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1

Introduction

The presence and motion of bubbles, drops, and particles in fluids impinge on
our everyday life to such an extent that it would be no exaggeration to say that
the phenomenon is ubiquitous. The water we drink has suspended particles and
dissolved gases in it, some contributing to our well being, while others have
harmful effects; the air we breathe has minute solid particles in it; the fizzy
drinks have gases dissolved in the form of small gas bubbles. Blood, a vital
element of life, immediately comes to mind with its red and while blood cells
suspended in plasma. One can think of numerous other examples involving
interaction between particles and fluids encountered literally everywhere in
everyday life and in technology. Thus, to understand the behavior of particles
in a fluid has been a challenge throughout the history of man.

From a technological standpoint, numerous operations in chemical and

processing industries involve fluid-particle systems. Fluidization technology
relies almost solely on fluid-particle (solid and bubbles) interactions. Hydraulic
and pneumatic transportation of particulate materials involves hydrodynamic
interactions between the conveying medium (liquid/gas) and the material to be
transported. Other examples of solid–fluid interactions include the filtration of
polymer melts, sewage sludges and paper coatings, sedimentation and thicken-
ing of slurries, disposal of wastes from mineral industries, interpretation of the
rheological behavior of suspensions, trickle bed, fixed bed, and slurry rectors,
etc. Less appreciated applications of fluid-particle systems include the motion of
red blood cells in capillary flow, chromatographic separations, electrophoresis,
separation of macromolecules according to their sizes, etc. The most common
method to affect gas/liquid contacting is to introduce the gas through a multi-
hole distributor in the form of tiny gas bubbles. Naturally, the rates of transfer
processes and chemical reactions are essentially governed by the flow field
established around the ascending gas bubbles. Introduction of an inert gas into
a pool of liquid to improve mixing is a commonly used process in metallurgical
and chemical industries. Dispersion of a liquid in another immiscible liquid,
resulting in an emulsion, is indeed a basis for the manufacture of a class of poly-
mers, cosmetics, personal care products, toiletries, foodstuffs, metallic foams,
and alcoholic beverages, etc. Thus there is no dearth of examples of industrial
relevance involving interactions between bubbles, drops, and particles and a
flowing or stagnant continuous fluid phase.

1

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2

Bubbles, Drops, and Particles in Fluids

Ever since the discovery of the laws of motion enunciated by Newton

more than three centuries ago, it has been a customary practice to accept the
Newtonian fluid model as a standard fluid behavior. Indeed the fluid mechanics
of Newtonian-fluid particle systems has witnessed remarkable years of progress
in recent times, especially with the advent of elegant numerical techniques and
super computers. While it has reached a respectable level of maturity, many
complex multiphase flows continue to pose challenges and defy predictions.
Yet it is a simple task to generate experiments that could never be explained
(even qualitatively) by the standard Newtonian fluid model (Boger and Walters,
1992)! Indeed, the Newtonian fluid behavior seems to be an exception rather
than the rule.

Over the past four to five decades, there has been an increasing recognition

of the fact that most materials of practical and industrial interest do not conform
to the simple Newtonian fluid behavior, and are accordingly known as “rheolo-
gically complex” or “non-Newtonian fluids,” (albeit the earliest reference to
non-Newtonian behavior dates back to 700

b.c. (Slattery, 1972). Indeed, many

interesting historical accounts (Markovitz, 1968, 1977, 1985; Scott Blair, 1982;
Joseph, 1986; Litt, 1989; Doraiswamy, 2002) show that not only the field of
rheology (in some form or the other) is an ancient science, but also has an incred-
ibly rich history (Tanner and Walters, 1998). The development and growth of
this discipline has been touched on by many pioneering individuals, including
Newton, Maxwell, Lord Kelvin, Boltzmann, Einstein (Physics Noble Prize,
1921), Flory (Chemistry Nobel Prize, 1974), de Gennes (Physics Nobel Prize,
1991), Chu (Physics Nobel Prize, 1997), Bingham and Reiner, to mention a few.

The simplest and also the commonest type of departure from the Newtonian

behavior is shear-thinning (or pseudoplasticity) wherein the apparent viscosity
(shear stress divided by shear rate) of a fluid decreases with increasing shear
rate. Besides, some of these materials behave like a viscous fluid in long-time
experiments, while the initial response to applied external stress is like that of
elastic solids. One can qualitatively explain this type of behavior by postulat-
ing that these visco-elastic materials possess a sort of “memory.” Yet another
example of non-Newtonian behavior is the so-called time dependence wherein
the value of the stress generated by an imposed velocity gradient varies with
the duration of experiment. Finally, it is not uncommon to encounter all these
(and many other) complexities in a single material and application under appro-
priate circumstances. While the early developments in this field were almost
exclusively motivated by the use of rubber, polymer, and plastics as poten-
tial substitutes for the traditional materials of construction, namely, glass and
metals, it is no longer so (White, 1990). The non-Newtonian fluid behavior
is much more widespread than it is generally perceived. For instance, as one
walks down the aisle of a supermarket, one passes a wide range of food products
(both manufactured and natural) that are either in the form of emulsions (or

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Introduction

3

suspensions) or have been in an emulsified form sometime during the course of
their production. Flavored milks, creams, salad dressings, dips, fruit yoghurts,
mayonnaise, for instance, immediately come to mind! Similarly, many phar-
maceutical products, including suspensions, laxatives, gels, and creams are all
multiphase mixtures that exhibit varying levels of complex rheological beha-
vior.

Table 1.1

provides a brief summary of diverse industrial settings where

rheolgocially complex behavior is encountered at some stage of processing.
It is virtually impossible to imagine modern life without emulsions, suspen-
sions (pastes) or dispersions. Thus, over the years, due to the ever increasing
needs and demands of consumers coupled with the advancements in technology,
chemical and process engineering applications extensively use a wide range of
rheologically complex materials including polymeric melts and solutions, and
multiphase mixtures (foams emulsions, dispersions).

In many of the applications involving bubbles/drops/particles — fluid sys-

tems cited in the foregoing, the liquid phase displays complex non-Newtonian
flow behavior. Typical examples with solid particles include the use of dilute
polymer solutions in enhanced oil recovery operations, in drag reduction applic-
ations, in pipeline transportation of coarse grains (Maciejewski et al., 1997;
Darby, 2000), as thickening agents in food processing (Holdsworth, 1992),
as vehicles in pharmaceutical formulations (Berney and Deasy, 1979; Miller
and Drabik, 1984), the use of drilling muds to provide lubrication as well as
to keep the debris in suspension, the handling and processing of fermentation
broths, emulsion polymerization, processing of filled polymers, downstream
processing in nuclear and biotechnological processes, hydraulic conveying of
large lumps of materials in carrier liquid containing fine particles, etc. Simil-
arly, the dynamics of bubbles in rheologially complex systems directly impinges
on the degassing of polymeric melts and freshly prepared concrete as well as
on the process of aeration used extensively in food processing applications. Litt
(1989) provided an exhaustive list of industrial examples where non-Newtonian
behavior is of central importance in determining the efficiency of a process.

There is no question that in real life applications, one often encounters

swarms or clusters of particles rather than an isolated single particle. Notwith-
standing the intrinsic complexity of particle–particle interactions, experience
has shown that an understanding of the behavior of single particles not only
provides useful insights into the underlying physical processes, but often also
serves as a launching pad for undertaking the modeling of these more realistic
applications. In addition to this, single particle studies are also of consider-
able theoretical interest in their own sight, as significant differences have been
observed, even at a macroscopic level, in non-Newtonian media as compared
to the behavior of single particles in linear fluids. For instance, the shapes of
bubbles and drops in free-fall/rise observed in a non-Newtonian medium are
entirely different from those encountered in a Newtonian liquid phase under

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Bubbles,

Drops,

and

Particles

in
Fluids

TABLE 1.1

Diversity of Systems Exhibiting Non-Newtonian Fluid Behavior

System

Reference

Dairy product and waste slurries

Hart et al. (1966); Staley et al. (1973); Prentice (1992)

Poultry waste slurries

Hashimoto and Chen (1976); Chen and Hashimoto (1976); Bjerkholt et al. (2005a, 2005b,

2005c)

Polymeric materials used as thickening agents in

pharmaceutical applications, creams, gels, etc.

Berney and Deasy (1979); Miller and Drabik (1984); Korhonen et al. (2000, 2002); Chang

et al. (2002)

Paper coating colors

Arzate et al. (2004); Higgins (1997)

Food stuffs (fruit yoghurts, butter, egg albumen, jams,

jellies, marmalades, fruit juices, salad dressing, ice cream

and cake toppings, cake mixes, ice creams, soups, etc.)

Sherman (1970); Tung et al. (1971); Rao et al. (2005); McClements (2004); Faridi (1989);

Borwankar and Shoemaker (1992); Rao (1999)

Food processing

Steffe (1996); Holdsworth (1992); Heldman and Lund (1992)

Polymers and polymeric solutions

Bird et al. (1987a, 1987b); Carreau et al. (1997); Larson (1998); Gupta (2000)

Cosmetics and toiletries, personal care products (lipsticks,

nail polish, creams and lotions, sun screens, etc.)

Laba (1993); Garcia-Morales et al. (2004); Brummer (2006)

Cement pastes and fresh concrete

Yaron and Ish-Shalom (1975); Tatersall (1983); Banfill (1991); Yahia and Khayat (2001);

Brower and Ferraris (2003)

Biological fluids (blood, synovial fluid, saliva, semen)

Gabelnick and Litt (1973); Briedis et al. (1980); Dunn and Picologlou (1977a, 1977b); Shi

et al. 2004); Chmiel and Walitza (1980); Rosentrater and Flores (1997)

Industrial polysacchrides

Lapasin and Pricl (1995)

Molten lava, magmas, rocks, and soils

Petford (2003); Hailemariam and Mulugeta (1998); Cristescu (1988); Saar et al. (2001)

Mineral slurries and mine tailings; agricultural chemicals,

fly-ash slurries

Schramm (1996); Usui et al. (2001); Hou et al. (2005)

Mud and debris flows

Coussot (1997, 2005); Bonadonna et al. (2005)

Lubricants

Davenport (1973); Jean (1989)

Paints and pigments; coatings and inks

Patton (1979)

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Introduction

5

nominally identical conditions. Additional motivation to study the flow of fluid
past single objects (sphere, cylinder, bubble, drop, for instance) stems from the
fact that these highly idealized shapes are also used to benchmark the suitability
of a constitutive relation or of numerics prior to using them for complex flow
problems. Conversely, the flow around a sphere or over a cylinder are also used
to explore the extent of molecular orientation (Haward and Odell, 2004), or to
ascertain other rheological properties (de Bruyn, 2004; Dollet et al., 2005), or
such highly idealized studies have direct relevance to their use as probe particles
in polymer solutions (Ye et al., 1998). Another interesting example of non-
Newtonian effects is the phenomenon of drag reduction in external flows such
as over a sphere or a cylinder or more realistic shapes for example, ship hulls.
Small doses of macromolecules added to water can reduce the skin friction by 30
to 40% that directly influence the power requirements for ships, submarines, etc.

This book aims to provide an upto date and critical account of the progress

made thus far in this field. At each stage, considerable effort has been made
to present the most reliable and generally accepted methods for performing
process engineering calculations.

1.1 SCOPE AND ORGANIZATION

Though this work is primarily concerned with the behavior of rigid and fluid
particles in rheologically complex systems, a terse background material on the
corresponding flow in Newtonian fluids is included in each chapter. This not
only facilitates the subsequent treatment for non-Newtonian fluids, but also
serves as a reference for qualitative comparisons between different types of flu-
ids. It is also important to mention the chief limitations of this work at the outset:
the only body force considered herein is that due to the earth’s gravitational field.
Similarly, the fluids are assumed to be incompressible. This is a good approxim-
ation for most non-Newtonian fluids referred to in this work, except for foams,
gaseous dispersions, and emulsions. Likewise, with the notable exception of

Chapter 9,

no temperature or concentration effects are included in any other

chapter. Finally, rheological behavior associated with the presence of electric
or magnetic fields is also not considered herein. Each chapter is concluded by
a short summary of the current state of the art and by presenting only those
predictive formulae that have been tested adequately.

Chapter 2

provides a brief exposition to the non-Newtonian flow character-

istics of fluids. Different types of non-Newtonian fluid behavior are described.
A selection of the widely used rheological equations along with their merits
and demerits is presented. By way of examples, typical values of model para-
meters are cited to give the reader a feel for numbers. This chapter is closed
by outlining the role of dimensional analysis in visco-elastic fluids, especially

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6

Bubbles, Drops, and Particles in Fluids

with regard to the definition of Deborah or Weissenberg number, as used in the
interpretation and correlation of experimental results.

The steady motion of rigid particles (a sphere, a cylinder and of other

nonspherical shapes) in purely viscous fluids, without a yield stress, is dealt with
in

Chapter 3.

Theoretical and experimental developments have been critically

examined for a variety of generalized Newtonian fluid models. In particular,
the influence of shear-rate-dependent viscosity on drag coefficient, the terminal
falling velocity and on the detailed flow field is presented and highlighted for a
sphere. The immense difficulties of quantifying the size, shape and orientation
of nonspherical particles during their free-fall are highlighted and these are
further accentuated in the case of non-Newtonian liquids.

The static equilibrium and dynamic behavior of rigid particles in visco-

plastic media is treated in

Chapter 4.

Notwithstanding the long and inconclusive

debate about the existence of a true yield stress, the capacity of these fluids to
support particles is used extensively in mineral, pharmaceutical, and personal
care product manufacturing industries. Starting with the issue of static equilib-
rium, attention is given to the size and shape of fluid-like zones, drag force,
and terminal settling behavior of spherical and nonspherical particles in visco-
plastic fluids. The roles of regularization schemes used in numerical simulations
and the difficulties of unambiguous measurement of yield stress in such exper-
iments appear to be the major sources which add to the degree of confusion
prevailing in the literature. The available scant information pertaining to the
hydrodynamics of nonspherical particles is also included here.

Chapter 5

addresses the analogous problem of a sphere and a cylinder

undergoing steady translation in visco-elastic media. In particular, a detailed
progress report on the numerical simulations and experimental developments
on the steady sedimentation of a sphere in a cylindrical tube is presented. Next,
the corresponding developments relating to the two dimensional cross flow of
visco-elastic liquids past a confined and an unconfined long circular cylinder
are considered. Many other interesting effects not reported hitherto are listed
here that are generally ascribed to the visco-elasticity of the liquid. Combined
effects of visco-elasticity and shear-thinning are also treated in this chapter. The
unusual phenomena such as migration and alignment of single and clusters of
particles, formation of shear-induced structures, etc. in a variety of flow fields,
are also discussed.

The behavior of fluid particles with mobile interface-bubbles and drops —

is treated in

Chapter 6.

Theoretical models for their formation from orifices and

nozzles are presented and discussed. The other methods of drop formation such
as by atomization and by breakup of liquid jets and sheets are also mentioned
in brief. Quantitatively different shapes of bubbles and drops are observed in
rheologically complex media. Terminal velocity–volume behavior, and drag
coefficient of single particles and their ensembles in free-fall conditions are

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Introduction

7

analyzed. Finally, consideration is given to the problem of coalescence and
breakage in non-Newtonian continuous phase under various flow conditions.

The hydrodynamics of non-Newtonian fluid flow on its own and together

with a gas in porous media and packed beds is dealt with in

Chapter 7.

An

overview of the wide variety of approaches employed to model fluid flow in
porous media is presented. In particular, the two commonly used frameworks
are described in detail: the capillary bundle approach and the submerged objects
approach. First, the flow of purely viscous fluids is analyzed followed by that of
visco-elastic systems. The main emphasis is on the delineation of flow regimes,
establishment of pressure loss-throughout relationships, followed by the effect
of particle shape and containing walls on the frictional pressure drop across
the bed. The flow of visco-elastic and of the so-called drag reducing polymer
solutions is also considered briefly. Included in this chapter is also the hydro-
dynamics of Newtonian and non-Newtonian media in fibrous beds, as that
encountered during the flow in tumors, and in polymer processing applications
aimed at the fabrication of fiber composites. In particular, consideration is given
to the prediction of permeability of the medium. The scant work dealing with
the simultaneous flow of a non-Newtonian liquid and a gas through a packed bed
is also included in this chapter. The chapter is concluded by presenting a brief
discussion on the variety of anomalous phenomena observed in packed beds
such as slip effects, adsorption, gel formation, differences between the bulk and
the in situ rheological characteristics, mechanical degradation of polymers in
flow conditions, etc.

Hydrodynamically similar problems of fluidization and sedimentation are

treated in

Chapter 8.

In particular, the influence of non-Newtonian char-

acteristics on the bed expansion dynamics, minimum fluidization velocity,
and the rate of sedimentation of concentrated suspensions in the initial con-
stant concentration region are examined in this chapter. These areas remain
virtually unexplored. Even less studied are the three phase fluidized bed sys-
tems involving a non-Newtonian liquid phase. Based on the available limited
information, an attempt has been made to develop suitable predictive expres-
sions for the macroscopic process parameters. The sparse literature relating to
the settling of fibrous suspensions is also included here.

Boundary layer flows and inter-phase heat/mass transport are considered

in

Chapter 9.

Starting with the three commonly used idealized geometries,

namely, a plate, a sphere or a cylinder, the laminar momentum, thermal, and
concentration boundary layers for power-law fluids are first considered, for
both free and forced convection regimes. Next, the role of visco-elasticity is
presented. Included in this chapter are also the scant numerical and experimental
correlations that allow the prediction of heat and mass transfer coefficients for
single particles at one end, and for packed and fluidized beds, tube bundles at
the other end of the spectrum of applications.

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8

Bubbles, Drops, and Particles in Fluids

The confining walls are known to influence the hydrodynamic behavior of

isolated rigid and fluid particles, both by exerting extra retardation force and by
altering the shape of fluid particles.

Chapter 10

examines such wall effects in

details. Starting with the extent of wall effects in Newtonian fluids, the available
scant body of knowledge for non-Newtonian fluids is reviewed here for rigid
and fluid particles. While the chapter is mainly limited to the sedimentation of
spherical and nonspherical particles in cylindrical domains, some key references
are also provided for the other shapes of confining boundaries.

Finally, in

Chapter 11,

attention is focused on the applicability of the falling

object rheometry for Newtonian and non-Newtonian systems. In particular, con-
sideration has been given to the methods based on the application of the falling
ball, the rolling ball, the rotating sphere, and the falling cylinder configurations.
While all these methods have proved to be of immense value (particularly at
high pressures and temperatures) for both qualitative and quantitative measure-
ments of viscosity and of product quality for Newtonian systems, their utility to
non-Newtonian fluids is severely limited. This is simply due to the complexity
of the flow field prevailing in these devices. However, under appropriate circum-
stances, it is possible to infer some useful rheological information, especially
about the zero-shear viscosity, yield stress, shear-dependent viscosity, etc. from
a knowledge of their settling velocity and geometrical configuration. In spite
of these limitations, most of these devices are extensively used in scores of
industries (food, paints, pharmaceuticals) for routine quality control purposes.

© 2007 by Taylor & Francis Group, LLC