Bubbles, Drops,

and Particles in

Non-Newtonian Fluids

SECOND EDITION

R. P. Chhabra

Indian Institute of Technology

Kanpur, India

DK3171_half-series-title 5/12/06 10:42 AM Page i

The photograph on the cover is based on the photographs kindly provided by Dr. Angus Paterson, Pater-

son and Cooke Consulting Engineers, Cape Town, South Africa; Professor I. Frigaard, University of Brit-

ish Columbia, Vancouver, Canada; and Professor I.L. Kliakhandler, Michigan Technological University,

Houghton, Michigan.

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Library of Congress Cataloging-in-Publication Data

Chhabra, R. P.

Bubbles, drops, and particles in non-Newtonian fluids / Raj P. Chhabra.-- 2nd

ed.

p. cm. -- (Chemical industries series ; 113)

Includes bibliographical references and indexes.

ISBN-13: 978-0-8247-2329-3 (acid-free paper)

ISBN-10: 0-8247-2329-5 (acid-free paper)

1. Non-Newtonian fluids--Textbooks. 2. Fluid mechanics--Textbooks. I. Title. II.

Chemical industries ; v. 113.

QC189.5.C55 2006

620.1’06--dc22

2006003734

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Bubbles, Drops, and Particles in Non-Newtonian Fluids,
Second Edition, R. P. Chhabra

DK3171_half-series-title 5/12/06 10:42 AM Page G

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To Neeru for her unconditional and

unlimited love; and to Millie and Rajat,

who wonder what I do all day

RAJ: “dk3171_c000” — 2006/6/8 — 23:02 — page vii — #7

Preface to
the Second Edition

Since the publication of its first edition in 1993, significant advances have
occurred in this field that impinge on almost all topics covered in this book. This,
coupled with the fact that the first edition has been out of print for a while now,
prompted me to prepare a new edition of this book. The goals and the structure
of this edition are the same as in the first edition: to provide a reference text for
graduate students and researchers by presenting a comprehensive and critical
evaluation of the available extensive literature relating to the non-Newtonian
effects in multiphase flows encompassing single particles at one extreme and
concentrated systems such as porous media at the other extreme, and to provide
reliable estimation methods required by practicing professionals to perform
day-to-day routine design calculations relating to the process technology of non-
Newtonian fluids, such as the prediction of the settling velocity of individual
particles and concentrated suspensions or the frictional pressure drop in porous
media flows.

Bearing in mind the independent reviews of the first edition and the sug-

gestions made by the readers of the first edition, the entire book has been
reviewed. Where the need was recognized, the presentation has been improved
by reorganizing the material for easier understanding, new material to facilitate
comprehension has been added, and the most current viewpoints and research
results have been incorporated in this revised edition. Apart from the general
updating of all chapters, the specific changes made from the first edition are
summarized as follows.

The introductory material in

Chapter 1

has been completely overhauled and

greatly expanded to illustrate both the pervasive and the ubiquitous nature of
non-Newtonian fluid behavior as encountered in everyday life and in techno-
logy. In

Chapter 2,

a short section on extensional flow has been added due

to its direct relevance in porous media flows and in the growth of gaseous
inclusions in polymer melts and other non-Newtonian media encountered in
food-processing and geological engineering applications. New research res-
ults relating to the flow of time-independent fluids (without a yield stress) past
axisymmetric particles like a cylinder or a circular disk or spheroidal particles
have been incorporated into

Chapter 3.

Likewise, the corresponding body of

experimental results for the free settling behavior of a range of nonspherical

RAJ: “dk3171_c000” — 2006/6/8 — 23:02 — page viii — #8

shapes of particles has been treated in detail in this chapter. Also, the interesting
behavior exhibited by ascending light spheres in quiescent polymer solutions
has now been introduced in this chapter. For added emphasis, the hydrodynam-
ics of rigid particles in visco-plastic media has now been treated separately in

Chapter 4.

Since the first edition, the significant advances made in the field of

particle motion in visco-elastic fluids, especially for the two benchmark prob-
lems of a sphere in a tube and a cylinder in a planar slit, are reflected in the
new look of

Chapter 5.

The treatment of the dynamics of bubbles and drops

and their ensembles in a wide variety of stagnant and moving non-Newtonian
media has been completely revised in

Chapter 6

to take into account the new

developments since the publication of the first edition. In particular, the meas-
urements of the detailed velocity fields close to the rising bubbles have been
included here. An extensive section on the flow of both Newtonian and non-
Newtonian liquids in fibrous media has been added to

Chapter 7.

The other

changes to this chapter include a slight expansion of the section on the use
of the volume averaging methods for modeling porous media flow and a new
section on the two-phase flow of a gas and a non-Newtonian liquid in packed
beds. The treatment of the role of visco-elasticity in liquid–solid fluidization and
in the sedimentation of concentrated suspensions of noninteracting particles has
been somewhat sharpened in

Chapter 8,

especially in view of the new evidence

showing the formation of preferred flow passages in these systems. The new

Chapter 9

presents an overview of the extensive literature on the non-Newtonian

effects in boundary layer flows and on the interphase heat/mass transfer from
particles immersed in a variety of non-Newtonian fluids. To highlight their sig-
nificance, the additional effects arising from the presence of confining walls on
the sedimentation of particles have now been treated on their own in a separate

Chapter 10. Chapter 11

has been considerably expanded by adding a whole

range of viscometers including the falling cylinder (needle) and the rolling ball
devices to evaluate a variety of non-Newtonian characteristics, all of which
are routinely used for monitoring the quality of a range of products during the
course of their manufacture. Finally, in this edition, the list of references cited
has been placed at the end of the text, for this format not only avoids duplication
between chapters, but it is also thought to be more convenient for the reader.

And lastly, I would like to thank the readers who have made such helpful

suggestions and have drawn my attention to the errors in the first edition, many
of which I would have never spotted myself. Please keep up the good work and
let me know if, no, not if, but when, you find errors in this edition.

R.P. Chhabra

Kanpur, India

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Preface to
the First Edition

For the past 300 years or so, the simple Newtonian fluid model has been accep-
ted as the standard fluid behavior. Though most gases and low molecular weight
substances do exhibit this kind of fluid behavior, in recent years, there has been
an increasing recognition of the importance of non-Newtonian flow character-
istics displayed by most materials encountered in everyday life, both in nature
(gums, proteins, biological fluids such as blood, synovial fluid, etc.) and in tech-
nology (polymers and plastics, emulsions, slurries, etc.). Indeed, so widespread
is the non-Newtonian behavior that it would be no exaggeration to say that the
Newtonian fluid behavior is an exception rather than the rule. Consequently, the
last three to four decades have witnessed a remarkable upsurge of interest and
research activity in the mechanics of non-Newtonian fluids. An unprecedented
increase in the number of research papers, books, and conference proceedings
testifies to the rapid growth of this field. There are even specialized serial public-
ations (Journal of Rheology, Rheologica Acta, and Journal of Non-Newtonian
Fluid Mechanics) devoted largely to the publication of new results in this area.

This book is neither about the non-Newtonian fluid behavior per se nor does

it purport even to touch on, let alone cover, all aspects of the non-Newtonian
fluid mechanics. Rather, it deals with a narrow but important class of prob-
lems involving the motion of small, rigid, and deformable particles (and their
ensembles) in a viscous medium. Although many years of research have been
devoted to the behavior of particles in Newtonian media, many challenging
problems in both theory and applications still remain unresolved. But even
more challenging is the subject of particle motion in non-Newtonian media,
such as the motion of oil drops in polymer solutions through a porous medium
during enhanced oil recovery, the dissolution of gas bubbles during ferment-
ation, the key role played by the bubbles encapsulated in the production of
foam plastics, etc. Equally important are the numerous applications involving
significant inter-particle interactions such as those encountered in the fluid flow
in packed and fluidized beds, bubble columns, slurry reactors, and in hindered
settling of concentrated suspensions. In spite of such overwhelming pragmatic
and fundamental significance, none of the existing books on non-Newtonian
fluid mechanics deals with these topics adequately. This book is an effort to

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fulfill this need. In particular, it has been written to accomplish the following
two specific goals:

1. To provide a reference text for research workers by way of presenting

a comprehensive and critical evaluation of the voluminous literat-
ure available on the particle motion in non-Newtonian media and
the related multiparticle process applications, as mentioned in the
foregoing.

2. To provide useful design information often required by practicing

scientists and engineers in day-to-day routine calculations such as
the estimation of the minimum fluidization velocity or the prediction
of pressure drop for flow through a fixed bed of particles, etc.

Aside from these two primary objectives, parts of the material presented herein
have been used at the Indian Institute of Technology in Kanpur in a first-year
graduate course on Engineering Applications of Rheology for the past 5 years.
Thus, depending upon one’s interest and taste, I believe that this material can
form a part of an elective course on Process Applications of Rheology for senior
undergraduate or first-year graduate students.

The limitations of this book should also be mentioned. The most important

of all is that it concentrates primarily on the free motion of spherical particles
under the influence of gravity. The scant work relating to the behavior of non-
spherical particles is included only by way of providing the pertinent references.
Likewise, no additional effects arising from the presence of electrical (electro-
rheological fluids) or magnetic fields, the compressibility of the fluids, etc. are
included unless specifically mentioned otherwise.

R.P. Chhabra

Kanpur, India

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Acknowledgments

Every book is in part the product of the knowledge and enthusiasm of
the author’s friends. This one is no exception. It is a pleasure to acknow-
ledge the assistance I have received from many individuals during the course
of the preparation of this book. While it is impossible to acknowledge all of
them, I am particularly indebted to my Ph.D. advisor, Dr. Peter Uhlherr (retired
professor from Monash University, Melbourne) for introducing me to the fas-
cinating world of non-Newtonian fluids, and to my postdoc mentor, Professor
Jack Richardson (Professor Emeritus, University of Wales, Swansea) not only
for further sharpening my interest in the mechanics of these peculiar substances,
but, more importantly, for keeping me from going astray.

Over the past 25 years, I have greatly benefited from my collaborations with

the following friends from all over the world: Professor Tam Sridhar (Dean of
Engineering, Monash University), Professor Pierre Carreau (Ecole Polytech-
nique de Montreal), Professor Daniel DeKee (Tulane University), Professor
Rakesh Gupta (West Virginia University), Professor R. Shankar Subramanian
(Clarkson University), Professor Jacques Comiti (GEPEA, St. Nazaire),
Professor Jose Ferreira (Universidade de Tras-os-Montes e Alto Douro,
Vila Real, Portugal), Professors Evelyne Mauret and Maurice Renaud (both of
EFPG, INPG, Grenoble), Professor T. Sundararajan (Indian Institute of Tech-
nology, Chennai), and Professor V. Eswaran (Indian Institute of Technology,
Kanpur). Many others have been gracious and generous in responding to my
requests for their experimental results and for some of the photographs that
appear in this work. Thoughtful suggestions have consistently come from my
past and present students via their critical comments and alternate points of
view on many issues. It is impossible to mention everyone, but I would like to
single out Ram Prakash, Amit Dhiman, Sunil Dhole, and Nanda Kishore for
their extensive help with the preparation of this revised edition. I would also
like to thank Roopam Agarwal and Vandana Bhatnagar for their expert help
with some of the artwork used in this edition. The failures that remain are, of
course, entirely my responsibility and show my stubbornness.

I would also like to thank Anita Lekhwani, Jill Jurgensen, Elise Oranges

(all of Taylor & Francis) and Mohan (at Newgen Imaging Systems, Chennai)
for their patience and help at every stage of this project.

To live in close intimacy with a person obsessed with a single idea is never

an easy task. My family has been subjected to my obsession with these bizarre
fluids ever since I was introduced to them about 30 years ago. This book is

RAJ: “dk3171_c000” — 2006/6/8 — 23:02 — page xii — #12

a small tribute to the patience and fortitude of my parents, my brother, and
his family, for their unquestioning support of my work and aspirations. It is not
customary to express gratitude to one’s wife in public, for she is considered to
be a part of the husband to the extent that her coauthorship is tacitly assumed
in any book her husband writes. There is little doubt that this book would not
have been possible without Neeru’s help.

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The Author

R.P. Chhabra is a professor of chemical engineering at the Indian Institute
of Technology, Kanpur. After receiving his B.S. and M.S. degrees in chem-
ical engineering from the University of Roorkee (now the Indian Institute of
Technology, Roorkee) and the Indian Institute of Science, Bangalore, respect-
ively, he obtained his Ph.D. from Monash University in Melbourne (Australia).
He has been a visiting professor at several universities, including the University
of New South Wales, Sydney; Clarkson University, Potsdam (NY); The State
University of New York at Buffalo; Ecole Polytechnique de Montreal; Uni-
versite de Nantes, Nantes; and the Cape Peninsula University of Technology,
Capetown.

Dr. Chhabra’s current teaching and research interests are in the general areas

of non-Newtonian fluid mechanics, multiphase flows, and transport properties
of molten metals and alloys. He has 6 books, 20 invited book chapters and
reviews, and over 200 technical papers to his credit. He is a recipient of the
Amar Dye Chem (1988) and the Herdillia (1996) awards (of the Indian Institute
of Chemical Engineers) for excellence in research. He is a Fellow of the Indian
National Academy of Engineering.