$. Then the granular temperature $T_2(r)$ is

$$T_2(r) = \left( \frac{i_0(\lambda_F r)}{i_0(\lambda_F r_0)} \right)^2 T_0. \qquad (3.50)$$

Again the temperature difference $\Delta T_g$ between case 1 and 2 is calculated at $r = r_I$,

$$T_1(r_I) = \left( \frac{\Theta_{AB}i_0(\lambda_F r_I) + k_0(\lambda_F r_I)}{\Theta_{AB}i_0(\lambda_F r_0) + k_0(\lambda_F r_0)} \right)^2 T_0, \qquad (3.51)$$
$$T_2(r_I) = \left( \frac{i_0(\lambda_F r_I)}{i_0(\lambda_F r_0)} \right)^2 T_0. \qquad (3.52)$$

Then, the temperature difference is

$$\Delta T_g = \left[ \left( \frac{\Theta_{AB}i_0(\lambda_F r_I) + k_0(\lambda_F r_I)}{\Theta_{AB}i_0(\lambda_F r_0) + k_0(\lambda_F r_0)} \right)^2 - \left( \frac{i_0(\lambda_F r_I)}{i_0(\lambda_F r_0)} \right)^2 \right] T_0 \quad (3.53)$$

in three dimensions.

### 3.1.4 Phenomenological aspects

#### Energy equipartition breakdown

<figure><img src="image_19.png" /><figcaption>**Fig. 3.2.** Ratio $\tau = T_1/T_2$ of the granular temperatures, showing non-equipartition of energy ($\tau \neq 1$) for different values of the coefficient of restitution $e$, using for example $e = 0.99, 0.90$ and $0.80$.</figcaption></figure>

Let us define the temperature ratio $\tau \equiv T_1/T_2$. In two dimension we have

$$\tau = \left( \frac{I_0(\lambda_F r_0)[\Theta_{AB}I_0(\lambda_F r_I) + K_0(\lambda_F r_I)]}{I_0(\lambda_F r_I)[\Theta_{AB}I_0(\lambda_F r_0) + K_0(\lambda_F r_0)]} \right)^2, \qquad (3.54)$$

and for three dimensions,

$$\tau = \left( \frac{i_0(\lambda_F r_0)[\Theta_{AB}i_0(\lambda_F r_I) + k_0(\lambda_F r_I)]}{i_0(\lambda_F r_I)[\Theta_{AB}i_0(\lambda_F r_0) + k_0(\lambda_F r_0)]} \right)^2 \quad (3.55)$$

since $\lambda_F > \lambda_I$ we can verify that $T_1 > T_2$, this means $\tau > 1$. So, the temperature ratio between the region with intruder and the region without intruder is not one. In the present model this lack of equipartition is due to a difference between the collisional dissipation rate related to the particle sizes. In the elastic limit $e \to 1$ the energy equipartition is restored $\tau \to 1$. In Fig. 3.2, we present the behavior of $\tau$ with the size ratio $\phi = r_I/r_F$, for different values of the coefficient $e$. The granular temperature difference increases with $\phi$ and depends on $e$. We can see that $\tau$ is nearly constant and close to unity when $e = 0.99$.
## 3.2 Thermal expansion

Granular materials are nonequilibrium systems and certainly they cannot be considered ergodic in the traditional sense. The system increases its energy as a result of external driving (e.g., vibration) while its decreases its energy by dissipation. There have been different attempts to define a statistical mechanics for granular media [66, 67, 98, 107, 175]. Recent studies suggest that the thermodynamic description proposed by Edwards [66] opens a door towards a statistical description of compact granular matter [21, 22, 165]. However, these findings are for weak driving and the generalization to stronger forcing is not evident.

The thermodynamic formulation proposed by Herrmann (see Ref. [98] for a detailed discussion) starts from the energy flux balance and the analog for the “equilibrium” is a steady state driven by the energy flux. If one allows for changes in the volume of the system the energy conservation will become $\Delta I = \Delta E_{int} + \Delta D + \Delta W$, where $\Delta I$ is the energy that was pumped into the system in a given time, $\Delta E_{int}$ is the change of “internal energy” (e.g., kinetic energy), $\Delta D$ is the energy dissipated in a given time and $\Delta W$ is the work done to change the volume. Theoretically we can derive the energy relaxation to the steady state for a driven granular medium [220]. The system can be considered to be in “equilibrium” when the excess of dissipated energy ($\Delta \tilde{D} = \Delta I - \Delta D$) should be zero [98]. Under this theoretical assumption and in the framework of the kinetic theory we proceed to calculate the thermal expansion coefficient for a granular fluid.

The change of mean energy of the system is basically due to a mechanical interaction with their external parameters (e.g., the amplitude $A_0$ and the frequency $\omega_0 = 2\pi f$ of vibration, the volume of the system $V$, and the pressure $p$). The work $W$ done to change the volume of the system from $V$ to a certain quantity $V + dV$ is equal to the change of its mean energy and its related to the mean pressure and volume by $dW = pdV + Vdp$. From the definition of granular temperature, the change of the granular temperature depends on the mean kinetic energy of the particles. A volume change $dV$ is related to a temperature change $dT_g$ by the equation of state (3.6).

We can express $V$ as a function of $T_g$ and $p$, $V = V(T_g, p)$. Thus, given infinitesimal changes in $T_g$ and $p$, we can write

$$ 
\begin{aligned}
dV &= \left(\frac{\partial V}{\partial T_g}\right)_p dT_g + \left(\frac{\partial V}{\partial p}\right)_{T_g} dp, \\
&= \alpha V dT_g - k_p V dp,
\end{aligned} 
\qquad (3.56) $$

where $\alpha$ is the thermal expansion coefficient defined as

$$ \alpha \equiv \frac{1}{V} \left(\frac{\partial V}{\partial T_g}\right)_p = -\frac{1}{n} \left(\frac{\partial n}{\partial T_g}\right)_{p,N}, \qquad (3.57) $$
and $k_p$ is the “isothermal compressibility” defined as

$$k_p \equiv -\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T_g} = \frac{1}{n} \left(\frac{\partial n}{\partial p}\right)_{T_g}. \qquad (3.58)$$

If in a first approximation we neglect the variations of the coefficients $\alpha$ and $k_p$, we can integrate Eq. (3.56) and find

$$\begin{aligned} V(T_g, p) &= V_0 \exp[\alpha \Delta T_g - k_p \Delta p], \\ &\approx V_0 [1 + \alpha \Delta T_g - k_p \Delta p]. \end{aligned} \qquad (3.59)$$

Now, under the assumption of negligible compressibility, the density changes are caused by temperature changes alone. From the temperature difference, Eqs. (3.41) and (3.53), the thermal expansion is

$$\tilde{\rho} = \rho(1 - \alpha \Delta T_g). \qquad (3.60)$$

Here, the constant density $\rho$ acts as a reference density corresponding to the reference temperature $T_0$, which can be taken to be the mean temperature in the flow. This is valid only in some average sense, when all the particles have the same density.

The thermal expansion coefficient can be derived from the equation of state (3.6) and definition (3.57). The general form of the coefficient of thermal expansion is

$$\alpha = \frac{1}{T_0} C(\nu), \qquad (3.61)$$

where $C(\nu)$ is a correction due to the density of the system. In the dilute limit $\nu \to 0$ one has $C(\nu) \to 1$, and the above expression tends to the expected value for a classical gas $\alpha = 1/T_0$. The explicit form of $C(\nu)$ is given in Appendix B.

## 3.3 Segregation forces

Buoyancy forces arise as a result of variations of density in a fluid subject to gravity. In the previous section we have introduced the change in the density of the granular fluid through the thermal expansion produced by the difference of granular temperatures calculated in Section 3. Now we propose that this density difference leads to a buoyancy force $f_b$, similar to the Archimedean force

$$f_b = \Delta\rho V_I g, \qquad (3.62)$$

where $\Delta\rho = -\alpha\rho\Delta T_g$, $V_I = \frac{\Omega_d}{d} r_I^d$ is the $d$-dimensional volume of the intruder and $g$ is the gravity field. Density variations driven by granular temperature gradients due to inelastic collisions, were observed by Ramirez et al., [201] in Molecular Dynamic
simulations, which in the presence of gravity, produces a buoyancy force driving the onset of convection cells. Recently, experimental evidence for this buoyancy-driven convection has been reported by Wildman et al., [232]. In the present model, the temperature gradient is obtained from the differences of the local density of dissipated energy between the region around the intruder and the region without intruder. The region with intruder is hotter than the region without intruder. This is a well established theoretical result shown in Section 3.1. It is important to note here that we are considering particles with equal material densities, so the buoyant force due to the material density differences $\mathbf{f}_A = (\rho_F - \rho)V_I\mathbf{g}$ does not play any role in the analysis carried out in this chapter and it will be treated in Chapter 4.

The intruder also experiences a viscous drag of the granular fluid. In the granular physics literature we find scarce studies of the forces on objects embedded inside granular flows [6, 238]. Experiments performed by Zik et al., reported measurements of the mobility/friction coefficients of a sphere dragged horizontally through a vertically vibrated granular system. They observed a linear dependence of the drag force on the sphere velocity. If the resistive force $\mathbf{f}_d$ is either linear or quadratic in the velocity, the problem admits an analytical solution. In this work the drag force $\mathbf{f}_d$ is considered to be linear in the velocity of segregation $\mathbf{u}(t)$, analogous to the Stokes' drag force.

$$ \mathbf{f}_d = -6\pi\mu r_I \mathbf{u}(t), \qquad (3.63) $$

where $\mu$ is the coefficient of viscosity of the granular fluid. The state-dependent viscosity possesses the general form [119, 125]

$$ \mu = \mu_0 \sqrt{T_g}, \qquad (3.64) $$

where the prefactor $\mu_0$ is a function of the fluid particle properties, and can be calculated using a Chapman–Enskog procedure for the solution of Boltzmann–Enskog transport equation. The explicit expressions of these prefactors are given in Appendix A.

Eq. (3.63) is assumed to be valid for the particle Reynolds numbers $Re = 2r_F\rho u/\mu$ less than unity. Calculating the settling velocity (see below) and the viscosity of the granular fluid (calculated in Appendix A), we can show that $Re \sim 0$ and that the Stokes law assumption should be valid.

Eqs. (3.62) and (3.63) express the acting forces in the segregation process

$$ \mathbf{f}_{seg} = \mathbf{f}_b + \mathbf{f}_d. \qquad (3.65) $$

Therefore, the equation of motion that governs the segregation process is

$$ \frac{\Omega_d}{d} r_I^d \rho \frac{d\mathbf{u}(t)}{dt} = \frac{\Omega_d}{d} r_I^d \Delta\rho \mathbf{g} - 6\pi\mu r_I \mathbf{u}(t). \qquad (3.66) $$
Now we suppose the granular system contained between two large parallel plates perpendicular to the gravitational field. We take the reference frame positive in the upward vertical direction. Arranging terms in Eq.(3.66) we find the following differential equation:

$$ \frac{du(t)}{dt} = \alpha \Delta T_g g - \frac{6\pi d\mu (\phi r_F)^{1-d}}{\Omega_d \rho} u(t), \quad (3.67) $$

where we have expressed the intruder's radius as function of the size ratio dependence $r_I = \phi r_F$, and the solution of this differential equation is the rise velocity of the intruder

$$ u(t) = \frac{\alpha \Delta T_g g t_0}{\phi^{1-d}} \left[ 1 - \exp \left( -\phi^{1-d} \frac{t}{t_0} \right) \right], \quad (3.68) $$

where the time-scale $t_0$ is given by

$$ t_0 = \frac{\Omega_d \rho}{6\pi d \mu r_F^{1-d}}. \quad (3.69) $$

The force balance between the drag force $f_d$ and the buoyant force $f_b$ gives the settling velocity $u_s$

$$ u_s = \frac{\alpha \Delta T_g g t_0}{\phi^{1-d}}. \quad (3.70) $$

The time-dependent intruder height $z(t)$ is

$$ z(t) = \frac{\alpha \Delta T_g g t_0}{\phi^{1-d}} \left[ t - t_0 \phi \left( 1 - \exp \left( -\phi^{1-d} \frac{t}{t_0} \right) \right) \right]. \quad (3.71) $$

To estimate the granular temperature $T_0$, we adopt the scaling relationship between the granular temperature and the experimental parameter $A_0$ and $\omega_0$ proposed by Sunthar and Kumaran for dense vibrofluidized granular systems [220]:

$$ T_0 = \frac{2\sqrt{2} m_F L (A_0 \omega_0)^2}{\pi N r_F (1 - e^2)}. \quad (3.72) $$

On a qualitative level our model satisfactorily reproduces the observed phenomenology: a large intruder migrates to the top of a vibrated bed, and the rise velocity increases with the intruder size. Solutions (3.68) and (3.71) are plotted in Figs. 3.3 and 3.4. Our results resembles the experimental intruder height time evolution described in Refs. [54, 65]. However the model cannot describe the intermittent ascent of the intruder since we calculate the mean velocities. Using the following model parameters: mass particle density of 2.7 g cm⁻³ (Aluminum), $r_F$ = 0.1 cm, $e$ = 0.95, $\nu$ = 0.7, $N$ = 5 × 10³, g = 100 cms⁻², $r_0$ = L/3 and $\Gamma$ = 1.33 we obtain that the order of magnitude of $z(t)$ (Fig. 3.4) coincides with the values reported by Cooke et al. [54].
<figure><img src="image_20.png" /><figcaption>**Fig. 3.3.** Intruder segregation velocity $u(t)$. The parameters are: mass particle density of 2.7 g cm$^{-3}$ (Aluminum), $r_F = 0.1$ cm, $e = 0.95$, $\nu = 0.7$, $N = 5 \times 10^3$, $g = 100$ cm s$^{-2}$, $r_0 = L/3$ and $\Gamma = 1.33$.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 3.4.** Intruder height time dependence $z(t)$. The parameters are: mass particle density of 2.7 g cm$^{-3}$ (Aluminum), $r_F = 0.1$ cm, $e = 0.95$, $\nu = 0.7$, $N = 5 \times 10^3$, $g = 100$ cm s$^{-2}$, $r_0 = L/3$ and $\Gamma = 1.33$. Inset: Measured intruder height (Fig. 3, Ref. [54]).</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. 3.5.** Intruder segregation velocity dependence on $\phi$. The parameters are: mass particle density of 2.7 g cm$^{-3}$ (Aluminum), $r_F$ = 0.75 cm, $e$ = 0.95, $\nu$ = 0.7, $N$ = 5 × 10$^3$, g = 100 c ms$^{-2}$, $r_0$ = L/3 and $\Gamma$ = 1.25. The experimental data points come from Ref. [65].</figcaption></figure>

From the settling velocity $u_s$ (3.70) we show explicitly the dependence on size. It is proportional to the size ratio $\phi$ and the granular temperature difference $\Delta T_g$ which also depends on the size ratio. Its agrees with the experimental fact that the larger the radius of the intruder, the faster is the ascent, reported by Duran et al. [65], The plotted solution (3.70) describes qualitatively the experimental results of Ref. [65], for $\phi > 4$, shown in Fig. 3.5. In this experiment, Duran et al., claim the experimental evidence of a segregation size threshold at $\phi_c$ = 3.3, below which the intruder does not exhibit any upward motion. Our model’s continuous aspect does not allow for the existence of this threshold. We argue that this discrepancy comes from the fact that experimental measures in this regime should be very difficult to carry out.

## 3.4 Conclusions

We derived a phenomenological continuum description for particle size segregation in granular media. We propose a buoyancy-driven segregation mechanism caused by the dissipative nature of the collisions between grains. The collisional dissipation rate naturally leads to a local temperature difference among the region around the intruder and the medium without intruder. In this model we proposed that the intruder’s presence develops a temperature gradient in the system which gives origin to a difference of densities. The granular temperature difference is due to the fact that the number of collisions on the surface increases with the size of the intruder, but
the local density of dissipated energy diminishes. So, the region around the intruder
is *hotter* than the region without intruder. From this temperature difference we can
conclude that there is a change in the density of the granular fluid. This leads to a
buoyancy force that is responsible for the intruder's upward movement.

In this work we made use of the tools of kinetic theory of gases to calculate the granular temperature. We observed a breakdown of the energy equipartition. This is in agreement with other reported experiments and models. In a certain sense our theory unifies the different aspects observed in the size segregation phenomenon. Explicit solutions of the dependence of height and velocity are calculated. The geometrical effect of a segregation threshold is not supported by our model. The intruder size dependence appears naturally in our model.

It seems that in most of the segregation experiments the granular convection is unavoidable [54, 65, 141, 146, 149, 179, 225]. It is also important to note that changes in the side walls can induce a transition to flow downward at the center of the container and upward along the boundaries [88, 139]. This situation was observed in the experiments of Knight et al., [141] where the intruder particle moves downward in the middle of a conical container. However, in spite of the convective flow that appeared in experimental setups, in the size-dependent regime there is not discernible convective flow in the center of the bed [54, 61, 64, 65, 149, 210, 211, 225], and we can conclude that convection had no influence (in this regime) on the intruder's motion.

Very recently it has been shown experimentally [232] and by computer simulations
[201] that the convection phenomenon in granular fluids comes from the effect of
spontaneous granular temperature gradients, due to the dissipative nature of the
collisions. These temperature gradient leads to a density variation. The convection
rolls are caused by buoyancy effects initiated by enhanced dissipation at the walls
and the tendency of the grains at the center to rise. So, this segregation mechanism
could be described in the hydrodynamic framework proposed in this work subject to
the appropriate boundary conditions. In the convection regime an additional drag
force should appear coupling the intruder's movement with the bulk convection
stream. Further investigation is required to deduce the forces associated with the
convection drag on the intruder and the role of the container geometry.

In this chapter we only considered the case of the size-dependence on the segrega-
tion of a single intruder in a granular medium. The interplay between the intruder's
size and material density dependence will be treated in Chapter 4.
# Appendix A. Transport coefficients

In this appendix the prefactors appearing in Eqs.(3.9) and (3.64) are derived. Using a Chapman–Enskog procedure for the solution of the Boltzmann–Enskog transport equation, the transport coefficients for nearly elastic particles have been derived in Refs. [125] and [119].

In 2d the thermal conductivity $\kappa$ is

$$ \kappa = 3nr_F \left(\frac{\pi}{m_F}\right)^{1/2} \left[1 + \frac{1}{3}\frac{1}{G} + \frac{3}{4}\left(1 + \frac{16}{9\pi}\right)G\right] T_g^{1/2}, \quad (3.73) $$

where $G$ is $\nu g_0$, $g_0$ is the 2d pair correlation function given in Eq.(3.7), and $\nu$ is the area fraction $\nu = n\pi r_F^2$. It is convenient to express Eq.(3.73) introducing the prefactor $\kappa_0$ defined as

$$ \kappa_0 \equiv 3nr_F \left(\frac{\pi}{m_F}\right)^{1/2} \left[1 + \frac{1}{3}\frac{1}{G} + \frac{3}{4}\left(1 + \frac{16}{9\pi}\right)G\right]. \quad (3.74) $$

The result (3.73) takes the form

$$ \kappa = \kappa_0 \sqrt{T_g}. \quad (3.75) $$

In 3d the thermal conductivity is

$$ \kappa = \frac{15}{8}nr_F \left(\frac{\pi}{m_F}\right)^{1/2} \left[1 + \frac{5}{24}\frac{1}{G} + \frac{6}{5}\left(1 + \frac{32}{9\pi}\right)G\right] T_g^{1/2}, \quad (3.76) $$

where $G$ is $\nu g_0$, $g_0$ is the 3D pair correlation function given in Eq.(3.8), and $\nu$ is in this case the volume fraction $\nu = 4\pi nr_F^3/3$. In 3D the prefactor $\kappa_0$ is defined as

$$ \kappa_0 \equiv \frac{15}{8}nr_F \left(\frac{\pi}{m_F}\right)^{1/2} \left[1 + \frac{5}{24}\frac{1}{G} + \frac{6}{5}\left(1 + \frac{32}{9\pi}\right)G\right]. \quad (3.77) $$

The shear viscosity $\mu$ in 2d is

$$ \mu = \frac{1}{4}nr_F(\pi m_F)^{1/2} \left[2 + \frac{1}{G} + \left(1 + \frac{8}{\pi}\right)G\right] T_g^{1/2}. \quad (3.78) $$

It is convenient to express Eq.(3.78) introducing the prefactor $\mu_0$ defined as

$$ \mu_0 = \frac{1}{4}nr_F(\pi m_F)^{1/2} \left[2 + \frac{1}{G} + \left(1 + \frac{8}{\pi}\right)G\right]. \quad (3.79) $$

So, the result (3.78) takes the form

$$ \mu = \mu_0 \sqrt{T_g}. \quad (3.80) $$
In 3d the shear viscosity is

$$ \mu = \frac{1}{3} n r_F (\pi m_F)^{1/2} \left[ 1 + \frac{5}{16} \frac{1}{G} + \frac{4}{5} \left( 1 + \frac{12}{\pi} \right) G \right] T_g^{1/2}, \quad (3.81) $$

and the prefactor $\mu_0$ in 3d is defined as

$$ \mu_0 = \frac{1}{3} n r_F (\pi m_F)^{1/2} \left[ 1 + \frac{5}{16} \frac{1}{G} + \frac{4}{5} \left( 1 + \frac{12}{\pi} \right) G \right]. \quad (3.82) $$

## Appendix B. Thermal expansion coefficient

We can consider the volume of the system as a function of the granular temperature and the pressure $V = V(T_g, p)$. A change in the granular temperature $dT_g$ and the pressure $dp$, leads to the corresponding change in the volume $dV$

$$ dV = \left(\frac{\partial V}{\partial T_g}\right)_p dT_g + \left(\frac{\partial V}{\partial p}\right)_{T_g} dp. \quad (3.83) $$

As we have supposed that the pressure of the system is more or less constant, we can approximate $dp \sim 0$. The increment of volume $dV$ with an increment of the granular temperature $dT_g$ is

$$ dV = \left(\frac{\partial V}{\partial T_g}\right)_p dT_g. \quad (3.84) $$

Thus,

$$ \frac{dV}{dT_g} = \left(\frac{\partial V}{\partial T_g}\right)_p, \quad (3.85) $$

or

$$ \left(\frac{\partial V}{\partial T_g}\right)_p = \left[\left(\frac{\partial T_g}{\partial V}\right)_p\right]^{-1}, \quad (3.86) $$

and in terms of the number density $n$, we have

$$ \left(\frac{\partial n}{\partial T_g}\right)_{p,N} = \left[\left(\frac{\partial T_g}{\partial n}\right)_{p,N}\right]^{-1}. \quad (3.87) $$

From the definition of the coefficient of thermal expansion Eq. (3.57), and from the above statement, we find

$$ \alpha = -\frac{1}{n} \left(\frac{\partial n}{\partial T_g}\right)_{p,N} = -\frac{1}{n} \left[\left(\frac{\partial T_g}{\partial n}\right)_{p,N}\right]^{-1}. \quad (3.88) $$

The partial derivative $(\partial T_g / \partial n)_{p,N}$ can be calculated from the equation of state (3.89). In 2D the equation of state is
$$p = nT_g \left[ 1 + (1 + e) \left( \frac{\nu}{1 - \nu} + \frac{9}{16} \frac{\nu^2}{(1 - \nu)^2} \right) \right]. \quad (3.89)$$

where $\nu = n\pi r_F^2$. So, an elementary calculation leads to

$$\left(\frac{\partial T_g}{\partial n}\right)_{p,N} = (\nu - 1) \frac{p}{n^2} \times \frac{\left[\frac{7}{16}(1+e) - 1\right](\nu^3 - 3\nu^2) - (1-2e)\nu + 1}{\left\{\left[\frac{7}{16}(1+e) - 1\right]\nu^2 + (1-e)\nu - 1\right\}^2} \quad (3.90)$$

From (3.88) one obtains:

$$\alpha = \frac{n}{(1 - \nu)p} \times \frac{\left\{ \left[ \frac{7}{16}(1 + e) - 1 \right] \nu^2 + (1 - e)\nu - 1 \right\}^2}{\left[ \frac{7}{16}(1 + e) - 1 \right] (\nu^3 - 3\nu^2) - (1 - 2e)\nu + 1} \quad (3.91)$$

Using the equation of state (3.89) we can express $\alpha$ in function of the granular temperature

$$\alpha = \frac{1}{T_g} \{(1 - \nu) [1 + (1 + e)G]\}^{-1} \times \frac{\left\{ \left[ \frac{7}{16}(1 + e) - 1 \right] \nu^2 + (1 - e)\nu - 1 \right\}^2}{\left[ \frac{7}{16}(1 + e) - 1 \right] (\nu^3 - 3\nu^2) - (1 - 2e)\nu + 1} \quad (3.92)$$

this is

$$\alpha = \frac{1}{T_g} C(\nu), \quad (3.93)$$

where the correction coefficient due to the density of the system is defined as

$$C(\nu) = \{(1 - \nu) [1 + (1 + e)G]\}^{-1} \times \frac{\left\{ \left[ \frac{7}{16}(1 + e) - 1 \right] \nu^2 + (1 - e)\nu - 1 \right\}^2}{\left[ \frac{7}{16}(1 + e) - 1 \right] (\nu^3 - 3\nu^2) - (1 - 2e)\nu + 1} \quad (3.94)$$

For three dimensions the equation of state is

$$p = nT_g \left[ 1 + (1 + e) \left( \frac{\nu}{1 - \nu} + \frac{9}{16} \frac{\nu^2}{(1 - \nu)^2} \left[ 3 + \frac{\nu}{2(1 - \nu)} \right] \right) \right]. \quad (3.95)$$

In a similar way we find for 3D that the coefficient of thermal expansion is

$$\alpha = \frac{1}{T_g} \{(1 - \nu)^2 [1 + 2(1 + e)G]\}^{-1} \times \frac{\{\nu^3 - (2 - e)\nu^2 + (1 - 2e)\nu - 1\}^2}{\nu^4 - 4\nu^3 - (5 - e)\nu^2 + 4e\nu + 1} \quad (3.96)$$
and the correction coefficient $C(\nu)$ in 3D is defined as

$$C(\nu) = \{(1 - \nu)^2 [1 + 2(1 + e)G]\}^{-1} \times \frac{\{\nu^3 - (2 - e)\nu^2 + (1 - 2e)\nu - 1\}^2}{\nu^4 - 4\nu^3 - (5 - e)\nu^2 + 4e\nu + 1} \quad (3.97)$$

where $\nu = 4n\pi r_F^3/3$.
# 4 The upward and downward movement

The intruder segregation dependence on size and density is investigated in the framework of the hydrodynamic model introduced in Chapter 3. We analyze the transition from the upward to downward intruder’s movement.

## Introduction

The hydrodynamic model developed in Chapter 3 refers only to systems where the intruder and the granular fluids have the same density and differ only in sizes. As already shown, the size segregation velocity can be explained by a *thermal buoyancy* mechanism. This chapter is devoted to a generalization of the precedent analysis including the interplay between the intruder size and density. Also, we address the problem of the upward to downward segregation transition and some comments related with previous experimental works.

## 4.1 Segregation dynamics: Competition between buoyancy and thermal forces

### 4.1.1 Preliminaries

If we consider an intruder particle with material density $ρ_I$ which differs from the density $ρ_F$ of the granular bed, the segregation mechanism should include the following forces:

1. The Archimedean buoyancy force $f_A$: due to the differences between the intruder material density $ρ_I$ and the bed density $ρ_F$ in presence of a gravitational field $g$,

$$f_A = (ρ_F - ρ_I)V_Ig, \quad (4.1)$$

where $V_I = \frac{\Omega_d}{d}r_I^d$ is the d-dimensional volume of an intruder with radius $r_I$, and remember that, $\Omega_d = 2\pi^{d/2}/\Gamma(d/2)$ is the surface area of a d-dimensional unit sphere.

2. The thermal buoyancy force $f_T$: arising from density variations due to the differences in the local “granular temperature” $\Delta T_g$ (Eq. (3.62), Sec. 3.3),
$$ \mathbf{f}_T = \Delta\rho_F V_I \mathbf{g}, \qquad (4.2) $$

where $\Delta\rho_F = \rho'_F - \rho_F = -\alpha\rho_F\Delta T_g$, with $\rho'_F = \rho_F(1 - \alpha\Delta T_g)$ the density variation caused by the process of thermal expansion.

3. The drag force $\mathbf{f}_d$: which is considered to be linear in the velocity of segregation $\mathbf{u}(t)$ [6, 238], and is like the Stokes' drag force (Eq. (3.63), Section 3.3),

$$ \mathbf{f}_d = -6\pi\mu r_I \mathbf{u}(t), \qquad (4.3) $$

where $\mu$ is the coefficient of viscosity (see Appendix A, Chapter 3.)

Therefore the equation of motion that governs the segregation process is

$$ \frac{\Omega_d}{d} r_I^d \rho_I \frac{\mathrm{d}\mathbf{u}(t)}{\mathrm{d}t} = \frac{\Omega_d}{d} r_I^d [\rho_I - \rho_F(1 + \alpha\Delta T_g)] \mathbf{g} - 6\pi\mu r_I \mathbf{u}(t). \qquad (4.4) $$

### 4.1.2 Time evolution and settling velocity

We take the reference frame positive in the upward vertical direction. Arranging terms in (3.66) we find the following differential equation

$$ \frac{\mathrm{d}u(t)}{\mathrm{d}t} = [\rho_F(1 + \alpha\Delta T_g) - \rho_I] \frac{\mathrm{g}}{\rho_I} - \frac{6\pi d\mu}{\Omega_d \rho_I r_I^{d-1}} u(t), \qquad (4.5) $$

and the solution is

$$ u(t) = t_0 g \left[ \frac{\rho_F}{\rho_I} (1 + \alpha \Delta T_g) - 1 \right] \left( 1 - e^{-t/t_0} \right), \qquad (4.6) $$

where the time-scale $t_0$ is

$$ t_0 = \frac{\Omega_d \rho_I r_I^{d-1}}{6\pi d \mu}. \qquad (4.7) $$

The drag force (4.3) always acts opposite to the intruder velocity. So, the intruder's upward/downward movement is exclusively due to the buoyancy forces. For our theoretical calculations we define the settling velocity (i.e, when $t/t_0 \gg 1$)

$$ u_s = t_0 g \left[ \frac{\rho_F}{\rho_I} (1 + \alpha \Delta T_g) - 1 \right]. \qquad (4.8) $$

When $u_s > 0$ the resulting movement is upward (BNP). On the other hand, if $u_s < 0$ the resulting movement is downward (RBNP). Our analysis reveals that if $(1 + \alpha\Delta T_g)\rho_F/\rho_I > 1$, the intruder ascends. The opposite occurs when $(1+\alpha\Delta T_g)\rho_F/\rho_I < 1$. When $u_s = 0$ there is neither an upward nor a downward movement.
### 4.1.3 Remarks

In Section 3.3. we have outlined a heuristic method to estimate the temperature in a granular fluid in the presence of an intruder. The temperature difference between a region with intruder and a region without intruder $\Delta T_g = T_1 - T_2$ in Eq. (4.8) contributes, together with the density ratio $\rho_F/\rho_I$, to the BNP/RBNP transition. Therefore, to complete the description we have to consider how the temperature ratio $T_1/T_2$ depends on the mass ratio $m_I/m_F$. Then, we will also need the following results from Section 3.3 (i.e., Eqs. (3.54) and (3.55)):

For two dimensions

$$ \frac{T_1}{T_2} = \left( \frac{I_0(\lambda_F r_0)[\Theta_{AB}I_0(\lambda_F r_I) + K_0(\lambda_F r_I)]}{I_0(\lambda_F r_I)[\Theta_{AB}I_0(\lambda_F r_0) + K_0(\lambda_F r_0)]} \right)^2, \quad (4.9) $$

and for three dimensions,

$$ \frac{T_1}{T_2} = \left( \frac{i_0(\lambda_F r_0)[\Theta_{AB}i_0(\lambda_F r_I) + k_0(\lambda_F r_I)]}{i_0(\lambda_F r_I)[\Theta_{AB}i_0(\lambda_F r_0) + k_0(\lambda_F r_0)]} \right)^2 \quad (4.10) $$

## 4.2 Analysis

### 4.2.1 Temperature and settling velocity

The temperatures of the region with intruder and the region without intruder are different. For $T_1/T_2 > 1$ the thermal buoyancy force favors the upward movement. When $T_1/T_2 < 1$ the thermal buoyancy force favors the downward movement.

The nonlinear form of Eqs. (4.9) and (4.10) does not allow us to calculate analytically the explicit dependence on the mass ratio $m_I/m_F$ and the size ratio $\phi = r_I/r_F$. Therefore, we shall now compute numerically the temperature ratio $T_1/T_2$ as a function of the mass ratio $m_I/m_F$ for different values of $\phi$. We set the number of particles $N = 5 \times 10^3$, the volume fraction $\nu = 0.75$, $r_0 = L/2$ and the restitution coefficient $e = 0.95$. We consider small particles with unitary mass $m_F = 1$, and we vary the intruder's mass like $m_I = x m_F$, with $x \simeq 0$ ($m_I \ll m_F$) to $x = 2$. Figure 4.1 shows the temperature ratio dependence as function of the mass ratio for two and three dimensions.

From the analysis of Fig. 4.1 we can conclude that always $T_1 > T_2$, and the upward to downward movement is basically controlled by the Archimedean buoyancy force.

Let us concentrate on the dynamics of the segregation process described by Eq.(4.5). From the settling velocity $u_s$ (Eq.(4.8)) we can analyze the dependence on size and density ratio. To estimate the granular temperature $T_0$ we adopt the same scaling relation used in Section 3.3:
<figure><img src="image_23.png" /><figcaption>Fig. 4.1. Granular temperature ratio $T_1/T_2$ as a function of the mass ratio $m_I/m_F$ for (a) two dimensions, and (b) three dimensions.</figcaption></figure>

$$T_0 = \frac{2\sqrt{2} m_F L (A_0 \omega_0)^2}{\pi N r_F (1 - e^2)}$$

and for three dimensions we estimate the granular temperature as $T_0 \sim m_F(A_0 \omega_0)^2$.

In order to calculate $u_s$ we use the following model parameters: mass particle density $\rho_F = 2.7 \text{ gcm}^{-3}$, $r_F = 0.1 \text{ cm}$, $e = 0.9$, $\nu = 0.75$, $N = 5 \times 10^3$, $g = 100 \text{ cms}^{-2}$, $r_0 = L/2$, $A_0 = 2r_F \text{ cm}$, $\omega_0 = 5.81 \sqrt{g/A_0}$ for two dimensions, and $\omega_0 = 0.7 \sqrt{g/A_0}$ for three dimensions. Figure 4.2 summarizes the results of our calculations for size ratio $\phi$ from 1 to 10 in two dimensions and three dimensions.

<figure><img src="image_24.png" /><figcaption>Fig. 4.2. Dimensionless intruder settling velocity $u_s/(A_0\omega_0)$, as a function of density ratio $\rho_I/\rho_F$ in (a) two dimensions, and (b) three dimensions.</figcaption></figure>

From these results we can observe that the segregation is rapid in systems in which small particles are denser than the intruder. For fixed values of $\phi$, when the density ratio increases the rise velocity diminishes, this implies an increment of the intruder rise time. Finally, when the intruder's density is bigger than the small particles density the intruder particle sinks $u_s < 0$. For a fixed value of the density ratio the segregation rate increases with the size ratio. These result shows the condition for the crossover from the upward to the downward intruder's motion.
### 4.2.2 Phase diagram

Now from the dependence of the segregation velocity (4.8) on the mass and size ratio we obtain the phase diagram for the BNP/RBNP transition

<figure><img src="image_25.png" /><figcaption>Fig. 4.3. Phase diagram determined for (a) two dimensions, and (b) three dimensions.</figcaption></figure>

## 4.3 Discussion

In summary, we have studied the segregation dependence on size and density for a single intruder in a fluidized bed in the framework presented in Chapter 3.

On a qualitative level our model is in agreement with the experimental phenomenology described in Refs.[2, 204]: the higher the density of the intruder the lower the tendency to rise. In Fig.(4.2) we can note that when the density of the particles increases, in the range where $ρ_I/ρ_F < 1$, the segregation velocity diminishes, so the intruder's rise time will be bigger. In general a lighter particle rises more quickly than a heavier particle of the same size in the range where $ρ_I/ρ_F < 1$. In the range where $ρ_I/ρ_F > 1$ the intruder sinks, and in this case the downward segregation velocity increases with density and size. Our model also agrees with the qualitative behavior reported in Ref.[219], where the intruder height decreases as the density ratio increases. In this case, for the range where $ρ_I/ρ_F < 1$, the height of a lighter particle will be bigger than the height of a heavier particle of the same size at the same time. This is due to the fact that the rise velocity of a lighter particle will be bigger than the rise velocity of a heavier particle of the same size.

In agreement with the theoretical results obtained in Refs.[108, 127] and the numerical simulations of Refs.[108, 219], our model predicts that in the case where the density ratio increases, the intruder particle will sink.

A direct comparison with experiments performed in Ref.[179] is not appropriate in this work since they investigate the intruder's density effect in presence of convection and interstitial air. Comparison is also difficult with the experiments
performed in Refs.[149,217], where the role of inertia and interstitial air may play
an important role. It is important to recall that the experiment performed by Shin-
brot and Muzzio [217], does not correspond to the typical experimental conditions
of the Brazil nut effect. In these experiments the intruders are placed at the surface
of a vibrated bed. They reported that the heavy intruder remained at the surface
while the lighter intruder sank. The boundary condition for an intruder in this ex-
perimental set-up is different and more complicated than the boundary condition
for an intruder immersed in the granular bed. So, in this experiment we can not
consider upward/downward transition in the sense outlined in Ref.[108].

In the experiment performed by Liffman et al., [149] they reported that the total
rise time of the intruder is inversely proportional to the density of the intruder.
This is, a priori, in contrast to our findings. In our case, if the segregation velocity
decreases with the intruder's density, then the rise time decreases. This apparent
contradictory fact can be understood from the procedure followed by Liffman et al.,
for the intruder's density variation. They studied the motion of disks placed at the
base of a vibrated granular bed. They varied the density of the disks with similar
size, drilling holes in the disks. The intruder is “transformed” from a disk to a ring.
The inertia of these two objects is different. Also, it is important to note that the
energy dissipation is different. Energy is dissipated during collision, among other
things, due to the excitation of the internal modes of the object. So, the coefficient
of restitution should be different. In terms of our model, based on the concept of
granular temperature, the density of dissipated energy should be bigger for a ring
than a disk. In this case the disk is “hotter” than a ring, and the buoyancy force
would favor the upward movement.
Notation

The following notation is used throughout the text in Chapters 5 and 6. Please note that some conventions on notation have changed.

<table><tr><td>d = 2, 3</td><td>dimension of the system</td></tr><tr><td>l</td><td>index which denotes large particles</td></tr><tr><td>s</td><td>index which denotes small particles</td></tr><tr><td>r<sub>l</sub></td><td>radius of the large particles</td></tr><tr><td>r<sub>s</sub></td><td>radius of the small particles</td></tr><tr><td>&phi;<sub>i</sub></td><td>species i packing fraction</td></tr><tr><td>&phi; = &phi;<sub>l</sub> + &phi;<sub>s</sub></td><td>mixture packing fraction</td></tr><tr><td>e</td><td>restitution coefficient</td></tr><tr><td>g</td><td>gravitational acceleration</td></tr><tr><td>g<sub>ij</sub></td><td>radial distribution function</td></tr><tr><td>K<sub>ij</sub></td><td>compressibility factor</td></tr><tr><td>p<sub>i</sub></td><td>partial pressure of species i</td></tr><tr><td>&rho;<sub>i</sub></td><td>species mass density</td></tr><tr><td>&rho;<sub>i</sub></td><td>material density of species i</td></tr><tr><td>n</td><td>total number density</td></tr><tr><td>n<sub>i</sub></td><td>number density of species i</td></tr><tr><td>c<sub>i</sub></td><td>particle velocity</td></tr><tr><td>u<sub>i</sub></td><td>mean velocity of species i</td></tr><tr><td>C<sub>i</sub></td><td>peculiar velocity</td></tr><tr><td>v<sub>i</sub></td><td>diffusion velocity of species i</td></tr><tr><td>T<sub>i</sub></td><td>species granular temperature</td></tr><tr><td>T</td><td>mixture granular temperature</td></tr><tr><td>F<sub>i</sub></td><td>external force</td></tr><tr><td>&Gamma;<sub>l</sub></td><td>momentum source term</td></tr><tr><td>f<sub>i</sub></td><td>velocity distribution function</td></tr><tr><td>D<sub>i</sub></td><td>species inelastic dissipation</td></tr><tr><td>F<sup>A</sup><sub>B</sub></td><td>Archimedean buoyancy force</td></tr><tr><td>F<sup>T</sup><sub>B</sub></td><td>thermal buoyancy force</td></tr><tr><td>F<sup>st</sup><sub>ge</sub></td><td>static compressive force</td></tr><tr><td>F<sup>dyn</sup><sub>ge</sub></td><td>dynamic tensile force</td></tr><tr><td>V<sub>i</sub></td><td>volume of a particle of species i</td></tr></table>
<figure><img src="image_26.png" /></figure>
# 5 Aspects of the kinetic theory for binary granular mixtures: The two-temperatures scenario

An overview of the kinetic theory formalism for smooth inelastic hard particles is introduced in the present chapter. We consider a binary granular mixture with unequal granular temperatures. The pair distribution function for two colliding spheres was assumed to be the product of Maxwellian velocity distributions for each species.

## Introduction and motivation

*A beautiful and correct theory can be disfigured by an approximation introduced to simplify a calculation, but there are also approximations that retain the elegance of the correct theory and provide new and deep physical insights<fnref n="1" />.*

A basis for the derivation of hydrodynamic equations and explicit expressions appearing in them is provided by the Boltzmann kinetic theory conveniently modified to account for inelastic binary collisions. The goal of this chapter is to introduce the hydrodynamic equations for a binary mixture of smooth inelastic hard particles with unequal kinetic energy. To keep the scope of this chapter manageable we will limit our discussion to the balance of kinetic energy and the collisional energy source (dissipation rate) obtained to first order in the spatial gradients of the hydrodynamic fields. We defer the discussion of the balance of momentum to Refs. [4, 83, 85] and the set of transport coefficients used in this thesis to Ref. [122, 123]. The problem of a lack of kinetic energy equipartition has been a subject of recent important discussions, we encourage the reader to consult the articles [3, 19, 20]. To derive the present kinetic theory formalism, we followed the discussion of Jenkins and Mancini [122, 123], Garzó and Dufty [84] and Barrat and Trizac [19]. A complementary formalism is also presented in [111, 112].

Before discussing the kinetic theory for inelastic particles, an important remark is necessary. In Chapters 3 and 4 we have considered the case of an intruder (impurity) in a granular fluid of identical inelastic particles. It is worth to noting that a similar problem was analyzed in Refs. [25, 212, 213] in the framework of the Boltzmann-Lorentz kinetic equation of an unforced granular fluid (homogeneous cooling state). These approaches provide a more complete description for the fluid and intruder

<footnote n="1">¹ Quoted from L. Spruch, Rev. Mod. Phys. **63** (1991) 151</footnote>
probability distribution functions. However, it is still difficult to outline the segregation dynamics from [25, 212, 213]. An important step forward was done by Jenkins and Yoon [127] based on [11, 122]. We think that in view of the agreement of [127] with the theory and Molecular Dynamics simulations carried out by Hong, Quinn and Luding [108], it is worth attempting to construct a kinetic theory framework for the segregation phenomenon.

# 5.1 Definitions and preliminaries

## 5.1.1 Hydrodynamic fields

As a mechanical model for granular fluids we consider a binary mixture of slightly inelastic, smooth particles (disks/spheres) with radii $r_i$ ($i = l, s$, where $l$ stands for large particles and $s$ for small), mass $m_i$ in two and three dimensions ($d = 2, 3$). The system contains a number $N_i$ of particles of species $i$ in a volume $V$. The volume is a constant regular shape, and no particles flow across the surface, so that the total number of particles is constant.

Mean values are calculated in terms of the single particle velocity distribution function $f_i(\mathbf{c}, \mathbf{r}, t)$ for each species. By definition $f_i(\mathbf{c}, \mathbf{r}, t)d\mathbf{c}\mathbf{d}\mathbf{r}$ is the number of particles which, at time $t$, have velocities in the interval $d\mathbf{c}$ centered at $\mathbf{c}$ and positions lying within a volume element $d\mathbf{r}$ centered at $\mathbf{r}$. We can express the information that there are a number $N_i$ of particles of species $i$ in the volume $V$ of the system through the normalization condition

$$ \int d\mathbf{c}\mathbf{d}\mathbf{r} f_i(\mathbf{c}, \mathbf{r}, t) = N_i(\mathbf{r}, t). \qquad (5.1) $$

If the particles are uniformly distributed in space, so that $f_i$ is independent of $\mathbf{r}$, then the number density $n_i(\mathbf{r}, t)$ of species $i$ is

$$ n_i(\mathbf{r}, t) = \int d\mathbf{c} f_i(\mathbf{c}, \mathbf{r}, t). \qquad (5.2) $$

The total number density $n$ is the sum over both species,

$$ n = n_l + n_s. \qquad (5.3) $$

The species mass density $\rho_i$ is defined by the product of $n_i$ and $m_i$, and the total mixture density is

$$ \rho = \rho_l + \rho_s = \rho_l\phi_l + \rho_s\phi_s, \qquad (5.4) $$

where $\rho_i$ is the material density of species $i$ and $\phi_i$ is the $d$-dimensional volume fraction for species $i$:
$$ \phi_i = \frac{\Omega_d}{d} n_i r_i^d, \qquad (5.5) $$

where $\Omega_d$ is the surface area of a $d$-dimensional unit sphere. The mean value of any quantity $\psi_i = \psi(\mathbf{c})$ of a particle species $i$, is

$$ \langle \psi_i(\mathbf{c}) \rangle = \frac{1}{n_i} \int d\mathbf{c} \psi_i(\mathbf{c}) f_i(\mathbf{c}). \qquad (5.6) $$

The mean velocity of species $i$ is $\mathbf{u}_i = \langle \mathbf{c}_i \rangle$. The mass average velocity (barycentric velocity) $\mathbf{u}$ of the mixture is defined by

$$ \mathbf{u} = \frac{1}{\rho} (\rho_l \mathbf{u}_l + \rho_s \mathbf{u}_s). \qquad (5.7) $$

The peculiar velocity $\mathbf{C}_i$ of an $i$-particle is its velocity relative to the barycentric velocity, so that

$$ \mathbf{C}_i = \mathbf{c}_i - \mathbf{u} \qquad (5.8) $$

and the diffusion velocity $\mathbf{v}_i$ of species $i$ is its relative mean motion, thus

$$ \mathbf{v}_i = \langle \mathbf{C}_i \rangle = \mathbf{u}_i - \mathbf{u}. \qquad (5.9) $$

The species granular temperature is defined proportional to the mean kinetic energy of species $i$

$$ T_i = \frac{1}{d} m_i \langle \mathbf{C}_i \cdot \mathbf{C}_i \rangle = \frac{1}{d} m_i \langle C_i^2 \rangle \qquad (5.10) $$

and the mixture temperature is

$$ T = \frac{1}{n} (n_l T_l + n_s T_s) \qquad (5.11) $$

The coefficient of restitution for collisions between particles $i$ and $j$ is denoted by $e_{ij}$, with $e_{ij} \le 1$ and $e_{ij} = e_{ji}$.

### 5.1.2 The Boltzmann–Enskog equation for a mixture

The distribution functions $f_i(\mathbf{c}_i, \mathbf{r}, t)$ for the two species are determined from the set of nonlinear Boltzmann–Enskog equations

$$ \left( \frac{\partial}{\partial t} + \mathbf{c}_i \cdot \nabla + \frac{\mathbf{F}_i}{m_i} \cdot \frac{\partial}{\partial \mathbf{c}_i} \right) f_i(\mathbf{c}_i, \mathbf{r}, t) = \sum_j J_{ij} [\mathbf{c}_i | f_i(\mathbf{c}_i), f_j(\mathbf{c}_j)], \qquad (5.12) $$

where $\mathbf{F}_i$ is the external force acting on a particle.

The Boltzmann–Enskog collision operator $J_{ij}[\mathbf{c}_i | f_i, f_j]$ describing the scattering of pairs of particles is
$$J_{ij}[\mathbf{c}_1|f_i(\mathbf{c}_1), f_j(\mathbf{c}_2)] \equiv g_{ij}r_{ij}^{d-1} \iint d\mathbf{c}_2 d\hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21})(\hat{\sigma} \cdot \mathbf{c}_{21}) \left[ \frac{1}{e_{ij}^2} f_i(\mathbf{c}'_1) f_j(\mathbf{c}'_2) - f_i(\mathbf{c}_1) f_j(\mathbf{c}_2) \right] \quad (5.13)$$

where $g_{ij}$ is the radial distribution function, $r_{ij} = r_i + r_j$, $\hat{\sigma}$ is the unit vector directed from the center of the particle of type i to the center of particle j (separated at contact by $r_{ij}$, $\Theta(x)$ is the Heaviside step function ($\Theta(x) = 0$, for $x < 0$ and $\Theta(x) = 1$, for $x > 0$). The post-collisional velocities $\mathbf{c}'_i$ are given in terms of the pre-collisional velocities $\mathbf{c}_i$ by

$$\mathbf{c}'_i = \mathbf{c}_i + M_{ji}(1 + e_{ij})(\hat{\sigma} \cdot \mathbf{c}_{ji})\hat{\sigma} \quad (5.14)$$

where $M_{ji} = m_j/m_{ij}$, $m_{ij} = m_i + m_j$, and $\mathbf{c}_{ji} = \mathbf{c}_j - \mathbf{c}_i$ is the relative velocity between particles.

Also, in the Boltzmann–Enskog collision operator we have used the Enskog assumption for dense gases

$$f_{i,j}^{(2)}(\mathbf{c}_1, \mathbf{c}_2) \rightarrow g_{ij} f_i^{(1)}(\mathbf{c}_1) f_j^{(1)}(\mathbf{c}_2) \quad (5.15)$$

for the complete pair velocity distribution function $f_{i,j}^{(2)}(\mathbf{c}_1, \mathbf{c}_2)$.

### 5.1.3 Analysis based on the Boltzmann–Enskog equation

To find the equation satisfied by $\langle \psi_i \rangle$ defined in (5.6), we multiply both sides of Eq. (5.12) by $\psi_i$ and then integrate over all velocities $\mathbf{c}$. Thus we get

$$\frac{\partial}{\partial t} \langle n_i \psi_i \rangle = -\nabla \cdot \langle n_i \mathbf{c}_i \psi_i \rangle + n_i \frac{\mathbf{F}_i}{m_i} \cdot \left\langle \frac{\partial \psi_i}{\partial \mathbf{c}_i} \right\rangle + \sum_j [\chi_{ij}(\psi_i) - \nabla \cdot \theta_{ij}(\psi_i)], \quad (5.16)$$

where $\chi_{ij}(\psi_i)$ is the collisional source integral

$$\chi_{ij}(\psi_i) = g_{ij} r_{ij}^{d-1} \iint \int d\mathbf{c}_1 d\mathbf{c}_2 d\hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21}) (\hat{\sigma} \cdot \mathbf{c}_{21}) [\psi_i' - \psi_i] \times [1 + (1/8) r_{ij}^2 (\hat{\sigma} \cdot \nabla)^2 + \dots] f_i(\mathbf{c}_1) f_j(\mathbf{c}_2) \quad (5.17)$$

and where $\theta_{ij}(\psi_i)$ is the collisional flux integral

$$\theta_{ij}(\psi_i) = \frac{1}{2} g_{ij} r_{ij}^d \iint \int d\mathbf{c}_1 d\mathbf{c}_2 d\hat{\sigma} \hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21}) (\hat{\sigma} \cdot \mathbf{c}_{21}) [\psi_i' - \psi_i] \times [1 + (1/8) r_{ij}^2 (\hat{\sigma} \cdot \nabla)^2 + \dots] f_i(\mathbf{c}_1) f_j(\mathbf{c}_2). \quad (5.18)$$

*Remark.* When $\psi_i(\mathbf{c}_i) \rightarrow \psi_i(\mathbf{C}_i)$, the balance equation may be written in the form

$$\frac{\partial}{\partial t} \langle n_i \psi_i \rangle = -\nabla \cdot \langle n_i \mathbf{c}_i \psi_i \rangle + \left\langle n_i \frac{\partial \mathbf{C}_i}{\partial t} \cdot \frac{\partial \psi_i}{\partial \mathbf{C}_i} \right\rangle + \sum_j \left[ \chi_{ij}(\psi_i) - \nabla \cdot \theta_{ij}(\psi_i) - \theta_{ij} \left( \frac{\partial \psi_i}{\partial \mathbf{C}_i} \right) : \nabla \mathbf{u} \right] \quad (5.19)$$
To investigate transport processes, we must solve the Boltzmann–Enskog equation (5.12), with given initial conditions, to obtain the velocity distribution function. Some rigorous properties of any solution of (5.12) may be obtained from the fact that in any particle collision there are dynamical quantities that are rigorously conserved. For a granular fluid the independent conserved properties are mass and momentum. The transformation (5.14) conserves momentum but, when $e < 1$, does not conserve energy. Therefore, in a binary collision the change of kinetic energy $\Delta E \neq 0$. In the lowest-order approximation we assume that the fluid has a local Maxwell–Boltzmann (Maxwellian) distribution

$$f_i(\mathbf{c}_i) \approx f_i^{(0)}(\mathbf{c}_i) = \pi^{d/2} n_i \left(\frac{m_i}{2T_i}\right)^{d/2} \exp \left[-\frac{m_i}{2T_i}(\mathbf{c}_i - \mathbf{u})^2\right] \quad (5.20)$$

## 5.2 Balance equations and hydrodynamics

### 5.2.1 Energy balance

As a pedagogical task we now apply the formalism introduced in the previous section to derive the balance equations for the mixture temperature.

**Balance equation for species temperatures**

Since the granular temperature (5.10) is a function of $C_i$, the balance law (5.19) must be employed. Then

$$\begin{aligned} \frac{\partial}{\partial t} \left\langle n_i \left( \frac{1}{2} m_i C_i^2 \right) \right\rangle = & - \nabla \cdot \left\langle n_i \mathbf{c}_i \left( \frac{1}{2} m_i C_i^2 \right) \right\rangle + \left\langle n_i \frac{\partial \mathbf{C}_i}{\partial t} \cdot \frac{\partial \left( \frac{1}{2} m_i C_i^2 \right)}{\partial \mathbf{C}_i} \right\rangle \\ & + \sum_j \left[ \chi_{ij} \left( \frac{1}{2} m_i C_i^2 \right) \nabla \cdot \theta_{ij} \left( \frac{1}{2} m_i C_i^2 \right) \right. \\ & \left. - \theta_{ij} \left( \frac{\partial \left( \frac{1}{2} m_i C_i^2 \right)}{\partial \mathbf{C}_i} \right) : \nabla \mathbf{u} \right]. \end{aligned} \quad (5.21)$$

From the left hand of the above equation, we have $\langle \frac{1}{2} n_i m_i C_i^2 \rangle = \frac{1}{2} n_i m_i \langle C_i^2 \rangle$, since $n_i$ is independent of the velocity (cf. Eq.(5.2)), and from the definition of the granular temperature (5.10) we obtain

$$\frac{\partial}{\partial t} \left\langle \frac{1}{2} n_i m_i C_i^2 \right\rangle = \frac{d}{2} \frac{\partial}{\partial t} (n_i T_i) \quad (5.22)$$

Since,
$$ 
\begin{aligned}
\left\langle \frac{1}{2} n_i m_i \mathbf{c}_i C_i^2 \right\rangle &= \frac{1}{2} \left[ \langle n_i m_i \mathbf{C}_i C_i^2 \rangle + \langle n_i m_i C_i^2 \mathbf{u} \rangle \right] \\
&= \frac{1}{2} \left[ \langle \rho_i \mathbf{C}_i C_i^2 \rangle + d n_i T_i \mathbf{u} \right] \\
&= \frac{1}{2} \left[ \langle \rho_i \mathbf{C}_i C_i^2 \rangle + d n_i T_i (\mathbf{u}_i - \mathbf{v}_i) \right],
\end{aligned}
\quad (5.23) $$

we can rewrite the first term at the right hand of Eq.(5.35) as

$$ \nabla \cdot \left\langle \frac{1}{2} n_i m_i \mathbf{c}_i C_i^2 \right\rangle = \nabla \cdot \left( \frac{1}{2} \left[ \langle \rho_i \mathbf{C}_i C_i^2 \rangle + d n_i T_i \mathbf{u}_i - d n_i T_i \mathbf{v}_i \right] \right). \quad (5.24) $$

From the balance of linear momentum (see Eq.(5.37) below) we have

$$ \frac{\partial \mathbf{C}_i}{\partial t} = \frac{\mathbf{F}_i}{m_i} - (\mathbf{C}_i \cdot \nabla)\mathbf{u} + \frac{1}{\rho}(\nabla \cdot \hat{\mathbf{P}} - n\mathbf{F}), \quad (5.25) $$

where the total pressure tensor $\hat{\mathbf{P}}$ is the sum of the conductive pressure tensors $\hat{\mathbf{P}}_i$ of each species:

$$ \hat{\mathbf{P}}_i \equiv \langle \rho_i \mathbf{C}_i \mathbf{C}_i \rangle + \sum_j \bar{\theta}_{ij} (m_i \mathbf{C}_i) \quad (5.26) $$

So, from (5.25) and taking into account that $\frac{\partial}{\partial \mathbf{C}_i} (\frac{1}{2} m_i C_i^2) = m_i \mathbf{C}_i$, we obtain the following expression

$$ \left\langle n_i \frac{\partial \mathbf{C}_i}{\partial t} \cdot \frac{\partial}{\partial \mathbf{C}_i} \left( \frac{1}{2} m_i C_i^2 \right) \right\rangle = n_i \mathbf{F}_i \cdot \mathbf{v}_i + \frac{\rho_i \mathbf{v}_i}{\rho} \cdot \left[ \nabla \cdot \hat{\mathbf{P}} - n \mathbf{F} \right] - \rho_i \langle \mathbf{C}_i \cdot (\mathbf{C}_i \cdot \nabla) \mathbf{u} \rangle. \quad (5.27) $$

We define the species kinetic energy flux by

$$ \mathbf{Q}_i \equiv -\frac{d}{2} n_i T_i \mathbf{v}_i + \frac{1}{2} \langle \rho_i \mathbf{C}_i C_i^2 \rangle + \sum_j \mathbf{q}_{ij}, \quad (5.28) $$

where

$$ \mathbf{q}_{ij} \equiv \bar{\theta}_{ij} \left( \frac{1}{2} m_i C_i^2 \right), \quad (5.29) $$

and the species inelastic dissipation by

$$ \mathcal{D}_i \equiv - \sum_j \chi_{ij} \left( \frac{1}{2} m_i C_i^2 \right) \quad (5.30) $$

Using (5.29) and (5.30), the final form of the species kinetic balance equations is then obtained after a few straightforward steps:

$$ \frac{d}{2} \frac{\partial}{\partial t} (n_i T_i) = - \nabla \cdot \left( \frac{d}{2} n_i T_i \mathbf{u}_i + \mathbf{Q}_i \right) - \hat{\mathbf{P}}_i : \nabla \mathbf{u} + n_i \mathbf{F}_i \cdot \mathbf{v}_i + \frac{\rho_i \mathbf{v}_i}{\rho} \cdot \left( \nabla \cdot \hat{\mathbf{P}} - n \mathbf{F} \right) + \mathcal{D}_i \quad (5.31) $$
Balance equation for the mixture temperature

Since $\rho_l v_l + \rho_s v_s = 0$, the sum of the species kinetic balance equations (5.31) results:

$$ 
\begin{aligned}
\frac{d}{2} \frac{\partial}{\partial t} (n_l T_l + n_s T_s) = & -\frac{d}{2} \nabla \cdot [\mathbf{u} (n_l T_l + n_s T_s)] \\
& - \nabla \cdot \left[ \frac{d}{2} (n_l T_l \mathbf{v}_l + n_s T_s \mathbf{v}_s) + (\mathbf{Q}_l + \mathbf{Q}_s) \right] \\
& - \hat{\mathbf{P}} : \nabla \mathbf{u} + (n_l \mathbf{F}_l \cdot \mathbf{v}_l + n_s \mathbf{F}_s \cdot \mathbf{v}_s) + \mathcal{D},
\end{aligned}
\quad (5.32) $$

where

$$ \mathcal{D} \equiv \mathcal{D}_l + \mathcal{D}_s. \quad (5.33) $$

We define the mixture energy flux by

$$ \mathbf{q} \equiv \sum_i (n_i T_i \mathbf{v}_i + \mathbf{Q}_i), \quad (5.34) $$

After some algebraic manipulations, the balance equation for the temperature of the mixture

$$ \frac{dn}{2} \frac{\mathrm{D}T}{\mathrm{D}t} = -\nabla \cdot \mathbf{q} + \frac{n}{2} T \nabla \cdot (n_l \mathbf{v}_l + n_s \mathbf{v}_s) - \hat{\mathbf{P}} : \nabla \mathbf{u} + (n_l \mathbf{v}_l \cdot \mathbf{F}_l + n_s \mathbf{v}_s \cdot \mathbf{F}_s) + \mathcal{D}, \quad (5.35) $$
with $\frac{\mathrm{D}}{\mathrm{D}t} \equiv \frac{\partial}{\partial t} + (\mathbf{u} \cdot \nabla)$.

### 5.2.2 Mass and momentum balance

We could, similarly go on to use (5.16) to derive the hydrodynamic equation for mass and momentum balance, but we shall not do this here. They are summarized below.

The diffusion equation for species $i$ is obtained by taking $\psi_i = m_i$ in Eq. (5.16). The equation of continuity for the mixture results when the species mass balances are summed:

$$ \frac{D\rho}{Dt} = -\rho \nabla \cdot \mathbf{u}. \quad (5.36) $$

The balance of linear momentum for the mixture is obtained on the same way by taking $\psi_i = m_i c_i$ in Eq. (5.16):

$$ \rho \frac{D\mathbf{u}}{Dt} = -\nabla \cdot \hat{\mathbf{P}} + n_l \mathbf{F}_l + n_s \mathbf{F}_s. \quad (5.37) $$

The balance equations (5.35), (5.36) and (5.37) are rigorous consequences of the Boltzmann–Enskog equation (5.16). Nevertheless, it is clear that, in order to obtain from them practical hydrodynamic equations, one must find explicit expressions for the collisional sources $\xi_{ij}(\psi_i)$ and the collisional fluxes $\theta_{ij}(\psi_i)$. For calculating these quantities the velocity distribution functions are needed. Hydrodynamic equations can thus be obtained to various orders of approximation. Details are discussed in the references [5, 12, 122, 123, 234].
## 5.3 Inelastic dissipation and temperature ratio

To conclude this chapter, we outline the calculation of the species i inelastic dissipation and the temperature ratio $T_l/T_s$, following Barrat and Trizac [19].

We consider that there is not net flow ($\mathbf{u} = 0$) and we retain only the first term in the Taylor expansion of the collisional source integral (5.17). On the other hand, we assume that $T_l \neq T_s$.

We proceed to integrate the following equation:

$$ \mathcal{D}_i = -\sum_j g_{ij} r_{ij}^{d-1} \frac{m_i}{2} \iint \int d\mathbf{c}_1 d\mathbf{c}_2 d\hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21}) (\hat{\sigma} \cdot \mathbf{c}_{21}) f_i(\mathbf{c}_1) f_j(\mathbf{c}_2) (c_i''^2 - c_i^2), \quad (5.38) $$

where $\mathbf{c}_i'' = \mathbf{c}_i - M_{ji}(1 + e_{ij})(\hat{\sigma} \cdot \mathbf{c}_{ji})\hat{\sigma} \Rightarrow c_i''^2 - c_i^2 = M_{ji}^2(1 + e_{ij})^2(\hat{\sigma} \cdot \mathbf{c}_{ji})^2 - 2M_{ji}(1 + e_{ij})(\hat{\sigma} \cdot \mathbf{c}_{ji})(\hat{\sigma} \cdot \mathbf{c}_i)$ and substituting in (5.38) we can define the following two integrals:

$$ I_{ij}^{[1]} \equiv g_{ij}(1+e_{ij})^2 r_{ij}^{d-1} M_{ji}^2 \frac{m_i}{2} \iint \int d\mathbf{c}_1 d\mathbf{c}_2 d\hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21}) (\hat{\sigma} \cdot \mathbf{c}_{21})^3 f_i(\mathbf{c}_1) f_j(\mathbf{c}_2) \quad (5.39) $$

$$ I_{ij}^{[2]} \equiv -g_{ij}(1+e_{ij})r_{ij}^{d-1} m_i M_{ji} \iint \int d\mathbf{c}_1 d\mathbf{c}_2 d\hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21}) (\hat{\sigma} \cdot \mathbf{c}_{21})^2 (\hat{\sigma} \cdot \mathbf{c}_1) f_i(\mathbf{c}_1) f_j(\mathbf{c}_2) \quad (5.40) $$

To integrate (5.39) and (5.40) over $\hat{\sigma}$, we need the angular integral

$$ \begin{aligned} \beta_n &\equiv \int d\hat{\sigma} \Theta(\hat{\sigma} \cdot \hat{\mathbf{c}}_{21}) (\hat{\sigma} \cdot \hat{\mathbf{c}}_{21})^n \\ &= \frac{1}{2} \Omega_d \frac{\int d\hat{\sigma} \Theta(\cos \theta) (\cos \theta)^n}{\int d\hat{\sigma} \Theta(\cos \theta)} \\ &= \frac{1}{2} \Omega_d \frac{\int_0^{\pi/2} d\theta (\sin \theta)^{d-2} (\cos \theta)^n}{\int_0^{\pi/2} d\theta (\sin \theta)^{d-2}} \\ &= \pi^{\frac{d-1}{2}} \frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n+d}{2})} \end{aligned} \quad (5.41) $$

where $\hat{\mathbf{c}}_{21} = \mathbf{c}_{21}/|\mathbf{c}_{21}|$, $\Omega_d = 2\pi^{d/2}/\Gamma(d/2)$ is the surface area of a d-dimensional unit sphere and $\Gamma(z)$ is the gamma function ($\Gamma(n) \equiv \int_0^\infty dx e^{-x} x^{n-1}$).

From (5.41) it follows that

$$ \int d\hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21}) (\hat{\sigma} \cdot \mathbf{c}_{21})^3 = \frac{\pi^{\frac{d-1}{2}}}{\Gamma(\frac{3+d}{2})} c_{21}^3, \quad (5.42) $$

$$ \int d\hat{\sigma} \Theta(\hat{\sigma} \cdot \mathbf{c}_{21}) (\hat{\sigma} \cdot \mathbf{c}_{21})^2 (\hat{\sigma} \cdot \mathbf{c}_1) = \frac{\pi^{\frac{d-1}{2}}}{\Gamma(\frac{3+d}{2})} (\mathbf{c}_1 \cdot \mathbf{c}_2) c_{21}. \quad (5.43) $$

Putting (5.42) and (5.43) into (5.39) and (5.40) respectively, we find
$$I_{ij}^{[1]} = g_{ij}(1 + e_{ij})^2 r_{ij}^{d-1} M_{ji}^2 \frac{m_i}{2} \frac{\pi^{\frac{d-1}{2}}}{\Gamma(\frac{3+d}{2})} \iint d\mathbf{c}_1 d\mathbf{c}_2 f_i(\mathbf{c}_1) f_j(\mathbf{c}_2) c_{21}^3$$

$$I_{ij}^{[2]} = -g_{ij}(1 + e_{ij}) r_{ij}^{d-1} m_i M_{ji} \frac{\pi^{\frac{d-1}{2}}}{\Gamma(\frac{3+d}{2})} \iint d\mathbf{c}_1 d\mathbf{c}_2 f_i(\mathbf{c}_1) f_j(\mathbf{c}_2) (\mathbf{c}_1 \cdot \mathbf{c}_2) c_{21}$$

Writing $\mathbf{c}_1 = M_{ji}\mathbf{c}_{12} + M_{ij}\mathbf{c}_1 + M_{ji}\mathbf{c}_2 = M_{ji}\mathbf{c}_{12} + \mathbf{C}_{ij}$ and rearranging terms we can express (5.38) as

$$\begin{aligned} \mathcal{D}_i = & -g_{ii} r_{ii}^{d-1} (1 - e_{ii}^2) \frac{m_i}{4} \frac{\pi^{\frac{d-1}{2}}}{\Gamma(\frac{3+d}{2})} \int d\mathbf{c}_1 d\mathbf{c}_2 f_i(\mathbf{c}_1) f_i(\mathbf{c}_2) c_{21}^3 \\ & -g_{ik} r_{ik}^{d-1} m_i \frac{\pi^{\frac{d-1}{2}}}{\Gamma(\frac{3+d}{2})} \int d\mathbf{c}_1 d\mathbf{c}_2 f_i(\mathbf{c}_1) f_k(\mathbf{c}_2) \\ & \times [M_{ki}^2 (1 - e_{ik}^2) c_{21}^3 + 2M_{ki}(1 + e_{ik}) c_{21} (\mathbf{c}_{21} \cdot \mathbf{C}_{ki})] \end{aligned}$$

with $k \neq i$ and integrating at the Maxwell-Boltzmann order we obtain (see Eq. (10) in Ref. [19]):

$$\begin{aligned} \mathcal{D}_i = & \sum_j \frac{\pi^{\frac{d-1}{2}}}{d\Gamma(d/2)} g_{ij} r_{ij}^{d-1} m_i n_i n_j M_{ji} \left[ M_{ji} (1 - e_{ij}^2) \left( \frac{2T_i}{m_i} + \frac{2T_j}{m_j} \right) \right. \\ & \left. + 4(1 + e_{ij}) \frac{T_i - T_j}{m_i + m_j} \right] \left( \frac{2T_i}{m_i} + \frac{2T_j}{m_j} \right)^{1/2} \end{aligned} \quad (5.44)$$

We can split (5.44) into two terms:

1. The inter-species collisional dissipation rate $\mathcal{D}_i^I$

$$\mathcal{D}_i^I := \frac{\sqrt{2d}}{\sqrt{\pi}\Omega_d} \sum_j g_{ij} R_{ij}^d (1 + R_{ji})^{d-1} M_{ji}^2 (1 - e_{ij}^2) \frac{\phi_i \phi_j T_i^{3/2}}{r_i^{d+1} m_i^{1/2}} \left(1 + \frac{m_i T_j}{m_j T_i}\right)^{3/2} \quad (5.45)$$

2. The exchange collisional dissipation rate $\mathcal{D}_i^E$

$$\mathcal{D}_i^E := \frac{\sqrt{32d}}{\sqrt{\pi}\Omega_d} g_{ik} R_{ik}^d (1+R_{ki})^{d-1} M_{ik} M_{ki} (1+e_{ik}) \frac{\phi_i \phi_k T_i^{3/2}}{r_i^{d+1} m_i^{1/2}} \left(1-\frac{T_k}{T_i}\right) \left(1+\frac{m_i T_k}{m_k T_i}\right)^{1/2} \quad (5.46)$$

Hence, we could assert that the exchange term $\mathcal{D}_i^E$ is a clear consequence of the non-equipartition assumption. Note that $\sum_{i=l,s} \mathcal{D}_i^E = 0$. On the other hand, with the assumption of equipartition of granular energy ($T_l = T_s = T$) $\mathcal{D}_i^E = 0$ and (5.45) reduces to [122, 234]:

$$\mathcal{D}_i = \frac{\sqrt{2d}}{\sqrt{\pi}\Omega_d} \sum_j g_{ij} R_{ij}^d (1 + R_{ji})^{d-1} M_{ji}^2 (1 - e_{ij}^2) \frac{\phi_i \phi_j}{m_i^{1/2} r_i^{d+1}} \left(\frac{T}{M_{ji}}\right)^{3/2} \quad (5.47)$$
We can conclude that physically the lack of energy equipartition is determined by the different dissipation rates (5.45) and (5.46).

Now we sketch an important result that we shall use in Chapter 6.

The way how the energy is injected into the system to the two species determines together with the dissipation rates the functional form of the temperature ratio.

Let us consider the case of an uniformly heated granular system. This idealized situation has been used to model rapid granular flows [223, 224]. Barrat and Trizac [19] have recently considered the case of a homogeneous heated binary granular mixture to address the problem of species with unequal granular temperatures.

To study this system we start from the stochastic equation of motion for a particle $m_i dc_i/dt = F_i^D + m_i \hat{\xi}_i$. In this equation $F_i^D$ is the force due to inelastic collisions and $\hat{\xi}_i$ is the random acceleration due to external forcing, which is assumed to be Gaussian white noise and uncorrelated for different particles (i.e., $\hat{\xi}_{i\alpha}(t)\hat{\xi}_{j\beta}(t') = \xi_0^2 \delta_{ij}\delta_{\alpha\beta}\delta(t-t')$, where Greek indices denote Cartesian components and $\xi_0^2$ is the strength of the correlation which is proportional to the rate of the energy input per unit mass. The overline indicates an average over the noise source). The validity of this equations are widely discussed in references [223] and [224]. When we consider the associated forcing term in the Boltzmann–Enskog equation, the balance equation for species temperatures satisfies $m_i \xi_0^2 = D_i$ (in the stationary state). Using this relation it is straightforward to derive the corresponding equation for the temperature ratio $T_l/T_s$ [19]:

$$C_1 \left(\frac{T_l}{T_s}\right)^{3/2} + C_2 \left(1 + \frac{m_s T_l}{m_l T_s}\right)^{3/2} + C_3 \left(1 + \frac{m_s T_l}{m_l T_s}\right)^{1/2} \left(\frac{T_l}{T_s} - 1\right) + C_4 = 0, \quad (5.48)$$

with

$$C_1 = 2^{d-1}(1 - e^2)R_{sl}^d \phi_l g_{ll} \left(\frac{m_s}{m_l}\right)^{3/2},$$

$$C_2 = \sqrt{2}(1 - e^2)(1 + R_{sl})^{d-1} (\phi_s M_{sl}^2 - \phi_l R_{sl}^d M_{ls}^2) g_{ls},$$

$$C_3 = 2\sqrt{2}(1 + e)(1 + R_{sl})^{d-1} M_{sl} (\phi_s M_{sl} - \phi_l R_{sl}^d M_{ls}) g_{ls},$$

$$C_4 = -2^{d-1}(1 - e^2)R_{sl}^d \phi_s g_{ss}.$$

It is important to note that this model is quite good to compare the values measured in the experiments reported in [75, 229].

We can suppose that this theoretical construction could describes (at least as a first order approximation) the bulk of a vibrated granular system. Under this assumption any further analysis should be restricted to regions far from the top and bottom boundaries. A heuristic justification of this assumption can be found
in the recent computer simulations carried out by Paolotti et al. [194], where their
numerical data is in agreement (in the bulk) with the theory of Barrat and Trizac
[19]. We will show in Chapter 6 that this assumption is quite good for a systematic
description of the BNP/RBNP transition.
<figure><img src="image_27.png" /></figure>
# 6 A novel mechanism: Competition between buoyancy and geometric forces

Starting from the hydrodynamic equations of binary granular mixtures, we derive an evolution equation for the relative velocity of the intruders, which is shown to be coupled to the inertia of the smaller particles. The onset of Brazil-nut segregation is explained as a competition between the buoyancy and geometric forces: the Archimedean buoyancy force, a buoyancy force due to the difference between the energies of two granular species, and two *geometric* forces, one *compressive* and the other-one *tensile* in nature, due to the *size-difference*. We show that inelastic dissipation strongly affects the phase diagram of the Brazil nut phenomenon.

## Introduction

Following the ideas introduced in Chapter 5 we investigate the Brazil-nut segregation in a dry fluidized granular mixture in the absence of bulk convection. The purpose of this chapter is three-fold: firstly, to derive a time-evolution equation for the relative velocity of a single intruder, taking into account the non-equipartition of granular energy; secondly, to explain the driving mechanism for Brazil-nut segregation in terms of the *buoyancy* and *geometric* forces. Lastly, based on a simple model for energy non-equipartition, we will show how the inelastic dissipation determines the regimes of BNP and RBNP.

## 6.1 Hydrodynamics of granular mixtures

### 6.1.1 Preliminaries

The validity of the hydrodynamic approach even in the dense granular flows has recently been justified via the comparison of theory with various experiments[159]—here one has to be careful in choosing the appropriate constitutive model for pressure, viscosity, dissipation, etc. The constitutive model that we have used has been validated by performing MD simulations of binary mixtures[4].

We consider a binary mixture of slightly inelastic, smooth particles (disks/spheres) with radii $r_i$ ($i = l, s$, where index $l$ stands for large and $s$ for small), mass $m_i$ and number density $n_i$. The species mass density is $\rho_i(\mathbf{x}, t) = m_i n_i = \rho_i \phi_i$, where $\rho_i$ is the material density of species $i$ and $\phi_i$ is its volume fraction. The total mass-density, $\rho(\mathbf{x}, t)$, and the total number-density, $n(\mathbf{x}, t)$, are just the sums over their respec-
tive species values. The dissipative nature of particle collisions is taken into account through the normal coefficient of restitution $e_{ij}$, with $e_{ij} = e_{ji}$ and $0 \le e_{ij} \le 1$.

### 6.1.2 Momentum balance

Assuming unidirectional flow ($\mathbf{u}_i = (0, u_i(y, t), 0)$; $\partial/\partial x = 0$, $\partial/\partial z = 0$ and $\partial/\partial y \neq 0$) and neglecting viscous stresses, the momentum balance equation for species $l$ and $s$ can be written as [122]

$$ \varrho_l \frac{\partial u_l}{\partial t} = -\frac{\partial p_l}{\partial y} - \varrho_l g + \Gamma_l, \qquad (6.1) $$

$$ \varrho_s \frac{\partial u_s}{\partial t} = -\frac{\partial p_s}{\partial y} - \varrho_s g + \Gamma_s. \qquad (6.2) $$

(Note that the mass balance equations are identically satisfied for unidirectional flows.) Here $p_i$ is the partial pressure of species $i$, and $g$ the gravitational acceleration acting along the negative $y$-axis; $\Gamma_i$ is the momentum source term which arises solely due to the interactions between unlike particles and $\sum_{i=l,s} \Gamma_i = 0$ [122]. The assumption of negligible viscous stresses is justified if there is no overall mean flow in the system, or if the spatial variation of $u_i(y, t)$ is small.

To obtain constitutive relations for partial pressures we take into account the breakdown of equipartition of energy between the two species (in the equation of state) as found in many recent theoretical and numerical studies [4, 19, 122] and also confirmed in vibrofluidized experiments [75]. We assume that the single particle velocity distribution function of species $i$ is a Maxwellian,

$$ f_i(\mathbf{c}, \mathbf{z}, t) = n_i \left( \frac{m_i}{2\pi T_i} \right)^{d/2} \exp \left( -\frac{m_i (\mathbf{c} - \mathbf{u}_i)^2}{2T_i} \right), \qquad (6.3) $$

at its own granular temperature $T_i$, where $T_i = \frac{m_i}{d} < \mathbf{C}_i \cdot \mathbf{C}_i >$, with $d = 2$ and 3 for disks and spheres, respectively, $\mathbf{C}_i = \mathbf{c}_i - \mathbf{u}_i$ being the peculiar velocity, $\mathbf{c}_i$ the instantaneous particle velocity and $\mathbf{u}_i = \varrho_i^{-1} \sum_{i=l,s} \varrho_i \mathbf{u}_i$ the mixture velocity.

The equation of state for the partial pressure of species $i$ can then be written as [122]:

$$ p_i = n_i Z_i T_i, \qquad (6.4) $$

where

$$ Z_i := (1 + \sum_{j=l,s} K_{ij}) \qquad (6.5) $$

is the compressibility factor of species $i$, and

$$ K_{ij} := \phi_j g_{ij} (1 + R_{ij})^d / 2, \qquad (6.6) $$
with $g_{ij}$ being the radial distribution function and $R_{ij} = r_i/r_j$ the size-ratio.

The radial distribution functions for disks is [226]

$$g_{ij} = \frac{1}{1-\phi} + \frac{9 \phi_l R_{il} + \phi_s R_{is}}{8 (1 + R_{ij})(1-\phi)^2},$$

and for spheres [166]

$$g_{ij} = \frac{1}{1-\phi} + \frac{\phi_l R_{il} + \phi_s R_{is}}{(1+R_{ij})(1-\phi)^2} \left[ 3 + \frac{2\phi_l R_{il} + \phi_s R_{is}}{(1+R_{ij})(1-\phi)} \right].$$

Note that the $K_{ij}$ are related to the collisional component of the partial pressure, and $Z_i \to 1$ in the dilute limit $\phi \to 0$. We shall return back to discuss more about the compressibility factor later.

An explicit expression for the momentum source term, $\Gamma_l$, can be obtained using the Maxwellian velocity distribution function [122].

$$\Gamma_l = n_l K_{ls} T \left[ \left( \frac{m_s - m_l}{m_{ls}} \right) \frac{\partial}{\partial y} (\ln T) + \frac{\partial}{\partial y} \left[ \ln \left( \frac{n_l}{n_s} \right) \right] + \frac{4}{r_{ls}} \left( \frac{2m_l m_s}{\pi m_{ls} T} \right)^{1/2} (u_s - u_l) \right], \quad (6.7)$$

where $T := n^{-1} \sum_{i=l,s} n_i T_i = \sum_{i=l,s} \xi_i T_i$ is the mixture granular energy, $\xi_i := n_i/n$ the number-fraction of species $i$, $m_{ls} := m_l + m_s$ and $r_{ls} := r_l + r_s$.

## 6.2 Evolution equation for the rise velocity of the intruder

Subtracting equations (6.1) and (6.2) we obtain the following

$$\varrho_l \frac{\partial u_l^r}{\partial t} = -\frac{\partial p_l}{\partial y} + \frac{\varrho_l}{\varrho_s} \frac{\partial p_s}{\partial y} \left(1 + \frac{\varrho_l}{\varrho_s}\right) \Gamma_l \quad (6.8)$$

where $u_l^r := u_l - u_s$ is the relative velocity of the larger particles. Now we need expressions for $\partial p_l/\partial y$ and $\partial p_s/\partial y$.

From the equation of state (6.4) and the relation for partial pressures,

$$\frac{p_l}{n_l Z_l T_l} = \frac{p_s}{n_s Z_s T_s}, \quad (6.9)$$

we obtain the following expression for $\partial_z p_l$

$$\frac{\partial p_l}{\partial y} = \frac{n_l}{n_s} \left( \frac{Z_l}{Z_s} \frac{T_l}{T_s} \right) \frac{\partial p_s}{\partial y} - p_l \frac{\partial}{\partial y} \left[ \ln \left( \frac{p_s}{p_l} \right) \right], \quad (6.10)$$

and relating $\partial p_s/\partial y$ from (6.2) we derive the following evolution equation for $u_l^r$:

$$\varrho_l \frac{\partial u_l^r}{\partial t} = n_l \left[ m_s \left( \frac{Z_l}{Z_s} \frac{T_l}{T_s} \right) - m_l \right] g + \left[ 1 + \frac{p_l}{p_s} \right] \Gamma_l + p_l \frac{\partial}{\partial y} \left[ \ln \left( \frac{p_s}{p_l} \right) \right] - \varrho_s \left( \frac{\varrho_l}{\varrho_s} - \frac{p_l}{p_s} \right) \frac{\partial u_s}{\partial t}. \quad (6.11)$$
With additional assumptions of weak gradients in species number densities and granular energy, and retaining terms of the same order in the single intruder limit ($n_l \ll n_s$), the evolution equation can be considerably simplified to

$$m_l \frac{du_l^r}{dt} = \left[ m_s \left( \frac{Z_l}{Z_s} \frac{T_l}{T_s} \right) - m_l \right] g - \frac{4K_{ls}T}{r_{ls}} \left( \frac{2m_l m_s}{\pi m_{ls} T} \right)^{1/2} u_l^r + \left[ m_s \left( \frac{Z_l}{Z_s} \frac{T_l}{T_s} \right) - m_l \right] \frac{du_s}{dt}, \quad (6.12)$$

This is our time-evolution equation for the relative velocity of a single intruder: the first term on the right hand side is the net gravitational force acting on the intruder, the second term is a ‘Stokesian-like’ drag force and the third term represents a weighted *coupling* with the inertia of the smaller particles. Thus, in typical situations where one can neglect the last term (e.g. if the intruder is much heavier than the smaller particles), we end up with the familiar evolution equation where the inertia of the intruder is being balanced by the net gravitational force and the drag force. Only in this case, the interplay between the gravitational and drag force will eventually decide whether the intruder rises or sinks. In this analysis, we are only interested in the *steady-state* solution of the above equation, and we shall see below that the coupling term does not influence the *onset* of segregation. Neglecting transient effects, the steady relative velocity of the intruder can be obtained from

$$u_l^r = \frac{r_{ls}g}{4K_{ls}} \left( \frac{\pi m_{ls}}{2m_l m_s T} \right)^{1/2} \left[ m_s \left( \frac{Z_l}{Z_s} \frac{T_l}{T_s} \right) - m_l \right]. \quad (6.13)$$

Setting this relative velocity to zero, we obtain the criterion for the *transition* from BNP to RBNP:

$$m_s \left( \frac{Z_l}{Z_s} \frac{T_l}{T_s} \right) - m_l = 0, \quad (6.14)$$

which agrees with the expression of Jenkins and Yoon[127] for the case of equal granular energies ($T_l = T_s$), i.e.,

$$m_s \frac{Z_l}{Z_s} - m_l = 0, \quad (6.15)$$

As such, it is not evident from (6.14) *what the driving mechanism for segregation is*. Thus, we need to answer several questions. Can we recast the segregation criterion in terms of the well-known Archimedean and thermal buoyancy forces? Is there any new force, and what could be the physical origin of such forces?

## 6.3 Driving mechanisms: segregation forces

To understand the *origin* of segregation in the present framework, we now decompose the net gravitational force in Eq. (6.12) (i.e., $F := [m_s(Z_l T_l)/(Z_s T_s) - m_l] g$) for the *single intruder limit* in the following manner:
$$F = g \left[ (\rho_s - \rho_l)V_l + m_s \left(\frac{T_l}{T_s} - 1\right) \frac{Z_l}{Z_s} + m_s \left(1 - \frac{V_l}{V_s}\right) + m_s \left(\frac{Z_l}{Z_s} - 1\right) \right], \quad (6.16)$$

where $V_l$ is the volume of a particle of species i. The first term,

$$F_B^A := V_l(\rho_s - \rho_l)g, \quad (6.17)$$

is the effective Archimedean buoyancy force which arises due to the weight of the *displaced* volume of the intruder ($V_l$). The second term,

$$F_B^T \propto (T_l - T_s), \quad (6.18)$$

represents the buoyancy force due to the difference between the two species granular energies. This, being an *analog* of the thermal buoyancy, may be termed the *pseudo-thermal* buoyancy force.

There are two more terms in Eq. (6.16) which do not appear to be related to standard buoyancy arguments. The third term is *negative definite*, and vanishes identically if the intruder and the smaller particles have the *same size*. Note that

$$\epsilon_v^{st} := (V_l/V_s - 1) \quad (6.19)$$

is the *volumetric strain*. Thus,

$$F_{ge}^{st} := -m_s g \epsilon_v^{st} \quad (6.20)$$

is a *static compressive force* to overcome the barrier of the *compressive volumetric strain* arising out of the *size-disparity* between the intruder and the smaller particles. The fourth term in Eq. (6.16),

$$\propto (Z_l/Z_s - 1), \quad (6.21)$$

vanishes in the dilute limit $\phi \to 0$. It can be verified that $(Z_l/Z_s - 1)$ also vanishes identically, irrespective of the total volume fraction, if the particles are of the same size. Thus, the *origin* of this force is also tied to the *size-disparity* as in the third term $F_{ge}^{st}$. An interesting *physical* interpretation can be made if we consider the dense limit with a single intruder ($\phi_l \ll \phi_s$):

$$(Z_l/Z_s - 1) \propto R_{ls}^d \quad (6.22)$$

for $R_{ls} \gg 1$. Hence

$$\epsilon_v^{dyn} := (Z_l/Z_s - 1) \ge 0 \quad (6.23)$$

can be associated with a weighted *volumetric strain*, *tensile* in nature. Thus,

$$F_{ge}^{dyn} := m_s g \epsilon_v^{dyn} \quad (6.24)$$
is a *dynamic* tensile force that arises from the *excess* pressure difference due to the *nonideal* (collisional) interactions between the intruder and the *displaced* smaller particles.

Thus, the *geometric* effects due to the *size-disparity* contribute two new types of segregation forces:

$$F_{ge} = F_{ge}^{st} + F_{ge}^{dyn} = -m_s (\epsilon_v^{st} - \epsilon_v^{dyn}) g, \qquad (6.25)$$

the former is a static, compressive force and the latter is a dynamic, tensile force. On the whole, the collisional interactions help to reduce the net compressive force that the intruder has to overcome.

A question naturally arises as to whether we could get back the standard Archimedes law from Eq. (6.16) if we take the corresponding fluid limit, i.e. a large particle being immersed in a sea of small particles with $r_l \gg r_s$. In this limit it immediately follows that

$$F_{ge}^{dyn} \rightarrow m_s(V_l/V_s - 1) = -F_{ge}^{st} \qquad (6.26)$$

and hence

$$F_{ge} \equiv 0. \qquad (6.27)$$

Thus, the net gravitational force on a particle falling/rising in an otherwise quiescent fluid (at the same temperature) is nothing but the standard Archimedean buoyancy force,

$$F = F_B^A = g(\rho_s - \rho_l)V_l. \qquad (6.28)$$

It is worth recalling that when there is no size-disparity ($r_l = r_s$), the geometric forces are identically zero. Hence the behavior of a heavier particle in a sea of equal-sized lighter particles is similar to that of a particle in a fluid.

For illustrative purposes, we have calculated the energy ratio, $T_l/T_s$, (see the lower inset in Fig. 6.1) from the model of Barrat and Trizac [19] who obtained an implicit nonlinear expression for the granular temperature ratio for a homogeneously driven system (See Eq. (5.48) in Sec. 5.3).

To clarify our segregation mechanism, we show the variations of different segregation forces with the size-ratio in Fig. 6.1 for the two-dimensional case of equal density particles ($\rho_l = \rho_s$) in the single intruder limit ($\phi_l/\phi_s = 10^{-8}$) at a total solid fraction of $\phi = 0.7$, with the restitution coefficient being set to 0.9. For this case, the Archimedean buoyancy force is identically zero, and the total geometric force remains negative, as seen from the upper inset in Fig. 6.1. The pseudo-thermal buoyancy force is, however, positive. Thus, the competition between the pseudo-thermal buoyancy force and the geometric forces leads to a critical size-ratio above which
the intruder will rise for this case. (For the corresponding purely elastic case ($e = 1$ and $F_B^T = 0$), the net force is $F \equiv F_{ge} < 0$ and hence the larger particle will sink to the bottom.) This mechanism holds also for the more general case ($\rho_l \neq \rho_s$ and $F_B^A \neq 0$) for which the total buoyancy force ($F_B = F_B^A + F_B^T$) competes with the geometric forces ($F_{ge} = F_{ge}^{st} + F_{ge}^{dyn}$) to determine the transition from BNP to RBNP; the inclusion of dissipation merely affects the location of the transition point (see Fig. 6.2 and the discussion below, for details).

<figure><img src="image_28.png" /><figcaption>**Fig. 6.1.** Variations of segregation forces ($F/m_s g$) with the size-ratio for $\rho_l = \rho_s$ at $e = 0.9$; see text for other details. The upper inset shows the corresponding static and dynamic contributions to the total geometric force ($F_{ge} = F_{ge}^{st} + F_{ge}^{dyn}$). The lower inset shows the variation of $T_l/T_s$ with the size-ratio[19].</figcaption></figure>

## 6.4 Phase diagram and discussion

A typical phase diagram in the single intruder limit ($\phi_l/\phi_s = 10^{-8}$), delineating the regimes between BNP and RBNP, is shown in Fig. 6.2 for the two-dimensional case, with other parameters as in Fig. 6.1. (The qualitative features of the corresponding phase diagram for the three-dimensional case are similar.) Focussing on the purely elastic case ($e = 1$), we note that a transition from BNP to RBNP can occur following two paths (denoted by two arrows), one along the *constant mass-ratio* with decreasing size-ratio and the other along the *constant size-ratio* with increasing mass
<figure><img src="image_29.png" /><figcaption>**Fig. 6.2.** Phase diagrams for BNP/RBNP in two dimensions: $\phi = 0.7$ and $\phi_l/\phi_s = 10^{-8}$. Left inset: phase diagram with $e = 0.9$, $\phi_l/\phi_s = 10^{-8}$ (solid curve) and $\phi_l/\phi_s = 1$ (dashed curve). Right inset: phase diagram with $e = 0.9$, $\phi_l/\phi_s = 1$, $\phi = 0.7$ (dashed curve) and $\phi = 0.4$ (dot-dashed curve).</figcaption></figure>

ratio. In both cases, the Archimedean buoyancy force balances the net geometric forces at the transition point.

Comparison between the elastic ($e = 1$) and inelastic ($e = 0.9$) cases in Fig. 6.2 clearly shows that the non-equipartition of granular energy, responsible for the pseudo-thermal buoyancy force $F_B^T$, has a dramatic effect in reducing the regime of RBNP, and decreasing the value of $e$ reduces the size of this regime further. For the case of a mixture with equal volume fractions ($\phi_l = \phi_s$), however, the regime of RBNP is much larger as seen from the left-inset of Fig. 6.2. This observation is in qualitative agreement with the recent experimental results of Breu et al.[33] who found that the ‘reverse Brazil-nut effect is completely destroyed if $\phi_s \gg \phi_l$’.

The right-inset of Fig. 6.2 shows that the size of the RBNP-regime also increases with decreasing overall mean volume fraction. Since in vibrated-bed experiments increasing the shaking amplitude is equivalent to decreasing the mean volume fraction, our observation explains another interesting result of Breu et al. [33] that for a given mixture with specified size- and mass-ratio, the final state is that of RBNP at sufficiently high accelerations (see Fig. 6.2 in Ref.[33]).
We need to point out that calculating the energy ratio ($T_l/T_s$, [19]) we made the assumption that $e_{ij} = e$. Using a variable restitution coefficient, our phase-diagram at large mass-ratios will be modified, but the proposed segregation mechanism and the qualitative features of the phase-diagram remain intact.

To compare our segregation mechanism with others, we note that the scaling of the geometric forces ($\propto R_{ls}^d$) suggests that they can be compared to the effective percolation force of Rosato et al.[211], and hence we have a competition between buoyancy and percolation forces. In the percolation-condensation mechanism of Hong et al.[108], the condensation is driven by the two-species having different *condensation temperatures*<fnref n="1" />.

In conclusion, we have identified four different types of segregation forces: apart from the Archimedean buoyancy force and an analog of the thermal buoyancy force, there are two additional forces, the origin of both is tied to the size-disparity between the intruder and the smaller particles. We have demonstrated that the competition between the buoyancy and geometric forces determines the onset of segregation in the present scenario, and the inclusion of the pseudo-thermal buoyancy force (due to inelastic dissipation) further enhances the possibility of BNP. While the possibility of RBNP is rather limited in the single intruder limit, even at moderate dissipation-levels, either increasing the relative volume fraction of the intruders or decreasing the mean volume fraction enhances its likeliness as in the experiments of Breu et al.[33].

<footnote n="1">¹ A remark should be made here: In [108] the mixture is in contact with a thermal reservoir at a global temperature *T*. The system is then held at the value of this global temperature. Therefore, the system satisfies the equipartition of kinetic energy as opposed to our case.</footnote>
<figure><img src="image_30.png" /></figure>
# 7 Conclusions and perspectives

In this thesis, we have provided a theoretical and phenomenological description of the segregation dynamics in vibrated granular materials. One of our motivations has been to explore whether the hydrodynamic-description of granular materials can be applied to the derivation of time-evolution equations for the segregation velocity of a single intruder. The main message emerging from our analysis is that the effect of dissipation (which is the major signature of a granular fluid) have crucial consequences on the segregation dynamics. In particular, the rise velocity of an intruder particle and the phase-diagram of the BNP/RBNP transition are affected by buoyancy forces and dissipation in an essential way. Our theory can explain qualitatively several experimental observations. However, more work is required to introduce the effect of convection and boundary conditions.

In Chapters 3 and 4 we have exploited the idea of a buoyancy-driven segregation mechanism and analyzed the effect of the dissipation on the size segregation velocity. The model can explain qualitatively the increasing of the segregation velocity with the size of the intruder particle. On the other hand, we have studied the upward to downward segregation transition taking into account the difference of densities between the intruder particle and the particles of the granular fluid. Besides the experimental data analyzed in Sec. 3.3, some support for the hydrodynamic picture introduced in Chapter 3 may be found in the recent experiment carried out by Geromichalos et al. [86]. These authors showed that a qualitative estimate of the size segregation effect (for a dry regime) can be obtained within the scheme introduced in Sec. 3.1.2 for the calculation of the local temperature differences for the granular media in the presence of an intruder particle. On the other hand, they found that the BNP is reduced when liquid is added due to a modification of the dissipation rate.
Confirmation of the thermal expansion process and the buoyancy force may be provided by suitable computer simulations. Further investigation is required to deduced the forces associated with the convection and fluid drag.

In Chapter 6 we have introduced a novel mechanism for segregation. We have shown explicitly that the onset of segregation can be explained as a competition between the buoyancy and geometric forces:

* The *Archimedean buoyancy* force due to the weight of the displaced volume of the intruder.

* The *thermal buoyancy* force due to the difference between the two species granular temperatures.

* The *static compressive* force due to the barrier of the compressive volumetric strain arising out of the size-disparity between the intruder and the smaller particles.

* The *dynamic tensile* force from the pressure difference due to the interactions between the intruder and the smaller particles.

There are no experimental evidences so far for this mechanism, but the theoretical description introduced in Chap. 6 might stimulate new important experiments in the future. All the segregation forces are measurable in a standard vibrated-bed setup, and thereby making it directly verifiable via experiments. For example, the thermal buoyancy force ($F_B^T$) can be measured by measuring the species granular energies ($T_l$ and $T_s$) from the snapshots of successive particle configurations. These snapshots can also be used to measure the pair correlation function ($g_{ij}$ and hence the compressibility factor $Z_i$) and the dynamic geometric force ($F_{ge}^{dyn}$). The other two forces ($F_B^A$ and $F_{ge}^{st}$) are then easily obtained. Thus, knowing all these forces from experiments, one could construct each term of our calculation independently.

In Chap. 6 we have analyzed the BNP/RBNP phase diagram with favorable comparison with the recent experimental results of Breu et al. [33]. In this respect our results are in agreement with the remarkable works of Hong, Quinn and Luding [108] and Jenkins and Yoon [127], but we have considered the dissipative nature of a granular media, unlike the earlier works mentioned. In this chapter we have calculate the ratio $T_l/T_s$ using the model of Barrat and Trizac [19] which is able to reproduce the $T_l/T_s$-data of vibrofluidized-bed experiments carried out by Feitosa
and Menon [75]. It is important to note that experiments of segregation are usually
carried out under strong vertical shaking. This implies density and energy vertical
gradients. However, the question about the appropriate boundary condition for a
binary granular mixture vertically vibrated in the presence of gravity remains unan-
swered yet. This is a whole subject into itself which is of great importance, and which
involves issues and scenarios similar to those we have considered here. Indeed, by
calculating the coupling with the inertia of smaller particles and the spatial varia-
tion in species granular temperature (see Eq.(6.12)), it is possible to approach the
question of determining the reverse buoyancy effect [217]. It will be important to
determine the frequency dependence of the segregation phenomenon. As showed in
[2, 54, 80, 149, 195, 219, 225] the segregation rate depends on the acceleration param-
eter Γ. On the other hand, the experiment [33] shows that the final state is always
RBNP at Γ >> 1. It is necessary to incorporate adequate boundary condition at the
bottom plate (which is the origin of frequency dependence) to derive the analytical
expression for the segregation criterion.

In the analysis presented in Chapter 6 we have considered the hypothesis of Maxwellian velocity distribution function. Let us remark that the leading-order solution of the Boltzmann–Enskog kinetic equation for granular mixtures is a Maxwellian (See Ref.[122]), and the non–Gaussian correction term remains relatively small. Therefore, our segregation mechanism and the qualitative features of the BNP/RBNP phase diagram remain intact. Moreover, the Archimedean buoyancy force ($F_B^A$) and the static geometric force ($F_{ge}^{st}$) do not depend on the distribution function, and the non–Gaussian correction does not affect the dynamic geometric force ($F_{ge}^{dyn}$). Only the thermal buoyancy force ($F_B^T$) can be expressed in terms of the velocity distribution function.

Beyond BNP

There are also other more general issues which, though crucial for the hydrody-
namic study of the segregation phenomenon, are amenable to study through the
phenomenological sorts of investigations we have been discussing in this thesis. For
example:

* An accurate picture of the pair distribution function $g_{ij}$ is required taking into account the inelastic nature of a granular material and high densities. For exam-
ple, Ref. [158] presents a detailed discussion applied to the homogeneous cooling state of granular fluids.

* The dynamical properties of fluidized granular mixtures are currently an active subject. The research-effort to derive self-consistently hydrodynamic equations for binary granular mixtures is relatively new (e.g., Ref. [85]).

* An important question, not yet fully understood, is the breakdown of kinetic energy equipartition [3, 19, 20, 75, 160, 229] and how to describe theoretically the qualitative behavior of the granular fluid in such situations [3]. Of a particular interest are the recent computer simulations carried out by Paolotti et al. [194] for a binary granular mixture vertically vibrated in the presence of gravity.

* An extension of the actual kinetic theory for multi-component mixtures with more than two species [237] taking into account the species kinetic energies of each species could be interesting for more realistic studies including polydispersity.

* From the theoretical alternative presented in this thesis, the idea of non-spherical particles and anisotropic granular temperatures could be developed to explain the experimental findings reported by Shinbrot and Muzzio for *egg-shaped* intruders [217].

* Further investigation is required to deduce the drag force acting on an intruder particle in a granular fluid<fnref n="1" />.

* Finally, the expansion process of a fluidized granular material and the determination of the coefficient of thermal expansion remains an unexplored subject.

In conclusion, we have seen that the granular hydrodynamic has the potential for accurately describing many properties of the observed segregation dynamics. Our results are not only technical but very close to the reported experimental phenomenology. Of particular beauty is that all the forces derived in this thesis are measurable via computer simulations and/or standard experimental setups.

<footnote n="1">¹ After we have concluded this thesis, it was pointed out to us that D'Anna et al. [56] proved the connection between random grain motion and viscosity in shaken granular materials. Their experiments show that a granular system with strong dissipation has a definable viscosity and approximately obeys the fluctuation-dissipation theorem [56, 222].</footnote>
# 8 Version Française abrégée

## Introduction

### Pourquoi étudier la ségrégation des milieux granulaires

Un système granulaire réel est toujours caractérisé par ses degrés de polydispersité en taille, en masse et en forme, lesquels mènent souvent à une ségrégation non désirée.

La ségrégation est un processus dans lequel un mélange homogène de particules d'espèces différentes devient spatialement non-uniforme. Il y a alors séparation des espèces suivant leur taille ou leur masse. Ce phénomène est omniprésent dans l'industrie lors du transport et de la manipulation des mélanges granulaires. Dans la plupart des cas, il est exigé de maintenir un mélange homogène pendant un traitement, la ségrégation devant être évitée. Cependant, la ségrégation se produit spontanément dans les milieux granulaires vibrés.

Le phénomène des noix du Brésil (BNP) –où les particules les plus grosses montent jusqu'au sommet de l'échantillon lorsqu'elles sont soumises à de fortes secousses verticales– est devenu un des problèmes les plus curieux de la recherche dans les materiaux granulaires. Des mécanismes différents ont été proposés pour expliquer ce phénomène: la percolation (filtration), l'effet d'arche, la convexion, l'inertie, la condensation, la flottabilité, etc.

Mais “Comment la noix du Brésil monte?” n’est pas la seule question à laquelle nous devons répondre. Récemment il a été découvert que le mouvement descendant d’un intrus peut se produire. Ce comportement a été baptisé phénomène de noix du Brésil inverse (RBNP). Par conséquent, la question actuelle serait plutôt “Comment la noix du Brésil monte ou descend?”

Nous nous sommes penchés sur la ségrégation des milieux granulaires parce que ces questions restent encore ouvertes! et un point de vue théorique du problème est d'une importance substantielle pour comprendre le mécanisme sous-jacent.

### Compétence de cette thèse et organisation du manuscrit

Cette thèse contient trois projets consacrés à l'étude de la ségrégation des materiaux granulaires et basés sur une approche théorique (cf la bibliographie plus bas). Nous présentons une nouvelle interprétation des observations expérimentales des BNP et RBNP, basée sur un formalisme hydrodynamique, et ceci pour des particules inélastiques.

Cette thèse s'organise comme suit: Dans le Chapitre 2 nous présentons l'état actuel des recherches consacrées au phénomène de ségrégation, ainsi que les suppositions principales et les énoncés. Dans le Chapitre 3 nous développons un modèle hydrodynamique pour la ségrégation et nous discutons des effets de taille de la dynamique. Dans le Chapitre 4 nous étendons le modèle introduit dans le Chapitre 2 pour étudier les effets de taille et
de densité de l'intrus sur la transition BNP/RBNP. Dans le Chapitre 5 nous présentons une introduction à la théorie cinétique des mélanges granulaires binaires. Dans le Chapitre 6 nous introduisons un nouveau mécanisme de ségrégation basé sur la compétition entre flottabilité et forces géométriques. Enfin, nous discutons dans le Chapitre 7 les résultats principaux de cette thèse ainsi que quelques implications expérimentales. Cette thèse est basée sur la bibliographie suivante:

* Hydrodynamic model for particle size segregation in granular media
  Trujillo, L. & Herrmann, H. J.
  Physica A (2003) (in press).

* A note on the upward and downward intruder segregation in granular media
  Trujillo, L. & Herrmann, H. J.
  *Granular Matter* **5** (2003) 85–89.

* Segregation in a Fluidized Binary Granular Mixture: Competition between Buoyancy and Geometric Forces
  Trujillo, L., Alam, M. & Herrmann, H. J.
  *Europhysics Letters* (2003) (in press).

## Le problème des noix du Brésil

Les résultats expérimentaux et numériques en rapport avec la ségrégation dans les mélanges granulaires sont examinés dans le Chapitre 2. Nous commençons avec une brève discussion sur la physique des matériaux granulaires, dans le contexte général des recherches actuelles, où nous passons en revue les modèle théoriques et les différents mécanismes qui ont été proposés ces dernières années. L'attention est portée sur des systèmes granulaires soumis à des vibrations verticales. Nous mentionnons aussi d'autres situations où la ségrégation survient. Nous expliquons les motivations de notre nouvelle approche. Nous insistons particulièrement sur la structure théorique basée sur une description hydrodynamique des systèmes granulaires ainsi que sur ses implications dans le phénomène de ségrégation. Enfin, quelques aspects industriels sont répertoriés et nous commentons brièvement la littérature existante.

## Théorie phénomènologique de la ségrégation

### L'hydrodynamique granulaire

La matière granulaire dans le régime *fluide* peut être décrite au niveau macroscopique par des équations similaires à celles de la mécanique hydrodynamique [34, 89, 93, 126, 200]. Ces modèles hydrodynamiques fournissent un bon support de compréhension du comportement des systèmes soumis à des vibrations. Ils s'appuient sur la correspondance entre le mouvement des molécules d'un gaz et le mouvement aléatoire des particules granulaires [31, 32, 95, 103, 148].

Contrairement à un gaz moléculaire à l'équilibre où l'énergie cinétique moyenne d'une molécule est proportionnelle à la température du gaz (température thermodynamique), l'état d'équilibre naturel d'un materiaux granulaire est une configuration statique résultant