Deformation and breakup of drops in simple shear flows
Citation for published version (APA):Bruijn, de, R. A. (1989). Deformation and breakup of drops in simple shear flows. Eindhoven: TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR318702
DOI:10.6100/IR318702
Document status and date:Published: 01/01/1989
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DEFORMATION AND BREAKUP OF DROPS IN SIMPLE SHEAR FLOWS
R.A. de Bruijn
DEFORMATION AND BREAKUP OF DROPS IN SIMPLE SHEAR FLOWS
DEFORMATION AND BR.EAKUP OF DROPS '
IN SIMPLE SHEAR FLOWS
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan
de Technische Universiteit Eindhoven, op ge
zag van de Ree tor Magnificus, prof. ir. M.
Tels, voor een commissie aangewezen door het
College van Dekanen in het openbaar te verde
digen op vrijdag 3 november 1989 te 14.00 uur
door
ROBERT ANTONIE DE BRUIJN
geboren te Heerhugowaard
Dit proefschrift is goedgekeurd door de promotoren
prof. dr. A.K. Chesters
en
prof. dr. ir. L. van Wijngaarden
CONTENTS
1. INTRODUCTION
2. SCALING IAWS FOR THE FLOY OF EMULSIONS
2.1 INTRODUCTION
page
1
5
5
2.2 EMULSIONS WITH CONSTANT INTERFACIAL TENSION 5
2.3 EMULSIONS WITH SURFACTANT ADSORPTION 12
2.3.1 Fundamental equations
2.3.2 Slowly varying flows
2.3.3 Rapidly varying flows
2.4 EMULSIONS CONTAINING SOLID PARTICLES
2.4.l Position of the particles
2.4.2 Particles in the fluid phases
2.4.3 Particles at the interface
2.5 EMULSIONS OF NON-NEWTONIAN LIQUIDS
2.6 CONCLUSIONS
2.7 REFERENCES
2.8 LIST OF SYMBOLS
3. NEWTONIAN DROP BREAKUP IN QUASI STEADY SIMPLE SHEAR
FLOWS
3.1 INTRODUCTION
3.2 LITERATURE AND THEORY
3.2.1 Introduction
3.2.2 Experimental results
3.2.3 Slender body theories
3.2.4 Numerical techniques
3.3 FLOW IN THE COUETTE DEVICE
3.4 EXPERIMENTAL
3.4.1 Description of apparatus
3.4.2 Experimental procedure
3.4.3 Results
3.5 CONCLUSIONS
12
18
20
21
21
22
22
23
28
29
30
33
33
34
34
35
36
37
38
47
47
48
53
59
4.
5.
CONTENTS
3.6
3.7
REFERENCES
LIST OF SYMBOLS
NON-NEWTONIAN DROP BREAKUP IN QUASI STEADY SIMPLE SHEAR.
FLOW'
4.1 INTRODUCTION
4.2 NON-NEWTONIAN FLUID MECHANICS
4.3 LITERATURE REVIEW
4.3.1 Theoretical results
4.3.2 Experimental results
4.3.3 Conclusions
4.4 BREAKUP OF INELASTIC, SHEAR THINNING DROPS
4.4.1 Model liquids
4.4.2 Drop breakup experiments
4.4.3 Discussion
4.4.4 Conclusions
4.5 BREAKUP OF VISCOELASTIC DROPS
4.5.1 Model liquids
4.5.2 Drop breakup experiments
4.5.3 Discussion
4.5.4 Conclusions
4.6 REFERENCES
4.7 LIST OF SYMBOLS
DEFORMATION AND BREAKUP OF NEWTONIAN DROPLETS IN
TRANSIENT SIMPLE SHEAR FLOWS
5.1 INTRODUCTION
5.2 PROBLEM STATEMENT AND BOUNDARY INTEGRAL METHOD
5.3 NUMERICAL METHOD
5.3.1 Choice of mesh
5.3.2 Evaluation of surf ace variables
5.3.3 Evaluation of surf ace integrals
5.3.4 Redistribution of the mesh
5.3.5 Numerical stability and convergence
page
59
61
63
63
63
67
67
70
81
81
81
85
92
100
102
102
110
113
127
127
129
131
131
132
140
141
142
143
145
147
CONTENTS page
5.4 EXPERIMENTS 152
5.4.1 Descript ion of the Couette device 152
5.4.2 Transient flow in the Couette device 153
5.4.3 Experimental procedure 155
5.5 NUMERICAL CALCULATIONS 157
5.6 EXPERIMENTAL AND NUMERICAL RESULTS 158
5.6.1 Step profile response 158
5.6.2 Triangle profile response 168
5.6.3 Sine profile response 175
5.7 DISCUSSION 175
5.7.1 Step profile experiments 175
5.7.2 Triangle profile experiments 186
5.7.3 Sine profile experiments 188
5.8 CONCLUSIONS 189
5.9 REFERENCES 190
5.10 LIST OF SYMBOLS 193
6. NEWTONIAN DROP BREAKUP IN SIMPLE SHEAR FLOW:
THE TIPSTREAMING PHENOMENON 147
6.1 INTRODUCTION 147
6.2 LITERATURE 198
6.3 EXPERIMENT AL 201
6.3.1 Introduction 201
6.3.2 Viscosity ratio 203
6.3.3 Time dependency 203
6.3.4 Acceleration 207
6.3.5 Surfactants 207
6.4 NUMERI CAL 214
6.5 DISCUSSION 218
6.6 CONCLUSIONS 225
6.7 REFERENCES 225
6.8 LIST OF SYMBOLS 226
CONTENTS page
SUMMARY 229
SAMENVATTING 233
ACKNOWLEDGEMENTS 237
CURRICULUM VITAE 239
APPENDIX A VISGOEIASTIC DROP BREAKUP IN LITERATURE 241
APPENDIX B DROP BREAKUP EXPERIMENTS 255
APPENDIX G EVALUATION OF SINGUIARITIES IN BOUNDARY
INTEGRAL METHOD 269
1. INTRODUCTION
This thesis is part of a long term investigation aimed at modelling
the operation of emulsifying devices in the food industry. In
these devices two imrniscible liquids are mixed to obtain a
distribution of droplets of one of the liquids in the other. Many
everyday products are on a micro-scale dispersions of one fluid in
another. Examples are margarine, an emulsion of water droplets in a
partly crystallised oil, mayonnaise, an emulsion of oil droplets in
a water phase and ice cream, which is a dispersion of gas bubbles
in a partly crystallised water phase. For all of these products the
averaged size and the size distribution of these droplets and
bubbles can affect many product properties. Hence there is
considerable interest in the modelling and optimising of these
emulsification processes. The results described in this thesis are
not restricted to the food industry. They can also be applied to
other emulsification processes e.g. polymer blending and oil
recovery, and even to flow phenomena that involve deformable
particles e.g. red blood cells.
The obvious approach to model the operation of emulsifying devices
is, on the one hand modelling flow and mixing in the devices
concerned and on the other hand drop break-up and coalescence as
local processes. Drop break-up and coalescence can be considered as
local processes when these phenomena are governed by the flow and
material properties in the immediate surroundings of the drops
involved. This is generally the case in emulsification processes.
When both approaches are integrated a full mathematica! model of
the emulsifying operation will result. The advantages of such an
approach are twofold. First the local processes need only be
modelled in a few well defined types of flow (e.g. laminar simple
shear flow, laminar extensional flow and turbulent flows) and the
results are thus applicable to a range of emulsifying devices
operating under the same types of flow. Second the modelling of the
flow and mixing in a particular device can be used for the
modelling of other physical processes as well (e.g. heat transfer
2
and crystallisation). This modular approach will only be successful
when both the flow and mixing in the emulsifying device and the
local drop break-up and coalescence processes can be modelled
accurately. With the take off of computational fluid dynamics
accurate modelling of flow and mixing in process units has recently
become possible. In the knowledge of the drop break-up and
coalescence processes, however, a number of very important aspects
are still not understood.
The work in this thesis aims to descrlbe some aspects of the
viscous break-up of droplets in slmple shear flows. Viscous break·
up is valid when the drop deformation and break-up is induced by
viscous shear stresses and inertial effects can be neglected. This
situation generally applies to the break-up of small droplets in
highly viscous llquids. Three of those aspects in the viscous
break-up of droplets in simple shear flows have been identif ied
which are poorly understood and are very important for the
modelling of emulsification processes. First the break-up of non
Newtonian drop phases. This is important since many food products
exhibit a markedly non Newtonian behaviour. The available data on
non-Newtonian drop break-up are however scarce and inconsistent.
Second the break-up of droplets in transient shear flows is not yet
well described. This behaviour must however be known if
emulsification processes are to be described in which droplets
experience changes in the flow conditions during time scales in
which the droplet can deform and break-up. Third the origin of the
tipstreaming phenomenon in simple shear flow is not clear, nor is
it known when this mode of break-up will occur. This knowledge is
however important since tipstreaming will result in a greatly
different drop size distribution from the normal mode of drop
break-up. The work described in this thesis is an attempt to
understand and describe these aspects of viscous drop break-up in
simple shear flow.
This thesis starts with an introductory chapter on basic equations
and sealing laws for the flow of emulsions (chapter 2). These
should be known for both theoretical and experimental studies in
this area. The following situations are considered: emulsions with
constant interfacial tension, emulsions with surfactant adsorption,
emulsions containing solid particles and emulsion of non-Newtonian
liquids. For each of these systems the practical limitations of
application of sealing laws to the local processes of drop break-up
and coalescence are considered. Next a Couette device, which was
developed for experimental investigation of drop break-up in simple
shear flow, is described (chapter 3). The Couette device operates
on the principle of two counter rotating cylinders. A stagnant
layer is thus created in the gap between the two cylinders which
permits statie observations of droplets in quasi steady and
transient simple shear flows. This device has been successfully
tested by measurement of Newtonian drop break-up in quasi steady
simple shear flow. In the fourth chapter of this thesis the Couette
device was used to study experimentally the break-up of non
Newtonian droplets in quasi steady simple shear flows. Two
particular types of non-Newtonian behaviour were studied: shear
thinning liquids with viscosities obeying the power law equation
but with negligible fluid elasticity and viscoelastic liquids with
substantial elasticity combined with a shear rate independent
viscosity. These particular types of non-Newtonian drop phases were
chosen in order to separate the effects of shear rate dependent
viscosities and fluid elasticity. For most liquids these effects
occur simultaneously, which makes it very difficult to separate
these effects. In chapter 5 of this thesis the deformation and
break-up of Newtonian droplets in transient simple shear-~s is
studied. This investigation is mainly of numerical nature but is
supported by experimental work. A computer programme is developed
which calculates the evolution of Newtonian droplets in any
transient shear flow. The programme is based on the boundary
integral method by which the creeping flow equations inside and
outside the droplet are transformed into a form that only involves
quantities at the drop interface. This method is based on the
Fourier solution to the creeping flow equations and uses volume
potential theory. This programme is used to calculate the shape of
3
4
droplets with various viscosity ratios as a function of time for
various shear rate profiles: step profiles, triangular profiles and
sinusoidal profiles. The work was supported by drop deformation
triangular profiles of the shear rate. In the final chapter of this
thesis a special mode of drop break-up, tipstreaming, is
investigated. Tipstreaming is an experimentally observed mode of
drop break-up in which the droplet takes, upon increasing the shear
rate a sigmoidal shape and a stream of very small droplets is
ruptured off the tips of the drop. The investigations in this
chapter are aimed at unravelling the causes of this phenomenon and
use both experimental and numerical techniques.
2. SCALING LAWS FOR THE FLOW OF EMULSIONS
2.1 INTRODUCTION
For both theoretical and experimental study of emulsification
processes sealing laws for multiphase flow should be known. On the
theoretica! front these laws indicate the number and the nature of
the dimensionless groups concerned, while on the experimental front
they indicate the feasability of conducting observations at
practically convenient time and space scales. In this introductory
chapter the sealing laws are derived from the fundamental equations
governing liquid flow for the following multiphase systems:
emulsions with constant interfacial tension
emulsions with surfactant adsorption
- emulsions containing solid particles
- emulsions of non-Newtonian liquids.
For each system the practical limitations of application of sealing
laws to the sealing of emulsion break-up and coalescence are
considered.
2.2 EMULSIONS WITH CONSTANT INTERFACIAL TENSION
In this section the sealing laws will be derived for emulsions with
constant interfacial tension in which colloidal effects may be
neglected. Although the results obtained are not new, see for
example Chesters (1975), the derivation is illustrative of the
approach required, which will be extended to more complex
situations (sections 2.3 and 2.4).
The equations governing isothermal fluid flow in emulsions are the
equations of motion in both phases and the kinematic and stress
boundary conditions at the droplet interface. For isothermal,
incompressible Newtonian fluids the equations of motion are given
by the Navier-Stokes equation and the continuity equation.
5
6
The first is given by:
Dy p
Dt
811 p - + p (y • V) l.! = - v p + '1 v2 y + Eext
8t [2-1)
in which Eext denotes the external body forces per unit volume. In
what follows the only external body force considered is that due to
gravity, Egrav = pg, but the approach can be extended, if relevant,
to other body forces such as the fictitious ones, (centrifugal,
Coriolis) in systems viewed with respect to rotating axes.
The latter is given by
V.J.!= 0 [2-2)
The kinematic interface condition, neglecting interfacial mass
transfer is:
Yc = l.!d = Yinterface [2-3]
yielding the kinematic condition for the displacement of a material
point in the interface in the absence of interphase mass transfer
D~interf ace
~~~~~- = Yinterface = Yc [2-4]
Dt
From the assumptions of a massless interface and a constant
interfacial tension the stress equilibrium equations reduce to a
balance between interfacial tension forces and viscous traction
exerted by the fluid phasès. The equilibrium of the tangential
stresses is given by
!.t,d = Q [2-5]
where !.t,c and !.t,d are the tangential stresses exerted by/on the
continuous and dispersed phase respectively. The equilibrium of the
normal stresses is given by
[2-6]
where Tn,c and rn,d are the normal stresses exerted by/on the
continuous and dispersed phase respectively. The fluid stresses L can be calculated from the flow pattern and the constitutive
equation. For an incompressible Newtonian liquid the constitutive
equation is given by
in which Óij is the Kronecker delta defined by Óij
ó ij = 0 ( i;o<j ) .
[2-7]
1 {i=j) and
Similarity criteria can be derived by making the equations
dimensionless. All variables can be made dimensionless by a
combination of U, Land p, where U and L denote a characteristic
velocity and length scale of the system.
u u=-
u u L
. t, p
L'il, p = pu2' 1 Ri x
T = ' x L L [2-8)
In absence of external forces other than gravity this leads to the
following dimensionless equations.
7
8
In each fluid:
aü -= + <li • v) 11 at
(g : unit vector in direction of g)
v • 11 0
At the interface:
À Lt,c · Lt,d Q
À rn,c
in which the following dimensionless groups appear.
Re - p U L q·l (Reynolds number)
Fr u2 g·l L·l (Froude number)
We p u2 L ~-1 (Weber number)
À 11d 11·1 (viscosity ratio)
[2-9)
[2-10]
[2-11)
[2-12)
[2-13]
[2-14]
Re is a measure of the relative importance of inertial to viscous
forces, Fr of the relative importance of inertial to gravitational
forces and We of the relative importance of inertial to interfacial
forces. Since equations [2-9) to [2-11] apply to both phases they
yield the dimensionless coefficients
(where no subscripts are used, the dimensionless parameters refer
to the continuous phase). The complete set of dimensionless
parameters describing the flow in and around a single emulsion
droplet is thus
Re, Red, Fr, Frd, À and We
This set can be reduced and modified into a more f requently used
set (since every parameter can be replaced by any combination of
itself with other parameters):
Re, Fr, We, À and~
in which
~ - Pd p-l (density ratio)
The advantage of the dimensionless description follows from the
fact that systems characterized by the same value of the
dimensionless parameters and having the same dimensionless initial
and boundary conditions will behave exactly the same, apart from a
sealing factor. In the case of time dependent flows these boundary
conditions should also include the externally applied time scales.
These are often described by the dimensionless Strouhal number:
St - L/UT. For any practical situation it is almost impossible to
satisfy all five similarity criteria simultaneously. This becomes
clear if we replace Re, Fr and We by the following three
independent dimensionless coefficients:
We Fr
9
10
M, the Morton number, is a very special dimensionless parameter
since, apart from the gravitational constant, it consists of fluid
properties only. Sealing at constant gravity is thus only possible
for liquids having the same value of M. From Table 2.1 it will be
clear that only a few liquids can be scaled completely. However it
is often possible to relax one or more of the similarity criteria
without significant effect on the flow properties. One of these
cases will be discussed briefly.
Many emulsions of practical interest consist of very small
droplets. As a consequence the Reynolds and the inverse of the
Froude number based on the droplets are very small, indicating that
neither the gravitational forces nor the inertial forces affect the
flow in and around a droplet. For such flows the exact values of Fr
and P are irrelevant. Then the only relevant dimensionless
coefficients are those combinations which do not contain p or g,
namely n and À, where
u ~ n = We Re-1 = ~o
(Capillary number)
Under these conditiQns it is often possible to find model liquids
which allow sealing. Generally this can be done by maintaining À
but increasing viscosities and decreasing the velocities and
keeping o approximately constant. Using these sealing criteria a
large increase in length and time scales can be obtained which
permits visualisation of the phenomena such as break-up which are
generally very difficult to monitor in practical emulsions due to
the very small time and length scales involved. For emulsions with
constant interfacial tension silicone oils are a very flexible
model liquid, since a large range of viscosities are available
commercially (see Table 2.1).
TABLE 2.1 Fluid properties
Fluid M [-J
Water 1. 0 103 1.0 10-3 7. 3 10-2 2.5 10-11
Glycerol 1.3 103 1. 8 6.3 10-2 3 .2 102
Corn. syrup (25% water) 1.3 103 1.2 10-1 2.9 10-2 6.4 10-2
Corn. syrup (10% water) 1. 4 103 2.8 3.5 10-2 1. 0 104
Ethanol 7 ,9 102 1.2 10-3 2,3 10-2 2.1 10-9
Mercury 1.4 104 1.6 10-3 4. 7 10-1 4.4 10-1•
Sunflower Oil 9.2 102 6.0 10-2 3.4 10-2 3.5 l0-3
Castor 011 9. 7 102 7. 7 10-l 3.6 10-2 7. 6 101
Silicone Oil 47 v1*) 8. l 102 1.0 l0-3 1.7 10-2 2.5 10-g
Silicone Oil 47 VlOOO*) 9.7 102 1. 0 2.1 10-2 1.1 103
Silicone Oil 47 v100.ooo*) 9. 7 102 l.O 102 2.1 io-2 1.1 1011
*) ex Rhone Poulenc
11
12
The sealing criteria derived above are only valid when colloidal
effects can be neglected. This is more restrictive than it at first
appears: in any rupture process, Van der Waals forces furnish the
final destabilization. This applies to the rupture of a filament of
liquid connecting a splitting drop and also to the rupture of a
thin film of liquid separating two coalescing drops. In the farmer
case the exact level of the Van der Waals farces will have
negligible influence on the rupture time since the final pinching
phase is so fast. In the case of coalescence, however, the last
stage of film thinning is typically the slowest and the magnitude
of the Van der Waals forces will then be of major influence on the
process. Consequently, while the preceding relations can be applied
with confidence to the sealing of drop break-up processes, this
will not in genera! be the case for coalescence processes.
2.3 EMULSIONS WITH SURFACTANT ADSORPTION
2.3.l Fundamental equations
The interfacial tension of an emulsion droplet may usually not be
regarded as constant when surfactants are present. This is
especially so when the fluids are in motion and the emulsion
droplets are being deformed. Droplet deforrnation leads to
enlargernent of the interfacial area thus tending to change
concentration of the adsorbed surfactant. Surfactant adsorption
will compensate such changes in concentration hut can be a much
slower process than droplet deforrnation, thus leading to non
equilibrium surfactant concentrations at the interface and to
surface tension gradients.
Such gradients rnodify the stress boundary conditions at the droplet
interface (equations (2-5] and [2-6]). The equations describing
fluid flow in these ernulsions are thus not only the equations of
motion in both phases (equation [2-1] and [2-2]) and the kinematic
and stress equilibrium conditions at the droplet interfaces (the
modified verslons of equation [2-3), [2-5) and [2-6]). When
surfactants are present one should also take into account the
surfactant mass conservation relations and the interfacial
equations of state. If it is asswned that the surfactant is soluble
in the continuous phase only, the mass conservation relations at
the interface and in the continuous phase should be valid. For the
continuous phase the dif fusion equation in a flowing material is
given by
De
[2-15]
Dt
in which c, denotes the surfactant concentration in moles
surfactant per m3 solvent in the bulk and V the surfactant
diffusion coefficient and use has been made of the diffusion law of
Fick:
diffusional mass flux - ID 'ï/c
The surfactant mass conservation relation at the interface is given
by
Dr l DA 8c
- r +ID [2-16]
Dt A Dt 8n
in which
r: surface concentration of surfactants [moles m-21
A: area of interfacial element.
The first right hand term in equation [2-16) denotes the change in
surface concentration due to interfacial dilation, while the second
term denotes the change due to mass transport between bulk and
interface. The bulk and surface mass conservation relations [2-15]
13
14
and [2-16] are connected by the condition that the surface
concentration is in instantaneous equilibrium with the bulk
concentration immediately adjacent to the interface (cs>· For low
surface concentration this relation can often be described by the
Langmuir adsorption isotherm:
r [2-17]
in which rw and c1 are material constants.
This relation is only valid when lateral interactions between the
adsorbed molecules can be neglected.
For rather dilute solutions of simple surfactants (r << r00 ) this
equation reduces to
r" r [2-18]
This equation is known as the two dimensional Henry equation.
The equations of state of the interface relate the surface
concentration of surfactant to its effect on the interfacial
tension. When the adsorption is given by the Langmuir equation [2-
17], the surface equation of state is often given by the Frumkin
equation which is derivable from the Langmuir adsorption equation.
[2-19]
r which for low adsorption ~ << 1 reduces to
roo
r a - RT r~ RT r [2-20]
which is the ideal surface equation of state. In the equations
above, a0 is the interfacial tension in the absence of adsorbed
surfactant. These equations lose their validity if the adsorbed
surfactant molecules start interacting. The validity range of these
equations is thus for proteins and ether macromolecules rather
small. For small surfactant molecules the validity range is usually
larger. For such molecules the error in the Frumkin equation is
often smaller than 15% as long as r;r00 < 0.25. However, the
decrease in interfacial tension is usually still very small in that
range. Typically:
- a < 5 m Nm-1.
The scope for application of sealing criteria is thus rather
limited due to the small range of conditions under which surfactant
behaviour can be modelled easily.
If the interfacial tension is not constant the tangential stress
equilibrium equation [2-5] should be modified to include
interfacial tension gradients:
Lt,c - Lt,d - Vs a [2-21]
These gradients can effectively irnrnobilise the droplet interface
and are responsible for the increased drag coefficient observed for
bubbles in the presence of surface active materials.
15
16
The basic equations governing the flow of emulsions with adsorbed
surfactants have now been given.
In order to derive similarity criteria, these should be made
dimensionless using equation [2-8] and additionally
ë c r n a - r =-
' n= and ä
Co r"' L ao [2-22]
Thus yielding:
aü <i! V) - -v P + Re·l v2 Fr·l -+ . ,!! ,!! + g
at [2-23]
(2-24]
ij • u 0 (2-25]
-Uc ud = Uïnterf ace [2-26)
À Ît,c - it,d = 0 Vs ä (2-27]
[2-28]
Dë [2-29]
Bt
Dr Dt
(2-30]
or r [2-31]
a = 1 + Nsp ln (1-r) or a = l · Nsp r [2-32]
Complete similarity will only be obtained if all dimensionless
parameters are equal and if the same dimensionless boundary and
initial eonditions apply. Besides the dimensionless parameters
eneountered in the previous seetion, viz. Re, Fr, n, p and À some
new parameters appear whieh are defined by
Pe U L
ID
(or equivalently
roo No
L.e00
NA e1
eoo
Ngp -RT r00
ao
(Peelet-number)
v Se - sinee Pe - Re• Se).
ID
(Distribution - number)
(Adsorption - number)
(Surfaee pressure - number)
Pe deseribes the relative importanee of eonveetive surfaetant
transport to diffusive transport.
No deseribes the distribution of surfactant over interface and
bulk.
NA is a parameter indieating the proximity to saturated
adsorption.
17
18
Nsp• finally, describes the lowering of the interfacial tension
(the so-called surface pressure) relative to the interfacial
tension of a clean interface, as a function of surfactant
adsorption.
In what fellows the sirnplification which was introduced in the
previous section will also be assumed to be valid, namely that
Re << 1 and Fr << 1. It was already discussed that for the flow of
emulsions these simplifications are usually allowed. The set of
dimensionless groups thus reduces to:
Full sealing of the flow of emulsions with surfactant adsorption is
very difficult. The scope for sealing is limited by the difficulty
involved in keeping both Q and Pe or equivalently Se constant
during scale-up. As was shown in the previous section upscaling
with Q constant can be done by increasing viscosities. To keep
Se v/ID constant ID should thus be increased by the same factor
as v, but unfortunately higher viscosities tend to decrease ID. It
is thus very difficult to scale the surfactant adsorption process
completely.
In the following some special cases will be considered which allow
all relevant similarity criteria to be fulfilled.
2.3.2 Slowly varying flows
The case of slowly varying flows applies when diffusion processes
dominate over convection processes: Pe << 1. In this situation the
surfactant concentration will at all times be uniform throughout
the bulk, including the region adjacent to the interface. Thus r
and a will be constant at the drop interface, albeit with a
reduced interfacial tension.
'When overall surfactant gradients are maintained the conditions Pe
<< 1 is very restrictive. It implies that
U L 'Y
Pe = --~ << 1
ID ID
in which the characteristic length scale L represents a global
length scale. In this situation the flow velocities need to be so
low that the droplet will hardly be deformed. This case is thus
hardly relevant to emulsification processes.
However when no overall surfactant gradients are maintained,
surfactant molecules only need to diffuse in a small layer around
the droplet to ensure uniform surfactant concentration. In this
situation the characteristic length scale can be given by the
layer thickness L that contains as much surfactant as is needed to
cover the interface with an equilibrium adsorption:
L r/c
The above relation implicitely assumes that the layer thickness is
small compared to the drop size. It will be seen that this
condition is usually satisfied. r and c are related via the
adsorption isotherm. In the dilute case, for example, [2-18]
yields
The value of c1 varies considerably from one surfactant to
another, but is typically such that L is of the order of microns
or less, which is often smaller than the drop size. At large
surfactant concentrations r does not increase linearly with c
anymore and consequently L becomes even smaller. For example,
consider a surfactant at high concentration, such that
19
20
where N denotes the Avogadro number and Am the area occupied by
one adsorbed surfactant molecule. The latter is generally of the
order of 50 ft2, yielding r~ - 3 • 10-6. For a bulk concentration
of 30 mole/m3 the diffusive length scala becomes 0.1 µm. Fora
typical diffusion coefficient of ID - io-10 m2s-l this implies that
the flow can be considered as 'slowly varying' for shear rates up
to io-4 s-1. Clearly there are many practical situations that
satisfy these conditions of slowly varying flows.
2.3.3 Rapidly varying flows
Flows will be termed rapidly varying when convective mass
transport dominates diffusion (Pe >> 1). In this case mass
transport between bulk and interface will hardly occur. Variations
in surface concentration and interfacial tension thus will only be
due to changes in the interfacial area. The exact values of Pe is
thus not important as long as Pe >> 1. If in addition the
surfactant adsorption can be described by a linear adsorption
isotherm, the values of NA and Nn are not important either. The
set of relevant dimensionless groups is therefore reduced to: n, À
and Ngp.
From the example described in the case of slowly varying flows
(section 2.3.2) the applicability of Pe >> 1 is restricted to
concentrated solutions and extremely high shear rates (~ >> lo-4).
For low surfactant concentrations combined with high shear rates
one can however often apply this simplification successfully.
2.4 EMULSIONS CONTAINING SOLID PARTICLES
In many emulsions solid particles are present in the bulk phase or
at the interface as fat crystals in fat spreads and creams.
In this section sealing laws will be derived for these systems to
be able to study the effects of these particles on droplet break-up
and coalescence processes.
2.4.1 ~Qsition of the particles
The equilibrium particle position is determined by minimal total
free energy. However, minimizing the interfacial free energy Fs
also gives correct equations. Fs is defined by
[2-33]
where A denotes the interfacial area, u the interfacial tension and
the subscripts p, c and d refer to particle, continuous and droplet
phase respectively. The particle position proves to be determined
by the adhesion constant C.
c [2-34]
Four ranges of equilibrium positions and corresponding C-values can
be discerned.
c < -1 particles are completely wetted by the continuous
phase.
-1 < C < 0: particles are positioned at the interface, largely
wetted by the continuous phase.
21
22
0 < c < 1
G > 1
particles are positioned at the interface, largely
wetted by the droplet phase.
particles are completely wetted by the droplet phase.
For !Cl < 1, the adhesion constant is equal to the eosine of the
contact angle, as follows from the Young's equation
Ucd • COS 0 [2-35)
The contact angle is the angle between the cd and pd interfaces,
measured in the liquid with the higher density.
2.4.2 Particles in fluid phases
Yhen the particles are completely wetted by one of the fluid phases
the additional sealing requirements can usually be fulfilled.
It will be obvious that scaled-up particles should have the same
shape, density, relativa size and concentration and should be
wetted by the same fluid phase as the original particles.
As we have seen both gravitational and inertial forces are often
negligible for emulsions. However, care should be taken that this
still holds for scaled-up emulsions containing solid particles.
Even if the gravitational and. inertial forces are negligible for
the original emulsion it is thus better to use neutrally buoyant
particles.
2.4.3 Particles at the interface
The contact angle determines, as has been argued, the position of
the particles in the interface. Yhen sealing these emulsions one
should not only fulfill the requirements described in the previous
section, but also keep the contact angle constant. Since
macroscopie particles can sometimes be coated to give the required
surface properties, this can probably be satisfied. However, when
sealing dynamic processes, particles may move relative to the
interface. Thus, not only statie contact angles, but also dynamic
angles may occur.
Indications are that differences between dynamic and statie contact
angles scale with the capillary number (E.B. Dussan V 1979, Hoffman
1975) and so to the first approximation will be covered by the
foregoing sealing requirements.
A certain additional influence of system scale is however to be
expected, which will exclude vigorous sealing (De Gennes, 1985).
Serious sealing difficulties may be expected if the process of
entry of particles into the interface from one of the liquids is to
be scaled, since this again involves the rupture of a thin film of
liquid (see section 2.2).
2.5 EMULSIONS OF NON-NEWTONIAN LIQUIDS
In this chapter we will consider the sealing laws for emulsions in
which one or bath of the phases are non-Newtonian. Sealing of such
emulsions is many times more difficult than for emulsions
consisting of Newtonian liquids. The description of the flow of
non-Newtonian liquids involves a modified equation of motion in
which generally a rather complicated constitutive equation is
inserted. The modified equation of motion is given by;
Dy
P--~ - VP+ V.z. +Eext
Dt
[2-36]
23
24
The extra stress tensor L is related to the flow field by the
constitutive equation. For a Newtonian liquid this constitutive
equation is given in equation [2-1) and can be characterised by a
single fluid property, the dynamic viscosity. For non·Newtonian
liquids one generally needs more than one fluid property to
characterise the constitutive equation. The sealing of such
liquids therefore becomes increasingly difficult. In this chapter
we will only consider a certain type of non-Newtonian behaviour,
which includes on the one hand purely viscous behaviour with a
shear rate dependent viscosity, described by the power law model,
and on the ether hand viscoelastic behaviour. This behaviour can be
modelled by the so·called Criminale-Erickson-Filbey (CEF)
constitutive equation (Schowalter (1978)). This constitutive
equation is a simplification of the retarded motion expansion,
valid for steady flows. The CEF constitutive equation is in general
given by:
q(,Y) N1(Ï') N2()') N1 ( )') 0
L =--Q + 2 ('1-- + --) !2 !l - --!l [2-37)
'Î .y2 -y2 72
In this equation !l is the rate of strain tensor defined by:
[2-38)
.Y is the magnitude of the rate of strain tensor, defined by:
-Y = 2 J<Il:Q) [2-39]
fi is the corotational time derivative of the rate of strain tensor
and is defined by:
[2-40]
In this equation E is the rate of spin tensor defined by:
[2-41]
In equation [2-37] r1, N1 and N2 are the viscometric material
functions. These functions can be obtained from rheometrical
experiments in simple shear flows. In such a flow r is the
tangential stress measured, N1 is the first normal stress
difference and N2 is the second normal stress difference. In this
chapter we will only consider the following forms of the
viscometric material functions:
[2-4la]
[2-4lb]
[2-4lc]
This form of these functions was chosen to include the well known
power law liquid for which ~-0. The power law liquid exhibits a
shear rate dependent apparent viscosity, but does not show any
normal stress differences. The functions also include the so-called
Boger type liquids which exhibit an apparent viscosity which is
almost shear rate independent and show a first normal stress
difference.
To obtain the sealing rules for a liquid obeying the CEF
consitutive equation with the above defined viscometric material
functions equation [2-37] should be inserted in equation [3-36] and
the resulting equation should be made dirnensionless. All variables
can be made dimensionless by a combination of U, Land p (see also
equation [2-7]).
u
L .Y--i'
u
p
P u2
-t u --t L
L -~ u
LV
- L W=-W - u-
[2-59]
25
26
In absence of external forces other than gravity this leads to the
following dimensionless equation:
aü ~ + <ii • v)j! at
- - l 7n-l 1 -m-2 ° -V P + - '1 R + -- ('7,;, (2,R·,R - ,R)
Re* Re** '
[2 51]
In this equation Re* and Re** are modified Reynolds numbers
describing the ratio of the viscous stresses to inertial stresses
and the ratio of elastic stresses to inertial stresses
respectively. These modified Reynolds numbers are defined by:
Re* -K (U/L)n
[2-52] Pd u2
Re** -ic(U/L)m
[2-53] Pd u2
If we only consider the droplet phase to be non-Newtonian, the
other dimensionless equations are given by the continuity equation
and the kinematic and stress equilibrium conditions at the droplet
interface:
v . .l,i o
!!interface
- * - - À** - - 0 2 Rt,c - 2 À Rt,d <R·R - R>t,d
2 ~n,c - 2 À* ~n,d
[2-54)
Q [2-55)
We Re·l (R1-l + R2-l)
[2-56]
In these equations the dimensionless parameters À* and À** denote
the modified viscosity ratios, describing the ratio of the viscous
forces in the drop to the viscous forces in the continuous phase
and the ratio of elastic forces in the drop to the viscous forces
in the continuous phase respectively. These dimensionless
parameters are defined by:
K (U/L)n-1
~
** ~ (U/L)m-1 À =
The following dimensionless parameters have thus appeared in the
above equations:
Re, Re*, Re**, Fr, À*, À**, We, m, n.
This set of parameters can be modified and reduced into a more
generally used set of dimensionless parameters (since every
parameter can be replaced by any combination of itself with other
parameters):
Re, Fr, We, SR, ~. À*, mand n.
As compared with the situation described in section 2.2 for the
sealing of Newtonian liquids, there are 3 extra dimensionless
parameters viz. n, m and SR, while instead of the viscosity ratio À
a modified viscosity ratio À* should be used. This very much
restricts the possibilities for sealing. Many emulsions of
practical interest consist of very small droplets, thus allowing
neglect of the inertial and gravity forces. In this situation the
set of dimensionless parameters reduce to:
Sealing of emulsions containing inelastic shear thinning droplets
is only possible (since SR m = 0) if liquids are used, which have
the same value for the power law index n. If such liquids are found
sealing will still be less straightforward than for emulsions
consisting of Newtonian liquids since the modified viscosity ratio
À* is not constant (as is the viscosity ratio À) but also depends
on the flow conditions.
Although fully inelastic shear thinning liquids do not exist, there
are several shear thinning liquids that are almost inelastic. 27
28
Examples are e.g. solutions of Carboxy Methyl Cellulose (CMC),
certain solutions of the poly acrylic acid Carbopol in water/corn
syrup mixtures pnd dispersions of the silica particles Aerosil in
mineral oils. These model liquids can be made with power law
indices varying in the range 1-0.3.
Sealing of emulsions containing visco-elastic droplets obeying the
CEF constitutive equation is only possible when liquids are used
which have the same values for the indices m and n. In general this
is almost impossible. However a special class of model liquids bas
been developed that exhibit an almost shear rate independent
viscosity (n = 1) combined with elastic effects described by m = 1.5 - 2. These so-called Boger liquids are made of solutions of
poly-acryl-amide in water/corn syrup mixtures (Boger 1977/1978,
Boger and Nguijen 1978 and De Bruijn chapter 4 of this thesis.
2.6 CONCLUSIONS
Sealing laws for emulsion break-up and coalescence processes have
been derived for:
- emulsions with constant interfacial tension
- emulsions with surfactant adsorption
- emulsions containing solid particles
- emulsions of non-Newtonian liquids.
Sealing laws can be applied without great difficulties to
emulsions with constant interfacial tension.
Sealing of emulsions with surfactant adsorption is usually only
possible in "slowly" varying flows in which surfactant diffusion
dominates convection resulting in an equilibrium surface tension
all over the droplet, or in dilute "rapidly" varying flows in which
adsorption and desorption processes may be neglected.
Sealing of emulsions of non-Newtonian liquids is usually only
possible when a homologous series of liquids is available with the
same eonstitutive equation (e.g. shear thinning liquids and Boger
liquids).
Sealing of emulsions eontaining solid particles will beeome
diffieult if the proeess of entry of partieles into the interface
from one of the liquids is to be sealed.
2.7 REFERENCES
1. D.V. Boger, A highly elastie eonstant-viscosity fluid, J.
Non-Newtonian Fluid Meeh. 3, 87-91, (1977/1978).
2. D.V. Boger and H. Nguyen, A model viseoelastie fluid, Polym.
Engng. Sei. 18, 1037-1043, (1978).
3. R.A. de Bruijn, Newtonian drop break-up in quasi steady
simple shear flow, chapter 3 of this thesis.
4. R.A. de Bruijn, Non-Newtonian drop break-up in quasi steady
simple shear flow, chapter 4 of this thesis.
5. A.K. Chesters, The applicability of dynamic similarity
criteria to the isothermal, liquid-gas two phase flows
without mass transfer, Int. J. Multiphase Flow 2, 191-212,
(1975).
6. E.B. Dussan V., On the spreading of liquids on solid
surfaces: statie and dynamic contact lines, Ann. Rev. Fluid
Mech. 11, 371-400, (1979).
7. P.G. de Gennes, Wetting: staties and dynamics, Rev. Modern
Physics 57,827-863, (1985).
8. R. Hoffman, A study of the advaneing interface. 1. Interface
shape in liquid gas systems, J. Colloid Interface Sci. 50,
228-241, (1975).
9. G.G. Ngan, and E.B. Dussan V., On the nature of the dynamic
contact angle: an experimental study, J. Fluid Mech. 118,
27-40, (1982).
10. W.R. Schowalter, Mechanics of non-Newtonian fluids, Pergamom
Press, Oxford, 1978.
29
30
2.8 LIST OF SYMBOLS
interfacial area
adhesion constant
material property in Langmuir
adsoprtion isotherm
c surfactant concentration
~ rate of strain tensor
ID diffision coefficient
E body force per unit volume
Fs interfacial free energy
Fr Froude number
g gravitational acceleration
K consistency index
k Boltzmann constant= 1.3806 io-23
L length scale
M Morton number
m elastic power law index
NA Adsorption number
No Distribution number
Nsp Surface pressure number
N1 first normal stress difference
N2 second normal stress dif ference
n power law index
n normal vector
P pressure
Pe Peclet number
R Gas constant = 8.3143
Ri radius of curvature
Re Reynolds number
SR stress ratio
St Strouhal number
T temperature
t time
U characteristic velocity
!.! velocity
[moles m·3]
[moles m·3]
[s" 1 J
[m2 s·l1
[N m·3]
[Nm]
[ l [m
[Pa.sn]
[JK-1]
[m]
[ J [. l [ ·]
[ - J
[ - l [Pa]
[Pa]
[. l
[ - l [Pa]
[ -1 [JK·l mo1·l1
[m]
[ -1 [ - l [ - 1 [KJ
[s]
[m s· 11 [m s·l1
LIST OF SYMBOLS (Continued)
H rate of spin tensor
We Weber number
x position
fi density ratio
r surface concentration of
surf actants
r 00 material constant in langmuir
adsorption isotherm
~ rate of shear
Óij Kronecker delta
q dynamic viscosity
U contact angle
K elasticity index
À viscosity ratio
p density
a interfacial tension
L stress tensor
TI tangential stress function
0 Capillary number
SUB SCRIPTS
c continuous phase
d disperse phase
ext external
grav gravitational
n normal
0 original
p particle
s surf ace
t tangential
[ l [ - l [rn)
[ - J
[moles rn- 2 J
[rnoles rn" 2 J
[s-1]
[ - l [Pa.s]
[rad]
[Pa.sm]
[ l [kg m-3)
[N rn·l]
[Pa]
[Pa]
[ - l
31
32
SUPERSCRIPTS
dimensionless quantity
*•** modified
3. NEWTONIAN DROP BREAKUP IN QUASI-STEADY SIMPLE SHEAR FLOWS
3.1 INTRODUCTION
The present chapter deals with the modelling of the break-up of
Newtonian droplets in a Newtonian continuous phase in quasi steady
simple shear flows. Break-up criteria for droplets in simple shear
flows are available from the literature (both theoretical and
experimental) provided both the droplet and the continuous phase
are Newtonian, the shear rate is increased very slowly (quasi
steady simple shear) and the interfacial tension is constant over
the entire drop interface. These results are surveyed in section
3.2 and can serve as a starting point for further studies. These
are considered necessary because several important effects which
are known to have a great influence on drop break-up are not yet
sufficiently understood. In many emulsifying devices regions exist
with strongly varying shear rates. An example is the flow in almost
any stirred vessel. When strongly varying shear rates are exerted
on the droplet within time scales which are of the order of the
characteristic droplet break-up time very different break-up
criteria and modes of break-up were observed (Grace, 1982). These
effects are very important for the description of emulsification in
process units. Droplet break-up is also greatly affected when the
droplet or continuous phases are markedly non-Newtonian. This is
often the case for food processing where markedly non-Newtonian
ingredients are used.
Therefore an experimental device was developed to study the break
up of droplets in simple shear flows experimentally.
The device works according to the Couette principle with two
concentric counter-rotating cylinders and is described in sections
3.3 and 3.4.
The Couette device has been tested by reproducing experimental
results described in the literature. The Couette device has also
been used to study the break-up of non-Newtonian drops in quasi
steady simple shear flows (chapter 4) and to study the break-up
33
34
of drops in transient simple shear flows (chapter 5). The devices
is also used the study the tipstreaming phenor.1enon observed in
simple shear flows (chapter 6).
3.2 LITERATURE AND THEORY
3.2.1 Introduction
Taylor (1932) laid the foundations for the theoretical description
of the deformation of a small viscous droplet in a viscous
suspending liquid. Following the work of Einstein (1906) on the
viscosity of a fluid containing solid particles Taylor derived a
theory on the viscosity of an emulsion. These theories are based on
the method derived by Lamb (1932) using spherical harmonies to
describe the solution of the creeping flow equations in and around
the drops. Taylor assumed that the radii of the droplets were small
enough for the creeping flow equations to be valid (Reynolds number
based on the droplet smaller than unity) and that the deformation
of the droplets is very small. Taylor irnposed tangential stress
continuity between the suspending and the dispersed fluid phase and
a discontinuity in the normal stresses balanced by the Laplace
pressure. Furthermore Taylor assumed no slip conditions at the
interface for the velocities. These boundary conditions are still
generally applied, although they are usually not valid in the
presence of surfactants, see chapter 2 of this thesis. From the
velocity and stress distribution in and around a spherical droplet
Taylor derived a linear dependency of the deformation of a droplet
on the dimensionless shear rate. This relation is based on the
Laplace pressure distribution needed to compensate the normal
stress differences across the spherical drop interface, assuming
that the (small) deformation does not affect the normal stress
distribution.
This small deformation theory was extended and generalized by Cox
(1969). Cox adjusted the normal stress boundary condition by
balancing the fluid stresses with the interfacial tension forces on
the deformed surface and proposed a problem formulation in terms of
the drop excentricity. Cox examined the general problem of drop
deformation in transient linear shear flows within the limit of
first order ellipsoidal deformation of the drops. A major gain of
Cox's work is the possibility to expand it to higher order in the
drop excentricity. This has been done by Frankel and Acrivos
(1970), Barthes-Biesel (1972) and Barthes-Biesel and Acrivos
(1973). They extended Cox's model to the second order in the drop
excentricity. The resulting relation for the deformation does not
have a solution above a certain critical capillary number.
This can be interpreted as a prediction of the critical capillary
number at which drop break-up occurs. The critical capillary
numbers predicted by Barthes-Biesel and Acrivos (1973) are given in
figure 3.3 and will be compared with our experimental results.
3.2.2 Experimental results
The first experiments regarding the break-up of Newtonian droplets
in quasi steady simple shear flows were performed by Taylor (1934),
using a parallel band apparatus. Most of the later investigators
used a Couette device consisting of two coaxial counterrotating
cylinders. The advantage of the this design is the simpler
experimentation. Experimental studies on drop break-up in simple
shear flow were among others performed by Rumscheidt and Mason
(1961), Torza et al. (1972), Karam and Bellinger (1968) and most
extensively by Grace (1982). Grace's results will be used to
compare with our results (see figure 3.3). Experimental results
show that above a viscosity ratio of about 4 break-up is impossible
in quasi steady simple shear flow. A minimum in the capillary
35
36
number at which break-up occurs of about 0.5 is obtained for
viscosity ratios slightly lower than unity. For low viscosity
ratios break-up becomes increasingly difficult.
In simple shear flows four different modes of drop deformation and
break-up can be observed. These were first classified by Rumscheidt
and Mason (1961). For all four modes the droplet will assume an
ellipsoidal shape at very low shear rates with the principal axis
at an angle of 45 degrees with the flow direction. At very low
viscosity ratios this ellipsoidal deformation generally passes into
an elongated shape with an almost cylindrical centre section. Such
droplets generally break-up in two almost identical and equisized
drops with some small satellite drops in between. At viscosity
ratios around unity the deforming drop will generally form a neck
in the middle, which will progressively thin until two identical
daughter drops and some much smaller satellite drops are formed. At
viscosity ratios exceeding 4 a third mode of drop deformation is
observed. Upon increasing the shear rate the principal axis of the
drop rotates until the droplet is alligned with the direction of
the flow and a steady deformation is maintained. Finally a fourth
mode of drop deformation and break-up can sometimes be observed for
low viscosity ratios. In this mode, called tipstreaming, the
droplet takes, upon further increasing the shear rate, a sigmoidal
shape with sharply pointed ends from which very small fragments are
released. This mode, however, is not believed to be the normal
break-up mechanism in simple shear, but is related to the presence
of interfacial tension gradients at the drop surface (chapter 6 of
this thesis).
3.2.3 Slender bogy theories
Taylor (1964) also initiated the use of slender body theory to
describe the low Reynolds behaviour of droplets with a very low
viscosity ratio, which are subjected to high shear rates. This
usage was based on experimental observations that such droplets can
become very long and slender. This analytica! technique represents
the effect of the droplet on the flow by a distribution of
singularities along the drop axis and calculates the type, the
strength and positions of these singularities, together with the
position of the drop axis, that match the boundary conditions.
Later the slender body theory was refined and extended by
Buckmaster (1972,1973), Acrivos and Lo (1978), Hinch (1980), Hinch
and Acrivos (1980) and by Khakhar and Ottino (1986). First this
theory was applied to the case of an inviscid drop positioned in an
axially symetric pure straining flow. Later the theory was extended
to low viscosity drops and other linear shear flows, hyperbolic
flow and simple shear flow. The slender body theory has also been
applied to higher Reynolds numbers to establish the effects of
inertia forces. These theories predict a strongly decreased
effective shear rate just outside the droplet due to the presence
of the relatively non viscous droplet. Therefore the viscous farces
exerted on the droplet are very much reduced and high critica!
capillary numbers will result. For sirnple shear flows Hinch and
Acrivos (1980) predicted Ocrit - 0.054 À2/3.
3.2.4 Nurnerical technigues
Numerical techniques have been used to describe the deforrnation of
droplets in linear sbear flows. These studies were generally based
on a boundary integral rnethod by wbich the creeptng flow equations
inside and outside the drop are transformed into a form that only
involves quantities at the drop surface. This technique, derived by
Ladhyzhenskaya (1963) and further described by Youngren and Acrivos
(1975), Rallison and Acrivos (1978) and by Rallison (1981), is
based on the Fourier solution of the creeping flow equations, with
the use of volume potentials. Mathematica! problems arise with this
method for very small or very large viscosity ratios, since the
37
38
boundary integral bas neutral eigensolutions for these two
extremes. For axisymmetric problems the calculations are very much
simplified, since the drop surface can then be described by a curve
and less grid points are needed to obtain an accurate solution.
Another simplification occurs for simple shear flow since then one
of the terms in the boundary integral vanishes and the velocities
can be solved without a matrix inversion, substantially reducing
the computational time needed. For this situation Rallison (1981)
predicted a critica! capillary number of 0.42. This numerical
technique has also been applied in chapter 5 of this thesis
resulting in a predicted critica! capillary number between 0.45 and
0.50 fora viscosity ratio of 1.
3.3 FLOW IN THE COUETTE DEVICE
For the drop break-up experiments the simple shear flow is
generated in a Couette device. The flow in this device is not an
exact simple shear flow due to the curvature of the cylinders and
due to their finite length. In the following both effects will be
described.
First, however, the quasi steady flow pattern in the gap between
two infinitely long cylinders will be calculated from the basic
hydrodynamic equations. In this configuration the Navier-Stokes
equations can best be expressed in cylindrical coördinates. Due to
the symmetry of the problem (8/Bt ( ) = 0, Vz 0, vr = 0, 8/Bz ( )
= 0) these equations reduce to only one relatively simple
equation:
2 a v _____.J. 1
+ 2 r
ar ar
v
_1!. - 0 2
r
(3-1]
A solution to this equation is given by
v (r) </>
C r -1 r
or in terms of the angular velocity w - vq,/r:
w (r) c 1
c -2.
2 r
The constants C1 and Cz can be calculated from the boundary
conditions (counterrotating cylinders) given by
[3-2a]
[3-2b]
[3-3a]
(3-3b]
in which the subscripts I and II denote the radius of the inner and
outer cylinder respectively. Note that wr and wrr are defined in
such a way that they have a positive value. This results in:
2 2 w R + w R
I I II II c
1 2 2 [3-4a]
R • R II I
2 2
c -2 2 2 [3-4b]
R - R II I
The loc al shear rate in the gap is given by:
8 v 2
(r) _j_
-y - r 8r r 2
[ 3-5]
r
39
40
Since this shear rate is not constant across the gap the shear rate
at the position of the drop should be calculated. In the
experiments the stagnant layer is adjusted to coincide with the
drop position. The position of this layer r 8 can be calculated from
v~ - 0 resulting in:
r [3-6] s
The local shear rate at the stagnant layer can thus be calculated:
2 2 R w + R w
I I II II 7 (r ) - 2 [3-7]
s 2 2 R - R II I
The variation of the shear rate over the finite radius of the
droplet, R, can be obtained from Eq. (3.5]:
r
7 (r + R) ~ s r
s
s 7 (r )
+ 2R s [3-8]
Since in our apparatus rs is approximately 4.5 cm and the droplet R
is always smal~er than 0.5 mm the variation in shear rate over the
droplet is always smaller than 2%. Larger droplets should
preferably not be used because when the drop diameter is larger
than about 10% of the gap width the flow around the droplet will be
substantially influenced by the vicinity of the cylinders.
The description of the flow, which is given above, is only valid
when the Reynolds number of the flow based on the gap width is
small enough to prevent the occurence of Taylor vortices
(Schlichting, 1968). The relevant Reynolds number for this flow
is:
Re -
p 'Y (R • R ) c II I
'1 c
2
< 10
taking typical values for our experiments
Pc io3 kg ro·3
'Y < 91 s·l
Rn Rr - 1.05 cm
'Ic > 1 Pa.s
[3-9]
The Taylor instability will thus not occur in the Couette device
because
80 [3-10]
The calculations above are valid for the flow in a gap between two
infinitely long cylinders, but the Couette device has only a finite
length. In the following the end effects of top and bottom will be
described. The top of the Couette device usually does not disturb
the flow pattern, since it is open to air. Only when the continuous
phase evaporates and forms a film on top of the liquid, top effects
will occur. Bottom effects always occur, since the bottom rotates
with the outer cylinder. In order to know to which height these end
effects disturb the flow pattern, we have calculated the flow
numerically for a simplified geometry of the Couette device since
the sizes are chosen by RI 4 cm - 5 cm, height - ~ and the
bottom is chosen to be flat. The equations will be solved for a
stationary inner cylinder and a rotating outer cylinder. In this
configuration the Navier-Stokes equations can best be expressed in
cylindrical coördinates. Due to the symmetry of the problem (8/8t
( ) 0, Vz - 0, vr - 0) these equations reduce to one equation:
2 a v 2 ~ l 8v y u
0 [3-11] + + 2 r ar 2 2
ar r 8z
41
42
The boundary conditions for vq, (r, z) are:
v </>
(r, o) - w II
r [3-12a]
v
"' (r, "") = c r [3-12b]
1 r
v (R , z) - 0 [3-12c] </> I
v (R . z) =w R [3-12d]
"' II II II
These equations can be made dimensionless with the length scale
RII-Rr
and the velocity scale wrr Rrr: Thus the equation to be solved is:
2 LJ. 1 ~ -1L
= 0 + + 2 ç a ç 2 2 a ç ç 8 ï
in which
v P.
v w R II II
r ç
R - R II I
z ï
R R II I
with the boundary conditions.
v (Ç, 0) -
1J (Ç • oo)
R - R II I
R II
w R II II
. ç
. ç + R w (R - R ) IIII II I
1 ç
[3-13]
[3-14a]
[3-14b]
0 [3-14c]
v cez, n - i [3-14d]
This equation is numerically solved by a difference scheme. The
discretization was done as follows:
u Vm,n
v v av m-1,n m+l,n
ae 2 h
2 v 2 v + v Q_:Q m-1,n ID,!! m+l,n
2 2 ae h
2 v - 2 v + v Q_:Q m,n-l m,n m,n+l
2 2 ar h
in which h is the step size of€ and\. Substitution of this
discretization in Eq. [29] and reorganizing the terms results in
v m,n+l
2 h_
(4+.,,2) v ~ m,n
-1l v - (l+
m,n-1 2€
-1l v - (1 - ) v
m+l,n 2€ m-1,n
[3-15]
This equation was numerically solved with h - 0.025 in 100
iterations. The results are shown in Fig. 3.1 in which the velocity
is plotted as a function of the height in the cylinder for various
positions in the gap. From these results it can be concluded that
bottom ef f ects have disappeared at a height equal to about one gap
width (error< 1%).
43
44
0.80
0.60
0.40
0.20
··.\" -\\", \ \ "' 1 . "' . \ ......
----~~---
l • ........
\ \ ----\ \ \ \ \ . \ \ \ ' l \
\ \ \ ''\ \ .,_ ' -\ ---------------------\ \
','-.,, , __ --. ... _____ " _________________ _
0.00 '------'-------'-------''------'-------' 0.00 0.45 0.90 1.35 1.80 2.25
--> Height (Dimless}
1/6 gap 2/3 gap
i/3 gap 516 gap
1/2 gap
Fig. 3.1.a Effect of bottom on flow pattern in Couette device for various posit1ons in the gap. Velocity ratio of cylinders: ·1:4
' .... '-\ ', ·. \ .... . \\ \ ....
l\ ". \ 0.80 '1 ' ... "'
\ . ' \ 1 •• '-
---
\ \ "·. '-, 1 \ ·. -.... ____________ _
\ . \ \
0.60 \ \
\ \ \ \\ \ .
0.40
\ \. \ ' " '-...,......._
\ ----'\ ----------------
\\",_
0.20 '-....._ ""....__ ______ ... " _______________ _
0.00 .~--J__,,
0.00 0.45 0.90 1.35 1.80
--> Heigth [Dimless}
1/6 gap 213 gap
1/3 gap 5/6 gap
1/2 gap
Fig. 3.1.b Effect of bottom on flow pattern in Couette device for various posit1ons in the gap. Velocity ratio of cylinders: -1:1
2.25
45
46
' ' ·.\ ', ·. \ '·. \
0.80 .\ · ... \ ,\ \ '' \
\ \ " ' \ "-. \' ' ~ \ '-...._
0.60 \ \ -----
\ ' ' \ \ \
0.40 ......... " .. ". " .. ··-···.
\ \ \ ' \ \ \ ' " '\. \ "'-,
\ '"'-......,._ \ ---\\, ----------------
''--, __
-------------~------------0.20
0.00 ~----~-____J_ ____ -1. _____ ._ ___ ___.
0.00 0.45 0.90 1.35 1.80 2.25
--> Height [Dim.less}
1/6 gap 213 gap
1/3 gap 516 gap
1/2 gap
Fig. 3.l.c Effect of bottom on flow pattern in Couette device for various positions in the gap. Velocity ratio of cylinders: -4:1
3.4 EXPERIMENTAL
3.4.1 Description of apparatus
Por the measurement of deformation and break up of individual
droplets in simple shear flow a Couette type apparatus has been
built, consisting of two concentric counterrotating cylinders.
Thus a stagnant layer is created which permits statie optical
observation of a droplet in a flow field (See Fig. 3.2).
The diameter of the inner and outer cylinder are respectively:
Rr - 39.75 mm and R11 50.25 nun. The height of the inner cylinder
is 50 mm. The inner cylinder is made from aluminium, while the
outer cylinder is made from precision-bore glass which permits
direct visual observation both from the side direction and from
beneath. The cylinders are mounted in ball bearings.
The cylinders are driven by ribbed belts from two permanent magnet
D.G. servomotors with tach (Electro Graft Corporation type E-652-
00-004) each with a reduction of 1:30. The motors are controlled by
a speed control unit (E-652-M) which also has a remote control
facility requiring a voltage range of 0-10 V with a resolution of
1:1000. We used this remote facility to control the motor speed
from a personal computer. The actual rotation speeds were measured
independently and were available in a digital read out. The maximum
attainable shear rate is 91.2 s-1 at a rotation speed of the
cylinders of 100 rev./min.
The drops were observed either by viewing in radial direction
through the cylinder wall or along the z-axis by viewing through a
mirror placed at an angle of 45° beneath the annular slit. In the
case of observation through the cylinder wall we used a small
cuvette with one curved surface and one flat surface to minimize
distortion effects by the curved glass surface of the outer
47
48
cylinder. The advantage of observation in radial direction is that
in the case that the densities of the two phases do not exactly
match, the drop is kept more easily in focus, but its orientation
in the gap between the cylinders is not exactly known. The drops
were illuminated with a cold light source (Schott type KL 500) with
flexible glassfiber light transmitters. We used a Zeiss stereo
microscope (type SV) with an objective with a focal length of 100
mm and a zoom mechanism. On the microscope a Sony camera (type DXC
1850P) and a Contax RTS-camera were mounted which made recording of
the experiment on video possible. A video timer monitored the time
elapse continuously. The overall maximum magnification of the
optica! system was 500 times with an accuracy on 1%.
3.4.2 Experimental procedure
For a drop break-up experiment the following procedure was used.
The Couette apparatus is filled with the continuous liquid phase. A
drop of the dispersed phase is inserted from a syringe (Pasteur
pipette) usually in the middle of the gap and halfway the heigth of
the inner cylinder. Because it appears to be difficult to insert
just one drop, all unwanted drops are sucked out of the continuous
phase by a syringe. One or both cylinders are slowly rotated until
the droplet is visible on the video screen. The drop radius is
measured from the video screen. Next the video recording is started
and the rotational speeds of both cylinders are slowly increased in
such a way that the droplet remains (almost) stagnant at the center
of the video screen. From the ratio between the inner and outer
cylinder speeds the place of the stationary drop in the gap can be
calculated and the speeds of the cylinders can be further
increased. When drop break-up occurs the rotational speeds from
the cylinders are taken from their digital output and the rotation
Figuur 3.2a
Fig. 3.2.a Couette device: top view
49
50
PARTICLE ü'lder
considerotion
{
\
CAM. ERA LENSE
-- j- ---~~JJSTABl.E l~ROR
Fig. 3, 2, b Couette device: front view
OUTER"' CILINa;:R~
and video recording are stopped. At that time the temperature of
the liquid in the gap is measured. After removing all unwanted
fragments by suction with the syringe a next experiment can be
started with one of the fragments or with a freshly inserted
droplet.
The viscosity as a function of shear rate of the Newtonian liquids
has been measured with a Haake type CV 100 viscometer using a
concentric cylinder geometry (type ZC 15). This apparatus was
thermostatted at 23°C and the shear rate range was from about 1 to
80 s·l.
Two different methods have been applied to measure the interf acial
tension between the used liquids;
a) The Wilhelmy plate method, where a silver plate with
circumference of 0.06 m is drawn from one liquid into the other
and the excess force, corrected for gravity effects, gives
directly the interfacial tension. This method is not suitable
for very low interfacial tensions and can have an accuracy of
0.1%.
b) The sessile drop method, which is based on the drop deformation
under gravity forces. After measuring height and width of the
droplets the interfacial tensions can be calculated with the aid
of the tables given by Bashforth and Adams (1883). An accuracy
is claimed of 0.1%. High interfacial tensions, however, are
difficult to measure.
All interfacial tensions were measured with respect to a silicone
oil with a viscosity of 1 Pas as being representative for all
silicone oils. This because the surface tension of these oils
against air is found to be constant at different viscosity values.
The droplet and continuous phases used are shown in Table 3.1. The
silicone oils (type Rhodorsil 47 ex Rhone-Poulenc Chimie Fine) were
made at different viscosity values by blending a high and a low
viscosity batch so covering a viscosity range from 0.9 to 65 Pas.
51
Table 3.1 Physical data of model liquids
nr. droplet pbase viscosity
[Pa.s)
Castor Oil 8. 7 • 10-l
2 Sunflower Oil 5. 7 • 10-2
3 Ethylene Glycol 1.6 • 10-2
4 Aniline 3. 7 10-3
5 Corn Syrup/Water (90/10) 2.8
6 Corn Syrup/Wat:.er (85/15) 7 .8 10-1
Corn Syrup/Water (80/20) 2.9 • 10-1
Corn Syrup/Water (75/25) 1.2 • io-1
9 Silicone Oil 2.1 lo1
10 Silicone Oil 1.6 • 101
11 Silicone Oil 1.4 • 101
12 Silicone Oil 1.2 • lol
13 Silicone Oil 9.3
14 Silicone Oil 6.4
nr. continuous phase viscosit.y
{Pa.sJ
15 Corn Syrup/Water (92/8) 6.0
16 Silicone Oil. 9.0 • 10-1
17 Silicon-e Oil 5. 7
18 Silicone Oil 6.0
19 Silicone Oil 1.1 • 101
20 Silicone Oil 1.6 lo1
21 Silicone OH 2. 7 • io1
22 Silicone Oil 3.8 101
23 Silicone OH 4.0 • io1
24 Silicone Oil 4.3 io1
25 Silicone Oil 6. 5 • 101
52
The corn syrup solutions were based on a concentrated corn syrup
(type Globe 01170 ex GPC Netherlands B.V.) which was diluted with
water so covering a viscosity range from 0.12-6.0 Pas. Further
ethylene glycol, aniline, castor oil and sunflower oil were used in
standard quality.
3.4.3 Results
Using the procedure as given in paragraph 3.4.2 the critica! shear
rate for drop break-up was measured. The results are given in Table
3.2. The results, averaged per combination of droplet and
continuous phase are visualized in Fig. 3.3. From this figure it
will be clear that there is a very good correlation between the
Capillary number 0 (the dimensionless shear rate at which drop
break-up occurs) and the viscosity ratio.
• • + + ++ •
--+
Fig. 3.3
LEGEND •"' own experi.rtentol results +-Grec", 1982 6. - Barthes-Bl.eseL, 1972 X• Rol.ll.son, 1981 17 • Hl.noh + Acr i, vos, l 980
+
+
IJl+
+
+ +~
+ +" + + +. _..+;+•
Comparison of drop breakup data
53
54
This correlation can be used as a break-up criterion to predict
maximum droplet sizes in simple shear flows. The correlation
obtained is in agreement with the one measured by Grace (1982).
This comparison shows that the scatter in Grace's results is far
larger than in our results (Fig. 3.3). The theoretica! calculations
by Hinch and Acrivos (1980) and Barthes-Biesel and Acrivos (1973)
appear to predict break-up at somewhat lower shear rates than
determined experimentally. Qualitatively, however, their
correspondence with experimental results is very good. The
numerical simulations by Rallison and those described in chapter 5
of this thesis are in very good agreement with the experimental
results.
From the break-up criteria obtained it will be clear that drop
break-up in simple shear flows is most efficient when the droplet
viscosity is a little lower than the continuous phase viscosity
At viscosity ratios above about 4 break-up in simple shear flow is
not possible at all (e.g. preparation of diluted, oil in water
emulsions is not possible in simple shear flows).
A simple 4 parameter function has been used to fit the experimental
data. The function features the theoretically predicted behaviour
Ocrit - Àa for low values of À, together with a maximum viscosity
ratio, Àmax• above which break-up will not occur:
log Ocrit C2
C1 + a log À + ~~~~~~~~ log À - log Àmax
[3-16]
Curve fitting resulted in a = -0.733, Amax 9.27, C1 -1.560 and
C2 -1.135 with a MSQ of 0.008. (See Fig.3.4). The fit however, is
not very good close to Àmax· A much better fit was obtained by a 5
parameter function
log Ocrit C1 + a log À + C3 (log A)2 + log À
C2
log Àmax [3-17]
with Àmax = 4.08, a = 0.0994, C1
0.124 and MSQ 0.002.
-0.506, C2 = -0.115 and C3 =
The 5 parameter function gives a much better representation close
to Àmax (see Fig.3.4) but to very low À is not justifiable.
For À - 1 this correlation predicts Ocrit 0.48.
Fig. 3.4
•
LEGEND •- own experi..ttenlal res:ulls V= 4 parameter fl,t A • 5 parameter f(, t
Drop breakup correlations
î
55
Table 3.2 Drop break-up experiments
drop cont. " R 7 0crit phase pbase {m Nm- 1 J [mml [s-11 1-1 [-)
20 4 .1 0.143 1.11 5.4 • 10-1 0. 84
0.120 1. 60 0. 75
0.102 1.97 0.79
0.095 1.98 o. 73
0.065 3. 01 0. 77
0.061 3.39 0. 81
23 4.1 0.227 0.60 2.2 • 10-2 1.3
0.148 0.99 1. 4
0.102 1. 31 1.3
0 .075 1. 59 1.2
0.062 2.04 1. 2
0.048 2.59 1.2
0.042 3.24 1.3
0,039 3.23 1.2
0 .021 6. 70 1.4
2 17 2.5 0.475 1.41 1. 0 • 10-2 1.5
0.324 1.63 1.2
0.256 2.14 1.2
0.200 2. 91 1.3
0.152 3. 74 1. 3
0.117 4. 70 1.3
0.092 6.26 1.3
2 20 2.5 0.338 1.20 3, 6 • 10-3 2,6
0 ,234 2.26 3.4
0.130 3.11 2.6
0.090 4 .15 2.4
0. 443 0,85 2.4
0.398 1.20 3.1
0.262 1. 54 2.6
2 23 2. 5 0.315 1.41 7 .1
o. 176 1. 94 5.5
0.113 3.00 5. 4
0.083 4 .17 5.5
17 13' 5 0' 542 19. 10 2.8 • 10-3 4. 3
0.484 18.54 3.8
0' 418 22. 71 4 .0
0,277 34. 78 4 '0
20 13. 5 0.559 13 .06 1.0 • 10-3 8. 7
0. 447 15.94 8. 4
0,308 23 .63 8.6
0. 345 32. 10 9,3
23 13. 5 0.533 15.35 4.0 10-4 24
0. 308 22.91 21
20 5. 0 0.301 33 .83 2. 3 • 10-4 33
18 35 0 452 5. 93 4. 7 10- 1 0.46
0. 352 8.06 0. 49
56
Table 3.2 Continued (a)
drop cont. q R 'r ). noi:it phase phase fm Nm-ll !nrnl [s-1] [-] [-]
5 19 35 0.390 4. 37 2.5 • 10-l 0.55
0.265 5.84 0.50
0.506 3.14 0.52
0.345 4.63 0.52
0.233 6.21 0. 47
21 35 0.410 2.17 1.1 • 10-1 0.68
0.265 3.24 0.65
5 24 35 0.330 1.89 6, 5 • 10-2 0. 77
16 33 0,464 36.57 8. 7 • 10-l 0,46
0.367 48.11 0.48
6 18 33 0.506 5.17 1.3 10-l 0 .57
0,486 7.36 0.52
0.303 9.27 0.51
0.235 11. 95 0.51
0.185 14. 78 0.50
19 33 0.412 4.97 6.9 • 10-2 0. 70
0.310 6.95 0. 74
0.220 9.18 0. 70
22 33 0.366 2.80 2.0 • 10-2 1.2
0.285 3.87 1.3
0.209 5.02 1.2
25 33 0.355 2.09 i.2 • io-2 1.5
0.280 2. 75 1.5
0.175 3.91 1.3
0.140 4.98 1.4
18 31 0.431 8.67 4.8 • 10-2 o. 72
0.339 11.06 o. 73
0.262 13.92 0.71
0.205 17 .26 0.68
19 31 0.383 4.32 2. 6 • 10-2 1. 3
0.261 12.31 1.2
22 31 0.383 4. 75 7 ,6 • 10-3 2.2
0,395 6,47 2.3
0.215 7. 75 2.1
25 31 0.429 3. 21 4.5 • 10-3 2.9
0,322 3 .94 2.6
0.190 6.54 2.6
16 29 0.512 32.93 l.3 • 10-l 0.52
18 29 0.461 13 .12 2.0 10-2 1.3
0.357 16.11 1.2
0.281 18.-5 1.1
0.217 23.35 1.1
0.173 31. 70 1.1
8 19 29 0.366 12.08 t. l • 10-2 1.7
0.295 15.90 1.8
0,227 21.15 1.9
57
Table 3.2 Continued (b)
drop cont. R " 0 crit phase phase [m Nm-1 ] [l1lll) r.-1 1 [-) [-J
22 29 o. 402 8.41 3.1. 10-3 4.4
0.322 9.09 4.2
0 .251 11.90 3.9 g 15 38 0.540 57 .96 3. 3 5.1
0.640 45. 93 4.8
0. 555 63.83 5.6
10 15 38 0.434 14.91 2.6 1.1
0.352 19. 77 1.2
0.285 25.38 1. 2
0.217 32.85 1.2
11 15 38 0.470 11.63 2.3 0.85
0.381 15.12 0.89
0.298 19.09 0.87
0.238 24.36 0 .88
12 15 38 0 .446 9.83 2.0 0,69
0.343 12.15 0.66
0.274 16. 00 0.69
0.214 20.01 0.68
0.187 21.21 0.62
13 15 38 0.393 8.64 1.5 0.55
0 .323 10.87 0. 57
0. 243 13.69 0.54
0.193 17 .80 0.55
0.154 22.65 0.56
14 15 38 0. 447 6.16 l.o 0.45
0.331 7 .98 0.43
0.270 9. 88 0.43
0.208 12.48 0.42
0.175 16.43 0. 46
58
3.5 CONCLUSIONS
Reliable break-up criteria are available to predict droplet break
up in quasi steady simple shear flow, provided both liquid phases
are Newtonian and surfactants are absent. The break-up criteria are
fully described by the drop capillary number 0 and the viscosity
ratio À.
Simple shear flow is found to be most efficient for the break-up of
droplets with a viscosity ratio in the range 0 1 < À < 1. For
droplets with a viscosity of more than 4 times the viscosity of the
continuous phase, break-up is not possible at all in simple shear
flows.
The experimental results obtained with the developed Couette
apparatus show a very small scatter and are in agreement with
results obtained elsewhere. It is thus concluded that a well
functioning apparatus has been developed that will be used to
investigate the effects of non-Newtonian drop phases on drop break
up (de Bruijn, chapter 4 of this thesis), to investigate the effect
of transient simple shear flow on drop deformation and break-up
(chapter 5 of this thesis) and to study the tipstreaming phenomenon
in simple shear flow (chapter 6 of this thesis).
3.6 REFERENCES
1. A. Acrivos and T.S. Lo, Deformation and break-up of a single
slender drop in an extensional flow, J. Fluid Mech. 86, 641-
672, (1978).
2. D. Barthes-Biesel, Deformation and burst of liquid droplets
and non-Newtonian effects in dilute emulsions, Thesis Stanford
University, Michigan, USA, (1972).
3. D. Barthes-Biesel and A. Acrivos, Deformation and burst of a
liquid droplet freely suspended in a linear shear field, J.
Fluid Mech., 61, 1-21, (1973).
59
60
4. F. Bashforth and J.C. Adams, An attempt to test the theories
of eapillary aetion, University Press, Cambridge, (1883).
5. R.A. de Bruijn, Sealing laws for the flow of emulsions,
ehapter 2 of this thesis.
6. R.A. de Bruijn, Non-Newtonian drop break-up in quasi steady
simple shear flows, ehapter 4 of this thesis.
7. R.A. de Bruijn, Deformation and break-up of Newtonian droplets
in transient simple shear flows, ehapter 5 of this thesis.
8. R.A. de Bruijn, Newtonian drop break-up in simple shear flows
the tipstreaming phenomenon, ehapter 6 of this thesis.
9. R.G. Cox, The deformation of a drop in a general timedependent
fluid flow, J. Fluid Meeh., 37, 601-623, (1969).
10. A. Einstein, Ann. Physik, 19, 289, (1906).
ll. N.A. Frankel and A. Aerivos, The eonstitutive equation for a
dilute emulsion, J. Fluid Mech., 44, 65-78, (1970).
12. H.P. Graee, Dispersion phenomena in high viscosity immiseible
fluid systems and applieation of statie mixers as dispersion
deviees in sueh systems, Chem. Eng. Commun. 14, 225-277,
(1982).
13. E.J. Hinch, The evolution of slender inviscid drops in an
axisymmetrie straining flow, J. Fluid Mech., 101, 545-553,
(1980).
14. E.J. Hinch and A. Aerivos, Long slender drops in a simple
shear flow, J. Fluid Mech. 98, 305-328, (1980).
15. H.J. Karam and J.C. Bellinger, Deformation and break-up of
liquid droplets in a simple shear field, Ind. Eng. Chem.
Fundam. 7, 576-581, (1968).
16. D.V. Khakkar and J.M. Ottino, Deformation and break-up of
slender drops in linear flows, J. Fluid Mech., 166, 265-285,
(1986).
17. O.A. Ladyzhenskaya, The mathematieal theory of viscous
incompressible flow, Gordon and Breach, New York, (1963).
18. H. Lamb, Hydrodynamics, 6th ed. Dover press, New York,
(1945).
19. J.M. Rallison, Note on the time dependent deformation of a
viscous drop which is almost spherical, J. Fluid, Mech., 98,
625-633, (1980).
20. J.M. Rallison, A numerical study of the deformation and burst
of a viscous drop in general shear flow, J. Fluid Mech., 109,
465-482, (1981).
21. J.M. Rallison, The deformation of small viscous drops and
bubbles in shear flows, Ann. Rev. Fluid Mech. 16, 45-66,
(1984).
22. J.M. Rallison and A. Acrivos, A numerical study of the
deformation and burst of a viscous drop in an extensional
flow, J. Fluid Mech., 89, 191-200, (1978).
23. F.D. Rumscheidt and S.G. Mason, Particle motions in sheared
suspensions, XII deformation and busrt of fluid drops in shear
and hyperbolic flow, J. Coll. Int. Sci. 16, 238-261, (1961).
24. H. Schlichting, Boundary layer theory, McGraw-Hill, New York,
1968.
25. G.I. Taylor, The viscosity of a fluid containing small drops
of another fluid, Proc. Roy Soc. A 138, 41-48, (1932).
26. G.I. Taylor, The formation of emulsions in definable fields of
flow, Proc. Roy Soc. A 146, 501-523, (1934).
27. G.I. Taylor, Conical free surfaces and fluid interfaces, Proc.
Int. Congr. Appl. Mech., llth, p 790-796, Munich (1964).
28. S. Torza, R.G. Cox and S.G. Mason, Particle motions in sheared
suspension, XXVII transient and steady deformation and burst of
liquid drops, J. Coll. Int. Sci. 38, 395-411, (1972).
29. G.K. Youngren and A. Acrivos, On the shape of a gas bubble in a
viscous extensional flow, J. Fluid Mech., 76, 433-442, (1975).
3.7 LIST OF SYMBOLS
C1 C2 C3: constants
h step size of Ç and Î
m Ç-position
[s-1, m2.s-l]
[ - ]
[ - J
61
62
LIST OF SYMBOLS (continued)
n
p
R
t
v
°' i'
ï
'ld· À
v
ç p
17
v
(l
w
'lc
r, ;/>, z
Fr
Re
ï-position
pressure
drop radius
radius of inner and outer cylinder
position of stagnant layer
time
velocity
constant
shear rate
normalized axial position
viscosity of drop and continuous
viscosity ratio
kinematic viscosity
normalized radial position
density
interfacial tension
dimensionless velocity
Capillary number
angular velocity
cylindrical coordinates
Froude number
Reynolds number
phase
[ - l [Pa]
[m]
[m]
[m]
[s] [m s-lj
[ l [s-1]
[ - ]
[Pa.s]
[ - ]
[m2.s-l]
[ - l [kg.m-3]
[N.m-1]
[ l [ - l [s-1]
4. NON-NEWTONIAN DROP BREAK-UP IN QUASI STEADY SIMPLE SHEAR FLOW
4.1 INTRODUCTION
The present chapter deals with the modelling of the break-up of
non-Newtonian droplets in a Newtonian continuous phase in quasi
steady simple shear flow. The break-up behaviour of Newtonian
droplets in a Newtonian continuous phase in quasi steady simple
shear has been described in the previous chapter. The break-up of
non-Newtonian droplets is especially relevant to food processing
where markedly non-Newtonian ingredients are used. In this chapter
theoretical and experimental results available in the external
literature are surveyed. These results, however, do not give a
complete understanding of non-Newtonian drop break-up. Therefore
the break-up of two types of non-Newtonian droplets was studied
experimentally in the Couette device described in chapter 3 of this
thesis: droplets which combine a strong shear-thinning behaviour
with very low normal stress differences (low elasticity) and
droplets which combine high elasticity with an almost constant
(i.e. shear-rate independent) viscosity. These two types of droplet
rheology were chosen to be able to separate the effects of shear
thinning behaviour and of droplet elasticity which usually occur in
a combined way.
4.2 NON-NEWTONIAN FLUID MECHANICS
The basic equations describing the flow of non-Newtonian droplets
in a Newtonian continuous phase are the equations of motion and the
mass conservation relations in both droplet and continuous phase
and the boundary conditions which apply to the droplet interface
complemented by the constitutive equation of the fluid. The
constitutive equation of a liquid relates the stresses in the
liquid to the deformation of the liquid. The constitutive equation
of a Newtonian liquid is relatively simple and the equation of
motion of a Newtonian liquid thus reduces to the well known Navier-
63
64
Stokes equations. Since there are many types of non-Newtonian
behaviour a large number of constitutive equations have been
formulated to describe these liquids. The basic equations
governing the flow are discussed in more detail in chapter 2 of
this thesis.
In this chapter we shall present the rheometrical data with a
simplified description of the constitutive equations that is only
valid for steady simple shear flows. The stresses in a liquid can
be described by the stress tensor !.
t" 7 12
'"J !. 7 21 7 22 7 23 7 31 7 32 7 33
[4-1]
The deviatoric stress tensor representing the deviation of the
stress from an isotropic state (described by a single scalar, the
pressure) as a consequence of flow is defined in such a way that
1"11 + 1"22 + T33 = 0. Since only the deviatoric stresses lead to
drop deformation attention will be focussed on these, omitting the
prefix deviatoric for convenience. Further it can be shown that the
stress tensor is symmetrie in the absence of extraneous moments, so
that Tij
that r31
Tji· For a simple shear flow it can further be shown
1"32 = 0. The stress tensor is thus fully determined by
only three independent material functions of the
shear rate -)<.
r12 7 21 7 11 - 7 22
7 22 - r33
1"1 ()')
Ni ()'2)
N2 ('Î2)
[4-2-a]
[4-2-b]
[4-2-c]
r1 ('Î) is the shear stress function, N1 ('Î 2 ) is usually called the
first normal stress difference and N2 C.Y 2 ) the second normal stress
difference; 1"1 (j) is an uneven function and Ni (1' 2 ) and N2 (j2 )
are even functions of i.
For a Newtonian liquid the material functions reduce to
[4-3-a]
[4-3-b]
in which the dynamic viscosity ~ is independent of the shear rate.
Inelastic, shear-thinning liquids can in a certain shear-rate
regime often be described by the power law equation:
[4-4-a]
[4-4-b]
in which K is the consistency index and n the power law index.
In most viscoelastic liquids the second normal stress difference is
much smaller than the first normal stress dif ference and can
therefore be neglected. This is for instance done in the upper
convected Maxwell model which can be described by the following
material functions
Tl - ~ i N1 - 2 8 ~ i2
N2 - o
(4-5-a]
[4-5-b]
in which ~ is called the dynamic viscosity and 8 the relaxation
time of the upper convected Maxwell fluid. However it is very
difficult to find model liquids which fully correspond to this
constitutive equation. In practice the ratio of r1/t is not exactly
constant and the first normal stress difference does not vary
exactly proportional to ~2. Thus more general material functions
are needed. We will therefore describe our fluids with the
following material functions.
65
66
o.
[4-6-a]
[4-6-b]
[4-6-c]
in which ~ is the elasticity index and m the elastic power law
index. This turns out to be a rather good description of the
viscometric data of the non-Newtonian model liquids used in our
studies. These material functions obey the Criminale-Erickson
Filbey constitutive equation, which is a simplification of the
retarded motion expansion, valid for steady state flows
(Schowalter, 1978). This constitutive equation is thus not suited
to describe transient phenomena, but it was adopted for the ease
with which the viscometric data of the model liquids used in
present and other studies could be described.
The use of dirnensionless groups in studying the dynamics of non
Newtonian liquids cannot be as straightforward as it is for
Newtonian liquids due to uncertainties in the constitutive
equations of practical liquids. However three dimensionless numbers
are aften used: the Weissenberg number Nwe• the stress ratio SR and
the Deborah number De. The Weissenberg number is a measure of the
relative importance of elastic and viscous effects:
N We
y
L [4-7]
in which 0 is a natural time, characteristic of the liquid, and V
and L are a characteristic velocity and length, respectively.
If the constitutive equation of the liquid considered is well known
e.g. the Maxwell upper convected liquid, the definition of the
natural time is straightforward and the relation with fluid
elasticity is clear. When normal stress differences and shear
stresses are only known in a limited shear rate regime it is aften
better to use the dimensionless stress ratio SR
s R T
[4-8]
12
When the stress ratio is plotted against shear stress one can
compare the relative importance of fluid elasticity of different
viscoelastic fluids. The Deborah number is the ratio of a natural
fluid time to a characteristic observation or process time. The
process time should describe the flow experienced by material
points, thus it is a time scale related to the movement in the
direction of the flow, whereas the process time scale in the
Weissenberg number, L/V, is generally related to the velocity
gradient perpendicular to the flow direction. When De << 1 the
material behaves in a fluid-like manner and when De >> 1 the
material behaves solid-like.
4.3 LITERATURE REVIEW
4.3.1 Theoretical results
Very few theoretical approaches to the break-up of non-Newtonian
droplets in simple shear flow have been pursued. This is probably
partly due to the great complexity of the problem because it
involves three dimensionality, free surfaces, non-stationary flow
and complex constitutive equations.
Van Oene (1972) used a modified interfacial tension to account for
the fluid elasticity. The scientific basis of this theory is very
weak. The model is based on an expression for the recoverable free
67
energy Fg for constrained recovery in shear flow which was
developed by Janeschitz-Kriegl (1969) from an expression for the
stress tensor in polymerie liquids. For an upper convected Maxwell
liquid in simple shear flow one obtains:
[4-9]
Van Oene's rather qualitative argument was that when a droplet is
formed in a continuous phase the recoverable free energy changes
by
[4-10]
in which R denotes the radius and the subscripts d and c denote the
drop and continuous phase respectively. This change in recoverable
free energy was taken into acccount as an additional contribution
to the interfacial free energy a, which is the free energy of
formation of the interface per unit area
R a* - a + - (N - N )
6 l,d l,c [4-11]
This model predicts that elasticity of the droplet phase stabilizes
the droplet. This conclusion at least qualitatively corresponds
with experimental evidence (see section 4.3.2). This model also
predicts that elasticity of the continuous phase destabilizes the
droplet. However for a very large continuous phase first normal
stress difference, the model predicts a negative modified
interfacial tension. This, however, will not occur since when a*
reduces smaller droplets will be broken up, so negative values for
a* will not be obtained.
More theoretica! results are available for the description of the
break-up process of droplets or jets which already possess an
extended cylindrical shape.
Chin and Han (1980) considered the effects of fluid elasticity
(Maxwell model) on the break-up in non-uniform shear flow. They
considered a long, neutrally buoyant, liquid cylinder in a
continuous phase subjected to a cylindrical Poiseuille flow. The
long droplet is located on the axis of the tube. They assumed
axisymmetric disturbances which are periodic in the flow direction
and which grow or decay exponentially in time. They performed a
linear hydrodynamic stability analysis to determine the stability
or instability of the disturbance. They found that the stability
region of such a cylindrical droplet increases when the capillary
number based on the droplet increases. This can be interpreted as
either the stabilizing effect of increased shear rate in the
Poiseuille flow or as the destabilizing effect of an increased
interfacial tension. The stability region increases when the
continuous phase elasticity increases. Chin and Han (1980) also
analysed the stability of elongated droplets when they are
disturbed by waves of all wave-lengths under the condition of
inertialess flow. They concluded that the growth rate of the
disturbance is higher for increased droplet elasticity and for
decreased capillary numbers. As the droplet phase viscosity becomes
greater than the continuous phase viscosity the growth rate of
disturbance becomes slower. If it is assumed that the fastest
growing wavelength is responsible for drop break-up it can be
concluded that the smaller the droplet phase viscosity and the
larger the droplet phase elasticity, the larger the broken droplets
will be.
Bousfield et al (1986) have described the surface tension driven
break-up of viscoelastic jets. They found that disturbances on
viscoelastic jets grow more rapidly at short times than on
Newtonian jets, but that the growth is retarded at longer times due
to the development of large extensional stresses. The formation of
satellite drops was found to be retarded by the fluid elasticity.
69
70
4.3.2 Experimental results
Gauthier et al (1971) were the first to report introductory
experimental results on the deformation and break-up of non
Newtonian drops in a Newtonian continuous phase in simple shear
flows (see Appendix A). For shear thinning droplets they did not
found any deviation from the Newtonian behaviour. For viscoelastic
droplets their experiments indicate an increase of the critical
capillary number which they attributed to the droplet elasticity.
Since they performed just a few experiments on non-Newtonian
droplets which they have documented very poorly their results are
not very conclusive.
Tavgac (1972) has performed experiments with viscoelastic drop
phases (mostly Poly Acryl Amide (P.A.A.) in water/glycerine) in
Newtonian silicone oils (See Appendix A). He used a Couette device
with a gap of 7.65 mm allowing shear rates up to 90 Tavgac
described the rheological behaviour of his model liquids by a
modified 5 parameter Bird-Carreau model with the following material
functions.
~
r1 (i) - ~ 2 2
[4-12a]
p=l 1 + À l,p
i
2 2 ~ À . 1
N <7) - E p 2.p
1 2 2 [4-12b]
p~l 1 + À i l,p
Nz (~) - 0 [4-12c]
in which
ri ri p 0 "' >.
1,k
_2_ >. >. "'1 l,p 1 (p + 1)
_2_ À >.
(p + 1)"'2 2,p 2
The critical capillary number at break-up will thus depend on 5
dimensionless numbers:
n crit [
I'/ 0 n -,
crit ric a ' 1 "' ' 2
>. 1 >1, (o' Do), ,:J [4-13]
For the solutions of P.A.A. in water/glycerine used by Tavgac the
ratios a2/a1 and >.1/-'2 are approximately constant. For those
liquids the functional dependence thus reduces to
0 crit
[4-14]
Tavgac presented his experimental data as plots of Ocrit vs De for
the various drop phases used. To compare his results with ours we
had to estimate the power law constants K, n, K and m from the 5
Bird-Carreau parameters. This can be done numerically since in the
relevant shear rate regime (0.1 s·l < t < 100 s·l) the shear
stresses and normal stress differences of the model liquids used by
îavgac vary with tn and tm. The model liquids exhibit strong shear
thinning behaviour (0.19 < n < 0.79) combined with considerable
fluid elasticity.
71
72
There are several inaccuracies in Tavgac's work, the most striking
experimental error being the large drop sizes. Although the gap
width of the apparatus is only 7.65 mm, he used droplet diameters
varying between 0.56 < D < 4.22 mm. Most experiments were done with
droplets occupying far more than 10% of the gap. This will lead to
wall effects. Especially the conclusions on the observations with
drop phase T7 are erratic due to these wall effects. For this drop
phase the critical capillary numbers were thought to be lower than
the Newtonian limit which was assumed to be Ocrit 0.8 for De < 1
with a minimum for De= 0.25. According to our results (chapter 3
of this thesis) the Newtonian limit is however much lower (Gerit
0.5). The data for De< 0.2, which show a decrease in critical drop
size with increasing elasticity are erratic due to wall effects.
Hence the critical capillary numbers smoothly increase with
increasing droplet elasticity. Further, Tavgac does not correct
for the shear rate variations in the gap. The average shear rate is
determined instead of the shear rate at the stationary layer, which
may result in an additional error of up to 14%.
Tavgac correlates his experimental results by
[4-15)
with C1 and 0 0 ,crit dependent on the fluid system. From this
correlation he predicts the existence of a critical drop size,
below which break-up can not occur, given by
C À a 1 1
R [4-16] crit 17
c
C1 was found to decrease with increasing viscosity ratio. Therefore
we also plotted rvs 7crit for each fluid system (See Fig. 4.1).
Fluid systems T3, T4 and T8 appear to indicate critical drop radii.
For the other fluid systems rcrit is expected to be much lower than
the drop sizes studied.
10
" "
î . (j)
~ 1 -e 8-'b .
.
Fig. 4.La
+
shear ra te [ 1 /sJ
Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate+ Drop phase: 0.1% Poly Acryl Amide
+ +
+
...
Elmendorp (1986) bas performed some experiments witb viscoelastic
drop pbases (PAA in water) in a Newtonian silicone oil (See
Appendix A). The viscoelastic model liquids sbowed botb a sbear
thinning behaviour and nonnal stress differences. He used a Couette
device for his experiments, with a gap width of 6 mm. The data were
presented in a plot of l/rcrit vs i. The experiments were done with
drop sizes between 0.8 < D < 3.4 mm. Wall effects due to too large
droplets will tbus be present. For these reasons bis results are
very inconclusive.
73
74
Fig. 4.1.b
shear rate ( 1/s)
Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 0.4% Poly Acryl Amide
Prabodh and Stroeve have performed experiments with a viscoelastic
drop phase (PAA in Corn Syrup) in Newtonian silicone oils (See
Appendix A). They used a cone and plate device allowing shear rates
up to 400 s-1, and determined the shear rate at which a droplet of
known initial size broke up. The viscoelastic model liquid they
used however probably degraded when subjected to shear rates above
60 s-1. They observed that some drops would not break-up during
shear and would become greatly extended. These droplets sometimes
did break-up when the shear rate was suddenly decreased to zero.
They have presented their experimental data in a plot of Ocrit vs 7
for various continuous phase viscosities and in a plot of Ocrit vs
10~.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~--.
" •
1 ; . .
Fig. 4.Lc
+ " + + ....
++* + ++
+ ++- +
shear rate [ 1 /s]
Viscoelastic drop breakup results obtained by Tavgac {1972). Critical capillary number vs. shear rate. Drop phase: 0.75% Poly Acryl Amide
À for various critical shear rates. From these two plots the
experimental results could be traced. From their data Prabodh and
Stroeve concluded that below a viscosity ratio of 0.5 the droplet
elasticity has a stabilizing effect but above this viscosity ratio
the viscoelastic drops are less stable than Newtonian drops. We do
not agree with that conclusion since our Newtonian drop break-up
data slightly differ from their data. At all viscosity ratio's the
capillary numbers for the viscoelastic drops are above the ones
obtained by ourselves for Newtonian droplets (chapter 3 of these
thesis).
75
76
.
Fig. 4.1.d
+ +
+
+* ++ ++++
shear rate [ 1 /s]
Viscoelastic drop breakup results obt:ained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 1.0% Poly Acryl Amide
A detailed analysis of their drop break-up data (see Appendix A)
throws up many questions. They state that for systems with a
viscosity ratio À < 1 the critical capillary number varies almost
linearly with the shear rate (Ocrit - Gonst. ~). This statement was
found to be based on observations of the break-up of equally sized
droplets: e.g. droplets with a radius of 21 µm dispersed in the
same continuous phase were found to break-up at shear rates varying
from 40 to 265 s-1. In our opinion this is either due to an
enormous scatter in their results or, more likely, due to the fact
that these data refer to observations of drop break-up after
reducing the shear rate to zero. At any rate these statements are
10 •
1 Il)
~ 1 -
e 8-..,
" -"
0.1 0.1
Fig. 4.1.e
++ +
+ ++
shear rate [ 1 /s)
10
Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 2.0% Poly Acryl Amide
misleading. For exactly the same reason their conclusion that the
capillary number increases with increasing first normal stress
difference for the drop phase, evaluated at the imposed critical
shear rate in the continuous phase, is misleading.
The data obtained by Prabodh and Stroeve do not give a broad view
on the effect of elasticity of the droplet phase on droplet break
up since the stress ratio SR only varies between 1.3 and 2.0. Their
results show a stabilizing effect of the droplet elasticity.
100
77
78
1-
î fl)
~ ' -f . 8- . -tJ
Fig. 4.Lf
+
" +
+
shear rate [1/s]
Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 2. 6% l'oly Acryl Amide
Mirmohammad-Sadeghi (1983) has performed experiments with a
viscoelastic drop phase (P.A.A. in Corn Syrup) in Newtonian
silicone oils (See Appendix A). He used the cone and plate device
which was also used by Prabodh and Stroeve. The experimental
procedure used by Mirmohammad-Sadeghi was as follows. A very dilute
but coarse emulsion was brought into the device, a slow shear rate
ramp was applied (ramp up times of 0.75 s, plateau stages of
5-120 s and ramp down times of 0-60 s). Afterwards the maximum drop
size in the emulsion was determined microscopically. The ramp up
and ramp down times were chosen such that they did not affect the
resulting drop sizes. Mirmohammad-Sadeghi observed that
1 ....
Fig. 4.1.g
+
l-+ ++
shear ra te ( 1/s]
Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary number vs. shear rate. Drop phase: 1.0% H.E.C.
viscoelastic droplets are stretched out to greater length prior to
break-up as compared with Newtonian droplets.
The experimental data were presented in a plot of 0 vs 7 for
various drop phases with increasing polymer concentration. From
this plot the experimental results could be traced. The data were
also plotted as Ocrit vs Ocrit • SR. The stress ratios varied with
polymer concentration from 0.05 to 1. All viscoelastic drop break
up experiments were performed at similar viscosity ratios (0.55 <À
< 0.77). From the data it was concluded that the critical capillary
number was not constant for each droplet phase, but varied with the
79
80
10
'" " "
~ . fl)
iè 1 -e 8--b .
Fig. 4.1.h
++ ++ +
•+ ++ +
shear rate [ 1 /s)
Viscoelastic drop breakup results obtained by Tavgac (1972). Critical capillary nwnber vs. shear rate . Drop phase: 1.5% Kelzan
shear rate. It was also concluded that at low shear rates droplet
elasticity destabilizes, but the opposite is true at high shear
rates.
Several question marks should be put here. The critical drop sizes
determined are sometimes as large as 300 µm (diameter) where as the
maximum width of the gap between cone and plate is about 450 µm.
These data are consequently obtained in inhomogeneous shear flows
since the droplet diameter should ideally be smaller then 1/10 of
the gap width to avoid wall effects. This means that many data are
thus unreliable. Only the data obtained at very low shear rates (<
10 s-1) show critical capillary numbers which are significantly
below the corresponding Newtonian values. Especially these data
correspond to far too large droplets. Consequently the conclusion
of the destabilizing effect of droplet elasticity is not correct.
4.3.3 Conclusions
Very few theoretical approaches to the break-up of viscoelastic
drops in simple shear flows are available. More extensive studies
are available for the break-up of long cylindrically shaped
droplets in axisymetric non uniform shear flows.
Several experimental studies on the break-up of viscoelastic drops
have been reported. These studies, however, are inconclusive due to
limited observations, experimental errors, and coupling of the
effects of shear rate dependent viscosity and droplet elasticity.
Contrary to the conclusions of some authors, the available data
indicate that droplet elasticity has a stabilizing effect on break
up though it may accelerate the initial growth of disturbances.
4.4 BREAK-UP OF INELASTIC, SHEAR-THINNING DROPS
4.4.1 Model liguids
Our investigations are aimed at characterizing the effects of non
Newtonian droplet rheology on the break-up of droplets in simple
shear flows. There are many types of non-Newtonian behaviour but we
have selected two typical behaviours: liquids that are shear
thinning (a strong dependence on the shear-rate of the viscosity)
but exhibit very slight normal stress differences (negligible
elasticity) and liquids that have a nearly constant viscosity
(shear rate independent) but exhibit pronounced elasticity at
increasing shear rates (large normal stress differences). These
liquids were chosen to discern between the effects of two
frequently encountered non-Newtonian phenomena: shear rate
dependent viscosities and normal stress differences. In the
81
82
previous section on available literature it was shown that these
effects are usually coupled. In this chapter the break-up of small
inelastic shear thinning droplets obeying Eq. [4-4] will be
discussed. As derived by dimensional analysis of the basic
equations in chapter 2 of this thesis the break-up behaviour of
such droplets in a Newtonian continuous phase can in most relevant
cases be described by just three dimensionless groups:
11 'Î R
the capillary number: (l -
_c __
n-1 Ki_
- an apparent viscosity ratio: A a
- the power law index: n
,., c
The break-up of viscoelastic droplets will be discussed in section
4.5.
To study the inelastic shear-thinning behaviour, model liquids for
the drop phase had to be developed exhibiting this behaviour in the
shear rate range 1-100 s-1, being the range applied in the Couette
device. Following Acharya et al. (1976) solutions of polyacrylic
acid were used, which are commercially available as Carbopol (ex
B.F. Goodrich). At neutral pH value these Carbopol solutions built
up a gel structure but at low or high pH value no gel structure is
formed and they then exhibit strong shear-thinning behaviour
combined with low elasticity. Carbopol 941 was used because this
type gives very high viscosity values. Four different types of
these solutions were used:
1. Solutions of Garbopol 941 in water at various concentrations
(1.0 wt%, 0.5 wt% and 0.25 wt%). These solutions had a low pH.
2. Solutions of Carbopol 941 in an aqueous solution of sodium
hydroxide (1.5 wt%) at various concentrations. (0.75 wt%, 0.64
wt%, 0.5 wt% and 0.35 wt%) These solutions had a high pH.
3. Solutions of Garbopol 941 in a mixture of glycerol (80 wt%) and
water (20 wt%) at various concentrations (1.0 wt%, 0.5 wt% and
0.25 wt%).
4. Solutions of Carbopol 941 in a mixture of corn syrup (type Globe
01170 ex CPC Netherlands BV) (± 50 wt%) and water (± 50 wt%) at
various concentrations (1.0 wt% and 0.75 wt%).
All solutions were made in batches of 200 g by stirring the
solutions for one hour at moderate agitation. To prevent
bacteriological deterioration pentachlorophenol was added.
In all experiments the continuous phase was a Newtonian liquid.
Silicone oils (type Rhodorsil 47 ex Rhone-Poulenc Chimie Fine) were
used. Different viscosities were obtained by blending a high and
low viscosity batch.
The rheological properties of the liquids were measured with a
Haake type CVlOO viscometer using a concentric cylinder geometry
(type ZC 15) and an Instron rotary rheometer model 3250 using a
cone and plate geometry (diameter 60 mm, cone angle 2.4°). The
apparatuses were thermostatted at 23°C. The shear rate range was
from about 1-80 s·l. With the Haake, one measures apparent
viscosities as a function of shear rate, while the Instron measures
simultaneously tangential and normal farces (i.e. both viscosity
and normal stress differences). The elastic properties of the
droplet phases were characterized by normal stress differences in
steady shear experiments since these (large) deformations are more
similar to the deformations occuring in the droplet during droplet
deformation and break-up than the (small) deformations occuring in
oscillatory shear experiments. The rheological measurements were
fitted to the material functions given in Eq. [4-6]. The
rheological properties of the model liquids are tabulated in Table
4-1. For some drop phases normal stress differences could be
83
84
measured and these liquids are thus not strictly inelastic. However
for all experiments the ratio of the normal stress difference to
the tangential stresses was always less than unity. In section 4.5
it will be shown that the effect of droplet elasticity on drop
break-up is then negligible. Where no values are given for the
normal stress difference measurements they were below the accuracy
limit of the instrument. The apparent viscosities of the shear
thinning drop phases are shown in Fig. 4.2. The rheological
properties of the drop phases did not change signif icantly in the
time over which the experiments were performed (less than 5%)
Two different methods have been applied to measure the interfacial
tension between the liquids used;
a) The Wilhelmy plate method, where a silver plate with
circumference of 0.06 m is drawn from one liquid into the other
and the excess force, corrected for gravity effects, gives
directly the interfacial tension. This method is not suitable
for very low interfacial tensions and can have an accuracy of
0.1%.
b) The sessile drop method, which is based on the drop deformation
under gravity forces. After measuring height and width of the
droplets the interfacial tensions can be calculated with the aid
of the tables given by Bashforth and Adams (1883). An accuracy
is claimed of 0.1%. High interfacial tensions, however, are
difficult to measure.
All interfacial tensions were measured with respect to a silicone
oil with a viscosity of 1 Pas as being representative for all
silicone oils used. This because the surface tension of these oils
against air is found to be independent of the viscosity.
Interfacial tension measurements are very difficult for shear
thinning liquids since they show very high viscosities at the very
low shear rates occuring during these measurements. Reproducible
measurements, however, were obtained when the Carbopol solutions
were first subjected to very high shear rates (- 104 s·l). This
shear treatment strongly reduced the apparent viscosities, thus
allowing interfacial tension measurements, and, hopefully, did not
greatly affect the interfacial tension. The measurements are
tabulated in Table 4.1.
4.4.2 Drop break-up experiments
The measurements of deformation and break-up of individual droplets
were performed in a Couette device with two counter-rotating
cylinders. The flow in this unit approximates simple shear flow.
The apparatus and the adopted procedure have been fully described
in chapter 3 of this thesis. The experimental results are tabulated
in Appendix B. Only these results are given where fracture of
droplets was observed. Tipstreaming (see chapter 6 of this thesis)
was also observed for the solutions of Carbopol in glycerol/water.
Tipstreaming was always found to occur at capillary numbers in the
range 0.4 < 0 < 0.8, but these data are not given in Appendix B.
For the various types of shear-thinning drop phases the drop break
up data are plotted in a dimensionless form in Fig. 4.3. In these
figures the critical capillary n1.llllber at drop break-up, 0, is
plotted versus the viscosity ratio Àa· As a reference a fit through
the Newtonian drop break-up data (chapter 3 of this thesis) is also
plotted (curve). This function is given by
log Ocrit
in which C1
Cz
C3
C4
c 3
C + C log À + 1 2 log À + C
1.560
0.733
1.135
0.967
[4-17]
4
85
TAB LE 4.1 Model liquids used for shear-thinning drop break-up
Dro:elet ;:ehase~
Nr. Description K n x m
[Pa.sn) [-] [Pa.sm) [-) [mNn- 1 J
ST.D.1 C941.Hz0 1.0% 10 0.25 0. 75 26.5
ST.D.2 C941.llz0 0.5% 4.8 0.29 0.5 0. 75 28
ST.D.3 C941.Hz0 0.25% 1. 95 0.34 29
ST.D.4 C941.HzO.NaOH 0. 75% 4. 0 0.42 1. 5 0.55 29
ST.0.5 C941.HzO.NaOH 0.64% 1.9 0. 44 0.55 29.5
ST.D.6 C941.HzO.NaOH 0. 5% 0 .64 0.45 0.5 0.55 30
ST.D.7 C941.llzO.NaOR 0.35% 0.35 0. 48 30.5
ST .D.8 C94l.Glyc.Hz0 1.0% 3. 7 0. 44 2.4 0.62 21
ST.D.9 C941.Glyc.ll2o o.5% 0.95 0.61 0. 4 0. 7 23
ST .D.10 C941.Glyc,H20 0.25% 0.33 0 .65 24
ST.D.11 C941.CS 50.l!zO 0. 75% 20 0.42 0.3 0.9 38
ST.D.12 C941.CS 49.HzO 1.0% 29 0.47 1. 7 0.8 35
ST .D.13 C941.CS 48.HzO 0. 75% 30 0. 48 0.3 1.0 35
Continuous phases
Nr. Desc.ription K n x m [Pa.sn) [-) [Pa.sm] [-J
ST.C. l Silicone Oil 0. 9
ST.C.2 Silicone Oil 2.6
ST.C.3 Silicone Oil 5.67
ST.C.4 Silicone Oil 6.3
ST.C.5 Silicone Oil 16.0
ST.C.8 Silicone Oil 40 .0
86
. ëi5 0 ü (j)
> Î
Shear thinning liquid Carbopol 94 î .H20
-+- 1.0% - -A- 0.5% -e- 0.25%
îEî
! 1 " i
1
îEO ~ ~
~ 1 r 1
î r
t
shear rate ( î /sec)
Fig. 4.2.a Rheological properties of shear thinning model liquids. l.0%, 0.5% and 0.25% Carbopol 941 in water
87
(ij cü û. ___..
::>-, +-' (j)
0 ü (j)
s
88
îEî
Î
Î 1
Shear thinning liquid Carbopol 94 î .NaOH
-+-- 0.64% --è.- 0.5% -e-- 0.35%
~ t
~ ~~ 1
~ &~ 1-
~ r G-..
' ',,& ~ r
'&... ........... ' "A-'s.. -
! ' -1:.. " f- '-s._ &.
~ ' ~il. t 's..
'G--s._
1E-J 0
1 i 1 j Il 1 1 j 1 !
Î îO îOO
shear rate ( î /sec)
Fig. 4.2.b Rheological properties of shear thinning model liquids. 0.64%, 0.5% and 0.35% Carbopol 941 in water/NaOH
>, +-' (j)
0 ü (j)
>
1Eî
1EO
1E-î
1E-2
Shear thinning liquid Carbopol 94 î .H20/glycerol
-!- î.0% --6.- 0.5% -e- 0.25%
1 10 100
shear rate ( î /sec)
Fig. 4.2.c Rheological properties of shear thinning model liquids. 1.0%, 0.5% and 0.25% Carbopol 941 in water/glycerol
89
-(j)
cv Q_ ..._...
:>-, +-' (j)
0 u (j)
>
90
100
10
1
0. Î
Shear thinning liquid Carbopol 94 î .H20/cornsyrup ---+- 0.75% - -&- 1.0% -e- - 0.75%
50%cs 49%cs 48%cs
~
' ' '" ~'..__ ' ' ''!è.
'o
1 10 100 1000
shear ra te ( 1 /sec)
Fig. 4.2.d Rheological properties of shear thinning model liquids. 0.75%, l.0% and 0.75% Carbopol 941 in water/corn syrup (50%/49%/48% corn syrup respectively)
àï 10
~ c » .._ Jg lï !U 0
1
I t:. J
/
viscosity ratio
+ 1.0% A 0.5% 0 0.25%
Fig. 4.3.a Drop breakup results for various shear thinning drop phases. 1.0%, 0.5% and 0.25% Carbopol 941 in water
To calculate an apparent viscosity ratio Àa, the apparent droplet
viscosity was taken to be that at the critica! shear rate icrit
(applied to the continuous phase at the moment of drop break-up)
n-1 K •
À [4-18] a '7
c
91
92
viscosity ratio
+ 0.75% A 0.64% 0 0.50% + 0.35%
Fig. 4.3.b Drop breakup results for various shear thinning drop phases. 0.75%, 0.64%, 0.5% and 0.35% Carbopol 941 in waterfNaOH
4.4.3 Discussion
The solutions of Carbopol in water and in the aqueous solution of
hydroxide usually showed normal drop break-up, while the solutions
of Carbopol in water/glycerol mixtures showed almost always
tipstreaming. The reason for this has not been studied, but it may
be related to the presence of surface active materials (see chapter
6 of this thesis), which is indicated by the rather low interfacial
tension. The solutions of Carbopol in water/corn syrup mixtures
were intended to extend the results to higher viscosity ratios.
--+-- 0.75% 50%cs
viscosity ratio
~ 1.0% 49%cs
~~ + + +
0 0.75% 48%cs
Fig. 4.3.c Drop breakup results for various shear thinning drop phases. 0.75%, 1.0% and 0.75% Carbopol 941 in water/corn syrup (50%/49%/48% corn syrup respectively)
The experimental results show the following:
1. There is a reasonably good correlation between the measured
critica! capillary number 0 and the apparent viscosity ratio
calculated with Eq. [4-18].
2. The scatter in the shear-thinning break-up data is larger than
the scatter in the Newtonian break-up data (chapter 3 of this
thesis).
3. At low viscosity ratio's the critica! capillary number varies
approximately by O - Àa-2/3, being the theoretically predicted
dependence for Newtonian droplets. 93
94
4. The value of the experimentally determined critical capillary
numbers are systematically higher f or the shear thinning
droplets than for the Newtonian droplets.
5. The deformation of the shear-thinning droplets prior to break-up
was generally somewhat larger than for the Newtonian droplets.
The mode of break-up, however, was very similar.
6. The critica! capillary number for apparent viscosity ratio's
above 0.5 increases rapidly.
The first observation is somewhat remarkable since from dimensional
analysis (chapter 2 of this thesis), for ideally inelastic shear
thinning liquids one predicts that the critical capillary number is
a function of two dimensionless groups: the apparent viscosity
ratio Àa and the power law index n. For these model liquids the
power law index varied between 0.25 ~ n ~ 0.48. On closer
examination, however, there appears to be a slight tendency for the
lower power law index drop phases to have a somewhat higher
capillary number.
The second observation is partly misleading because in the
graphical representation of the Newtonian drop break-up data the
critical capillary numbers obtained for a given combination of a
droplet and continuous phase were averaged. For shear thinning
droplets this is not allowed since the apparent viscosity ratio is
not constant for one combination of liquids.
The fourth observation cannot only be interpreted by a higher shear
rate at which drop break-up occurs, but can also be interpreted by
an apparent viscosity ratio which is too high (i.e. the data points
have been shifted to the right). This interpretation implies the
presence of effective shear rates within the droplet, that are
higher than those applied to the continuous phase. To obtain an
empirical correlation for these break-up results we have taken the
ratio of the internal shear rate ~int and the applied shear rate 'l
to be constant:
'lint - C ~ (4-19)
The modified viscosity ratio is thus given by
n-1 n-1 K .
"int K -Y
n-1 n-1 À c c À [4-20] ml '1 '1 a
c d
The constant C has been determined by statistica! analysis of the
break-up data of the solutions of Carbopol in water and in the
aqueous solution of sodium hydroxide. A least squares method for
non-linear functions based on a Marquardt algorithm was used. The
best fit was obtained for
c - 5.1 [4-21]
The thus calculated modified viscosities are tabulated in Appendix
B and the results are graphically presented in Fig. 4.4 for all
drop phases. Below a modified viscosity ratio of 0.1 the shear
thinning drop break-up results can be well described by the
Newtonian drop break-up criteria, provided the modified viscosity
ratio is used. Above this modified viscosity there appears to be a
discrepancy between the shear thinning drop break-up data and the
Newtonian drop break-up criteria. This discrepancy is even more
pronounced for the results with the solutions of Carbopol in the
water/corn syrup mixtures.
For spherical Newtonian droplets the flow inside the droplet can be
obtained analytically with the method described by Taylor {1932).
This analysis was done by Bartok and Mason. For a droplet placed
in a simple shear flow with the velocity components
u -y y
v-o
w - 0
[4-22a]
[4-22b]
[4-22c]
95
96
they derived the following internal velocity field.
2 2 2 2 (x + y + z ) x
u = "I 5 - 4-+ (2>. - 1) 4 (À + 1) 2 2
R R
[4-23a]
2 2 2 2 x (x + y + z )
4 L_ + v = "I 5 (2.x + 5) 4 (À + 1) 2 2
R R
[4-23b]
z xy w = "I 4-
4 (À + 1) 2 [4-23a]
R
To f ind the internal shear rates one should look at the rate of
strain tensor ~. which is defined as ~ = ~ (1 + in which 1 is
the velocity gradient tensor defined as 1 = 8y/84.
2 2 2 2 xy Sx + 8 y + 5 z 3zy
-3 2 2 2
R R R
2 2 2 "/ 8 x + 8y + 5 z 2 xy 3zx
~ = -3 --4 (). + 1) 2 2 2
R R R
3 zy 3 zx 4 xy
2 2 2 R R R
(4.24]
Generally the flow in the droplet is thus not a simple shear flow.
Only for some special positions in the droplet the flow reduces to
simple shear. The maximum rate of shear in a simple shear flow is
• 5 • "( =--y int 2
[4·25]
which is obtained for: z = o, y = o, x = ± R and À = o. Generally
the strength of a flow field is given by the norm of the rate of
strain tensor: IRI. The norm will be defined by
IRI = )2 R !l.1
[4-26]
For simple shear flow the norm thus defined is equal to the shear
rate. The strength of the internal flow field varies in the
droplet. As a measure of this strength the maximum norm in the
droplet was chosen. The maximum in the norm can be found by
standard analytica! techniques to find the extremes of a function
under a constraint. After tedious calculation it can be shown that
the global maximum of the norm of the velocity gradient tensor of
the internal flow field is, for all viscosity ratios, obtained for
x - ± l:t !2îl. y ± l:t /2îl and z - 0 [4-27]
The maximum norm is given by
1121 max
- .:y (Hl)
J7 [4-28]
Elmendorp (1986) has also given an analysis of the rate of shear in
ellipsoidal drops. In the limit of spherical droplets, however, his
analysis predicts zero rate of shear since his analysis assumes
rigid rotation for spherical droplets. Especially for low viscosity
ratios this assumption is wrong.
97
98
The maximum norm of the velocity gradient tensor will be used to
calculate a theoretically modified viscosity ratio Àm2 by taking
the droplet viscosity at the shear rate given by 111max:
À m2
By
is
K ( 111 )
combining
obtained:
n
max
I'/ c
Eq.
- ---1 - n
À + m2
n-1
[4-29)
[4-28] and [4-29] an implicit expression for Àm2
1
[:cl 1 ----
1 - n 1 n À "Î J7 ~ 0 [4-30]
m2
The value of Àm2 was numerically calculated for each break-up
experiment with a standard Newton-Raphson zero point technique.
The results are tabulated in Appendix B and are graphically
presented in Fig. 4.4. From these results it will be clear that
when Ocrit is plotted versus Àm2 there is much better
correspondence to the Newtonian break-up criteria than when the
apparent viscosity ratio Àa is used. For viscosity ratios close to
unity the description in terms of Àm2 is also in better agreement
than the description in terms of Àml· This is due to the prediction
of Eq. [4-13] that for high viscosity ratio 1 s (À> J7 - 1 ~ 1.6)
the maximum norm of the velocity gradient tensor becomes smaller
than the applied shear rate.
The theoretically obtained shear rates in spherical Newtonian
droplets are lower than the empirically determined internal shear
rate based on the best fitting modified viscosity ratio. This is
probably due to the shear thinning characteristics of the droplet
phases. For shear thinning liquids it is well established (e.g.
Crochet et al. 1984) that the velocity changes are restricted to a
smaller distance than for Newtonian liquids, resulting in higher
(ij 10 _Q
E 2
0
viscosity ratio
Fig. 4.4.a Drop breakup results for shear thinning drop phases plotted for Apparent viscosit:y ratio À•
maximum shear rates. It is thus very likely that the shear rate in
the droplet near the interface, which is probably the most relevant
shear rate, is higher for a shear thinning droplet than for a
Newtonian droplet.
Although the analysis of the flow within the droplet may offer an
explanation for the shear rate at which break-up occurs it has not
explained why break-up occurs at larger critical drop deforma
tions. This effect might be due to the elongational viscosity of
the drop phase. Many non-Newtonian liquids are known to have a
ratio of the elongational to simple shear viscosity that is (much)
larger than 3 as it is for Newtonian liquids. During the final
stages of pinch-off and break-up, the elongational component of the
99
100
viscosity ratio
Fig. 4.4.b Drop breakup results for shear thinning drop phases plotted for Modified viscosity ratio Àm1
flow in the droplet will become very important. When the
elongational viscosity is much higher, larger drop deformations
prior to break-up are thus expected.
4.4.4 Gonclusions
Solutions of Garbopol 941 in water and in aqueous solutions of
sodium hydroxide are appropriate model liquids exhibiting strong
shear thinning behaviour in the shear rate range 1-100 combined
with very low normal stress differences.
For viscosity ratios below 0.1 the critica! capillary number at
which drop break-up occurs for shear-thinning droplets can be
correlated by the Newtonian break-up criterion (Eq. [4-17))
viscosity ratio
Fig. 4.4.c Drop breakup results for shear thinning drop phases plotted for Modified viscosity ratio À""'
provided a modified viscosity ratio Àml (Eq. [4-20]) is used to
account for the internal shear rate in the droplet, which is higher
than the applied shear rate.
For viscosity ratios of the order of unity the critical capillary
number at which break-up occurs rises with increasing viscosity
ratio. This rise starts for shear thinning droplets at lower
viscosity ratios than for Newtonian droplets. This is probably due
to the fact that for viscosity ratios greater than unity the
internal shear rate becomes considerably lower than the applied
shear rate.
101
102
4.5 BREAK-UP OF VISCOELASTIC DROPS
4.5.l Model liguids
As discussed in the review of the literature on non-Newtonian drop
break-up, there are only a few studies available on the break-up of
viscoelastic drops in simple shear flows. None of them is very
conclusive. The current studies were aimed at discerning the
effects on droplet break-up in simple shear flows of two frequently
encountered types of non-Newtonian behaviour of the droplet phase:
shear rate dependent viscosities and normal stress differences. The
first study was discussed in section 4.4. In this section a study
of viscoelastic droplet break-up will be discussed in which model
liquids for the droplet phases were used that exhibit hardly any
shear rate dependency for the viscosity combined with considerable
normal stress differences (obeying Eq. [4-6]). As derived by
dimensional analysis of the basic equations in chapter 2 of this
thesis, the break-up behaviour of such droplets in a Newtonian
continuous phase can in most relevant cases be described by five
dimensionless groups:
'7 'Î R
- the capillary number: 0 _c __
a
- an apparent viscosity ratio: À
- the power law index: n
- the elastic power law index: m
- the stress ratio: S R
a
n-1 K...i.._
'7 c
Our present studies are aimed at further reducing this number by
taking n = 1 and m is approximately equal for all droplet phases.
For such model liquids the choice of the shear rate at which the
apparent viscosity ratio should be determined is much less
ambiguous than it was for the inelastic shear thinning droplets.
Hence only three dimensionless groups will be left n, Àa and SR.
For the study of viscoelastic drop break-up, model liquids for the
drop phases had to be developed that show high normal stress
differences combined with shear rate independent viscosities in the
shear rate range 1-100 s·l, being the range applied in the Couette
device, Such liquids are called Boger-liquids since Boger (1977,
1978) was the first to report such liquids. Boger suggested the use
of small amounts of the polyacrylamide Separan (ex Dow Chemical
Company) in corn syrup. Choplin et al. (1983) prefered the use of
solutions of the polyacrylamide Pusher (ex Dow Chemica! Company) in
mixtures of glycerine and water. They reported several
disadvantages of the separan/corn syrup mixtures: flocculation of
the polymer, crystallization of the syrup and irreproducable
results from batch to batch. In the studies of shear thinning drop
break-up, however, glycerine solutions resulted in tipstreaming,
which is outside the scope of the present studies. We circumvented
all these problems by using small amounts of Pusher in water/corn
syrup mixtures. The viscosity of the water/corn syrup mixture was
adjusted by adding various amounts of water to a corn syrup (type
Globe 01170 ex CPC Netherlands B.V.). The elasticity of the model
liquids was varied by adding various amounts of polyacrylamide
(type Pusher 700 ex Dow Chemical Company). To prevent bacteriologi
cal deterioration sodium azide (NaN3) was added at 0.02 wt%. The
solutions were made homogeneous, after adding the polyacrylamide to
the corn syrup/water mixture, by alternately stirring very gently
and stopping. This procedure was continued for one week.
103
104
In all experiments the continuous phase was a Newtonian liquid.
silicone oils (type Rhodorsil 47 ex Rhone Poulenc Chimie Fine) were
used to have little problems with traces of surface active
materials. Different viscosities were obtained by blending a high
and a low viscosity batch.
The rheological characterization was performed in the same way as
described in section 4.4.l for the shear thinning drop break-up
experiments. To measure the high normal stress differences a
careful and systematic rheometric method was applied. Since it is
known that real overshoot behaviour (Lockyer and Walters, 1976) can
be encountered and that the waiting time between two measurements
with one sample affects this overshoot behaviour (Stratton and
Butcher, 1973) the following approach was adopted. After a sample
was introduced between the cone and plate the sample was
periodically subjected to 60 s rest and 60 s steady shear with
shear rates increasing per step. After 60 s of steady shear the
overshoot was generally levelled off (See also Michele, 1978} and
hence the readings were taken after 60 s of steady simple shear
flow. The rheological properties of the model liquids were found
to be very stable in time, provided they were made according to the
previously described procedure. They are tabulated in Table 4.2,
and graphically represented in Fig. 4.5. The smaller the water
content of the corn syrup/water mixture and the lower the
polyacrylamide level, the more the model liquid has a shear rate
independent viscosity. When 0.5% polyacrylamide is added the
viscosity is clearly shear rate dependent. The elastic power law
index m varies between 1.2 and 1.7 but does not systematically vary
with the corn syrup/water mixture viscosity or the polyacrylamide
level.
The interfacial properties were measured with the Wilhelmy plate
method as described in section 4.4.1.
TAB LE 4.2 Model liquids used for viscoelastic drop break-up
Droplet phases
Nr. Desoription K n " m " [Pa.sn] [-) [Pa.sm] [-] [mNm-1 l
VE.D.l 75. 25/ 0.12 29
VE.D.2 /0.05% 0.175 0.01 1.47 29
VE.D.3 /0 .1% 0,29 0.95 0.06 l.57 29
VE.D.4 /0.2% 0.85 0 .84 0.6 1.4 28
VE.D.5 /0,5% 4 .15 0.65 8 1.2 27
VE.D.6 BO. 20/ 0.29 l 31
VE.D.7 /0. 05% 0.53 0.88 0.19 1.48 31
VE.D.8 /0 .1% 0. 78 1 0.51 1.55 31
VE.0.9 /0.2% 1.67 0,84 2.1 l.47 30
VE.D.10 /0. 5% 5.5 0.64 16 1.19 29
VE.D.11 85.15/ o. 78 l 33
VE.D.12 /0. 05% 1.15 1.05 0.16 1.37 33
VE.D.13 /0.1% 1.2 1.05 0.48 1. 40 33
VE.D.14 /0.2% 2 0.99 1.2 1. 57 32
VE.D.15 /0. 5% ll. 7 0.67 55 1.21 31
VE.D.16 90.10/ 2.8 1 35
VE.D.17 /0.1% 4.1 0.98 l. l 1. 58 35
VE.D.18 /0 .2% 8.15 0.90 12 1.40 34
VE.D.19 /0.4% 5.6 0.98 3.9 1. 75 33
Continuous phases
Nr. Description K n " m
rea .• n1 [-) [Pa.sm) [-)
VE.C.1 Silicone 011 0.9
VE.C.2 Silicone Oil 6.0
VE.C.3 Silicone Oi l 11.36
VE.C.4 Silicone Oil 26.60
VE.C.5 Silicone OH 38.11
VE.C.6 Silicone Oil 43.2
VE.C.7 Silicone Oil 64.6
105
106
100
10
viscoelastic 1iquid PAA in 75/25 CS/H20
shear rate ( î /sec}
···+·· visc. 0.05%
·A" visc. 0.1%
O·· visc. 0.2%
+ ·· visc 0.5%
--.-.. norm. 0.05%
-it- norm. 0.1%
--+--norm. 0.2%
-6.- norm. 0.5%
Fig. 4.5.a Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (75%)/water mixture
-ro fb ({) ({) (])
.P ({)
ro E '-0 c
viscoelastic liquid PAA in 80/20 CS/H20
1000 ....------
100
10
."+" visc. 0.05%
""b. · visc. 0.1%
··O··· visc. 0.2%
".+·· visc 0.5%
_,._norm. 0.05%
-e- norm. 0.1%
norm. 0.2%
-±.- norm. 0.5%
100 1000
shear rate ( î /sec)
Fig. 4.5.b Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (80%)/water mixture
107
108
~ ~ (f) (f) <l.J !o... +-' (f)
ro E !o...
0 c -
viscoelastic lîquid PAA in 85/ î 5 CS/H20
1
0.1 '--1-'-'-u.u.M.~..J....J..lfl!tl 1 1!1111!1
' " "...1
···+·· visc. 0.05%
···/::.· · visc. 0.1%
O·-- visc. 0.2%
··+·· visc 0.5%
_,.,._norm. 0.05%
-.-norm. 0.1%
-+-norm. 0.2%
--A- norm. 0.5%
0.1 10 100 1000
shear rate ( î /sec)
Fig. 4.5.c Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (85%)/water mixture
ro § 0 c -
viscoelastic liquid PAA in 90/10 CS/H20
shear rate ( î /sec)
".+" visc. 0.1%
""t;,.". visc. 0.2%
".O··· visc. 0.5%
".+··· visc 0.4%
-....-.. norm. 0.1%
_._norm. 0.2%
.........,__norm. 0.5%
-~- norm 0.4%
Fig. 4.5.d Rheological properties of viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (90%)/water mixture
109
110
4.5.2 Drop break-up experiments
The measurements of deformation and break-up of individual droplets
were performed in a Couette device with two counter-rotating
cylinders. The apparatus and the adopted procedure have been fully
described in chapter 3 of this thesis. In first instance drop
break-up experiments were performed with a wide range of combina
tions of 19 droplet phases and 7 continuous phases.
The experimental results are tabulated in Appendix B, where the
apparent viscosity ratio Àa and the stress ratio SR are calculated
at the critical shear rate at which droplet break-up occured.
To compare these data with the behaviour of Newtonian droplets, the
critical capillary number at which break-up occured has been
plotted versus the apparent viscosity ratio in Fig. 4.6 together
with the 5 parameter fit through the Newtonian drop break-up data
(chapter 3 of this thesis).
log 0 crit
in which
C1 -0.506
C2 -0.0994
C3 0.124
C4 -0.115
C5 -0.611
c 2
C + C log À + C (log À) + 1 2 3 log À + C
5
[4-31]
For some droplet phases, especially those with a high corn syrup
level and a high polyacrylamide level it was observed that break-up
in stationary simple shear flow was impossible below a certain drop
radius. Therefore additional systematic experiments were performed
(j; 10 .0 E :J c
» '--
~ u (IJ u
viscosity ratio
Fig. 4.6.a Drop breakup results for viscoelastic model liquids.
Fig. 4.6.b
0.5%, 0.2%, 0.1% and 0.05% in corn syrup (75%)/water mixture used as drop phase
+ + •
viscosity ratio
Drop breakup results for viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (80%)/water mixture used as drop phase
1E1
111
,_ 10 Q)
.Q
§ c
>-,_ ~ ëi ro ü
112
viscosity ratio
Fig. 4.6.c Drop breakup results for viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (85%)/water mixture used as drop phase
+ +
viscosity ratio
Fig. 4.6.d Drop breakup results for viscoelastic model liquids. 0.5%, 0.2%, 0.1% and 0.05% in corn syrup (90%)/water mixture used as drop phase
to determine this critical drop radius for various combinations of
droplet and continuous phase. These results are also tabulated in
Appendix B. Some of the results have been visualized in Fig. 4.7.
4.5.3 Discussion
The experimental results on viscoelastic drop break-up show the
following features:
1. The critical capillary number at which break-up occurs does
clearly not only depend on the apparent viscosity ratio, as for
Newtonian droplets, but also on droplet elasticity (See Fig.
4.6).
2. The critical capillary numbers for the viscoelastic droplets are
higher (sometimes much higher) than for Newtonian droplets:
droplet elasticity clearly stabilizes the droplet against break
up. No systematic observations were made of critical capillary
numbers below the Newtonian drop break-up measurements (See Fig.
4.6). This is contrary to (erroneous) conclusions drawn by
Tavgac (1972), Prabodh and Stroeve and Mirmohammed-Sadeghi.
3. The deformations of the viscoelastic droplets prior to break-up
are often much larger than of Newtonian droplets, especially for
the rather elastic droplets. Stable droplets or rather
threadlike objects with a length to width ratio well above 100
have been observed in steady simple shear flows.
4. Droplets with a large length to width ratio, which were stable
in steady simple shear flow, often were unstable when the flow
was stopped abruptly. Droplet break-up according to a Taylor
instability like mechanism occured, resulting in a large number
of equal sized fragments. Such observations however, have not
been included in the tabulated experimental results as drop
break-up measurements, since only droplet break-up in quasi
steady increasing shear flows was being studied.
113
114
Ê s -~ 0 "'O 1E-1 (tl i...
~
shear rate (1/sec)
• break~ observed
o no break~ observect
Fig. 4.7.a Determination of critical drop radius for viscoelastic drop phase VE.D.28 in continuous phase VE.C.35
1E-1
•
0
shear rate ( 1 /sec)
• break-'-'> observed
o no break-1.4' observed
Fig. 4. 7 .b Determination of critical drop radius for viscoelastic drop phase VE.D.28 in continuous phase VE.C.37
115
116
5. For a combination of a certain droplet and continuous phase a
series of measurements generally resulted in one point, or a
vertical series of points, in the plot of Ocrit vs Àa· This is
due to the almost shear rate independent droplet viscosity.
The critical capillary number for such a combination increases
for decreasing droplet sizes and hence increasing rate of shear.
Since m > 1 for all droplet phases, the critical capillary rises
also with increasing stress ratio.
6. The increase in the logarithm of the critical capillary number
seems to be exponential with the logarithm of the stress ratio
and it seerns to becorne larger with increasing viscosity ratio.
This observation was the basis of the function that was used to
correlàte the viscoelastic drop break-up data: Ocrit = f (Àa,
SR). For Newtonian drop break-up, SR o, this function was
taken to be equal to the 5 parameter fit through our Newtonian
drop break-up data (Eq. [4-31]. To account for the effect of non
zero stress ratio the sirnplest possible exponential function was
chosen that yields, for constant stress ratio, a critical
capillary nurnber that rises with increasing apparent viscosity
ratio and that yields for all non zero stress ratios critical
capillary nurnbers higher than for SR o
log Ocrit =log 00 ,crit + C6 exp ((C7 +Cg log À) 2•log SR]
(4-32]
The constants C6, C7 and Cs have been determined by statistical
analysis of the break-up data given in Appendix B. Only those
data in which droplet break-up was actually observed in
increasing quasi steady simple shear flow, were used. A least
square method for non linear functions based on a Marquardt
algorithrn was used. The best fit was obtained for
c6 0.121
C7 1.474
Cs 0.355
Visco-elastic drop break-up in Newtonian continuous phase
fitted functlan OM
1.16
0.66
0.17
-8.37 ll._J~J__JL_J_~j___J~J...._J~J_..J[___J_~1-~-io~.soïlö".....1.~.J..........1:....-...i............,.o.3!I
101110 (p ) 101110 (SrJ
-2.60 -L&a P -1.00
la 10 (a• • rit
Fig. 4.8 llrop breakup for viscoelastic drops as a function of viscosity ratio and stress ratio
'-(J)
.D E
118
:J c
• break-up observed
.. ··-· .. ..... . •••••••
stress ratio
Fig. 4.9.a Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.003 < .l. < 0.01
This function is shown in Fig. 4.8. In Fig. 4.9 the break-up
data are shown together with the empirical correlation for
various ranges of the apparent viscosity ratio. The empirical
correlation is plotted for both the lower and the upper limit of
the apparent viscosity ratio range. Both the data at which
break-up was observed and those at which break-up was not
observed up to the maximum shear rate used are plotted in these
figures. From these figures it follows that the empirica!
correlation is a reasonable representation of the break-up
data.
• break-up observed
stress ratio
Fig. 4.9.b Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.01 < À < 0.02
7. For the droplet phases with the highest polyacrylamide
concentration and for the droplet phases with the highest corn
syrup concentration critica! drop radii have been observed.
10
For these droplet phases the experimental data have been plotted
in Fig. 4.7 as droplet radius vs critica! shear rate at which
break-up occured. The solid circles denote actually observed
break-up conditions. The open circles denote that break-up did
not occur up to the given rate of shear. These figures show that
for the larger drop radii and the lower shear rates the critical
119
120
• break-up observed
0 no break-up observed
• • " .... -~~· . ••• ~rP-.-• -.... . -· •.. ·-------.-.--
Fig. 4.9.c
stress ratio
Drop breakup results for viscoelastic drops as a function of the stress ratio for 0. 02 < À < 0. 05
capillary number is constant (slope = -1) or rises slowly with
decreasing droplet size (or equivalently the capillary number
rises with increasing stress ratio). Ata certain drop size,
however, there appears to be an abrupt change in the behaviour
of the droplets. Even when rnuch higher shear rates are applied
to such droplets, break-up will not occur.
© _Q
E ::l c
• break-up observed
n no break-up observed
stress ratio
Fig. 4.9.d Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.05 <À< 0.1
•
The drop radius at which this change occured were determined
from Fig. 4.7. The critica! drop sizes are tabulated in Table
4.5. The critica! drop sizes tend to increase with the apparent
viscosity ratio, but do not correlate at all with the stress
ratio (See Fig. 4.10). Such critica! drop sizes have been
predicted, on empirica! arguments by Tavgac (1972) (See section
4.3.2) but have to the author's knowledge not yet been observed
or explained theoretically. 121
L QJ
.D E :::J c
122
• break-·up 0 no break-up observed observed
10
/
/ ~ 0
/00 8Sl2> 0 CQ) 0
0 0 0 0 00
0
•o
stress ratio
Fig. 4.9.e Drop breakup results for viscoelastic drops as a function of the stress ratio for 0 .1 < >. < 0. 2
A very qualitative argument predicting critical drop radii is
given below. For Newtonian drops break-up will occur when the
viscous shear stress exerted by the continuous phase is larger
than the interf acial tension f orces or
fl c
7 > !!. or O R
11 'Î R c --->ü
a crit [4-33]
•
10
L (IJ Ll E ::J c
• b:eak-up observed
o no brea'-\-·up observed
/
/! 0
0 0
/. / 0 D
. / 0 CJ 0 0
!/ 0 ~/ 0 0
/. .·: . • z!-_ •
80 • • • . . ____ ...:;. . . . . . .. __.... .--.- .~ ... ·=.=:::=-~ ..... ~ ..... "- _ ....
0.1 0.1
Fig. 4.9.f
• •
stress ratio
Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.2 < À < 0.5
.~
0
For viscosity ratios between 0.1 <À< 1, which is the range in
which most of the data on critical drop sizes were obtained, we
know that Ocrit = 0.5 0 0 ,crit· For viscoelastic drop break-up
the viscous shear stresses exerted by the continuous phase
should also overcome the elastic forces in the droplet, which
are given by
123
w .D E :::J c
124
• break-up observed
•
- l t . . 1 • l J ·~ 1
o no break-up observed
0
8 •
stress ratio
... 0 •
0 0
Fig. 4.9.g Drop breakup results for viscoelastic drops as a function of the stress ratio for 0.5 < À< 2
Let us assume that the viscous shear stresses exerted by the
continuous phase should simply be greater than the sum of the
interfacial forces and the elastic forces inside the droplet:
a m
" .y > n -+ c . /Ç t c o,crit R
[4-34a]
or equivalently
n 1 >
n 1 - Gonst. SR . À o,crit
[4-34b]
However, in some situations the right hand term in Eq. 4-34
increases more rapidly with i than the left hand term. This will
result in the inexistence of a solution. Break-up will thus
only be possible when
R>
0 (J
o,crit
m 'I i-C.iç,Y
c
The critical drop size
such that 'Ic .y - Gonst.
l
" m:T c
.y c
/Ç m
The critical drop size
R crit
·te /Ç":] " c
[4-35]
will be obtained when the shear rate is
iç .ym has a maximum:
[4-36]
is thus obtained for
0 (J
o, crit (4-37]
ntr [c": m] ~ - Const. K.
Although these arguments qualitatively predict critical drop
sizes, the predictions are quantitatively not-correct.
This must be due to the over simplification of Eq. [4-34].
Prabodh and Stroeve have done viscoelastic drop break-up
experiments using high rates of shear and small droplet sizes
but have not reported any phenomenon indicating the presence of
critical drop sizes. This is probably due to the rather low
elastic power law index m of the drop phase used by Prabodh and
Stroeve: m - 1.06 (See Appendix A). Critical drop sizes can
probably only be observed when the elastic forces in the droplet
increase more rapidly with the shear rate than the applied shear
stresses (m > 1).
125
1EO
E' ..s ({) :J 'ö ro ,_
0.. 1E-1
0 ,_ -0
(ij u
:;::;
·= ü
Fig. 4.10
126
• • + +
" 'Î"! 0 • + VED.5 • . ,.. t:. VED.10
+ •• + 0 VE.D.15
+ VE.D.17
+ • VED.18
• VE.D.19
1EO
viscosity ratio
Critical drop radii observed for viscoelastic drop breakup in quasi steady simple shear flow as a function of the viscosity ratio
4.5.4 Conclusions
Droplet elasticity results in higher critical capillary numbers, at
which break-up occurs, than for Newtonian droplets.
This increase rises with droplet elasticity and is stronger for
viscosity ratio's of order unity than for lower viscosity ratio's.
Droplet elasticity results in much larger droplet deformations
prior to break-up. These long cylindrical drops can be stable in
steady shear flow but break up after cessation of the flow.
Critical drop sizes have been observed below which break-up is
impossible irrespective of the magnitude of the shear rate.
This is probably a result of the fact that for the liquids used the
elastic forces in the droplet increase more rapidly with the shear
rate than the shear stress exerted by the continuous phase. The
critical radius increases with the viscosity ratio.
4.6 REFERENCES
1. A. Acharya, R.A. Mashelkar and J. Ulbrecht, Flow of inelastic
and viscoelastic fluids past a sphere. 1. Drag coefficient in
creeping and boundary layer flows, Rheol. Acta 15, 454-470,
(1976)
2. W. Bartok and S.G. Mason, Particle motions in sheared
suspensions. 7. Internal circulation in fluid droplets
(theoretical), J. Colloid Sci. 13, 293-307, (1958)
3. F. Bashforth and J.C. Adams, An attempt to test the theories of
capillary action, University Press, Cambridge, (1883)
4. D.V. Boger, A highly elastic constant-viscosity fluid, J. Non
Newtonian Fluid Mech. 3, 87-91, (1977/1978)
5. D.V. Boger and H. Nguyen, A model viscoelastic fluid, Polym.
Engng. Sci. 18, 1037 1043, (1978)
127
128
6. D.W. Bousfield, R. Keunings, G. Marucci and M.M. Denn, Non
linear analysis of the surface tension driven break-up of
viscoelastic filaments, J. Non Newt. Fluid Mech. 21, 79-97,
(1986)
7. R.A. de Bruijn, Sealing laws for the flow of emulsions,
chapter 2 of this thesis.
8. R.A. de Bruijn, Newtonian drop break-up in quasi steady simple
shear flow, chapter 3 of this thesis.
9. R.A. de Bruijn, Newtonian drop break-up in simple shear flow
the tipstreaming phenomenon, chapter 6 of this thesis.
10. H.B. Chin and C.D. Han, Studies on droplet deformation and
break-up. 2. Droplet deformation in non uniform shear flow, J.
Rheol. 24, 1-37, (1980)
ll. L. Choplin, P.J. Carreau and A. Ait Kadi, Highly elastic
constant viscosity fluids, Polym. Eng. Sci. 23, 459-464,
(1983)
12. M.J. Crochet, A.R. Davies, and K. Walters, Numerical
simulation of non-Newtonian flow, Elsevier, Amsterdam, (1984)
13. J.J. Elmendorp, A study on polymer blending microrheology,
Ph.D. thesis, Delft University of Technology, (1986)
14. F. Gauthier, H.L. Goldsmith and S.G. Mason, Particle motions
in non-Newtonian media, 1. Couette flow, Rheol. Acta 10, 344-
364, (1971)
15. H. Janeschitz-Kriegel, Flow birefringence of elastico-viscous
polymer systems, Adv. Polym. Sci. 6, 170-318, (1969)
16. M.A. Lockyer and K. Walters, Stress overshoot: real and
apparent, Rheol. Acta 15, 179-188, (1976)
17. J. Michele, Zur Rheometrie viskoelastischer Fluïde mit der
Kegel-Platte Anordnung, Rheol. Acta 17, 42-58, (1978)
18. V. Mirmohammed-Sadeghi, An experimental study of the break-up
of model viscoelastic drops in simple shear flow, M. Sc.
thesis, University of California, (1983)
19. H. van Oene, Modes of dispersion of viscoelastic fluids in
flow, J. Colloid Interface Sci. 40, 448-467, (1972)
20. P. Prabodh and P. Stroeve, Break-up of model viscoelastic
drops in uniform shear flow, pers. conununication by P.
Stroeve, Dep. Chem. Eng., University of California
21. W.R. Schowalter, Mechanics of non-Newtonian fluids, Pergamom
Press, Oxford, 1978
22. R.A. Stratton and A.F. Butcher, Stress relaxation upon
cessation of steady flow and the overshoot effect of polymer
solutions, J. Polym. Sci. Polym. Phys. Ed. 11, 1747-1758,
(1973)
23. T. Tavgac, Drop deformation and break-up in shear fields, Ph.
D. thesis, University of Houston, (1972)
24. G.I. Taylor, The viscosity of a fluid containing small drops
of another fluid, Proc. Roy. Soc. A 138, 41-48, (1932)
4.7 LIST OF SYMBOLS
c. D
~ De
Fg
K
k
L
,k
m
N1
N2
Nwe
n
p
R
C1, .. Cg
u,v,w
constant
drop diameter
rate of strain tensor
Deborah number
recoverable free energy
consistency index
summation index
length scale
velocity gradient tensor
elastic power law index
first normal stress difference
second normal stress difference
Weissenberg number
power law index
summation index
drop radius
critical drop radius
stress ratio
velocity components
[ - l [ml
[ s - l]
[ - l
[J]
[Pa.sn]
[ - ]
[m]
[s-1]
[ - J [Pa]
[Pa]
[ - J
[ - J
[ - J
[m]
[m]
[ - l [m s- 11
129
130
LIST OF SYMBOLS (continued)
v x,y,z
"(
hnt
J'/
J'/c, J'/d
J'/o, J'/p
8
À
Àl, À2,
Àl,p· À2,p
Àa
Àml• Àm2
a
a*
velocity scale [m s-1]
Cartesian coordinates [m]
parameters of Bird-Carreau model (-]
shear rate (s-1]
internal shear rate [s-l]
dynamic viscosity [Pa.s]
dynamic viscosity of continuous/drop phase [Pa.s]
parameters of Bird-Carreau model [Pa.s]
relaxation time [s]
elasticity index
viscosity ratio
parameters on Bird-Carreau model
apparent viscosity ratio
modified viscosity ratio
interfacial tension
modified interfacial tension
shear stress
stress tensor
capillary number
critical capillary number
critical capillary number in Newtonian
limit
[Pa.sm]
[ - l
[s]
[ - l [ - l [N m-1]
[N m-1]
[Pa]
[Pa]
[ - 1
[ - l
[ - l
"
5. DEFORMATION AND BREAKUP OF NEWTONIAN DROPLETS IN TRANSIENT
SIMPLE SHEAR FLOWS
5.1 INTRODUCTION
Most investigations regarding drop deformation and breakup refer
to quasi steady state conditions. These are reviewed in chapter 3.
Very few investigations, however, are available on drop
deformation and breakup in transient simple shear flows. Grace
{1982) reported measurements on drop deformations prior to
break-up for drops subjected to shear rates well above their
critical shear rate and Torza et al. {1972) reported some
measurements on deformation and orientation as a function of time
for a few viscosity ratios. More detailed measurements are
available for other types of linear shear flows (Stone et al.,
1986).
The present investigation is aimed at studying both experimentally
and numerically the behaviour of Newtonian drops suspended in a
Newtonian liquid and subjected to transient simple shear flows at
low Reynolds number. This is relevant to emulsification processes
that involve flows in which a droplet experiences rapidly varying
flow conditions. The investigation involves the development of a
computer program to evaluate the shape of a droplet in a general
transient shear flow. This program is based on the boundary
integral method by which the creeping flow equations inside and
outside the drop are transformed into a form that only involves
quantities at the drop surface. This technique, derived by
Ladyzhenskaya (1963) and further described by Youngren and Acrivos
{1975), Rallison and Acrivos (1978) and by Rallison {1980, 1981),
is based on the Fourier solution of the creeping flow equations,
with the use of volume potentials. Mathematical problems arise
with this method for very small or very large viscosity ratios,
since the boundary integral has neutral eigensolutions for these
131
132
two extremes. For axisymmetric problems the calculations are very
much simplified, since the drop surface can then be described by a
curve and less grid points are needed to obtain an accurate
solution. Another simplification occurs for simple shear flow
since then one of the terms in the boundary integral vanishes and
the velocities can be solved without a matrix inversion,
substantially reducing the computational time needed.
The programme was applied to droplets with viscosity ratios
ranging between 0.5 and 5 and was used to calculate the shape of
the droplet as a function of time for various shear rate profiles,
step profiles, triangular profiles and sinusoidal profiles. The
present investigation also involves experimental work on drop
deformation and break-up in simple shear flows. Experiments have
been performed with droplets with viscosity ratios ranging from
0.01 to S. These droplets were subjected to step and triangular
shear rate profiles.
5.2 PROBLEM STATEMENT AND BOUNDARY INTEGRAL METHOD
The problem under consideration is that of droplets of an
incompressible Newtonian liquid with a viscosity nd' suspended in
another immiscible incompressible liquid with a viscosity nc' The
drop has an interfacial tension G and is subjected to a linear
shear flow at infinity with a velocity ~inf = !·!· Provided the
droplet is small enough, the flow in and around the droplet will
be dominated by the viscosity and inertia and gravity effects can
be neglected. This is allowed when the Reynolds numbers based on
the drop phase and the suspending phase are both small. Fluid flow
can then be described as a quasi steady state problem by the
Stokes or creeping flow equations. The velocities and pressures
are thus governed at each time by the following equations.
div u = 0 everywhere [5.la]
v 't = 0 x not E S [5.lbj
.! P I + l\ v u + v u T € vout [5. lc] !
't = - p I + lld 'il u + 'il u T E vin [5.ld] !
with ~ the velocity, .! the stress tensor with the isotropic part p
the statie pressure and the non isotropic part the flux of
momentum due to a velocity gradient. The boundary conditions for
the deformable droplet are given by
u" = u -u -c
1! 1 -+ inf
! ES
with n defined as the outward normal vector and where V ~
denotes the surface curvature.
[5.2a]
[5.2b]
[5.2c]
To find a general solution for these equations we follow the
approach described by Ladyzhenskaya (1969) and first solve the
Stokes equations for the flow due to a point source of force in
the k-direction applied in a certain point i:
11 v2 ~k(~,l) - (V q(~,l))k
div uk = 0
[5.3a]
[5.3b]
where x is an arbitrary point vector and q a scalar force term, S
the Dirac-S function and ~k a unit vector along the k-th
coordinate axis. All differentiations are carried out with respect
to ! and the point l• the point where the applied force is
concentrated, plays the role of a parameter. Fourier transforms
can be used to find the solutions of these equations. With Q(~)
and Q(~) the Fourier transform of ~(!) and q(~) respectively, this
results in the following set of equations.
133
134
1 - 11 «2 U. k - i « Qk = SJ.k
J j (2n)3/2
k «j uj o
with «Z (~·~>· The subscript j represents the
velocity. Unique solutions of Ujk and Qk can be
equations quite straightforwardly:
k 1
uj <~> n
Q\!l:> i~
(2n)3/2 2 ()l
[5.4a]
[5.4b]
component of the
derived from these
[5.Sa]
[5.Sb]
The inverse Fourier transform of equation [5.5] will give the
fundamental unique solution to the original flow problem.
k u. (x,v) J - "-
In these
in point
the k-th
[5.6a]
[5.6b]
equations u.k(x,v) represents the j-th velocity component J - "-
x, due to a point source of force in point l• acting in
~irection. qk(!,l_) represents the associated pressure
contribution in point x. Ye note for future use that the solutions
~k(!,l_) and qk also sa~isfy equation [5.3] when all the
differentiations are carried out with respect tol instead of~·
The above derived solutions ~k(~ 1 l'._) and qk are Greens functions
and can be used to define the following volume potentials
[5.7a}
[5.7b]
These volume potentials obey the inhomogeneous Stokes equations
div U = 0
[5.8a}
[5.8b]
in which !(~) represents the distribution function of the external
sources of force. In our case !(~) is determined by the
interfacial tension forces which can be described by a layer of
Stokeslets on the drop interface with a strength an V.n.
Next a solution for g will be sought which obeys these
inhomogeneous Stokes equations. Therefore Gauss's theorem is
applied to vector fields of the type u. ~ .. (U) to obtain Greens l lJ -
formula to the Stokes problem, using the following identity
[5.9]
in which the fact is used that both ~ and g are solenoidal (i.e.
divergence is zero). Formula [5.9] can be integrated over a domain
V, which gives
[5.10]
135
136
Now Gauss's theorem is applied to the vector field uk. •· .(U) - U. * k * 1 lJ - k 1 •·· (u) in which •·· (U) is given by the stress tensor•· .(u) lJ - lJ - lJ -
after interchanging ~k and g, interchanging qk and -P and
differentiating with respect to z. This will result in:
k
I [ [ 11 v2 u1 - ~ ] u\ - ui [ n v/ u\ + ~ ] ] dV v a xi a y1
= Is [ 'ij<!!> u\ nj - 'ij *c.!h Ui nj ] dS (5.11)
If we identify U with the volume potential which obeys equation
[5.8a] and ~k w~th the fundamental singular solution (5.6a],
equation [5.11) can be rewritten to:
Uk(!)= Jv ukl.(~,z> f.(z) dV + I ... *(uk) U.(x) n.(z) dS l s 1J - l - J
- Is uk1<!•l> 'ij(!!) nj(X) dS
P(!) Jv qk(!•l> fk(r) dV - Is qk(!•l> 'ij<!!> nj(Z) ds
J a lcx,x>
- 2 ll - Uk(~) n.(}:'.) dS s Cl x. J
J
From the solution [5.6] for ~k it follows that
* k •·. (u (x,v)) lJ - - "-
_:__ (xi-yi)(xj-yj)(xk-yk)
4 n l!-rl5
Using this relation together with the fundamental singular
solution [5.6], equation [5.12a] can be rewritten
[5.12a]
(5.12b]
[5.13]
Uk(~) = Jv uki(~ 1 l) fi(l) dV - IsKijk(E) Ui(l) nj(X) dS
- _1_ J J.k(~) i:.j(X) n.(l) dS 8 n n s J 1 1
with:
[5.14]
This equation can be used to derive the boundary integral equation
for the velocity of the points at the droplet interface by
applying this equation to the exterior of the droplet and to the
interior of the droplet. The exterior equation is formulated with . * inf . the disturbance veloc1ty Q = Q - Q = Q E.x. The interior
equation is formulated with the total velocity Q. In that case the
volume integrals vanish since external forces are applied neither
at infinity nor anywhere else outside the droplet interface. The
surface integrals at infinity vanish as well. Thus only surface
integrals just outside and just inside the interface remain.
Uk(~) E.x - Is(e) Kijk(E) U1<x> nj(X) dS
-1- I J 'k(r) i: .. (y) n1 <x> dS x e: S(e) [5.15a]
8 n n S(e) J - lJ -c
Uk(~) = - J K .. k(E) U1.(X) nJ.(X) dS S( i) lJ
- _1_ J Jjk(!) i: .. (X) ni(X) dS 8 n l"ld S(i) iJ
x e: S(i) [S.15b]
Since velocity continuity is prescribed by the boundary conditions
[5.2b] the velocity approaching the drop interface from the
137
138
exterior or from the interior must be identical. The J and K
integrals are however discontinuous at the drop interface. The
discontinuity in the J-integral is given by the normal stress
boundary condition [5.2c}. The discontinuity in the K-integral is
given by
JS(e) Kijk(E) Ui(X) nj(X) as Is Kijk(E) U1<x> nj<x> dS
- 112 Uk(~)
Is(i) Kijk(E) U1(l) nj(l) dS Is Kijk(E) Ui(X) nj(l) dS
[5.16a)
[5.16b]
Inserting these boundary conditions in equation [5.15) results in
the boundary integral formulation for the velocity of a point x on
the drop interface:
2 2(1-À)
Uk(!9 E.x - I K. 'k(r) ui <;O nj <x> ds l+À l+À s lJ -
11 I J.k(r) n.(y) v. n dS (l+À) 4 Jt nc s J - J -
[5.17]
Vhen À = 1 the equation is very much simplified because the
K-integral term, which itself involves interface velocity terms,
will then vanish. The resulting equation is an explicit expression
for the surface velocities Q(~), which can be solved after
discretising the drop interface with N points. This simplification
has been used by Rallison (1978,1981). In its full generality
solving the velocities at the points on the drop interface
involves the solution of a linear set of 3N equations. Equation
[5.17] has been made dimensionless to facilitate the numerical
solution. The positions have been made dimensionless with the drop
radius R, the velocities with u/nc and the deformation gradient
tensor~ by the characteristic velocity gradient y. The resulting
equation, in which all quantities represent dimensionless
quantities, is:
2 Q
Uk(~} = - E.x l+À.
2(1-À) I -- K. 'k(r} U.(z:} nj(l) dS
l+À. s lJ - l
1
I J 'k(r) n.(v) 'il • n dS J - J L -s 4 lt ( l+À.} [5.18]
Equation [5.18] is only valid for the points at the drop interface
but contains all information necessary to evaluate the full three
dimensional shape of a droplet as a function of time. The time
dependency does not stem from the inertia terms in the
Navier-Stokes equations, since they were neglected in the low
Reynolds assumption. However they stem from the changing boundary
conditions at the drop interface following a change in the applied
undisturbed flow field. The boundary integral formulation can be
extended to yield the velocity in any point inside or outside the
droplet expressed in quantities to be evaluated at the drop
interface. The only additional problem is to find expressions for
the full J-integral terms (for the interfacial velocities only the
jump in the J-integral terms was needed). If the interfacial
velocities have been calculated with equation [5.18), it is
however possible to evaluate the unknown terms éij(y).ni(l)= Tj(l)
for all the points at the drop interface, by solving a set of 3N
linear equations in Tj(l}· These equations are obtainable from
equation [5.15a]. Once these terms have been evaluated equation
[5.15] will give the following expressions for the velocity in a
point outside or inside the droplet.
139
140
2
3
12
+ 12
1 for ~ € vout
[5.19a]
I K .. k(r) U.(l_) n.(y) dS s lJ - 1 J -
1 Is Jjk<E> Tj<l> ds
Jt À '1c Cf I J.k(E) n.(l) V.!!. dS for ~ € Vout
Jt À nc s J J [5.19b]
where l. denotes a point on the drop interface, the integrations
are performed on the drop interface S and the terms Tj(l.) are
defined just outside the drop interface.
5.3 THE NUMERICAL METHOD
A numerical scheme has been developed to solve equation [5.18] for
the velocities at the drop interface. This scheme allows
calculation of the velocities at the drop interface for droplets
subjected to any type of linear flow field. For this scheme the
drop surface S(t) has been discretised by N collocation points x1•
Two meshes have been used in the present investigation, consisting
164 and 266 points distributed over the drop surface. The
evaluation of the surface variables, the normal and curvature of
the surface, is carried out by fitting the drop interface in the
locality of each collocation point by a second order curved
surface of which the surface variables can easily determined. A
method is developed to redistribute the points of the mesh at each
time step, in order to prevent clustering of the mesh points
during deformation of the mesh.
5.3.1 Mesh definition
Two similar meshes have been defined, one consisting of 164 points
(9 rows of 18 points, plus 2 points on the poles), and one with
266 points (11 rows of 24 points, plus 2). The initial
distribution of the collocation points over the spherical droplet
is shown in figure 5.1. The latter mesh has been created in order
to evaluate S(t) more accurately when the surface area of S(t)
increases and the total number of points per surface area
decreases accordingly. Two meshes have been used to test the mesh
dependency and the stability of the numerical method. The number
of triangular regions enclosed by the points is 324 and 528,
respectively, giving a surface area per triangular region of less
than 0.5 % of the total surface area. The points of the mesh are
spaced such that all triangles made up by those points have about
! 0
~ A 1 18
36
Fig. 5.1 Representation in spherical coordinates of the mesh with
266 collocation points that was used in numerical
calculations
141
142
the same surface area which is necessary for numerical stability.
The choice of less points on the first row resulted in a growing
periodical disturbance in the first time steps which proved to be
catastrophic for larger time steps. Both meshes consist of more
collocation points than the mesh used by Rallison (1981).
5.3.2 Evaluation of surface variables
To evaluate the surface variables, the drop interface is fitted in
the locality of each collocation point to a second order curved
surface which is point symmetrie with respect to the origin. This
curved surface can be represented by:
i i 2 i i 2 1 1 2 i i i i i i i i i a 1x l +a 2x z +a 3x 3 +a 4x 1x 2+a 5x 1x 3+a 6x 2x 3-1 Û
g(xi)
[5.20)
The coef ficients of the curved surface are determined for each
collocation point by fitting the curved surface through the point
under consideration and five neighbour points. The normal in each
collocation point can be found by:
nj a g(xi)
i a x j [5.21]
The local curvature is given by:
[5.22]
with R1 and R2 the two principal radii of curvature. These two
radii of curvature can be calculated if the intersections of two
planes, both perpendicular to the tangent plane in x1, and the
. i i second order curved surface, given by g(x1 )=0 with a 1 •• a 6 , are
known. To calculate the intersections a new orthonormal base is
defined. One of the new base vectors is the already known normal i n (Eq. [5.21]) and the other two base vectors can be found from
the tangent plane in xi. Both xi and g(x1) can be rewritten with
respect to this new base and are given i i by y and h(y )
respectively. The principal curvatures (the reciprocal values of
the radii of curvature) can then be calculated by:
- 2h i i h i h i y ly 3 y 1 y 3
+ h . . h2 . l l l
y 3Y 3 y 1 [5.23)
in which the subscript y1 . at the function h(y1) denotes that J
partial differentiation is carried out with respect to the
collocation point y1j. The subscript at j of k1j denotes the
direction in which the curvature can be found. The other
curvatures can be found by interchanging the subscripts 1, 2 and 3
cyclically.
This method for the evaluation of the surface variables will
always result in exact solutions for the normal and curvature for
ellipsoidal spheroids even in the case of negative curvatures.
This method was therefore preferred above the iterative method
used by Rallison (1981), which resulted in errors up to 2%.
5.3.3 Evaluation of the surface integral
The contribution to the J-integral of each collocation point xi is
calculated as follows. First the contribution of the triangles,
which do not have a vertex at x1, is calculated using the
trapezoidal rule. The contribution of these triangles can be
rewritten as contributions of the vertex points xj of the
143
triangle, to the point xi, since the centre of gravity of each
triangle is given by:
xtriangle =113 ( E xvertex ) [5.24]
Those triangles which do have a vertex at xi have to be dealt with
separately due to the 1/r singularity in J for r=O. The integral
however is finite when r approaches zero. An analytical expression
for the contribution of these triangles is given in Appendix Cl.
Once the numerical value for the contribution of such a singular
triangle is known, it can simply be added to the contributions of
all other points.
If the viscosity ratio is 1, the K-integral can be omitted, and
the velocity v1 in each point can be calculated directly by adding
the contribution of the external flow field to the value of the
J-integral. If the viscosity ratio is not equal to 1, the
K-integral has to be evaluated for each component of the velocity
vi. Since the velocity itself is present within the integral, a
system of linear equations with the integral contributions of the
velocity components of all collocation points equaling the sum of
the J-integral contributions and external flow field contributions
has to be solved. This involves matrix solving techniques. To
compose the elements of this matrix, an analytica! expression
similar to the one for the J-integral has to be found for the
K-integral. This expression is given in Appendix C2. Point
symmetry with respect to the origin has been used to reduce the
matrix by a factor 4 and to gain a factor 16 in computational
speed. The complete system of 399 linear equations with 399
unknowns is solved using an iterative Gauss-Seidel matrix solver
technique, with an accuracy of lE-6. The iterative Gauss-Seidel
technique was chosen because of its efficiency for problems of
which the solution changes little per time step and the dominance
of the diagonal elements of the matrix.
5.3.4 Redistribution of the mesh
The collocation points of the mesh behave during the deformation
of the droplet as material points on the surface. Hence they move
away from their original positions. The new positions can be
calculated as follows:
~new = ~old + y(~) at [5.25]
This movement results however, in clustering of points at the ends
of the deformed surface and a depletion at both sides (see
figure (5.2a)). This distribution of points will finally lead to
numerical instabilities at highly deformed surfaces (deformations
greater than 0.5 for a mesh with 164 points, see figure (5.2b)).
2-
-2L -2
Fig. 5.2.a
1 -1 0
1 2
Droplet contours in simple shear flov vith mesh correction
145
146
4.00-
2.40-
0.80-
-0.80-
-2.40
-4.00L -4
Fig. 5.2.b
1 -3
r - -
1 -1 0
- ...,
__ J
1 2
1 3
Droplet contours in simple shear flow without mesh
correct ion
1 4
A better method is to let the collocation points move in the
normal direction:
[S.26]
Vith this method however, the collocation points will also become
unevenly distributed over the drop surface when the droplet is
highly deformed. To prevent clustering of the collocation points
on the surface, the points of the mesh are redistributed at each
time-step. First, all points, which initially had the same
z-coordinate, are given the same relative z-coordinate with
respect to the z-coordinate of the top point using the equation of
the curved surface in the locality of the point. Second, the
points in the z plane of the droplet, are redistributed in such a
manner that they have about the same distance from one another,
making use of the coefficients aij of Eq. [5.20]. Finally all
collocation points on contours having the same z-coordinate are
redistributed similar to the points in the z-plane. Once the mesh
is redistributed, the surface variables are reevaluated.
5.3.5 Numerical stability and convergence
The numerical method described in the previous sections has been
tested on numerical stability and convergence both in the presence
and in the absence of external shear flow fields. In the absence
of an external flow field it was checked that the droplet remained
spherical. It was observed that this depended on the mesh
distribution. Optima! results were obtained with the mesh given in
figure 5.1. Yhen a mesh distribution as in figure 5.3 was used (as
was done by Rallison, 1981) wave-like disturbances were observed,
especially on the two rows of points close to the top points.
These wave-like disturbances showed themselves in inward movement
of some of the points on these rows, combined with an outward
movement of other points. This effect was related to the
asymmetry in such a grid. Some points on the third row are namely
surrounded by 5 triangles with comparable surface area, while
others are surrounded by 5 relatively small triangles and 1
relatively large triangle. The observed wave periodicity coincided
with the mesh periodicity. These wave-like disturbances were
especially noticeable at relatively large time steps. When the
droplet was subjected toa shear flow the wave-like instability
was suppressed by the applied flow. For the present mesh type the
sphericity was very well maintained. The equilibrium deformations
in x-y and x-z direction were for all viscosity ratios very small:
Dxy < 0.0004 and Dxz < 0.003. The somewhat larger deformation Dxz
147
148
is due to an inward displacement of the top points. Vhen an
equilibrium shape is obtained, all points tend to move inward
simultaneously. The relative decrease of the drop radius in a unit
dimensionless time was however quite small: 0.6%. A correction has
been built in to preserve the drop volume by inflating the droplet
radially after every time step.
0 , --- -- --- - --- ' , ' , ' , '
0 i M ~ w n H u '
8
Fig. 5.3 Representation of the mesh used by Rallison (1981)
Effects of time step and mesh point density on numerical
convergence have been investigated in relaxation experiments. For
these experiments the initial shape of the droplet was obtained by
multiplication of the z or the x coordinates by a certain factor
and the relaxation of the resulting ellipsoid was calculated as a
function of the time step size. Axis ratios ranging from 1:1:10 to
1:1:2 have been used. Examples are shown in figures 5.4 and 5.5.
From these results it became clear that a time step size of 0.1 is
small enough for À=l, for x~z a somewhat larger time step can even
be used, hut for X:0.5 a smaller time step is necessary: ót=0.05.
3.20-
1.60
-1.60-
-3.20 L -3.20
Fig. 5.4
Fig. 5.5
1 -1.60 0
1 1.60
1 3.20
Droplet contours at various times during the relaxation in
the absence of externally applied flow of an initially
ellipsoidal drop with an axis ratio of 3:1:1
Droplet contour during the relaxation in the absence of
externally applied flow of an initially ellipsoidal drop
with an axis ratio of 10:1:1
149
150
llhen smaller time steps were used the differences in Dxy and Dxz
were always smaller than 0.001. These required time step sizes are
in line with the relaxation times of the droplets. As a rule of
the thumb time steps of 0.01/À should be used in future
calculations. Yith respect to the mesh density the following can
be concluded from figure 5.6. Since some differences in
deformation (up to 0.03) between the 164 grid point mesh and the
266 grid point mesh were observed, reliable calculations need at
least a 266 grid point mesh. The effect of mesh distribution has
been investigated for X=l by comparison of the relaxation of
ellipsoids with their main axis in the z or the x direction. Only
small differences were observed. For the 164 grid point mesh the
difference in deformation was always smaller than 0.02 and for the
266 grid point mesh they were even smaller than 0.007. These
results indicate a good independence of the numerical solution of
the mesh distribution, especially for the 266 mesh point grid and
it is concluded that for these deformations a numerical scheme has
been developed with which mesh and time step independent results
can be obtained.
Numerical stability and convergence have also been studied for X=l
in step response experiments for the 164 and the 266 grid point
mesh. From the results in figure 5.7 the following can be
concluded. At small deformations (D<0.3) there is only a small
difference between the results obtained with the 164 and the 266
grid point mesh. These data confirm the results obtained in the
relaxation experiments. At larger deformations there is an
increasing discrepancy showing the deficiency of 164 grid point
mesh for larger deformations. This is especially clear from the
fact that above 2=0.40 no stable final deformations could be
calculated with the 164 grid point mesh. This is showed in figure
5.7 by the steady increase of the deformation with time. For the
266 grid point mesh stable deformations could be calculated up to
2=0.4. At large deformations (D>0.4) there should be some doubt
... ó
C"' 0 . ·- 0
ö E 1...
J2 N (J) • 00
Fig. 5.6.a
In
"' ci
Cl ':
"' "' ci
.Q ~ Qó E 1... U1
J2 (J) '"
0
8 ci
"' c ci Cl c;
0.0
Fig. 5.6.b
D = >.-1, z-oxLs><3, 266 polnls 0 ),=1) x-axLsx3, 266 pocnts " >.= 1, z-ax:Î...s*3, 164 pol,nt.s + >. = 1' x-axcs*3, 164 pocnt.s
Time
Relaxation of droplets vith an ellipsoidal initial shape.
Bffects of number and distribution of collocation points
for droplets vith an initial axis ratio of 1:1:3.
5.0 10.0
Time
o = i.=2, z-axl,s><2, 266 pocnts o = >-1 z-axcs><2, 266 pocnts 6 - ~-o:s~ z-axLs~2J 266 poLnts
lS.O
Relaxation of droplets vith an ellipsoidal initial shape.
Effects of viscosity ratio for droplets vith an initial
axis ratio of 1:1:2.
151
152
about the 266 grid point mesh as well and even higher mesh
densities are required to perform reliable calculations.
"! 0
": 0
"' à
:z "' 30 '""' a: ... ::c . er:. 0
a t....M w. c:i"'
N
0
à 0
c
o.o
Fig. 5. 7
5.0 10.0 15.0 TIME
'Y - 0-0.1, • 0-0.2, Il 0-0.3, l'l - 0-0. • 0-0. 0 - 0-0.1, " - 0-0.2, + - O=O. 3, x - 0-0.1, " - 0-0.45,
166 MESH POINTS 166 MESH POINTS 166 MESH POINTS 166 MESH POINTS 166 MESH POINTS 266 MESH POINTS 266 MESH POINTS 266 MESH POINTS 266 MESH POINTS 266 MESH POINTS
Mesh dependency of the results for the deformation
response of droplets to a step like shear rate profile for
X=l.
5.4 EXPERIMENTS
5.4.1 Description of the Couette device
Deformation and breakup experiments were carried out in a Couette
device. This device bas been described in section 3.4.1 of this
thesis. The Couette device consists of two counter rotating
cylinder~. In the gap between the cylinders a linear simple shear
field is generated (see figures (3.2)). The two cylinders are belt
driven. The motor controls can be operated both manually and
automatically. To drive the AC-motors automatically, a personal
computer has been used. The bottom and wall of the outer cylinder
are made of glass in order to visualise the flow phenomena.
5.4.2 Transient flow in Couette device
To evaluate the capabilities of the Couette device, to be used to
measure drop deformation and breakup in transient shear flows, the
response of the fluid flow in the gap between the two rotating
cylinders following a change in rotational speed of the cylinders,
has been calculated numerically. These calculations were performed
to ensure that inertia had no effect on the phenomena observed
between the cylinders. Bottom effects were neglected in this
analysis. The Navier-Stokes equations for this particularly simple
geometry reduce to:
[5.27]
in which U+ represents the velocity in the tangential direction
and r the radial position in the gap. The response to a step like
change in the velocity has been calculated, using the following
initial and boundary conditions:
u+ (r ,t) U+ (Rl,t) U.p (R2, t)
for t < 0 for t ~ 0 for t ~ 0
[5.28a] [5.28b] [5.28c]
These equations were solved numerically by an explicit centra!
differential scheme. This scheme has been used to calculate the
development of the velocity with time and the time required to
obtain 99% of the steady state velocity. The calculations carried
out for various speeds of the cylinders. The development of the
velocity in time is given in figure 5.8 fora velocity ratio of
-1:4. All the calculations show that the velocity is everywhere
within 99% of the steady state velocity after:
[5.29)
153
1.00 Veloai
0.80
0.60
0.40
0.20
Fig. 5.8
154
t - 0.1
t • 0.2
t • 0.5
/~i' / '/ •:
/ 1/:/1 / ///11
// /;'::// / , ' 1
/ / / ..- /1 /'; ! ' ." / 1
/ / I . 11 / / , . I
/ / I l 1
/ / / / 1
" / I / 1 / / • 1
/" / I I 1
"" / / I ' .n2'" "" / • I
1
~ " I I ' "" / I 1 " , . I i
" / / I ' / / / / /
// "/ ./ / ,' / / /./ /1
/ /' • / I
// " ./ ··" / /
/ // ,,./···· --~- .... /'
4.19 4.40 4.61 4.82 5.03
--> RadtJS {Cl'l'I/
Development of the velocity profile in the Couette device
after a step like change of the velocities of the
cylinders. Velocity ratio betveen inner and outer cylinder
is -1:4
For experiments using a silicone oil of 10 Pas as a continuous
phase this implies that the bulk fluid responds within 5 msec. In
the triangle profile experiments in the Couette device actual ramp
times were always more than 100 times the fluid response time. So
inertial effects due to fluid motion do not interfere with the
drop deformation and breakup reported in this investigation.
5.4.3 Experimental procedure
During an experiment a small fluid droplet ( R < 0.5 mm ) is
brought into the gap between the cylinders in the Couette device,
which is filled with another fluid with about the same density as
the droplet. This droplet is held at one place between the two
rotating cylinders in the stagnant zone (i.e. a zone where
U+(r)=O) by adjusting the rotational speeds of the cylinders
manually.
To keep the droplet positioned in the stagnant zone during the
experiments, a preliminary experiment is carried out to determine
the position of the droplet in the gap. This is done by
determining the ratio of the velocities of the 2 cylinders, at
which the stagnant layer coincides with the drop position. This
ratio is used to keep that droplet positioned in this stagnant
layer in subsequent experiments. The deformation of the droplet is
recorded on video together with a display of the rotational speeds
and the time and is analysed afterwards. In order to measure the
deformation using the video image of the particle as accurately as
possible, a high contrast illumination of the particle is chosen.
No distortion of the video image has been observed in any
direction. The error in the deformation measurements varied from
0.1% to 5 % depending on the quality of the video image and the
magnification of the particle, with the errors of most
measurements in the lower range.
155
156
In the experiments two types of fluids have been used.
Polymethylsiloxanate (Rhodorsil silicone oil 47, Rhone-Poulenc)
has been chosen for its Newtonian behaviour, high viscosity and
low dependency of its viscosity on temperature. Corn syrup (Globe
01170 of Cerestar Benelux BV.) has been chosen for its high
viscosity and Newtonian behaviour as well, but its viscosity
temperature dependency is rather strong. A high viscosity (10 Pas)
was necessary for both fluids since otherwise at lower viscosities
the time scales would be too small for the video registration. The
fluid systems used can be found in table (5.1).
TABLE 5.1: Fluid properties
Continuous phase Discrete phase interfacial tension
descript ion viscosity description viscosity [mNm-1 ) [Pas] [Pas]
CS/water 97 so 105 36 CS/water 10.8 so 11.0 37 CS/water 4.8 so 24.0 30 so 24.0 glyc./water 4.8 23
Silicone oil has been chosen as the high viscosity drop phase,
because corn syrup drops tend to become rigid. This may be due to
crystallisation of the droplet resulting from dissolution of the
water phase of the corn syrup droplet in the oil phase. The
viscosity temperature relation was measured with a Haake
rotational viscometer (SV12) and the interfacial tension with the
Yilhelmy plate method. Occasionally the drop phase has been
coloured with Sudan red for photographic reasons. The same
additive ( the fraction which has not been dissolved) could be
used as a tracer material in the drop phase to visualise the flow
pattern in the drop. Three viscosity ratios have been used for the
experiments: 0.01, 1 and 5. For the transient deformation
measurements (experiments 2,3 and 4) the value of the maximum
capillary number has been varied up to 5 times the critica!
capillary numbers concerned, while the value of the time to reach
that maximum capillary number has been varied from 0 to 10
dimensionless time units.
The following experiments have been carried out:
1. The deformation was measured as a function of the shear
rate, which was slowly increased, giving the droplet enough time
to reach its equilibrium deformation at all shear rates (quasi
steady state deformation experiments}.
2. The deformation was measured as a function of time
following a sudden, step like increase of the shear rate (step
response deformation experiments).
3. The deformation was measured as a function of the shear
rate and time, during a triangular shear rate ramp. (triangle
response deformation experiments).
4. The breakup of droplets in transient experiments was
observed and the number of fragments was counted. (breakup
experiments)
5.5 NUMERICAL CALCULATIONS
The program, written in Pascal, was run on a VAX 8530 machine. The
program required 20 seconds of CPU time per time step for the
simpler case of a viscosity ratio equal to 1 and 80 seconds per
time step for other viscosity ratios. Both figures refer to the
denser mesh consisting of 266 collocation points. To test the
program with respect to accuracy and reliability the following
experiments have been carried out, using both meshes under
consideration:
1. Relaxation of an ellipsoidal droplet in the absence of an
applied flow, with time steps 0.5, 0.1, 0.05, 0.01 and 0.005, for
157
158
oblate and prolate ellipsoids with the major axis in x and
z-direction (this results in 2 different meshes due to the
configuration of the mesh)
2. Deformation of the droplet in response to a step like
change in the shear rate for various time steps.
When above tests proved to be satisfactory other simulations were
carried out:
3. Step response experiments for various viscosity ratios for
capillary numbers up to the critical capillary number, with and
without mesh correction.
4. Triangle response experiments for the same set of
viscosity ratios.
S. Sine response experiments for various viscosity ratios.
5.6 EXPERIMENTAL AND NUMERICAL RESULTS
5.6.1 Step profile response
In this section the results of drop deformation experiments in
step profile simple shear flows will be described. The shape of
the droplets will be described by the deformation D=(L-B)/(L+B)
and the orientation of the droplet, given by the angle between the
longest axis of the droplet and the direction of the flow.
For a viscosity ratio of 1 the deformation and orientation of
drops are shown in figure 5.9 as a function of the dimensionless
time for capillary numbers ranging from 0.1 to 0.5. The results
show very good agreement between the experimentally observed
deformations and the numerically calculated ones especially when
the capillary number is smaller than 0.4. At higher capillary
numbers some deviation occurs and the numerical model predicts
unstable droplets where experimental measurements indicate stable
deformations up to 0.68, corresponding with a critical capillary
number of 0.48. Note that the experimental data for 2':0.486 relate
to an unstable situation; break-up occurred after t=36. At high
capillary numbers and shorter times the numerically calculated
deformation is somewhat smaller than the experimentally observed
deformation, but at longer times the numerically calculated
deformations become larger due to the instability of the drop. The
numerically calculated orientation angles are somewhat smaller
than the experimentally observed final deformations.
The general shape of the droplet is even at high capillary numbers
in very in good agreement with the experimentally observed droplet
shape as is shown in figure 5.10 where the grid points with a zero
or positive z-coordinate are projected on the x-y plane for
capillary numbers of 0.45 and 0.60 at various times. Both
experiments and calculations show that droplets attain an
ellipsoidal shape at small deformations followed by a sigmoidal
shape at deformations above 0.6.
For viscosity ratios of 0.5 and 2.0 the numerical results on drop
deformation and orientation as a function of time for various
capillary numbers ranging from 0.1 to 0.4 are given in figures
5.11 and 5.12 respectively. For a viscosity ratios of 5 and 0.01
the numerical and experimental results on drop deformation and
orientation as a function of time are presented in figure 5.13 and
5.14. The effect of the viscosity ratio on the step response is
shown in figure 5.15 fora capillary number of 0.40.
The final drop deformations have also been measured as a function
of the capillary number for the viscosity ratios 0.01, 1 and 5.
The results are given in figure 5.16. For each viscosity ratio the
results refer to a number of different drop sizes.
159
"' 0
" ei
'" 0
z"' s ci E--< cr: ... :c . cc: 0
0
(.._ "' w. 00
"' "' ei 0
0
o.o s.o 10.0
TIME
"'= 0=0.094, EXPERINENTAL • O·O. 194, EX PER IMENTAL ri = 0=0.284, EXPERIMENTAL ~ = 0=0.389, EXPERIMENTAL • - 0=0. 431, EX PER IMENTAL • 0=0. 486, EXPER IMENTAL
15.0
0
~] 2· Ç)
.,; "
z I~ QO ~"' E--<o cr: • E--< "' z"" S:lo ;sîi
0
tR 0
ó "' q
"' ... 0.0
0 = 0-0.1, "'= Cl=0.2, + - n-o .3, x o-o. 4, <:i = 0=0.45,
s.o TIME
NUMERI CAL NUMERI CAL NUMERI CAL NUMERI CAL NUMERI CAL
Fig. 5. 9 Numerical and experimental results for the response of
droplets to a step like shear rate profile for À=l.
•
10.0 15.0
Fig. 5.10
..... 0) .....
Ca•0.45 -> Ca•0.60 ->
t•0.50-1.76-4.27-19.60. t•0.75-3.52-5.78-11.31.
Contours of droplet (J.,:1) at various capillary numbers. Numerical calculations for '2=0.45 and 2-0.60
,,, c:i
0.0 2.0 ~.o s.o TIME
o 0-0 .1, NUMERI CAL " - 0-0.2, NUMERICAL + 0-0.4, NUMERICAL
8'.0 10.0
<; 0
" 0
tri "'
Zo 0.
~~ a: e-:z wc; -"' 0:::"' Ci
t:? 0
"' 0
* 0.0 2.0 4.0 6.0 TIME
Fig. 5.11 Numerical results for the response of droplets to a step
like shear rate profile for >--0.5.
B.O 10.0
~ 0
"' ei
.... zo 0 -f-.. a:"" >= • 0-:0 0 "-WN oei
0
0 0.0 5.0
T!ME
o - 0-0 .1, NUMERI CAL "' - 0-0.2, NUMERICAL + 0-0. 3, NUNERICAL
0-0.4, NUMERICAL
10.0
c: "' " c: R
0
z~ 0
f-.. 0 a: . f-..O z«> w -0-:o 0.
"' "' 0
ei lf)
c: ~
15.0
TIME
Fig. 5.12 Numerical results for the response of droplets to a step
like shear rate profile for >.=2.0.
0
ö "' 0 Il// •• _.,,__,
~---~~~~-,-~~~~-..~~~~--,
TIME
o - 0-0.18, EXPERIMENTAL v - 0-0.40, EXPERIMENTAL " - 0-0 . 80 , EXPER I MENTAL
o.o
o - n-o .1, "' - n-0.2, + - 0-0.4, x 0-0.8,
!O.O
TINE
NUMERI CAL NUMERI CAL NUMERI CAL NUMERI CAL
Fig. 5.13 Numerical and experimental results for the response of
droplets to a step like shear rate profile for À=5.
20.0 30.0
"' ei
N
ei
ei 0
ei_,._~,-~,.---,~~~~~~~~~~~~~ 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7 .0 8.0 9.0 10.0
Time
o = 0=0.38, experimental a = 0=0.76, experimental + - 0::::0.93, experimental " = 0=1.39, experimental x - 0=1.67, experimental
Fig. 5.14 Experimental results for the response of droplets to a
step like shear rate profile for A=0.01.
'1!
"': 0
w ci
z U>
0 ci !--< a:"' >:: • O'.::o 0 {~"' w. o"'
~
ci
"'! ~ *.-..~~~~~~~~~~~~~~~~
0.0 s.o TIME
o - À-0.5, NUMERICAL 8 - À-0.8, NUMERICAL + - }..-1.0, NUMERICAL x - À-2.0, NUMERICAL o }..-5.0, NUMERICAL
o.o 5.0
TINE
Fig. 5.15 Effect of viscosity ratio on the response of droplets to a
step like shear rate profile for ~0.4.
10.0 15.0
(D
;l "' 0:
z"' ~ci ,___, a:: ... >::: • e>::o 0 L_"' w. 00
0
o.o O.l 0.2 0.3 0.4 o.s 0.0 0.2 0.4 0.6 0.8 !.O 1.2 l.4 !.6 !.8 2.0
CAPILLARY NUMBER CAPILLARY NUMBER
o - l\=1.0, experimental " À=1.0, numerical
Fig. 5.16 Final deformation of droplets as a function of the
capillary number for various viscosity ratios.
168
5.6.2 Triangular profile response
In this section the response of a droplet to a triangular shear
rate profile is analysed. In these experiments the shear rate
increases linearly in a time ~inc until it reaches a maximum
value, ymax' followed by a linear decrease of the shear rate in a
time ~dec· The results of two numerical simulations are shown in
the figures 5.17 for a viscosity ratio 1 and for two different
values of i , y •t and 2y it respectively. In these figures max cri er the time is normalised with the value of ~inc for the different
curves in one plot. From the first set of curves (figure 5.17a) it
is clear that for values of ~inc smaller than 5 the droplet will
relax to its spherical shape without breakup. Even for values of
2ycrit the droplet will not break up, provided ~inc is short
enough. For longer times the droplet becomes unstable if ~inc
reaches a value of 5.
The same experiments have been carried out in the Couette device.
These results are plotted in figures 5.18 and 5.19. In these
figures the curves of a very simple first order model of the
droplet response for small deformations (see section 5.6.1) are
plotted as well.
Additional experiments have been carried out in the Couette device
for three values of the viscosity-ratio, respectively 5, 1 and
0.01. For these experiments attention was paid to the number of
fragments obtained after breakup of the droplet. The results for
the viscosity ratios 1 and 0.01 are plotted in figure 5.20. For
the viscosity ratio 5, no results are plotted due to the fact that
in only one case breakup had been observed, which could not be
reproduced.
0.50 \ '• )\ = 1.0 0 = Ocrit '• " •• 1
0.40
t;nc 5.0
0.30 i
1 0.20
0.10
0.5
0.00 0 2 4 6 8 10
->~tm.ft/fh:J
1.00 )\ = 1.0 0 = 2.0*0crit
0.80
0.60 i ! <'l
0.40
0.20
t1nc=0.5
0 2 4 6 8 10
->~tlmlt(r/tlnl':J
Fig. 5.17 Effect of the ramp time Tine on the response of a droplet (À=l) to a triangular shear rate profile (numerical calculations).
169
+
+
Fig. 5.18
170
"" +
1?..00
Response of a droplet (X=l) to a triangular shear rate
profile for 'inc=2 for various values of the capillary
number (experimental observations). The solid line
describes the first order model (section 6.7.2) and the +
describe the experimental deformation.
1.00 kYfflallOI?
,,.
+ 0.96
o.32 +
t Fig. 5.18
f: 0 = 4.00*0crit
240
+
·~o
Response of a droplet ()1..1) to a triangular shear rate
profile for ~inc=2 for various values of the capillary number (experimental observations). The solid line
describes the first order model {section 6.7.2) and the+
describe the experimental deformation
171
"""
t OM
Fig. 5.19
172
O.&O OIPfrnmWI
a:Tinc 1. 1 1 b:Tinc = 1.38
T dec 2.21 t Tdec = 2.49 0.6•
•2 ,. •2 ,. - ---
c: Tine= 1.so Tdec=2.91 t
d: Tine 1.94
T dec= 3.32
Response of a droplet (À=l) to a triangular shear rate
profile for ymax = ycrit for various values shear rate
ramp time ~inc (experimental observations). The solid line describes the first order model (section 6.7.2) and the+
describes the equilibrium deformation.
,.
+
OA&
Fig. 5.19
e: Tine = 2.77
Tciec=3.88
t :T1nc 6.92
T 00c= 7.75 ().64
,. 20 ---Response of a droplet (X.:1) to a triangular shear rate
profile for +max ycrit for various values shear rate
ramp time ~inc (experimental observations). The solid line
describes the first order model (section 6.7.2) and the+
describes the equilibrium deformation.
173
174
Fig. 5.20.a Number of fragments observed after breakup of a drop for
various viscosity ratios as a function of the maximum shear rate and the ramp time. À • 0.01
." S20
" . ••• " " 10
" " ... "
Fig. 5.20.b Number of fragments observed after breakup of a drop for
various viscosi ty ratios as a function of the maximum
shear rate and the ramp time. À • 1.0
••
"
5.6.3 Sine profile response
As an example of the behaviour of droplets in oscillatory flows,
the response of a droplet to a sine like shear rate profile was
studied. The profile used is given by:
2(t)
2(t)
A [l+sin(2nt/T-n/2)]
0
t>O
t<O [5.30]
with T the period of the sine function and A the amplitude. These
calculations were carried out for two viscosity ratios, 1 and 5
and for different values of the period Tand amplitude A. The
results are shown in the figure 5.21, where the deformation
response is given for various values of the capillary number and
various periods T. The time of the periods bas been made
dimensionless with the characteristic drop deformation time )1R/ 11.
5.7 DISCUSSION
5.7.1 Step profile experiments
The experimental results in figures 5.9 and 5.16 on step response
experiments and final deformation experiments at a viscosity ratio
of 1 show stable deformations up to a capillary number of 0.48,
corresponding with a maximum stable deformation of 0.68. These
experimental measurements can be compared with some data available
in the literature. The final stable deformation prior to break-up
measured by Rumscheidt and Mason (1961) was D=0.71 and the
critica! capillary number was found to be 0.45. Grace (1982)
reported a final stable deformation of D=0.72 and a critica!
capillary number of around 0.5. Taylor (1934) reported for a
viscosity ratio of 0.9 a maximum stable deformation of 0.79 and a
critical capillary number of 0.55.
175
176
0.50 À=1
0.40
T= 1.0
~ 0.30
j 0.20
0.10
0.00 0 4 6 12 16 20
---~
0,50
À=5
A = 0.5*Ücrit 0.40
0.30 ~
1 0.20
0.10
0.00 ""'---'----'----'----'---__J
0
Fig. 5.21
4 8 12 16 20
Response of drops of various viscosity ratios to a
sinusoidal shear rate profile for various values of the
amplitude and the period of the sine.
All these data are in close agreement with the present
measurements. At smaller deformations (up to about 0.25) the
experimental results show good agreement with the theories of
Taylor (1934) and Cox (1969), the experimental results by
Rumscheidt and Hason (1961) and the numerical results by Rallison
(1981). These results are compared with the present data in
figures 5.22 and 5.23. At small deformations the predictions by
Cox' formula coincide almost exactly with our experiments.
Comparison with the numerical calculations reveals that there is
very good agreement between the calculations and the experiments
for deformations up to about 0.5. For these deformations not only
the final deformations hut the entire transient response can be
calculated accurately. At higher deformations the numerical scheme
predicts unstable droplets whereas the experiments still show
stable droplets up to D=0.68. It is believed that this instability
is a numerical effect since at these deformations the mesh point
density is too small to describe the shape of the droplet
accurately. This is particularly notable in the x-y plane where
the deformation is largest. At these deformations the
circumference at the x-y plane is much enlarged. This instability
would probably be reduced when a mesh with more grid points,
especially in the circumferential direction in the x-y plane is
used.
These numerical calculations can also be compared with the
calculations by Rallison (1981) who used the same boundary
integral method to develop a similar numerical scheme for drops
having the same viscosity as the surrounding fluid. Rallison used
a grid with 117 mesh points. Rallison has used this scheme to find
the final deformation and orientations of a droplet in a simple
shear flow as a function of the capillary number.
0.11
Fig. 5.22
0.45
0.15
Fig. 5.23
178
Ralllspn
--
0.36
Comparison of numerical drop deformation results with the
data of Taylor (1934), Cox (1969), Barthes-Biesel (1972)
and Rallison (1981)
Comparison of final deformation measurements and
calculations with data from Rumscheidt and Mason (1961)
His calculations predict that stable drop deformations can be
obtained up to a critical capillary number of 0.42. The maximum
stable corresponding deformation is about 0.53, which is not as
close to the experimentally observed values, as is claimed by
Rallison (see figure 5.22, note that the dimensionless time scale
used by Rallison is l/4n smaller than the scale used in this
investigation and the capillary number used by Rallison is 4n
times the definition in this investigation). Some observations by
Rallison are not confirmed by the present investigation. Rallison
claims that smaller time steps were needed to maintain stable
drops when the deformation of the droplet increased. This was said
to be due to the fact that the mesh points come closer to one
another when the droplet becomes heavily distorted. As was already
shown in section 5.3.5 these effects were not observed for the
present scheme. Rallison (1981) also reports drop instability for
capillary numbers below the critica! value, provided the flow is
applied with sufficient suddenness. In the present investigation
nor, as far as we are aware, in any other investigation have these
effects ever been observed for a viscosity ratio of 1. The
experimental results on the response to a step profile for À=l,
some of which are given in figure 5.9, do not show any sign of
"overshoot" behaviour. For this viscosity ratio the deformation in
a step response experiment is a monotonously increasing function
of time. The numerical calculations for step profile responses do
not show any sign of this overshoot behaviour for the capillary
numbers at which stable final deformations were observed. It is
thus believed that this effect observed by Rallison (1981) is a
numerical artefact. The observations on the different modes of
break-up for capillary numbers just above and well above the
critica! value were confirmed in the present investigation.
For low capillary numbers it can be shown (see figure 5.24) that
Shanks transforms (Bender and Orzag, 1978) can be used very
successfully to estimate the final deformation.
179
180
O.OO Shaffls Transtarm
0.48
Q.36
o.24
0.12
Fig. 5.24
Ca•0.35 Shanks Transform
. . . . . . . . . . . . . . ' -.
Calcuiated deformation
Ca•0.20
Calculated deformation
4 8 12 16 20
-> Time /Dltrlle$S)
First order Shanks transforms performed on numerical
calculations at various shear rates for À = 1.0
This is done by extrapolation from the deformation vs time curve
shortly after the start of deformation. Shanks transforms can be
used to determine the sum of a slowly converging series (-z)k.
Yhen three successive terms A of such a series are known the
following relation will produce another series which often
converges much faster:
An+l An-1 - An2
An+l An-1 2An [5.31]
Convergence can sometimes further be speeded up by using higher
order Shanks transforms which are obtained by applying the
transform to the resulting series. Thus third order Shanks
transforms have been used to estimate the final deformations for
those capillary numbers where the numerical scheme becomes
unstable above a deformation of 0.5. The results are given in
table 5.2. The predictions can be compared with the experimentally
observed values for the maximum stable drop deformation of 0.68,
which is obtained for Sè:0.48. Theses predictions have been used
in the comparison of the data in figure 5.16.
TABLE 5.2 Predicted final deformations for À=l
Q D
0.40 0.54 ± 0.05
0.45 0.50 0.64 ± 0.05 0.72 ± 0.12
0.60 1.0 ± 0.3
Fora viscosity ratio of 1 in the step profile experiments the
deformation response indicates a first order behaviour of the
type:
-t1t0 D(t) = Dfinal ( 1- e [5.32]
182
The best fitted values for Dfinal and t 0 are given in table 5.3
and show that the time constant is not very dependent on the
capillary number. A comparison between the experimental and
numerical results and the first order predictions is shown in
figure 5.25.
TABLE 5.3 First order representation of step profile experiments
1 1 1 1 1 1 1 1 1 1 2 2 2 2 5 5 5 Q.8 0.5 0.5 0.5 0.01 0.01 0.01 0.01 0.01
2
0.1 0.2 0.3 0.4 0.45 0.097 0.194 0.284 0.389 0.431 0.1 0.2 0.3 0.4 0.1 0.2 0.4 0.4 0.1 0.2 0.4 0.38 0.74 0.93 1.39 1.67
Dfinal
0.11 0.21 0.32 0.51 0.59 0.10 0.23 0.36 0.52 0.62 0.11 0.23 0.35 0.43 0.11 0.23 0.53 0.47 0.11 0.19 0.46 0.29 0.56 0.67 0.81 0.88
2.2 2.1 2.2 3.2 3.9 1.3 2.4 2.9 2.4 2.8 3.3 3.5 3.8 3.4 6.4 7.2 9.6 2.5 1.6 1.2 2.0 1.2 1.0 1.2 1.0 1.0
type
num. num. num. num. num. exp. exp. exp. exp. exp. num. num. num. num. num. num. num. num. num. num. num. exp. exp. exp. exp. exp.
First order behaviour is also predicted by Cox' theory for the
transient behaviour of droplet in simple shear subjected to low
capillary numbers. Cox derived:
0.60 Oeformation /Dimfess]
----i 0.46~
1
0.36
0.24
Fig. 5.25
~L 4
+··
-> Tme {Dimless]
--- Cox' data
Numerical results
Experlmental results
- ··~ First-order model
Comparison of drop deformation calculations vith simple
first order behaviour
183
184
D [ -40t/19À2 [ 20 ]
Dfinal 1 + e + 2 19 À cos(t-T) - ~ sin(t-T)
[ [ 20 ]
2 2 ]-'h
• ; + (19 À) e -20t/(19À2)]
[5.33]
with 5 ( 19 À + 16)
4 À + 1) [ (20/2)2 + (19À) 2 l
T • tan -1 [-20 ] 19 À 2
which reduces to a simple first order behaviour for small
capillary numbers (2<0.15). For higher values of Q the first order
model is still a very good representation of the results. This
first order representation will also be used for the description
of the response of droplets to other shear rate profiles such as
the triangle profile.
The orientation results are compared with the calculations by Cox
(1969) and Rallison (1981) in figure 5.26. The comparison shows
close correspondence between the experimental observations and the
numerical results from Rallison and the present investigation. The
theoretical results by Cox deviate however clearly, even though
the deformation results are in very good agreement (see figure
5.22).
Fora viscosity ratio of 0.5 the numerical calculations show
deformations that are only marginally smaller than for À=l. The
orientation of the drops is somewhat closer to the streamlines
than for À.al. From table 5.2 it follows that the response times
are clearly faster than for À=l. For À-2 opposite effects are
observed. Note that the effect of increasing the viscosity ratio
on the drop response time is larger than the effect of decreasing
it. This is due to the fact that the drop response time is mainly
determined by the most viscous of the two fluid phases, whereas
the timescale is always made dimensionless with nc.
0
0
"' 0
:li "'! Cl
"' 0
!!; Co 0 •
]~ c: 0 <l> • ·r: 1Z Oo
ó .,, 0
"' "' 0
c:i "!
* 0.0 O.l 0.2 0.3 0.4 0.5
Capillary number
o - own numerical results "' - own experimental results + - theoreticol results by cox, 1969 " - numerical resulls by rollison, 1981
Fig. 5.26 Comparison of droplet orientation results vith the
resultsof Cox (1969), Rumsche1dt and Mason (1969) and
Rallison (1981).
The experimental results for a viscosity ratio of 0.01 show a
critical capillary number of 1.9 and a maximum stable drop
deformation of 0.91. These measurements agree well with
experimental observations by Grace (1982). The response times are
given in table 5.2 and are only a little bit faster than for the
higher viscosity ratios 0.5 and 1 as was expected.
The experimental and numerical results for À=5 (see fig. 5.13) are
in close agreement for small deformations only. A striking result
185
186
is the oscillation in the deformation observed for 9=0.8. For this
situation overshoot is clearly noticeable. These overshoot effects
cannot be due to fluid inertia. As was shown in section 5.4.2 the
shear rates following a step profile change in shear rate, can
initially be higher than the final shear rate. These inertial
effects, however, have died away after 0.5(R1-R2 )R2t~ seconds. The
overshoot behaviour is noticeable after 5nR/a. For these
experiments the overshoot in deformation thus occurs long after
the inertial effects have died away. Similar overshoot effects
have also been observed by Torza et al. (1972) for a viscosity
ratio of 25. These effects occur together with oscillations in the
orientation angle and are caused by the interaction between drop
deformation, drop rotation and energy storage at the interface. In
these situations the time scales related to drop rotation are
smaller than those related to drop deformation.
5.7.2 Triangle profile experiments
In these experiments it is observed (see figures 5.17-5.20) that
droplets will not breakup, even at very high shear rates (a few
times the critical shear rate), if the time in which the high
shear rate is experienced is very small. On the other hand, the
droplet will breakup at its critical shear rate if it gets enough
time to reach its critical deformation. Yhen a droplet was
subjected to a triangular shear rate profile with a maximum shear
rate of twice the critical value no breakup was observed for ramp
times of twice the characteristic drop deformation time nR/a. For
ramp times of 5 times the characteristic drop deformation time at
twice the critica! capillary number breakup was observed. Yhen a
droplet was subjected to a maximum shear rate of 4 times the
critical shear rate at a ramp time of 2.4 times the characteristic
drop deformation time breakup was also observed. Breakup thus
occurs faster when a droplet is subjected to higher shear rates.
The triangle profile experiments also indicate, just as was
observed for the step profile experiments, that the deformation
can be described by a simple first order model provided the
capillary number Q and the shear rate y are not too large. The
first order model response to a triangular shear rate profile has
been calculated from the step profile response using Fourier
transforms. The response of the droplet can be calculated from the
pulse response of the drop and the triangular pulse via
multiplication in the Fourier domain.
w(t) ~(t).s(t) [5.34]
where s(t) the triangular pulse and ~(t) the pulse response of the
drop and ~(t) the result of the convolution of the signals ~(t)
and s(t). In Fourier space this simplifies:
!(j) B(j).S(j) and ~(t) [5.35]
where the capitals denote the Fourier transforms of the signals.
In this case a simple multiplication between the
Fourier-transforms of the signal has to be carried out. It can be
seen from figure 5.18 that provided the capillary number and the
viscosity ratio are not too high, this first order model holds
well. This first order model can thus be used to calculate the
deformation in time of a droplet as a response to any kind of
shear rate profile. For small and medium droplet deformations the
actual drop deformation can thus be predicted for arbitrary shear
rate profiles using this first order model, without performing cpu
time intensive calculations.
The transient breakup experiments show that droplets tend to
breakup into 2,4,8,16,32 .•. fragments, even when the shear rates
are increased rapidly. Numbers of fragment in between this series
187
188
were rather rare. Yhen the shear rate is increased very slowly
this "binary" breakup behaviour can of course be expected, since
in quasi steady drop breakup in simple shear flow a drop generally
breaks up into two approximately equisized fragments. It is
however surprising that this "binary" breakup behaviour is also
observed in transient flows. Using this observation it is however
possible to predict the number of fragments, N, following a shear
rate profile with a maximum shear rate of Ymax·
3 ln(y /y .t)/ln2 N = 2 max cri [5.36]
Por the viscosity ratio 0.01 the "binary" breakup model does not
exactly hold, there must be a viscosity ratio effect which is not
included in the above description. This can be related to the
effect observed for low viscosity droplets that they not always
break into two almost identical fragments, but sometimes break
into three fragments.
5.7.3 Sinusoidal profile experiments
The results in figure 5.21 show some examples of the behaviour of
drops in oscillatory flows. It can be noticed that when the
droplets have been subjected to a sinusoidal shear rate profile
for a sustained period of time, the time averaged drop deformation
corresponds very well to the final deformation of a drop subjected
to a constant shear rate, equal to the time averaged shear rate of
the sinusoidal profile. This result was observed for all
frequencies of the oscillation. The maximum deformation, however,
was found to be strongly dependent on the frequency of the
oscillation. Por À ~ 1 and a period of the oscillation of •def'
the maximum deformation was found to be more than 12% higher than
the time averaged deformation. Yhen oscillatory shear rate
profiles with higher frequencies are applied the maximum
deformation is closer to the time averaged deformation, whereas
for lower frequencies the maKimum deformation will be larger.
Since breakup generally occurs when the drop deformation exceeds a
certain critica! value, it is thus expected that drops will
breakup in oscillatory flows with a time averaged shear rate well
below the shear rate required for breakup in quasi steady simple
shear flows, provided the period of the oscillation is smaller
than the characteristic drop deformation time \1Rla. For
oscillations with a much lower frequency the maximum shear rate
will become dominant. The results for a viscosity ratio of 5 show
a similar behaviour. These observations should be taken into
account when predicting drop breakup processes in flows in which
the droplet experiences rapidly oscillating flow conditions.
5.8 CONCLUSIONS
A numerical scheme has been developed, which allows prediction of
drop shapes in any transient linear flow field. The scheme is
based on the boundary integral method and uses 266 collocation
points to describe the drop surface. The scheme has been applied
to droplets with various viscosity ratios (0.5 < À< 5) subjected
to simple shear flows. Up to a drop deformation of 0.5 very good
agreement with experimental data was observed. At higher
deformations a larger number of collocation points is required,
although the predicted shape of the droplet prior to breakup is
already qualitatively correct.
In contrast to what has been suggested by other authors, the
critica! shear rate at which breakup occurs was experimentally
found not to differ between quasi steady shear situations and step
like shear rate profiles for viscosity ratios of 1 and smaller.
The deformation in these step profile experiments was always found
to be a monotonously increasing function of time. For viscosity
189
190
ratios greater than 1, overshoot behaviour in the deformation as a
function of time has been observed.
Yhen a drop is subject to a triangular shear rate profile with a
maximum shear rate which is higher than the critical shear rate in
quasi steady shear flows, breakup will only occur if this
triangular ramp stretches over a sufficiently long time. Yhen the
shear rate is applied for a very short time only, the drop will
relax to its spherical shape without breakup.
Yhen a droplet with À=l undergoes a shear rate which is much
higher than the critical shear rate in quasi steady si~ple shear,
the drop will break into a large number of fragments. The numbers
are very close the geometrical series with ratio 2. Lower
viscosity droplets generally breakup into a larger number of
fragments.
Yhen a droplet is subject to an oscillatory shear rate profile the
time averaged deformation was found to agree well with the final
deformation in a quasi steady shear flow with a shear rate equal
to the time averaged shear rate of the oscillatory profile.
Breakup in oscillatory flows is expected to be possible even when
the time averaged shear rate is smaller than the critical shear
rate, provided that the period of the oscillation is longer than
the characteristic drop deformation time.
5.9 REFERENCES
1 A. Acrivos and T.S. Lo, Deformation and break-up of a single
slender drop in an extensional flow, J. Fluid Mech. 86,
641-672, (1978)
2 D. Barthes-Biesel, Deformation and burst of liquid droplets
and non-Newtonian effects in dilute emulsions, Thesis Stanford
University, Michigan, USA, (1972)
3 D. Barthes-Biesel and A. Acrivos, Deformation and burst of a
liquid droplet freely suspended in a linear shear field, J.
Fluid Mech., 61, 1-21, (1973)
4 v. Bartok and S.G. Mason, Particle motion in sheared
suspensions. VIII singlets and doublets of fluid spheres, J.
Coll. Sci. 14, 13-26, (1959)
5 C.M. Bender, and S.A. Orszag, Advanced mathematica! methods
for scientists and engineers, McGraw-Hill, 1978
6 B.J. Bentley and L.G. Leal, A computer controlled four roll
mill for investigations of particle and drop dynamics in two
dimensional shear flows, J. Fluid Mech., 167, 219-240, (1986)
7 R.A. de Bruijn, Sealing laws for the flow of emulsions,
chapter 2 of this thesis
8 R.A. de Bruijn Newtonian drop break-up in quasi steady simple
shear flows, chapter 3 of this thesis
9 R.A. de Bruijn, Newtonian drop break-up in simple shear flows.
the tipstreaming phenomenon, chapter 6 of this thesis
10 J.D. Buckmaster, Pointed bubbles in slow viscous flow, J.
Fluid Mech. 55, 385-400, (1972)
11 J.D. Buckmaster, The bursting of slow viscous drops in slow
viscous flow, J. Appl. Mech.,40, 18-24, (1973)
12 R.G. Cox, The deformation of a drop in a general timedependent
fluid flow, J. Fluid Mech., 37, 601-623, (1969)
13 A. Einstein, Ann. Physik, 19, 289, (1906)
14 N.A. Frankel and A. Acrivos, The constitutive equation for a
dilute emulsion, J. Fluid Mech., 44,65-78, (1970)
15 H.P. Grace, Dispersion phenomena in high viscosity immiscible
fluid systems and application of statie mixers as dispersion
devices in such systems, Chem. Eng. Commun. 14, 225-277,
(1982)
16 E.J. Hinch, The evolution of slender inviscid drops in an
axisymmetric straining flow, J. Fluid Mech., 101, 545-553,
(1980)
192
17 D.V. Khak.kar and J.M. Ottino, Deformation and break-up of
slender drops in linear flows, J. Fluid Mech., 166, 265-285,
(1986)
18 O.A. Ladyzhenskaya, The mathematical theory of viscous
incompressible flow, Gordon and Breach, New York, (1963)
20 H. Lamb, Hydrodynamics, 6th ed. Dover press, NewYork, (1945)
21 J.M. Rallison, Note on the time dependent deformation of a
viscous drop which is almost spherical, J. Fluid Mech., 98,
625-633, (1980)
22 J.M. Rallison, A numerical study of the deformation and burst
of a viscous drop in general shear flows, J. Fluid Mech. 109,
465-482, (1981)
23 J.M. Rallison, The deformation of small viscous drops and
bubbles in shear flows, Ann Rev Fluid Mech 16, 45-66, (1984)
24 J.M. Rallison and A. Acrivos, A numerical study of the
deformation and burst of a viscous drop in an extensional
flow, J. Fluid Mech., 89, 191-200, (1978)
25 F.D. Rumscheidt and S.G. Mason, Particle motions in sheared
suspensions. XII deformation and burst of fluid drops in shear
and hyperbolic flow, J. Coll. Int. Sci. 16, 238-261, (1961)
26 H.A. Stone, B.J. Bentley and L.G. Leal, An experimental study
of transient effects in the break-up of viscous drops, J. Fluid Mech. 173, 131-158, (1986)
27 G.I. Taylor, The viscosity of a fluid containing small drops
of another fluid, Proc. Roy Soc. A 138, 41-48, (1932)
28 G.I. Taylor, The formation of emulsions in definable fields of
flow, Proc. Roy Soc. A 146, 501-523, (1934)
29 G.I. Taylor, Conical free surfaces and fluid interfaces
Proc. Int. Congr. Appl. Mech., llth, p790-796, Munich (1964)
30 S. Torza, R.G. Cox and S.G. Mason, Particle motions in sheared
suspensions. XXVII transient and steady deformation and burst
of liquid drops, J. Coll. Int. Sci. 38, 395-411, (1972)
31 G.K. Youngren and A. Acrivos, On the shape of a gas bubble in
a viscous extensional flow, J. Fluid Mech. 76, 433-442, (1975)
5.10 LIST OF SYMBOLS
B
D
terms of a series
set of coefficients describing a second order
curved surface around collocation point i
width of droplet
deformation: (L-B)/(L+B)
velocity gradient tensor
unit vector in k-th direction
f distribution function of external sources of
force
second order curved surface around point i
second order curved surface around point i with
respect to transformed coordinate system R1
~ rjrk
r r 3
3 r 1rjrk
4n r 5
curvature tensor
local curvature: tr ki
length of droplet
number of collocation points
3 dimensional space in neighbourhood of S
outward normal vector
pressure
scalar force (Fourier transform of q)
scalar force
drop radius
radii of curvature
orthogonal coordinate system in point i,
with one of the directions normal to the drop
surf ace
terms of Shanks series
shear rate ramp function
[-]
[m-2]
[m]
[-]
[s-1]
[-)
-1 [Pa m ]
[-]
[-]
[m-1)
[m-1]
[m-1]
[m]
[-]
[m]
(-] [Pa] [Pa] [Pa]
[m] [m-1]
[m]
[-)
[-]
193
194
LIST OF SYMBOLS (continued)
"
normal stress: ~.n
time
time constant
velocity (Fourier transform of ~)
disturbance velocity
velocity
posi tion
position of point source of force or position
at drop interface
Fourier transform of position ~
rate of shear
Dirac delta function
dynamic viscosity
viscosity ratio: nd/nc
kinematic viscosity
Fourier transform of t first order response function
interfacial tension
time
stress tensor
modified stress tensor
Fourier transform of ~
convolution of t and s
capillary number: nYR/a
subscripts
c continuous phase
crit critical
d droplet phase
dec decrease
i ,j ,k index of component
inc increase
[Pa]
[s)
[s) [m s-11 [m s-11
-1 [ms ]
[m]
[m]
[m]
[s-1] [-]
[Pa s]
[-]
[m2 s-11 [-]
[-] [N m-1]
[s]
[Pa]
[Pa]
l-1 [-]
[-]
LIST OF SYMBOLS (continued)
max maximum
s surf ace
S(e) limit approaching surf ace s from exterior
S(i) limit approaching surface s from interior
v volume
+ tangential direction
superscripts
in inside
inf at infinity
out outside
195
196
6. NEWTONIAN DROP BREAK-UP IN SIMPLE SHEAR FLOW: THE TIPSTREAMI
PHENOMENON
6.1 INTRODUCTION
Although most of the behaviour of small Newtonian drops in simple
shear flows is reasonably well established, the phenomenon of
tipstreaming is still poorly understood (Rallison, 1984).
Tipstreaming is an experimentally observed mode of drop break-up
(eg. Taylor, 1934 and Grace, 1982), in which the droplet takes upon
increasing the shear rate, a sigmoidal shape and a stream of very
small droplets is ruptured off the tips of the drop (figure 6.1).
This break-up behaviour is potentially very important since the
shear rates required for this type of break-up can be orders of
magnitudes lower than for the normal type of break-up, in which the
droplet is broken in two or three almost equally sized droplets
with a few tiny satellite drops in between (figure 6.1) and the
resulting droplets can be much smaller. Another area of potential
applicability is the emulsification of a droplet phase containing a
third phase (either solid or liquid) since the tipstreaming
phenomenon may be useful for separation processes (Srinivasan and
Stroeve, 1986 and Smith and van de Ven, 1985).
oo tipstreaming
/7 //000
C/ oook/
00c:P 0 •
fracture Fig. 6.1 Modes of drop breakup observed in simple shear
197
198
In this chapter an attempt is made to unravel the causes of the
tipstreaming phenomenon. In section 6.2 the literature on
tipstreaming will be surveyed and the suggested causes for the
phenomenon will be listed. In section 6.3 and 6.4 an experimental
and numerical investigation is reported, followed by a discussion
in section 6.5.
6.2 LITERATURE
Taylor (1934) was the first to report the tipstreaming phenomen in
simple shear flows. The materials he used and the operating
conditions are given in table 6.1. Taylor called it a transient
phenomenon since it disappeared when the shear rate was further
increased. Tipstreaming occurred at a capillary number of
0-0.71, whereas the expected critical capillary number for this
viscosity ratio (À=3•10-4) is 0-21 (chapter 3 of this thesis).
In agreement with Taylors observations Bartok and Mason (1959)
reported that when the shear rate was further increased a
tipstreaming droplet would resume an ellipsoidal shape. The
critica! capillary numbers for tipstreaming were reported to be
independent of drop sizes as for normal drop break-up.
Rumscheidt and Mason (1961) did observations of tipstreaming for
various fluid combinations (see table 6.1), all having in common
the low viscosity ratio l.3•lo-4 < À < 0.19. The critical
capillary numbers at which tipstreaming began were reported to be
almost constant: 0=0.5 with a standard deviation of 0.1. They
reported tip drops of about 50 µm, using mother drops of 0.5-2 mm
and inhibition of internal circulation for many of the fluid
combinations, including all those showing tipstreaming. When
inhibition occurs the observed phenomena can be expected to be
independent of the viscosity ratio. They observed that when a
droplet had been tipstreaming for some time at a certain shear rate
it would stop, but tipstreaming could restart when the shear rate
was further increased. Rumscheidt and Mason ascribed this effect to
a reduction of the drop volume, resulting in a drop of the
TABLE 6. 1 1.i terature observations o! tipstreaming
number reference drop pha.se cont; ~c ~d À r " 0
phase [Pas] [Pas] [-] (nm] CmN/ml [-]
Taylor (1934) CC14/parafin oil cs/w 11 3. 3E-3 3.0E-4 l. 57 23 0.71 1)
2 Bart.ok and CC14 es 9.4E-4 l.3E-4 0,332
Mason (1959} CC14 es 9.4E-4 l. 3E-4 0.698
CC14 es 7 9.4E-4 l. 3E-4 0.750
5 Rumscheidt and dibuthylphtalate 50 5.26 2E-2 4E-3 0.5-2 2.5 0.5+/-0.l
6 Mason (1961) ethylene glycol so 5.26 2E-2 4E-3 15. 0
7 distilled water so 5.26 lE-3 2E-4 38
water + O. 005% Tween 20 5.26 lE-3 2E-4 20
9 water + 0. 5% Tween 20 so 5.26 lE-3 2E-4 6.6
10 air so 5.26 1. 8E-5 3 .4E-6 20.9
11 silicone oil oco 6.00 1.1 0.19 4.5
12 glycerol oco 6.00 0.84 0 .14 10.4
13 silicone oil oco 6.00 2E-3 3E-4 2.7
14 air oco 6.00 l. 8E-5 3E-6 38. l
15 silicone oil g 0.8 3E-3 4E-3 26.3
16 CC14 es 9.0 1.2E-3 1.3E-4 38
17 Grace ( 1982) v 281.5 2.8E-3 1. OE-5 0.62
18 v 281.5 3. 7E-2 l. 3E-4 0. 55
19 4.55 1.0E-3 2.8E-4 0.58
20 oco 4. 55 1.3E-3 2.8E-4 0.65
21 oco 4. 55 3.0E-3 6. 6E-4 0.65
22 oco 4.55 5. 9E-3 1. 3E-3 0.69
23 281.5 0. 45 1. 6E-3 0. 93
24 v 281. 5 0.58 2.0E-3 0.69
25 oco 4.55 3.8E-2 8.3E-3 0.62
26 oco 4.55 7. 7E-2 1. 7E-2 0. 65
27 4.55 0.10 2.1E-Z 0.89
28 oco 4.55 0.15 3. 4E-2 0.58
29 oco 4. 55 0.25 5.4E-2 0.58
30 Smi tb and van water + TRS 1080 50 1.0 lE-3 lE-4 1.17 2.5 2.2 2 )
de Ven ( 1985)
abbreviations; cs/w corn sy:rup/water
corn sy:rup
sa silicone oil
oxidized castor oil
g glycerol
vorite 125
note: l) sbear ra te in this experiment: 0.95 .-1
2) 0. 49 .-1
199
200
capillary number below the critical one for tipstreaming. The
transition in droplet shape from rounded ends to pointed ends on
increasing the shear rate was observed to be sudden.
Torza, Cox and Mason (1972) only reported their tipstreaming
results qualitatively. They observed that the rate of increasing
the shear rate affected the break-up behaviour. For small values
of d 1/d t (2•10-4 s-2) the droplet did assume a sigmoidal shape,
but tipstreaming did not occur, whereas for higher values of
d ~/d t tip drops were released.
The results obtained by Grace (1982) have unfortunately not been
well documented. They however appear to confirm that tipstreaming
occurs at low viscosity ratios (À<<l) at almost constant capillary
numbers (0=0.65, standard deviation of 0.1). He probably also
observed that when the shear rate was further increased the droplet
resumed an ellipsoidal shape and could be broken at much higher
shear rates via a normal fracture mode.
Smith and van de Ven (1985) reported that an aqueous droplet
saturated with surface active material (see table 6.1) showed
tipstreaming behaviour with tip drops of about 10 µmin diameter,
although a droplet of distilled water did not show this behaviour.
Their main study concerned however the deformation and break-up of
droplets containing spherical solid particles. At low concentration
of these particles in the drop phase (3%) they observed that as the
shear rate was increased, the droplet became elongated and the
particles concentrated at the tips of the droplet. The particles
ruptured from the drops as singlets or doublets surrounded by
liquid of the drop phase. This phenomenon was observed at shear
rates that were too low to fracture the droplet itself. When given
sufficient time all particles were seen to leave the droplet from
the tips.
Tipstreaming related phenomena have also been observed in ether
type of flows, e.g. elongational flow (Taylor, 1934 and Sherwood,
1984) and the flow into a capillary (Carroll and Lucassen, 1976).
This literature survey shows that the tipstreaming phenomenon is
not understood although several factors have been shown to affect
it:
- the viscosity ratio
- the rate of increase of the shear rate
- the presence of surf actants
6.3 EXPERIMENTAL
6.3.1 Introduction
The experimental programme was aimed at unravelling the suggested
causes for the tipstreaming phenomenon, namely the viscosity ratio,
the rate of increase of the shear rate and the presence of
surfactants. Accordingly various sets of experiments were
performed. In the first set the viscosity ratio was varied from
2•10-4 to 0.1 for four different pairs of fluid combinations in
order to see the the effect of the viscosity ratio and the type of
fluid combination on the occurrence of tipstreaming (section
6.3.2). In the second set the tipstreaming phenomenon itself was
subject to a closer examination and the time dependency of the
phenomenon was studied (section 6.3.3). In the third set the effect
of the rate of increase of the rate of shear was studied (section
6.3.4). In the fourth set the effects of surfactants were studied
systematically by adding various levels of a surface active
material to a fluid combination that did originally not show any
tipstreaming.
The experiments were performed in the Couette device described in
chapter 3 of this thesis. The rheological properties of the model
liquids were measured with a Haake viscometer (type CVlOO) using a
concentric cylinder geometry (type ZClS). The interfacial
properties were either measured with a Wilhelmy plate or by the
201
202
method of drop deformation measurements in simple shear flow. The
latter method was generally applied when surfactants were added or
when the interfacial tension of a specific droplet in the Couette
device was to be measured. This method is based on the linear
dependency of the dimensionless drop deformation D = (L-B)/(L+B),
with L and B the length and the width of the droplet, on the
capillary number 0. This linear dependency is valid for small
drop deformations and constant interfacial tension. This method was
first described for viscosity ratios close to unity by Taylor
(1934), who derived a theoretical relation valid to the first order
in the deformation.
19 À + 16
16 À + 16 (6-1] D = 0
This relation was extended by Cox to be valid to the second order
in the deformation. Experimental data on the proportionality
constant have been obtained by Torza, Cox and Mason (1972) and
in chapter 5 of this thesis (see table 6.2).
TABLE 6.2 Steady drop deformations in simple shear flow
viscosity measured prediction by rat.ic proportionali ty Taylor
constant
8. 0 E-3 0.95 2)
8.0 E-2 1. 08 1). 1.01 0.25 1,02 1) 1. 04 0.55 1.00 1) l. 07 0. 82 1.20 1) 1.08 1. 0 1.12 2) 1.09 1.2 1. 02 1) 1.10 1. 7 1.20 1) 1.12 2. 6 0.93 ll l.14 3.6 1. 29 l) 10.2 0 .15 1)
18.6 0.085 1)
l) measurements from Torza et al, 1972 2) masurements from cba:pter S of this thesia
6.3.2 Viscosity ratio
To scan the effect of the viscosity ratio on the occurrence of
tipstreaming, a matrix of several types of fluids was used (see
table 6.3). The silicone oils were Rhodorsil oils type 47 (ex Rhone
Poulenc), the corn syrups were prepared by mixing corn syrups (type
Globe 01170 ex CPC Netherlands) with distilled water and the esters
were obtained by diluting Uraplast esters (ex DSM) with methyl
ethyl-ketone. The experiments were done in a quasi steady way.
The results in table 6.3 show that tipstreaming was not observed
for the corn syrup and ester droplets in silicone oil, even though
the viscosity ratio was varied between 0.1 and 2•lo-4. For ester
droplets in corn syrup, however, tipstreaming was observed for each
of these viscosity ratios. The silicone oil droplets in corn syrup
showed generally no tipstreaming. Only for a 5 mPas silicone oil
tipstreaming was observed if the standard batch Rhodorsil 47VS
was used. If a 5 mPas silicone oil was made by mixing the standard
batches of 1 mPas and 10 mPas silicone oil, tipstreaming did not
occur.
These results indicate that the type of the fluids is very crucial
for tipstreaming to occur. The viscosity ratio is not very
important provided it is much smaller than unity. The S mPas
silicone oil drop phase results indicate that minor components in
the liquids can determine whether or not tipstreaming will occur.
6.3.3 Time dependency
In this section the tipstreaming phenomenon itself is subject to a
closer examination. Especially the time dependency and the history
effects involved in tipstreaming will be studied.
When a droplet was inserted in the Couette device and the shear
rate was slowly increased, the droplet at first assumed an
ellipsoidal shape. At higher shear rates a sigrnoidal shape was
assumed and at capillary numbers of around 0.5 tipstreaming
occurred. When this shear rate was maintained it was observed that 203
TAB LE 6.3 Occurrence of tipstreaming
6-3A: 5. 0 Pas silicone oil as 6-3C: 5. O Pas corn syrup as
continuous phase continuous phase
drop phase À tip- drop phase À tip-
streaming streaming
100 mPas corn syrup 2.0E-2 no 100 mPas silicone oil 2.0E-2 no
50 mPas corn syrup l.OE-2 no 50 mPas silicone oil 1.0E-2 no
10 mPas com syrup 2.0E-3 no 10 mPas silicone oil 2.0E-3 no
5 mPas corn syrup 1. OE-3 no 5 mPas silicone oil 1. OE-3 yes/no
1 mPas silicone oil 5. OE-4 no 100 mPas ester 2.0E-2 100 mPas ester 2. OE-2 yes
50 mPas ester l.OE-2 no 50 mPas ester 1.0E-2 yes
10 mPas ester 2. OE-3 no 10 mPas ester 2.0E-3 yes
5 mPas ester l.OE-3 no 5 mPas estér 1.0E-3 yes
1 mPas ester 2.0E-4 no 1 mPas ester 2. OE-4 yes
6-3B: 1. 0 Pas silicone oil as 6-30: 1.0 Pas corn syrup as
continuous phase continuous phase
100 mPas corn syrup 1. OE-1 no 100 mPas silicone oil 1. OE-l no
50 mPas corn syrup 5.0E-2 no 50 mPas silicone oil 5.0E-2 no
10 mPas corn syrup l.OE-2 no 10 mPas silicone oil l.OE-2 no
5 mPas corn syrup 5.0E-3 no 5 mPas silicone oil 5.0E-3 yes/no
l mPas silicone oil l. OE-3 no
100 mPas ester 1. OE-1 no 100 mPas ester 1. OE-1 yes
50 mPas ester 5.0E-2 50 mPas ester 5. OE-2 yes
10 mPas ester l.OE-2 10 mPas ester 1.0E-2 yes
5 mPas ester 5.0E-3 no 5 mPas estèr 5.0E-3 yes
1 mPas est.er 1. OE-3 no 1 mPas ester 1. OE-3 yes
204
tipstreaming ended after a certain period, varying between 10
seconds and a few minutes. However when the shear rate was further
increased tipstreaming often started again at a higher shear rate
and the same process could be repeated until the shear rate became
too high and the droplet assumed an ellipsoidal shape again. At
higher shear rates a normal mode of droplet fracture was observed.
Contrary to the statement by Rumscheidt and Mason (1961), who
ascribed the ending of tipstreaming to a decrease in drop volume
and a subsequent lower capillary number, the observed ending and
restarting of tipstreaming was not merely due to a reduction of the
capillary number. When ending of tipstreaming was observed, the
drop volume was decreased very little, the decrease was usually
hardly detectable, while the necessary capillary number to restart
tipstreaming was substantially larger.
When a droplet was subjected to a slowly increasing shear rate and
tipstreaming had started at a certain shear rate it generally ended
again at a higher shear rate. When the shear rate was then reduced
to zero and the experiment was repeated it was observed that
tipstreaming would start at significantly higher shear rates than
before, although the ending occurred at comparable shear rates. A
typical sequence of such experiments is given in Table 6.4. When a
droplet was left at rest after tipstreaming for a period of 10-30
minutes, tipstreaming would start again at the initia! critica!
shear rate.
It was further observed that tipstreaming never occurred when the
shear rate was slowly decreased from close to the critical shear
rate for normal fracture type drop break-up.
These observations of the tipstreaming phenomena indicate that
there appears to be some sort of depletion effect and the
occurrence of tipstreaming depends on the history of the droplet.
205
TABLE 6.4 History effects on tipstreaming
drop phase: 100 mPas esther
continuous phase: Pal$ corn syrup
interfacial tension: 26. 7 mNm-1
viscosity ratio: 0.024
drop radius shear rate capillary number remarks
[mm] [s-1] [-]
0.28 start of experiment
5. 4 0.25 start. af tipstre:aming
10. 4 0.48 end of tipstreaming
0.28 0 back to rest
7. 5 0 .34 start of tipstreaming
9.8 0. 45 end of tipstreaming
0.28 back to rest
9.0 0.41 start of tipstreaming
10. l 0.46 end of tipstreaming
13.6 0.62 fracture of droplet
0.35 0 start of experiment
4. 3 0,25 start of tipstreaming
a.z o. 47 end of tipstreaming
12.3 0.71 fracture of droplet
206
6.3.4 Acceleration
To examine the effects of the rate of increase of the shear rate,
experiments were performed with model liquids, that had shown
tipstreaming in the previous experiments, and with model liquids
that had not exhibited any tipstreaming in the previous quasi
steady experiments.
Several sets of the tipstreaming experiments are given in Table
6.5. These experiments were performed at various constant rates of
increase of the shear rate ~. In this table the shear rates at
which tipstreaming first occurred are given, together with the
corresponding capillary number and the ratio r of the
characteristic drop deformation time (tdef - ~c r/a) to the time of
increase of the shear rate up to the moment tipstreaming started
tramp· The results show that there is a strong tendency for the
critical shear rate to increase at higher accelerations.
The systems that did not show tipstreaming in the steady state
experiments did not show it in these acceleration experiments
either.
6.3.5 Surfactants
To study the effects of surfactants on the tipstreaming phenomenon
surfactants were added to the dispersed phase of a fluid
combination that did not show any tipstreaming, namely a 5 Pas corn
syrup/water mixture as the continuous phase and low viscosity
silicone oils (10 and 50 mPas) as dispersed phases. It was found
that glycerol-1-mono-oleate was a good surface active material for
these fluids. At saturation concentration the interfacial tension
was reduced to 2-3 mN/m. For both fluid combinations a set of
experiments was performed with a range of surfactant concentrations
added to the droplet phase. The results are given in Table 6.6. The
interfacial tension values given in this table for the non zero
levels of surfactant were obtained via the drop deformation
method. 207
TABLE 6.5 Effect of accéleration ( cont.inuous phae.e is corn syrup)
drop phase 'Id "• À q r 1 'Y tramp tdef T ll
[Pas! [Pas] [-] [mN/ml [mm] [l/sJ [l/s2J [•) [s] (-] [-]
ester 0.1 10 0.01 27 .9 0.51 3.9 l.O 3.9 0.18 O.Oli6 0.62
0.49 3.1 0.5 6.2 0.18 0.029 0.47
0.48 2.3 0.1 23 0.18 0. 0078 0.35
0.46 2.1 0.05 42 0.18 0. 0043 0.29
silicone oil 0.005 5 0 .0011 26.7 0.39 4.9 1.0 4.9 0. 07 0 .014 0.36
0.26 5. 7 0.5 11.4 0.05 0 .004 0.27
0.56 2.8 O.l 28 0.10 0. 004 0.28
0. 54 2. 7 0.05 54 0.10 0.002 0.26
ester 0.1 10 0 .01 33. 0 0.52 3.9 1. 0 3.9 0.16 0 .04 0 .49
0.63 3. 7 1.0 3. 7 0.19 0.05 0. 57
0.63 3.6 1.0 3.6 0 .19 0.05 0. 54
0.63 3.8 1.0 3.8 0 .19 0.05 0.57
0.59 2.9 0.5 5.8 0.16 0.03 0. 42
0 .60 3. 7 0.5 7 .4 0.18 0.03 0.52
0. 72 2.6 0.5 5.2 0.22 0. 04 0.46
0.69 2.6 0. 5 5.Z 0.21 0.04 0.43
0 .65 1.9 0.1 19 0.20 0.01 0.29
0.63 2.2 0.1 22 0 .19 0.009 0.33
208
TAB LE 6.6 Effect of surfactants on tipstreaming
TABLE 6,6A: drop phase 50 mPas silicone oil with various l&Vèls of
glycerol-l-mono-oleate
continuous phase: 5 Pas corn syrup
viscosi ty ratio : 0 .01
No. surfaotant " 1 n break-up rt.ip "tip [%] [mN/mJ Cnml [l/•l [-] mode [µm] [N/ml
30 0.391 30.1 2.0
0. 430 27.4 l. 7 f
0.560 21.7 1. 7 f
4 0. 0001 30 0 .361 >16.2 >1.0 n.t
5 0.435 >12.9 >1.0 n.t
6 0.360 >19.4 >1.2 n.t
7 0.554 19.9 1. 6 f
8 0.357 22.2 1.3 f
0.0005 30 0.305 >14.5 >0.8 n.t
10 0.374 >14.9 >1.0 n.t
11 0.586 5. 7 0.59 t 10
12 20.1 2.1 f after t
13 0.001 26 0.416 >17 .5 >1.2 n.t
14 0.483 7 .2 0.61 t 15 o. 736 5.6 0.69 t 10 5
16 0.854 3.6 0.52 t 20 2
17 0.550 6.7 o. 71 t
16 0.594 6.9 0.80 t
19 19.l 2.2 f aft.er t
20 0. 502 7 .l 0.69 t 6
21 0.573 6.3 0.69 t 6
22 0.565 6.0 0.68 t
23 20.5 2.3 f after t
24 30 0.392 29.1 1.9 f*
25 26 0.416 6.5 0 .64 t
26 0.463 10.0 0 .87 t
27 24.4 2.1 f after t
28 0.270 10.9 0.52 t
29 0 .431 6.0 0.45 t
30 0.567 5.6 0,55 t 10
Abbreviations: f =fracture t - tipstreaming n.t =no tipstreaming f af ter t fracture af ter
* drop is fragment of exp. 23 tipstreaming td - tipdropping t af ter r tipstreaming after rest
209
TABLE 6,6A: continued
No. surfactant " r ~ n break-up rtip "t..ip [%] CmN/mJ [Dl!l] [l/sJ [-] mode [/Jlll) [N/ml
31 0.005 22 0.386 6.81 0.58 t
32 0.370 6.5 0.53 t
33 0.349 7.2 0.55 t 8
34 0.197 >20.1 >0.84 n.t..
35 0 .214 13. 7 0.57 t
36 0.227 11. 8 o. 46 t
37 0.2419.00.56t
38 0.250 9.1 0.59 t
39 0.263 6.6 0.46 t
40 0.271 7 ,5· 0.53 t.. 10
41 0.276 10 .5 0.62 t
42 0.334 5.3 0 .49 t 10
43 0.338 7 .3 0. 53 5-10
44 o. 354 7 .6 o. 47 t
45 o. 516 5.1 0.46 t 10
46 0.620 3. 4 0.51 20
47 0.01 11 0.472 3.3 0. 70 t
48 0.449 3.0 0.62 t 18
49 0.05 0.359 1. 3 0.59 f
50 0.319 1. s 0,60 f
51 0.302 1.4 0.55 f
52 0.080 4.8 0. 49 f
53 0. 042 6.7 0 .36 f
54 0.237 1.9 0.57 f
55 0.221 3.1 0.87 f
56 0.1 3, 6 0. 440 1.3 0 .59 f
57 0.281 2.0 0.59 f
sa 0 .197 2.4 0.48 f
59 0.152 4. 3 0.67 f
60 o.s 2.~ 0.324 1.2 o. 75 f
61 0. 214 1.6 0 .66 f
62 0.250 1,4 0.68 f
63 0 .178 1. 7 0.61 f
64 0.119 2.4 0.56 f
65 0.197 1. 9 o. 73 f
210
TABLE 6.6B: drop phase 10 mPas" silicone oil wi th various levels of
glycerol-1-mono-oleate
continuous phase: 5 Pas corn syrup
viscosi ty ratio ; 0.002
No. surfactant <1 7 (l break-up rtip qtip [%] [mN/mJ [mm] [l/•l [-] mode [jllll] [N/m]
l 30 0 .341 >14.6 >1. 0 n.t
2 0.350 >19.2 >1.3 n.t
0.280 >18. l >1.0 n.t
0. 0001 30 0.398 >25.3 >1.8 n.t
0.411 >19.1 >l.8 n.t
0.430 >15.4 >1.2 n.t
0.484 >11.4 >l.'4 n.t
8 0.387 >23.8 >1.5 n.t
9 0.496 >15.0 >l.4 n.t
10 0. 0005 30 0.398 >17 .1 >1.2 n.t
ll 0.385 >17 .6 >1.2 n.t
12 0.318 >13.3 >0.8 n.t
13 0. 001 30 0.340 9. 0 0.58 td
14 9.5 0.61 t 24
15 0. 338 >13. 7 >0.74 n.t
16 0. 395 >ll.6 >0.73 n.t
17 0.335 10.2 0 .59 td
lB 0.334 10.2 0.59 td
19 11.1 0.64 t
20 0.005 22 0. 567 5.1 0.73 t 34 10
21 0. 606 4 .4 0.67 t 22 0 .537 4.6 0 .58 t
23 0. 718 5.0 0. 71 t 43 15
24 0.308 7. 0 0.54 t 10
25 0.322 7 .2 0,58 t
26 0.320 7. 0 0.56 t
27 0.319 7. 5 0.60 t
211
TABLE 6.6B: continued
No. surfact.ant Il' r :, Il break-up rtip "tip (%] [mN/mJ [mnJ [1/s] [-J mode [µml [N/ml
28 0.01 16 0.339 4.3 0.51 t 15
29 0.356 5.0 0.62
30 0.321 6.0 0.63 t 10
31 0.261 6.9 0.58 t 10
32 0.271 7 .0 0.62 t
33 0.510 3.6 0.59 t 8
34 0.218 8.1 0.58 t 35 0.407 4,6 0.60 t
36 0.299 6.3 0.61 t 37 0.450 4. 7 0.53 t 20 2.3
38 0.498 4.2 0.62 t 20 2.2
39 0.05 3.5 0.393 l.8 l.O t
40 0.393 2.0 1.1 t 10
41 0.359 2.4 l.3 t 10
42 0.312 2.5 l.1 t
43 0.1 2.2 o. 353 l.6 1.3 t 10 44 0.350 1.5 1.2 t aft.er r 10
45 1,8 1.5 t aft.er r 10
46 2.2 1. 8 t after r 10
47 0.5 1.7 0.343 1.8 2.1 f
48 0,078 2.9 o. 74 f
49 0.185 1.5 0.94 f
212
For bath the 10 mPas and the 50 mPas silicone oil drop phases the
results show the same trend. When no or extremely low levels of
glycerol-1-mono-oleate were added the systems do not show any
tipstreaming. At a certain level, however, tipstreaming is
observed, whereas at much higher levels of the surfactant no
tipstreaming was observed anymore and the droplets could only be
braken up via a normal fracture mode of break-up. For the 10 mPas
silicone oil drop phase, tipstreaming was not observed at
surfactant levels below 0.001%(wt). At 0.001%(wt) sometimes no
tipstreaming could be observed at all and only droplet fracture was
observed, whereas for other droplets tipstreaming did occur,
although sometimes the tip droplets were not emitted continuously
(tipstreaming) but more intermittently (tipdropping). For the
surfactant levels between 0.005%(wt) and 0.1%(wt) tipstreaming was
observed in all experiments. At the highest level of surfactant,
0.5%(wt), a normal fracture mode of drop break-up was observed,
without passing through a tipstreaming stage. Higher levels of
surfactants were not tried because of supersaturation. The radius
of the emitted tip droplets was found to vary between 8 µm and 25
µm. For the 50 mPas silicone oil drop phase a very similar trend
was observed although some variations were observed. At surfactant
level below 0.001%(wt) no tipstreaming was ever observed. At higher
levels tipstreaming was generally observed, although some droplets
exhibited drop fracture without passing through a tipstreaming
stage. At surfactant levels of 0.05%(wt) and higher tipstreaming
was never observed anymore.
For several emitted tip droplets the interfacial tension was
estimated from the deformation method. Even though the results are
not very accurate because the tip droplets were very small and the
droplet deformations could not be determined very accurately, these
experiments gave some striking results. The interfacial tension of
the tip droplets were invariably lower (usually much lower) than
the interfacial tension of the mother droplet. The interfacial
tension of the tip droplets was often close to the saturation
value.
213
214
The results described above indicate that tipstreaming can occur
if an interf acial tension gradient can be developed in the droplet
surface at a certain surfactant concentration range with a lower
interfacial tension near the tips of the droplet.
6.4 NUMERICAL
The numerical technique described in chapter 5 of this thesis has
been used to calculate the position of the rnaterial points on the
rnidplane of the drop surface, when the droplet is subjected to a
sirnple shear flow. The nurnerical calculations are based on a
surface integral rnethod, which rewrites the Stokes-equations in and
around the droplet, via Fourier transforrns, toa surface integral.
Thus one only needs values of quantities at the drop surface to
calculate the velocity of a point at the surface. This surface
integral was solved after discretizing the surface of the droplet.
The rnethod was slightly modified for the calculations under
discussion. The calculation of the velocity of each point of the
surface was identical, but the new position of the point was now
sirnply calculated by:
x x + v • dt -new -old
[6-2]
In chapter 5 of this thesis this sirnplest rnethod to calculate the
new point positions was not used, but a more sophisticated
redistribution technique to enhance numerical stability. Here,
however, it was desired to follow the position of a material point
on the drop surface, so that this basic method had to be applied.
Calculations were performed for droplets characterised by viscosity
ratios of 0.5, 1.0 and 2.0, that were subjected to a step profile
sirnple shear flow with capillary nurnbers ranging from 0.1 to 0.4.
The results are presented in figure 6.2 where the contours of the
rnidplane of the droplets are given at various time intervals after
the onset of the flow. The results clearly show that once the
TABLE 6.7 Concentration of material points for various viscosity ratios
Table 6.7A: Comparison at constant time, T-0.05
À D Dm in Dmax [ - l [ - l [ - l [ - l
2.0 0.046 0.96 1.04 1.0 0.067 0.92 1.07 0.5 0.082 0.82 1.16
Table 6. 7B: Comparison at constant drop deformation, D=0.05
>. T Droin Dmax [ - l [ l [ - ] [ l
2.0 0.55 0.96 1.04 1.0 0.36 0.94 1.05 0.5 0.29 0.90 1.09
TABLE 6.8 Viscous and interfacial stresses in tipstreaming experiments
'Ic "m "t l:!.u r Vstr intstr [Pas] [l/s] [mN/m] [mN/m] (mN/m] [mm] [Pa] [Pa]
5. 0.736 28. 28.5 5.0 3.6 26.0 2.0 24.0 0.854 18.0 28.1 5.0 3.4 22.0 2.0 20.0 0.620 17.0 32.2 5.0 5.1 22.0 10.0 12.0 0.567 25.5 21.2 5.0 5.0 22.0 15.0 7.0 0.718 25.0 9.7 5.0 4.7 16.0 2.3 13. 7 0.450 23.5 30.4 5.0 4.2 16.0 2.2 13.8 0.498 21.0 27.7
215
Time = 0.00 {dim.less) Time = 0.25 {dim.less) Time = 0.50 (dim.less)
Time == 0.75 (dim.lessl Time = 1.00 (dim.lessl Time == 0.00 (dim.less)
Fig. 6.2.a Droplet contours in simple shear flow as a function of the dimensionless time for a viscosity ratio of 0.5 and a Capillary number of 0.3
216
droplets obtain an ellipsoidal shape, the concentration of the
material points around the points of the ellipsoid increases,
indicating a contraction of the drop surface, whereas along the
long side of the disk the concentration decreases, indicating an
expansion of the drop surface. This effect is more pronounced for
lower viscosity ratios. Closer examination of the contours on
figure 6.2 reveals that the surface is most contracted just over
the pointed ends of the ellipse. This corresponds exactly with the
position of the tips during tipstreaming, These effect have been
quantified in figures 6.3 and 6.4. Figure 6.3 gives the
dimensionless drop deformation as a function of time for each of
the experiments. Figure 6.4 gives the distances between the two
neighbouring grid points at the midplane of the droplets that are
Time = 0.00 ldim.lessl Time = 0.50 (dlmJessl Time = 1.00 (dlm.lessl
Time = t50 (dimJessl Time = 2.00 (dîm.less) Time = 3.00 (dim.lessl
Fig. 6.2.b Droplet contours in simple shear flow as a function of the dimensionless time for a viscosity ratio of 1.0 and a Capillary number of 0.3
closest to one another and that are the farthest apart. These
distances have been normalized with the initial spacing between the
grid points at the midplane. These results show that for a
capillary number of 0.3 and a dimensionless time of 0.5 the
concentration effects are about four times more pronounced for the
viscosity ratio of 0.5 than for 2.0. (see Table 6.7). These
calculations were performed assuming a constant interfacial
tension. This assumption is only valid for small times.
217
218
1 ~'
( \ ' · .TT; \ . /
'
Time = 0 00 (dim.lessl Time = 0.50 (dimJess) Time == 1.00 (dim.less)
Time = 2.00 (dim.less) Time " 3.00 (dim.less) Time = 5.00 (dimJessl
Fig. 6.2.c Droplet contours in simple shear flow as a function of the dimensionless time for a viscosity ratio of 2.0 and a Gapillary number of 0.3
For larger deformations the numerical calculations are not anymore
appropriate, since interfacial tension gradients may than become
important and these effects are not accounted for in these
calculations.
6.5 DISGUSSION
The experimental and numerical results seem to indicate that
tipstreaming is a result of the built up of interfacial tension
gradients over a droplet resulting in low interfacial tensions
À= 1.00, Time= 0.50 À -= 2.00, Time = 0.50
À -= 0.50, Time-= t.00 À -= 1.00, Time -= 1.00 À = 2.00, Time = 1.00
Fig. 6.2.d Droplet contours in simple shear flow as a function of the viscosity ratio for dimensionless times of 0.5 and 1.0 and a Capillary number of 0.3
close to the points of the ellipsoids. Such interfacial tension
gradients make the drop surface less mobile allowing the shear
stresses exerted by the continuous phase to pull out a stream of
tip droplets. This explanation however can only hold when the shear
stresses exerted by the continuous phase are large enough to
maintain such an interfacial tension gradient and when the
diffusion of the surface active material from the droplet to the
surface is slow enough not to interfere with the build up of the
interfacial tension gradients. The restarting of tipstreaming after
a droplet was brought back to rest and a new experiment was begun,
219
Ul IJ)"' (1) • __,o E _ _, n
vc::i
c QN
jc::i 0 E L-Oo
<+-(j) Do
tf)
0
({)
({)"' ID •
__,o
c _g ~ _,..)
0 E L Oo
<+-(])
Do
ei+'-----,.---,-----,----,-----, D-4f:'---~---.------~--~
0.0 1.0 2.0 3.0 4.0 Eme !&m.Less)
LEGEND ei =Curve l t;, =Curve 2 x =Curve 3 o •Curve 4
n - 0.10 & >. - o.5o 0 - 0.20 &. À = 0.50
0 - 0.30 &. À = 0.50
tl - 0.40 & /, - 0.50
5.0 0.0 1.0 2.0 3.0 4.0 T~me (&m. Less l
LEGEND @=Curve o- 0.10 Il/.. - LOO
"_ 2 0 = 0.20 &: ;\ • 1,00
"'""' 3 0 = 0.30 & À= 1.00
•= 4 n• o.40 & /, - 1.00
•= 5 o- 0.30 & 1.- 2.00
5.0
Fig. 6.3 Drop deformations as a function of time for various Capillary numbers and viscosity ratios
220
indicate that in about a minute the concentration of surface active
material can be restored.
An order of magnitude calculation to see whether or not the shear
stresses exerted by the continuous phase will be able to maintain
an interfacial tension gradient on the droplet can be performed by
comparing both stresses for actual experiments. The viscous
(j) u c
Cl
t'Î
0 "' :;; à . -' D
c; Ü-J-~~~~~~~~~~~~T·~--,
0.0 1.0 2.0 3.0 TL.me [ dlm. Less l
LEGEND o Curve l li =Curve 2 x = 3
4
o- 0.10 tl.,.._ 0.60
n = 0.20 &. >. = 0.60
0 = 0.30 e. 'A - 0.50
0: - 0.40 &. /\. - 0.50
Cl
N
(IJ (IJ tl)
ID_: -'
E .J u c;
ID u c 0 tf)
(;i ci . -' D
Eme (di..m.Lessl
LEGEND ©-= Cut""ve 1 11 =Curve 2 "=Curve 3 •=Curve 4 •=Curve 5
0 - 0.10 " À - 1.00 0 - 0,20 6 À - 1.00
(l - 0.30 & À= 1.00
o - 0.40 & À - 1.00 a-o.3o a >.-a.oo
5.0
Fig. 6.4 Minimum and maximum distances between material points at the drop surface as a function of time for various Capillary numbers and viscosity ratios
stresses will be of the order of: ~c7• whereas the interfacial
gradient stresses will be of the order of Aa/r, in which Aa is
the difference in interfacial tension between the mother droplet
and the tip droplet. In table 6.8 these order of magnitude
calculations are given for all the experiments for which the
interfacial tension of the tip droplet was determined. The results
show that the two stresses are reasonably well of the same order of
magnitude. This was already expected beforehand since the critica!
capillary number was of the order 1 and Aa was of the order of a. 221
222
Whether or not surfactant diffusion occurs on these time scales is
somewhat more difficult to estimate, since it involves an
estimation of the diffusion coefficient of glycerol-1-mono-oleate
in silicone oils. Tipstreaming was found to occur typically at a
concentration, w, of 0.005%(wt) in the droplet phase. The number of
surfactant molecules in a droplet is thus given by
(6-3)
where p is the density of the drop phase, p - 1000 kg m·3, Nav the
Avogadro nurnber and M the molecular weight of glycerol-1-mono
oleate, M-348, resulting in Nd - 4.5•1012 fora 0.5 mm droplet.
The number of molecules needed to cover the undeformed drop surface
completely is given by
(6-4]
where Am is the area of a glycerol-1-mono-oleate molecule in the
drop interface, Am-38A2, resulting in Ns - 8.3•1012. Thus
approximately a f if th of the total number of surf actant molecules
in the droplet suffices to cover the droplet surface completely. To
estimate the time scales involved in surfactant diffusion in the
droplet, one can use the standard solutions to transient diffusion
in a sphere. These show that complete equilibrium is obtained when
Fo- IDt/r2 - 0.5 [6-5)
where Fo is the Fourier number. At a Fourier number of about 0.01
already a substantial amount of diffusion from a layer of thickness
r/10 has taken place (such a layer contains already enough
surfactant molecules to cover the drop surface completely).
A reasonable guess of the diffusion coefficient, taking into
account the high viscosity, seems to be
ID [6-6]
This would imply that complete equilibrium by diffusion from a
thin layer near the drop surface can, for a droplet with r-0.5 mm,
already occur after 10 minutes. The time scale for equilibrium
corresponds reasonably well to experimental time scales showing
that when the droplet has been brought to rest after a tipstreaming
experiment and an experiment was started again after about a minute
that tipstreaming would take place, but at higher capillary numbers
and that when a droplet was brought back to rest after a
tipstreaming experiment and an experiment was started after 15
minutes, tipstreaming occurred again at the original capillary
number.
If the hypothesis is correct that tipstreaming can only occur when
interfacial tension gradients are present, the tipstreaming
conditions must also fulfil the rapidly varying flow criteria
(chapter 2 at this thesis). These criteria state that interfacial
tension gradients can only be present when the Peclet number based
on a layer from which diffusion must occur to equilibrate the
interfacial tension, is smaller than 1.
Pe Ud/ ~ [6-7]
An estimation of the layer thickness d for low concentrations of
surfactants can be obtained from the lower limit of the Langmuir
adsorption isotherm, such that there are as many surfactant
molecules present in this layer as there are needed for equilibrium
adsorption:
d [6-8]
223
224
with r 00 the maximum surface concentration of the surfactant and
r~/C1 the slope at low concentrations of the Langmuir adsorption
isotherm. r 00 can be estimated from the size of the surfactant
molecules:
[6-9)
with N the Avogadro number and Am the area of the adsorbed
molecule. A typical value for Am is 4•lo-19 m2, resulting in
r~ = 4.10-6 molesm·2. The order of magnitude for C1 follows from
the interfacial tension measurements at several surfactant
concentration levels (see table 6.6). A typical value for C1 is
given by 0.005 wt% - 0.1 moles/kg. This implies that Pe = 1 is
reached for a shear rate of
[6-10]
Since tipstreaming is only thought to be possible if Pe>>l, which
is the rapidly varying flow condition and tipstreaming was observed
for shear rates typically between 1 and 10, (see section 2.3.2 and
2.3.3), the above order of magnitude calculations confirm the
hypothesis that tipstreaming in simple shear flows is due to
interfacial tension gradients near the tip of the droplet.
Using this hypothesis it is also possible to predict the presence
of tipstreaming under different emulsification conditions. The
main differences are the smaller initial droplets, higher shear
rates and lower viscosities, resulting in higher diffusivities.
When a typical value ofID-5•lO·lO is taken for the diffusivity of
surfactants in a low viscosity liquids, say 50 mPas, the shear
rate that corresponds with Pe=l becomes 100 s-1. Since the applied
shear rates applied during emulsification in liquids of about 50
mPas are generally in excess of this shear rate, the flow
conditions can be termed rapidly varying. It is thus likely that
tipstreaming can also occur during emulsification in lower
viscosity liquids. When however higher surfactant concentrations
are present, the layer thickness d becomes much smaller than
described by r 00/C1, resulting in much higher shear rates required
for tipstreaming.
6.6 CONCLUSIONS
Tipstreaming can occur when interfacial tension gradients can
develop, resulting in low interfacial tension at the tips and a
higher tension elsewhere.
Tipstreaming will not occur at extremely low surface active
material levels, when the interfacial tension can not even be
lowered locally, nor at high levels where there is so much surface
active material present that the interfacial tension will be low
all over the droplet.
Sealing laws based on the above hypothesis have been formulated.
These allow prediction of the occurrence of tipstreaming under
different emulsification conditions.
6.7 REFERENCES
1. W. Bartok and S.G. Mason, Particle motion in sheared
suspensions. VIII singlets and doublets of fluid spheres, J.
Coll. Sci. 14, 13-26, {1959)
2. R.A. de Bruijn, Newtonian drop break-up in quasi steady simple
shear flows, Chapter 3 of this thesis.
3. R.A. de Bruijn, Sealing laws for the flow of emulsions, Chapter
2 of this thesis.
4. R.A. de Bruijn, Deformation and break-up of newtonian drops in
transient simple flows, Chapter 5 of this thesis.
5. B.J. Carroll and J. Lucassen, in Theory and practica of
emulsion technology, Chap 1 Smith, A.L. ed. Academie, London,
(1976)
225
226
6, R.P. Grace, Dispersion phenomena in high viscosity immiscible
fluid systems and application of statie mixers as dispersion
devices in such systems Ghem. Eng. Commun. 14, 225-277, (1982)
7. J.M. Rallison, The deformation of small viscous drops and
bubbles in shear flows Ann Rev Fluid Mech 16, 45-66, (1984)
8. F.D. Rumscheidt and S.G. Mason, Particle motions in sheared
suspensions XII deformation and burst of fluid drops in shear
and hyperbolic flow, J. Coll. Int. Sci. 16, 238-261, (1961)
9. J.D. Sherwood, Tipstreaming from slender drops in a non-linear
extensional flow, J. Fluid Mech. 144, 281-295, (1984)
10. P.G. Smith and T.G.M. van de Ven, Shear induced deformation and
rupture of suspended solid/liquid clusters, Coll. and Surf. 15,
191- 210, (1985)
ll. M.P. Srinivasan and P. Stroeve, Subdrop ejection from double
emulsion drops in shear flow, J. of Membrane Sci. 26,231-236,
(1986)
12. G.I. Taylor, The formation of emulsions in definable fields of
flow, Proc. Roy Soc. A 146, 501-523, (1934)
13. S. Torza, R.G. Cox and S.G. Mason, Particle motions in sheared
suspensions. XXVII transient and steady deformation and burst
of liquid drops, J. Coll. Int. Sci. 38, 395-411, (1972)
6.8 LIST OF SYMBOLS
~
B
C1
D
ID d
Fo
L
M
Nd
area of adsorbed molecule
width of droplet
material property in Langmuir adsorption
isotherm
drop deformation
diffusion coefficient
[ l
diffusional layer thickness [m]
Fourier number [-)
length of droplet [m]
molecular weight [kg]
number of surfactant molecules in drop [-]
LIST OF SYMBOLS (continued)
Pe
r
t
v
x
w
p
a
T
0
Avogadro number, 6.0231 1023
number of molecules required to cover
the undeformed droplet with a monolayer
Peclet nurnber
drop radius
time
characteristic drop deformation time
ramp time of shear rate profile
velocity
position
weight concentration
material constant in Langmuir adsorption
isotherm
shear rate
rate change of shear rate
dynamic viscosity
viscosity ratio, Àd/Àc
density
interfacial tension
time ratio, tdef/tramp
capillary nurnber
[ J [ - l [m]
[s]
[s]
[s]
[m·s-1]
[m]
[ - 1
[moles•m-2]
[m·s-1]
[m·s-2]
[Pa•s]
[ - ]
[kg·m-3]
[N•ml]
[ -1 [ - ]
227
228
SUMMARY
This thesis is part of a long term investigation aimed at modelling
the operation of emulsifying devices in the food industry. The
approach is to model on the one hand the flow and mixing in the
devices concerned and on the other hand to model drop breakup and
coalescence as local processes. The work in this thesis aims to
describe some aspects of the viscous breakup of drops in simple
shear flows that are not well understood, namely the breakup of
non-Newtonian drops, the deformation and breakup of drops in
transient shear flows and the origin of the tipstreaming phenomenon
observed in simple shear flow.
In chapter 2 the basic equations and sealing laws for the flow of
emulsions are presented for use in experimental and numerical
investigations. Fora number of situations the practical
limitations of application of sealing laws to the local processes
of drop deformation and break-up are considered. It is concluded
that sealing laws can be applied without great difficulties to
emulsions with constant interfacial tension. Sealing of emulsions
with surfactant adsorption is usually only possible in "slowly"
varying flows in which surfactant diffusion dominates convection,
resulting in an equilibrium surface tension all over the droplet,
or in dilute "rapidly" varying flows in which adsorption and
desorption processes may be neglected. Sealing of emulsions of non
Newtonian liquids is usually only possible when a homologous series
of liquids is available with the same type of constitutive equation
(e.g. shear thinning liquids and Boger liquids).
In Chapter 3 a Couette device, which was developed for experimental
investigation of drop break-up in simple shear flow, is described.
The Couette device operates on the principle of two counter
rotating cylinders. A stagnant layer is thus created between the
two cylinders which permits statie observations of droplets in
quasi steady and transient simple shear flows. This device has been
successfully tested by measurement of Newtonian drop breakup in
229
230
quasi simple shear flow and in the absence of surface active
materials. Reliable breakup criteria have been formulated in terms
of the critical capillary number, which only depends on the
viscosity ratio.
Chapter 4 describes the use of the Couette device to study the
breakup of non-Newtonian droplets in quasi steady simple shear
flows. Two particular types of non-Newtonian behaviour were
studied: shear thinning liquids with viscosities obeying the power
law equation but with negligible fluid elasticity and viscoelastic
liquids with substantial elasticity combined with a shear rate
independent viscosity. These particular types of non-Newtonian drop
phases were chosen in order to separate the eff ects of shear rate
dependent viscosities and fluid elasticity.
For shear thinning drops it is concluded that below a viscosity
ratio of 0.1 the drop breakup criteria can be described by the
breakup criteria for Newtonian drops, provided a modified viscosity
ratio is used to account for the internal shear rate in the
droplet, which is higher than the applied shear rate. For higher
viscosity ratios shear thinning drops are more difficult to break
up than Newtonian drops. This is probably due to the fact that the
internal shear rate then becomes considerably lower than the
applied shear rate.
For viscoelastic drops it is concluded (in contrast with certain
statements in the literature) that drop elasticity impedes breakup.
This effect increases with drop elasticity and is rnuch stronger for
viscosity ratios of order unity than for low viscosity ratios. The
deformation of viscoelastic drops prior to breakup was often
observed to be much larger than for Newtonian drops. It was
observed further that critical drop sizes exist, below which
breakup is irnpossible in quasi steady sirnple shear, irrespective of
the magnitude of the shear rate. This probably sterns frorn the more
rapid increase with shear rate of the elastic forces inside the
droplet than of the shear stresses exerted by the continuous
phase.
In chapter 5 of this thesis the deformation and breakup of
Newtonian droplets in transient simple shear flows is studied. This
investigation is mainly of numerical nature but is supported by
experimental work. A computer programme is developed which
calculates the evolution of Newtonian droplets in any transient
shear flow. The programma is based on the boundary integral method
by which the creeping flow equations inside and outside the droplet
are transformed into a form that only involves quantities at the
drop interface. This programma is used to calculate the shape of
droplets as a function of time with viscosity ratios ranging
between 0.5 and 5 and various shear rate profiles: step profiles,
triangular profiles and sinusoidal profiles. The computed results
correspond very well with experimental data up to drop deformations
of 0.5. For equiviscous drops it was observed that the critical
shear rate at which breakup occurs was the same for quasi steady
shear situations and step like shear rate profiles. For viscosity
ratios greater than 1 "overshoot" in the deformation as a function
of time bas been observed in the step profile experiments. When a
droplet is subject to a triangular shear rate profile with a
maximum shear rate which is larger than the critica! shear rate,
breakup will only occur if this triangular ramp stretches over a
long enough time. When a droplet suddenly undergoes a shear rate
which is much larger than the critical shear rate, it will breakup
into many fragments. The number of fragments has been determined
for two viscosity ratios as a function of the maximum shear rate
and the duration of the triangular ramp. When droplets are subject
to oscillatory flows it was observed that the time averaged
deformation corresponds well with the final deformation in a quasi
steady shear flow with a shear rate equal to the time averaged
shear rate of the oscillatory profile. Breakup in oscillatory flows
is possible when the time averaged shear rate is smaller than the
critical shear rate, provided that the period of the oscillation is
longer than the characteristic drop deformation time.
In the final chapter a special mode of drop breakup, tipstreaming,
is investigated. Tipstreaming is an experimentally observed mode of
231
232
drop breakup in which the droplet takes, upon increasing the shear
rate a sigmoidal shape and a stream of very small droplets is
ruptured off the tips of the drop. The investigations in this
chapter are aimed at unravelling the causes of this phenomenon by
use of both experimental and numerical techniques. It is concluded
that tipstreaming can only occur when interfacial tension gradients
(associated with surface active contaminants) can develop,
resulting in reduced interfacial tension at the tips. Tipstreaming
will not occur at extremely low surfant concentrations, nor at high
concentrations where the interfacial tension is low all over the
droplet.
SAMENVATTING
Dit proefschrift maakt deel uit van een langlopend onderzoek
gericht op de modellering van de werking van emulgeerapparatuur in
de levensmiddelenindustrie. Hierbij wordt enerzijds de stroming en
menging in de betreffende apparaten gemodelleerd en anderzijds
worden het opbreken en coalesceren van druppels als locale
processen gemodelleerd. Het werk in dit proefschrift heeft tot doel
enige aspecten van het visceuze opbreken van druppels in
enkelvoudige afschuifstromingen te besèhrijven, die nog niet goed
begrepen zijn, namelijk het opbreken van niet-Newtonse druppels, de
deformatie en het opbreken van druppels in tijdsafhankelijke
afschuifstromingen en de reden van het verschijnsel "tipstreaming"
dat optreedt in enkelvoudige afschuifstromingen.
In hoofdstuk 2 worden de basisvergelijkingen en schalingsregels
voor de stroming van emulsies gepresenteerd voor gebruik in
experimenteel en numeriek onderzoek. Voor een aantal situaties
worden de praktische begrenzingen voor het toepassen van schalings
regels op de locale processen van deformeren en opbreken van
druppels beschouwd. Er wordt geconcludeerd dat schalingsregels
zonder al te grote moeilijkheden toegepast kunnen worden op
emulsies met constante oppervlaktespanning. Schaling van emulsies
met adsorptie van surfactants is over het algemeen alleen mogelijk
in "langzaam" variërende stromingen, waarin diffusie van surfactant
domineert over convectie, resulterend in een evenwicht van de
oppervlaktespanning over de gehele druppel, of in verdunde "snel"
varierende stromingen, waarin adsorptie- en desorptie-processen
verwaarloosd kunnen worden. Schaling van emulsies van niet-Newtonse
vloeistoffen is over het algemeen alleen mogelijk wanneer een
homologe reeks vloeistoffen beschikbaar is met eenzelfde type
constitutieve vergelijking (b.v. afschuifsnelheid-verdunnende
vloeistoffen en Boger-vloeistoffen).
In hoofdstuk 3 wordt een Couette-apparaat beschreven dat ontwikkeld
is voor experimenteel onderzoek naar het opbreken van druppels in
233
234
enkelvoudige afschuifstromingen. Het Couette-apparaat werkt volgens
het principe van twee tegen elkaar indraaiende cilinders. Zo
ontstaat er een stilstaande laag vloeistof, waardoor het mogelijk
is druppels in quasi-stationaire en tijdsafhankelijke afschuif
stromingen stilstaand waar te nemen. Het apparaat is met succes
getest door het opbreekgedrag van Newtonse druppels te meten in
quasi-stationaire afschuifstromingen en in de afwezigheid van
oppervlakte-actieve materialen. Betrouwbare criteria voor opbreken
zijn geformuleerd in de vorm van een kritisch capillair getal dat
alleen afhankelijk is van de viscositeitsverhouding.
Hoofdstuk 4 beschrijft het gebruik van het Couette-apparaat om het
opbreek gedrag van niet-Newtonse druppels te bestuderen in quasi
stationaire enkelvoudige af~chuifstromingen. Twee bijzondere typen
niet Newtons gedrag zijn onderzocht: afschuifsnelheid-verdunnende
vloeistoffen, met viscositeiten die zich gedragen volgens het
machtwet-model maar met verwaarloosbare elasticiteit in de
vloeistof en viscoelastische vloeistoffen met een aanmerkelijke
elasticiteit in de vloeistof, gecombineerd met een viscositeit die
onafhankelijk is van de afschuifsnelheid. Deze bijzondere typen van
niet-Newtons gedrag zijn gekozen om onderscheid te kunnen maken
tussen effecten veroorzaakt door afschuif snelheid-afhankelijke
viscositeit en elasticiteit van de vloeistof. Voor afschuif
snelheid-verdunnende druppels wordt geconcludeerd dat beneden een
viscositeitsverhouding van 0.1 de criteria voor het opbreken van
druppels beschreven kunnen worden door die voor Newtonse druppels,
vooropgesteld dat een aangepaste viscositeits-verhouding gebruikt
wordt om de af schuifsnelheid in de druppel in rekening te brengen,
die hoger is dan de opgelegde afschuifsnelheid. Bij hogere
viscositeitsverhoudingen zijn afschuifsnelheid-verdunnende druppels
moeilijker op te breken dan Newtonse druppels. De oorzaak hiervan
ligt waarschijnlijk in het feit dat de afschuifsnelheid in de
druppel dan aanmerkelijk lager is dan de opgelegde afschuif
snelheid.
Voor viscoelastische druppels wordt geconcludeerd (in tegenstelling
tot enige uitspraken in de literatuur) dat druppelelasticiteit het
opbreken bemoeilijkt. Dit effect neemt toe met elasticiteit van de
druppel en is veel sterker voor viscositeitsverhoudingen van orde 1
dan voor kleine viscositeitsverhoudingen. De deformatie van
viscoelastische druppels vlak voor het moment van opbreken bleek
vaak veel groter te zijn dan voor Newtonse druppels. Verder is
waargenomen dat er kritische druppelgroottes bestaan, waar beneden
opbreken van druppels in quasi-stationaire enkelvoudige afschuif
stromingen onmogelijk is, onafhankelijk van de grootte van de
afschuifsnelheid. Dit is er waarschijnlijk de oorzaak van dat de
elastische spanningen in de druppel sneller toenemen met de
afschuifsnelheid dan de afschuifspanningen uitgeoefend door de
continue fase.
In hoofdstuk 5 van dit proefschrift wordt het deformatie- en
opbreek-gedrag van tijdsafhankelijke enkelvoudige afschuif
stromingen bestudeerd. Dit onderzoek heeft een overwegend numeriek
karakter, maar wordt ondersteund door experimenteel werk. Er is een
computer-programma ontwikkeld dat de ontwikkeling van de vorm van
Newtonse druppels in een willekeurige tijdsafhankelijke
afschuifstroming berekent. Dit programma is gebaseerd op de
oppervlakte-integraal-methode door welke de vergelijkingen voor de
kruipstroming in en om de druppel getransformeerd worden tot een
vorm die alleen grootheden op het druppel-oppervlak bevat. Dit
programma is gebruikt om de vorm van druppels als een functie van
de tijd te berekenen voor viscositeitsverhoudingen varierend tussen
0.5 en 5 en voor diverse profielen van de afschuifsnelheid: stap
profielen, driehoek-profielen en sinusvormige profielen. De
berekende resultaten komen erg goed overeen met de experimentele
data tot aan deformaties van 0.5. Voor equivisceuze druppels werd
waargenomen dat de kritische afschuif snelheid waarbij opbreken
plaats vond gelijk is voor quasi-stationaire en stap-profiel
experimenten. Voor viscositeitsverhoudingen groter dan 1 is
"overshoot" in de deformatie als functie van tijd waargenomen in de
stap-profiel"experimenten. Wanneer een druppel onderworpen is aan
een driehoeksvormig afschuifsnelheidsprofiel met een maximale
afschuifsnelheid welke groter is dan de kritische afschuifsnelheid,
dan zal opbreken alleen plaatsvinden wanneer dit driehoeksvormig 235
236
profiel voldoende lang aangehouden wordt. Wanneer een druppel
plotseling een afschuifsnelheid ondergaat die veel groter is dan de
kritische afschuifsnelheid, zal deze in veel fragmenten opbreken.
Het aantal fragmenten is voor een tweetal viscositeitsverhoudingen
bepaald als functie van de maximale afschuifsnelheid en de
tijdsduur van het driehoeksvormige afschuifsnelheidsprofiel.
Wanneer druppels onderworpen zijn aan een oscillerende stroming,
wordt waargenomen dat de tijdsgemiddelde deformatie goed overeen
komt met de eind-deformatie in een quasi-stationaire afschuifstro
ming met een afschuifsnelheid gelijk aan de tijdsgemiddelde af
schuifsnelheid van het oscillerende profiel. Opbreken in oscil
lerende stromingen is mogelijk wanneer de tijdsgemiddelde afschuif
snelheid kleiner is dan de kritische afschuifsnelheid, vooropge
steld dat de periode van de oscillatie langer is dan de druppel
deformatietijd.
In het laatste hoofdstuk van dit proefschrift wordt een bijzondere
vorm van opbreken, "tipstreamlng", onderzocht. Tipstreaming is een
vorm van druppel opbreken, die experimenteel waargenomen is.
Hierbij neemt de druppel, als deze vervormt, een sigmoidale vorm
aan en breekt een stroom zeer kleine druppels aan de uiteinden van
de druppel af. Het onderzoek in dit hoofdstuk is erop gericht
de oorzaken van dit verschijnsel op te sporen met gebruikmaking
van zowel experimentele als numerieke methoden. Er wordt
geconcludeerd dat tipstreaming alleen plaats kan vinden wanneer
oppervlaktespanningsgradienten (verbonden met oppervlakteactieve
verontreiniging) ontstaan, resulterend in verlaagde oppervlakte
spanning aan de uiteinden. Tipstreaming zal niet plaats vinden bij
zeer lage concentraties van surfactant en ook niet bij hoge
concentraties, waarbij de oppervlaktespanning over de gehele
druppel laag is.
ACKNOWLEDGEMENTS
The work described in this thesis was carried out in the period
1984-1988 while I was employed in the section surface chemistry and
rheology of the Unilever Research Laboratory in Vlaardingen (NL).
Part of this work was subsidlsed by the European Community under
the BRITE project RIIB.0085.UK(H) on computational fluid dynarnics,
narned Development of experirnentally tested 3-D computer codes for
fundarnental design of process equipment involving non-Newtonian
multi-phase turbulent fields. This was a joint project which
involved the New Science group of ICI, based in Runcorn (UK), the
Department of Mechanical Engineering of Imperial College, London
(UK) and Unilever Research in Vlaardingen (NL). I would like to
acknowledge the management of Unilever Research for perrnission to
use this part of my research for my thesis. Thanks are also due for
the support and facilities I received during the completion of this
thesis.
I would also like to thank a number of people without whom this
could not have been cornpleted. First of all I would like to thank
my promotor, Allan Chesters, who initiated this work when he was
section manager of the section surface chemistry and rheology of
the Unilever Research Laboratory in Vlaardingen. Thanks for all
the valuable suggestions on the direction of this work, for all the
things I learned from you and for the pleasant cooperation. Thanks
are also due to Nico Hoekstra for his contribution in the
development of the Couette device and his continuous, friendly and
helpful support on the experimental parts of this thesis. Further I
would like to thank Reinier Baker, Ab Boon and Quint van Voorst
Vader for their valuable contributions to various parts of this
thesis. Finally I would like to thank Wim Tjaberinga for his
competent contribution to the investigation into transient drop
deformation and break-up, first as a student at Delft University of
Technology and later as a colleague at the Unilever Research
Laboratory in Vlaardingen.
237
238
CURRICULUM VITAE
De schrijver van dit proefschrift is op 1 augustus 1960 te
Heerhugowaard geboren. De middelbare school heeft hij deels aan het
Christelijk Lyceum te Alphen aan de Rijn en deels aan de
Christelijke Scholengemeenschap te Aalten gevolgd. Na het behalen
van het VWO-diploma begon hij in 1978 met de studie Technische
Natuurkunde aan de Universiteit Twente. In 1983 studeerde hij af
als natuurkundig ingenieur bij prof. dr. P.F. van der Wallen
Mijnlieff. Het afstudeerwerk, betreffende het lineair
viscoelastisch gedrag van vesicles, werd uitgevoerd onder
begeleiding van dr. J. Mellema. Van september 1983 tot maart 1988
was de auteur werkzaam bij het Unilever Research Laboratorium in
Vlaardingen in de sectie Oppervlakte Chemie en Reologie. Dit
proefschrift geeft een verslag van een deel van het onderzoek dat
de auteur tijdens zijn werk aldaar heeft verricht. Dit onderzoek
werd begeleid door prof. dr. A.K. Chesters, eerst als sectieleider
en later als hoogleraar aan de Technische Universiteit Eindhoven.
Vanaf maart 1988 is de auteur werkzaam bij het Unilever Research
Laboratory Golworth House te Sharnbrook, UK.
239
240
APPENDIX A: VISCOEIASTIC DROP BREAK-UP IN LITERATURE
TABLE A.l Model liquids used by Gauthier et al 1971
Drop pbases
No. Description K n m
Gl Poly propylene glycol 0.024 1
G2 Carbopol in PPG 2.29 0.71 ? ?
G3 Carbopol in PPG 0.11 0.97 ? ?
G4 Water 0.001 1
GS 1.5% PAA in water ? ? ? ?
G6 4.0% PAA in water ? ? ? ?
Continuous Phase
No. Description Viscosity [Pa.s]
G7 Silicone oil 5.31
241
242
TABLE A.2 Drop break-up experiments performed by Gauthier et al 1971
Drop Cont. a
phase phase [m N m-1]
Gl G7 8.6
G2 G7 8.6
G3 G7 8.6
G4 G7 21.2
G5 G7 19.4
G6 G7 16.5
R
[mm]
0.28 1.06
0.42-1.16
0.31-0.80
0.43-1.14
0.33-0.83
0.54-0.83
0.96-4.11
0.87-2.48
1.02-2.81
1.79-5.18
3.04-8.20
N.o.
À
[ - l
0.004
0.41-0.56
0.021
0.0002
0.22
2.84
0
?
?
0
?
?
flcrit
[ - J
0.59±0.12
0.62±0.03
0.57±0.07
0.54±0.05
0.71±0.03
TABLE A.3 Model liquids used by Tavgac
Drop phases
No. Description K n m
Tl 0.1% PAA in 20% glyc 0.082 0.65 0.021 1.16
T2 0.4% PAA in 20% glyc 1.4 0.43 3.5 0.82
T3 0.75% PAA in 20% glyc 3.7 0.36 13 0.61
T4 1.0% PAA in 20% glyc 5.6 0.33 20 0.67
T5 2.0% PAA in 20% glyc 3.3 0.25 190 0 49
T6 2.6% PAA in 20% glyc 56 0.3 330 0.61
T7 1.0% HEC in water 6.5 0.79 0.24 1. 62
T8 1. 5% Kelzan in water 14 0.19 16 0.47
Continuous phase
No. Description Viscosity [Pas]
T9 Silicone oil 30.8
243
TABLE A.4 Drop break-up experiments performed by Tavgac
Drop Cont. (}' R .y À SR 0 phase phase [mN/m] [mm] [l/s] [ - J [ - J [ - ]
Tl T9 23.5 0.63 58.00 6.4E-04 2.03 48.00 0.66 57.00 6.5E-04 2.01 49.00 1.01 34.00 7.7E-04 1.55 45.00 1.24 24.00 8.8E-04 1.30 39.00 1.49 21.00 9.2E-04 1.21 41.00 2.11 13.00 l. lE-03 0.95 36.00
T2 T9 20 1.00 4.50 1. 9E-02 4.49 7.45 0.94 4.20 2.0E-02 4.38 6.54 0.89 4.00 2.lE-02 4.29 5.90 1.10 3.80 2.lE-02 4.21 6.92 0.84 4.00 2.lE-02 4.29 5.56 1.10 3.30 2.3E-02 3.98 6.01 1. 30 3.20 2.3E-02 3.94 6.89 0.89 4.00 2.lE-02 4.29 5.90 1.10 2.40 2.8E-02 3.52 4.37 1.20 2.10 3.0E-02 3.34 4.17 0.92 2.00 3.lE-02 3.28 3.05
T3 T9 18.6 0.66 3.51 5.4E-02 4.81 3.82 0. 77 1.67 8.7E-02 3.99 2.13 1.03 0.91 1 3E-Ol 3.43 1.55 0.49 3.11 5.8E-02 4.67 2.54 1.23 0.74 1. 5E-Ol 3.26 1.51 0. 71 1. 72 8.5E-02 4.03 2.01 0.49 5.45 4.lE-02 5.37 4.42 0.39 11.54 2.5E-02 6.48 7.53 0.72 1.65 8.7E-02 3.98 1. 97 0.43 7.95 3.2E-02 5.90 5.63 0.51 4.81 4.4E-02 5.20 4.05 1.40 0.59 1. 7E-Ol 3.08 1. 36 0.65 3.81 5.lE-02 4.91 4.13 0.42 8.69 3.0E-02 6.03 6.10 1.19 0.76 l.4E-Ol 3.28 1.49 0.82 1.29 l.OE-01 3.75 1. 76 0.58 2.36 6.9E-02 4.36 2.29 0.62 2.84 6.2E-02 4.56 2.92 0.42 6.62 3.6E-02 5.64 4.58
T4 T9 14.9 0.40 7.00 4.9E-02 6.92 5.80 0.41 6.20 5.4E-02 6.64 5.20 0.42 5.50 5.8E-02 6.38 4.80 0.43 4.70 6.4E-02 6.04 4.20 0.47 3.40 8.0E-02 5.41 3.30 0.47 3.00 8.7E-02 5.19 2.90 0.48 1.90 1. 2E-Ol 4.44 1.90 0.48 1. 80 1.2E-Ol 4.36 1. 80 0.55 1.50 l.4E-Ol 4.10 1. 70
244
TABLE A.4 Continued
Drop Cant. (J R 'Î' À SR n phase phase [mN/m] [mm] [l/s] [ - 1 [ -1 [ - 1
T4 T9 14.9 0.82 1.00 1. 8E-Ol 3.57 1.70 1.00 0.63 2.5E-Ol 3.05 1. 30 1.27 0.42 3.3E-01 2.66 1.10
T5 T9 9.1 0.37 4.10 3.7E-02 80.78 5.10 0.38 3.50 4.2E-02 77. 77 4.50 0.45 2.90 4.8E-02 74.34 4.40 0.62 2.20 5.9E-02 69.57 4.60 0.75 1.50 7.9E-02 63.46 3.80 0.86 1.20 9.3E-02 60.15 3.50
T6 T9 6.6 0.29 6.40 5.0E-01 10.48 8.70 0.34 5.10 5.8E-01 9. 77 8.20 0.56 2.70 9.lE-01 8.02 7.00 0.56 2.60 9.3E-01 7.92 6.80 0.67 2.10 l.lE+OO 7.42 6.60
T7 T9 38.1 0.28 3.25 1. 6E-Ol 0.10 0.75 1.28 0.55 2.4E-01 0.02 0.56 1.04 0.58 2.4E-Ol 0.02 0.49 0.81 0.71 2.3E-01 0.03 0.46 0.41 1.43 2.0E-01 0.05 0.47 0.65 0.89 2.2E-01 0.03 0.47 0.53 1.12 2.lE-01 0.04 0.48 1.23 0.59 2.4E-01 0.02 0.58 0.35 1.85 1. 9E-01 0.06 0.52 0.67 0.78 2.2E-Ol 0.03 0.42 0.99 0.62 2.3E-Ol 0.02 0.50 0.52 1.09 2.lE-01 0.04 0.45 0.30 2.27 1. 8E-Ol 0.07 0.55 0.28 2.86 1. 7E-01 0.09 0.64 0.47 1.31 2.0E-01 0.05 0.50 0.38 1. 78 1. 9E-Ol 0.06 0.54 1.37 0.52 2.4E-Ol 0.02 0.58 1. 95 0.38 2.6E-Ol 0.02 0.60
T8 T9 20.9 0.41 9.81 7.2E-02 2.17 6.58 1.05 1. 69 3.0E-01 1. 32 2.94 0.79 1. 89 2.7E-01 1. 37 2.47 0.75 2.22 2.4E-Ol 1.43 2.75 1.16 1.41 3.4E-01 1.26 2.69 0.45 5.43 1. 2E-Ol 1.83 4.04 0.41 8.21 8.3E-02 2.06 5.63 0.50 4.74 1. 3E-01 1. 77 3.96 0.50 4.61 1. 3E-Ol 1. 75 3.82 0. 77 7.62 8.8E-02 2.02 9.68 0.35 8.39 8.lE-02 2.07 4.86
245
246
TABLE A.5 Critica! drop size predictions by Tavgac
drop
phase
Tl
T2
T3
T4
TS
T6
T7
T8
3.5
0.37
0.094
0.088
0.022
0.012
0.24
0.05
Àl
[s]
0.08
3.9
7.6
8.5
26
41
0.33
6.3
a
[mNm]
23.5
20
18.6
14.9
9.1
6.6
38.l
20.9
f'/c
[Pas]
30.8
30.8
30.8
30.8
30.8
30.8
30.8
30.8
Rcrit
[mm]
0.21
0.94
0.43
0.36
0.17
0.11
0.10
0.21
TABLE A.6 Model liquids used by Elmendorp
Drop phases
No. Descript ion
El PAA in water
(1% Separan AP 273)
E2 PAA in water
(l.5% Separan AP 30)
E3 PAA in water
0.5% Separan AP 273)
E4 PAA in water
(0.75% Separan AP 30)
Gontinuous phase
No. Description
ES Silicone oil
K n
7.9 0.3
7.9 0.4
1.2 0.4
1.4 0.5
Viscosity
[Pa.s]
5.5
m
5.4 0.8
1. 7 1.5
1.0 1.1
0.6 1.0
247
248
TABLE A.7 Drop break-up experirnents perforrned by Elrnendorp
Drop Cont.
phase phase
El ES
E2 ES
E3 ES
E4 ES
22
22
22
22
R
[mm)
1. 7
1.0
0.7
1. s 1.0
0.7
1.4
1.0
0.7
0.6
0.5
1.4
0.9
0.7
0.4
1. 9
3.5
S.l
1.5
1. 9
2.9
1. 3
1. 9
2.5
3.1
3.6
1.0
1. 5
2.0
3.1
À
[ - J [ - )
0.78 0.99 0.81
0.Sl 1.3 0.88
0.36 1. 7 0.89
0.96 0.33 0.54
0.78 0.47 0.48
0.64 0.60 O.Sl
0.18 1.0 0.46
0.14 1. 3 0.48
0.11 1. 6 0.44
0.096 2.0 0.47
0.084 2.3 0.45
0.22 0.44 0.35
0.18 0.54 0.34
0.16 0.63 0.35
0.13 0. 77 0.31
TABLE A.8 Model liquids used by Prabodh and Stroeve
Drop phase
No.
PSl
Description
PAA in corn syrup
(0.0275% Separan 30)
Continuous phases
No. Description
PS2 Silicone oil
PS3 Silicone oil
PS4 Silicone oil
PS5 Silicone oil
PS6 Indopol H25
PS7 Silicone oil
k n
2.7 1.0
Viscosity
[Pa.s]
57.7
25.5
12.0
6.2
2.3
1. 7
m
4.1 1.06
249
TABLE A.9 Drop break-up experiments performed by Prabodh and Stroeve
Drop Cont. a R 1' À SR flcrit phase phase [mNm-1] [µm] [s-1] [ - J [ - l [ - J
PSl PS2 31.8 37 24 0.47 1. 8 1.6 37 12 1.8 0.8 30 104 1. 9 5.6 25 85 2.0 3.8 24 146 1.6 6.4 22 104 1. 9 4.1 21 265 1. 3 9.9 21 43 1. 9 1.8 21 170 1.5 6.5 21 130 1. 7 5.0 21 93 2.0 3.5 21 75 2.0 2.6 21 58 1. 9 1.8 21 40 1. 9 1. 5 20 67 2.0 2.4 20 49 1. 9 1. 8 19 265 1.3 9.0 19 104 1.9 3.6 19 104 1. 9 3.5 18 61 1. 9 2.0 17 186 1. 5 5.7 16 104 1. 9 3.0 15 104 1. 9 2.8 15 61 1. 9 1. 7 15 43 1. 9 1. 8 14 104 1. 9 2.7 14 43 1. 9 1. 7 11 40 1.9 0.8
8 40 1. 9 0.6 PSl PS3 31. 7 34 37 0.11 1. 9 1.0
31 58 1. 9 1.2 31 40 1. 9 1.0 28 265 1. 3 5.9 27 46 1. 9 1.0 27 210 1.4 4.5 27 170 0.11 1. 5 3.7 27 130 1. 7 2.8 27 75 2.0 1.5 26 104 1. 9 2.2 26 67 2.0 1.8 26 93 2.0 1.9 24 104 1. 9 2.2 23 265 1.3 1.0
250
TABLE A.9 Continued
Drop Cont. a R ,y À SR Ocrit phase phase [mNm-1] [,um] [s-1] [ - J [ - l [ - l
PSl PS4 35.8 73 37 0.23 1. 9 0.9 71 55 1. 9 1. 3 68 61 1. 9 1.4 60 40 1.9 0.8 57 104 1.9 2.0 57 75 2.0 1.3 56 58 1. 9 0.9 51 93 1. 9 1.6 50 130 1. 7 2.2 47 170 1.5 2.7 46 156 1.6 2.4 44 217 1.4 3.2 43 210 1.5 3.0
PSl PS5 17 .2 58 58 0.44 1. 9 1.0 45 75 2.0 1.1 45 61 1.9 1.0 40 61 1. 9 0.9 39 93 1. 9 1. 3 37 104 1. 9 1.4 35 104 1. 9 1.3 30 128 1. 7 1.4 30 130 1. 7 1.4 28 128 1. 7 1. 3 27 104 1.9 1.0 24 170 1. 5 1. 5 22 210 1. 5 1. 7 22 180 1. 5 1. 5 20 217 0.44 1.4 1.6 17 259 1.3 1.6
PSl PS6 33.3 142 104 1. 2 1. 9 1.0 77 104 1. 9 0.6 75 150 1. 6 0.8 64 140 1.6 0.6 58 147 1. 6 0.6 57 217 1.4 0.9 54 104 1. 9 0.4 44 311 1. 2 0.9 39 217 1.4 0.6 34 315 1.2 0.8 27 320 1. 2 0.6
PSl PS? 35.5 107 220 1.6 1.4 1. 2 88 223 1.4 1.0 70 210 1. 5 0.7 70 223 1.4 0.8 49 323 1. 2 0.8
251
252
TABLE A.10 Model liquids used by Mirmohammad-Sadeghi
Drop phases
No. Description
MSl eorn syrup (CS)
MS2 0.001% PAA in es
MS3 0.002% PAA in es
MS4 0.005% PAA in es
MSS 0.01% PAA in es
MS6 0.02% PAA in es
Continuous phases
No.
MS7
MS8
Description
Silicone oil
Silicone oil
K
3.4
9.2
9.7
9.7
11.2
10.3
ri [Pa.s]
12.1
12.9
n
1
0.96
0.98
0.89
0.93
0.89
m
0.48 0.97
1.0 1.06
0.89 1.27
5.1 0.97
3.3 1. 26
TABLE A.11 Drop break-up experiments performed by Mirmo-hammad-Sadeghi
Drop Cont. a R 'Y >. SR Ocrit phase phase [mNm-1] [µm] [s-1] [ - l [ -1 [ - l
MSl MS8 50 154 14 0.26 0.56 127 23 0.75 126 18 0.57 123 22 0.70 107 24 0.66
91 29 0.68 85 31 0.68 74 39 0.75 68 34 0.59 56 47 0.68
MS2 MS7 33 109 17 0.67 0.05 0.68 63 25 0.05 0.58 59 27 0.05 0.58 47 36 0.05 0.62 30 62 0.05 0.68 29 64 0.05 0.67
MS3 MS7 33 53 11 0. 77 0.1 0.21 46 20 0.1 0.34 35 29 0.1 0.37 34 46 0.1 0.57 30 57 0.1 0.62 30 57 0.1 0.63 28 76 0.1 0.78
MS4 MS7 33 75 9 0.55 0.2 0.24 49 23 0.3 0.41 38 29 0.3 0.41 31 52 0.4 0.59 28 66 0.5 0.68 27 71 0.5 0.69 22 80 0.5 0.65
MS5 MS7 33.5 54 9 0.73 0.5 0.18 52 10 0.5 0.19 48 17 0.5 0.29 43 37 0.5 0.57 43 30 0.5 0.46
253
TABLE A.11 Continued
Drop Cont. a
phase phase [mNm-1]
MS5 MS7 33.5
MS6 MS7 33.5
254
R
[µm]
41
41
41
39
38
38
35
32
23
23
136
113
91
82
76
68
67
63
61
42
À SR Ücrit
[ - l [ - J [ - l
36 0.73 0.5 0.57
31 0.5 0.46
30 0.5 0.44
39 0.5 0.55
36 0.5 0.49
34 0.5 0.47
44 0.5 0.55
41 0.5 0.48
72 0.5 0.60
72 0.5 0.60
6 0.61 0.6 0.31
8 0.7 0.34
15 0.9 0.48
19 0.9 0.55
21 1.0 0.58
27 1.1 0.66
29 1.1 0.70
22 1. 0 0.50
26 1.1 0.57
52 1.4 0.79
APPENDIX B DROP BREAK-UP EXPERIMENTS
TABLE B.l Viscoelastic drop break-up.
Drop Cont. " R .y ), 6& 0 crit phase phase [mNm-1 J [mmJ (•-ll [-J r-1 r-1
VE,D,l VE.C.1 29 0.648 23. 75 0 .13 0.48
0.512 32. 93 0 .13 0.52
VE.D.1 VE.C.2 29 0.461 13.12 0.020 1.3
0.357 16. 11 0.020 1.2
0.281 18.05 0.020 1.1
0.217 23.35 0.020 1.1
0.173 31. 70 0.020 1.1
VE.D.1 VE.C.3 29 0. 533 8,33 0.011 1. 7
0.366 12.08 0 '011 1. 7
0.295 15.90 0.011 1.8
0.227 21. 15 0.011 1.9
VE.D.l VE.C.5 29 0.402 8.41 0.0031 - 4.4
0.322 9.09 0.0031 - 4.2
0.251 11.90 0.0031 - 3.9
VE.D.2 VE.C.l 29 0.59 21 0.20 0.23 0.39
0.47 31 0.20 0.28 0.45
0.37 45 0.20 0.33 0.51
0,29 58 0.20 0,37 0 .53
0.22 51 0.20 0.35 0 .20
VE.D.2 VE.C.2 29 0. 50 9. 87 0.030 0.16 l. 0
0. 40 12.32 0.030 0.18 1,0
0.32 15. 73 0.030 0.20 1.0
0.24 20.34 0 .030 0.23 1.0
0.18 30.62 0 .030 0.28 1.1
0, 14 53. 79 0.030 0.36 1.6
VE.D.2 VE.C.3 29 0.53 4.11 0.016 0.11 0. 85
0.40 6. 05 0.016 0.13 0.95
0.29 12.22 0.016 0.18 l.4
0.22 20.09 0. 016 0.23 1. 7
0.16 28.77 0 .016 0.27 1.8
VE.0.2 VE.C. 5 29 0,48 4. 77 0. 0047 0.12 3.0
0.37 7 .12 0.0047 0.14 3.5
0.26 9.80 0.0047 0.16 3 .4
0,20 12.59 0.0047 0.18 3 .3
0' 15 15. 87 0.0047 0,20 3.1
VE.0.2 VE.C.6 29 0.625 4.02 0.0042 0.11 3. 7
0 .470 4.95 0.0042 0.12 3.5
0.381 5.89 0.0042 0.13 3.2
0.289 8. 74 0.0042 0.15 3.8
0.208 10 .27 0.0042 0.17 3 .2
0 .154 14. 63 0,0042 0.20 3. 4
0 .108 18.99 0.0042 0.22 3.1
255
TABLE B.l (Continued)
Drop Cont. 17 R 7 À 8R °'crit. phase phase CmNm- 11 (mm] [s-11 (-1 c-1 [-]
VE,D.3 VE.C.1 29 0. 476 >64 0.54 2.8 >LO
0.609 >64 0.54 2.8 >1.2
0.428 >64 0.54 2.8 >0.9
VE.D.3 VE.C.2 29 0.515 8.16 0.043 0.77 0.87
0.395 9.90 0.043 0.86 0.81
0.308 13.60 0.042 1.1 0. 87
0.236 22.33 0,042 1.4 1.1
0.491 8.35 0.043 0. 78 0.85
0.383 11.24 0.043 0.94 0.89
0.293 14.62 0.042 1.1 0.89
0.26 26.09 0.042 1.6 1.2
0.172 39.93 0.040 2.1 1. 4
VE.D.3 VE.C.3 29 0.686 4.34 0.024 0.52 1.2
0 .527 5.84 0.024 0.62 1.2
o. 414 8.12 0.023 o. 76 1. 3
0.316 11.46 0.023 0.95 1. 4
0.217 16.31 0.022 l.2 1.4
0.166 22.72 0.022 1.5 1.5
VE.D.3 VE.C.5 29 0.536 3.58 0.0071 0.46 2.5
0.407 4. 71 0. 0071 0.55 2.5
0.312 6.65 o. 0068 0.68 2.7
0.233 S.86 0. 0068 0.81 2.7
VE.D.3 VE.C.6 29 0.678 2. 76 0.0065 o. 39 2.8
0.536 3.84 0.0063 0.48 3 .1
0.407 5.37 0. 0063 0.59 3.3
0.293 6.63 0. 0060 0.67 2.9
0.223 8.81 0.0060 0.80 2.9
0.169 11.92 0.0060 0.97 3.0
0.136 13.99 0.0058 1.1 2.8
VE.D.4 VE.C.2 28 0.678 5.27 0.11 1.8 0. 77
o. 434 11.99 0.095 2.8 1.1
0.308 19. 30 0.088 3.7 1.3
VE.D.4 VE.C.3 28 0.559 3.86 0 .060 1. 5 0.88
0.434 5.07 0.058 1. 8 0.89
0.342 6.91 0.055 1. 9 0.96
0.259 13.96 0.049 3.1 l. 5
0.199 18.26 o. 047 3.6 1. 5
0.628 3.54 0. 061 1. 4 0.90
0. 464 5.62 0.056 1. 9 l.1
0. 316 8. 43 0.053 2.3 1.1
0.226 17 .15 0. 048 3.5 1. 6
0.176 20.45 0 .046 3.8 1.5
256
TABLE B.l (Continued)
Drop Cont. " R ~ À ~ 0crit phase phase (mNm- 1 J (lllll) (s-11 (-] (-] [-)
VE.D.4 VE.C.5 28 0.609 2.34 0.019 1.1 1.9
0.431 3.41 0,018 1. 4 2.0
0.310 5.02 0.017 1. 7 2.1
0.211 8.10 0.016 2.3 2.3
0.157 13. 75 0.015 3.1 2.9
VE.D.4 VE.C.6 28 0.452 2.84 0.017 1. 3 2.0
0.354 3.90 0.016 1.5 2.13
0.261 5,33 0.015 1.8 2.2
0.190 7 .30 0.014 2.2 2.1
0.139 12.53 0 .013 2.9 2.7
VE.D.5 VE.C.2 27 0.640 >16 0.26 8.8 >2.3
0.455 >16 0.26 8.8 >1.6
VE.D.5 VE.C.3 27 0.613 3. 70 0.23 4.0 0.95
0.448 5.79 0.20 5.1 1.1
0.398 6.10 0.19 5.2 1.0
0.446 1.81 0.088 2.7 1.1
0.256 3.60 0.070 3.9 1.3
0,193 >9 0.051 6.3 >2.4
0,613 1.23 0.089 2.2 1.2
0.298 1.99 0.075 2.8 0,95
0.218 3.92 0.059 4 .l 1. 4
0.166 >14 0.039 7 .9 >3.5
0.253 >19 0.13 9.6 >2,0
VE.D.5 VE.C.5 27 0.582 1. 72 0.090 2.6 1. 4
0.352 2.09 0.084 2.9 1. 0
0.211 3.20 0,072 3. 7 0. 95
0. 704 0.72 0.12 1.6 0.72
0.829 0.72 0.12 1.6 0.84
VE.D.6 VE.C.2 31 0.536 6.90 0.048 0.71
0.431 8.67 0.048 0.72
0.339 11.06 0.048 0. 73
0.262 13.92 0.048 0.71
0.205 17.26 0.048 0.68
VE.D.6 VE,C,3 31 0.548 6.18 0.026 l. 2
0. 383 9.32 0.026 1. 3
0.261 12. 31 0.026 1.2
VE.D.6 VE.C.5 31 o. 383 4. 75 0.0076 - 2.2
0.295 6. 47 0.0076 - 2.3
0.215 7. 75 0.0076 - 2.1
VE.D.6 VE.C.7 31 0.655 2.59 0. 0045 - 3.5
0.429 3.21 0. 0045 - 2.1
0.322 3.94 0.0045 - 2.6
0.190 6.54 0.0045 - 2.6
257
TABLE B.l (Continued)
Drop Cant. (f R ; 8R ncrit phase phaae [rnllm-l] [lllll] [a-l] [-] [-] [-]
VE.D.7 VE.C.2 31 0.569 6.21 0.072 1.1 0.68
0.400 8.69 0.068 1.3 0. 67
0.312 18. 76 0,082 1.1
0.253 >41.65 o. 057 3.4 >2.0
VE.D.7 VE.C.3 31 0.621 3.98 0. 040 0.82 0.9
0.455 5.37 0.038 0.98 0.9
0 .343 8.04 0.036 1. 3 1.0
0 .238 14 .91 0.033 l. 8 1.2
0.182 20.20 0.033 2.2 1.3
VE.D.7 VE.C.5 31 0.443 3 .67 0.012 o. 78 2.0
0.301 4 .96 0.012 0.93 1.8
0.215 7 .30 0.011 1.2 l.9
0.148 9.93 0.010 1. 4 1.8
0.042 31.59 0.0092 2.9 1.6
0.449 3.25 0.012 0.73 1.8
0.355 5.04 0.012 0.95 2.2
0.241 6.52 0.011 1.1 l. g
0.175 8.05 0.011 l.3 1.3
0.137 10.01 o. 010 1. 4 1. 7
0.175 17.82 0.069 3.2 1.1
0.69 1. 94 0.020 0.94 1. 7
0.50 2. 76 0.020 1.1 1. 7
0.36 3.99 0.020 1.4 1. 8
0.25 4.93 0.020 1.6 1.5
0.20 6.69 0.020 1.9 1.6
0.15 8.97 0.020 2.2 1. 7
0.13 11.09 0.020 2.5 1.8
VE.D.8 VE.C.2 31 0.551 s. 79 0.13 1. 7 0.62
0.440 9. 74 0.13 2.3 0.83
0.328 15.52 0 .13 3.0 0.99
0.262 16. 47 0 .13 3.1 0.84
0.202 >54 0.13 5.6 >2.l
VE.D.8 VE.C.3 31 0.640 3.45 0.069 1.3 0 .81
o. 443 5.31 0.069 1.6 0.86
0.299 6.89 0.069 1. 9 0. 75
0.230 9.66 0.069 2.3 0.81
VE.D.8 VE.C.6 31 0,694 1. 76 0.018 0.89 l. 7
0.503 2.87 0. 018 1.2 2.0
0 .335 4.34 0.018 1. 5 2.0
0.229 5.22 0.018 l.6 1. 7
0.172 7 .10 0.018 l. 9 1. 7
VE.D.9 VE.C.2 30 0.622 4.99 0.22 3.5 0.62
0.482 6.82 0.21 4.2 0.66
0.347 14.11 0.18 6. 7 0.98
0.266 >17 .43 0 .18 7 .6 >0.93
0.202 >25.95 0.17 9. 8 >1.1
0.461 >14 0.18 6.6 >l.3
258
TABLE B.1 (Continued)
Drop Cont, " R 7 ~ ~ 0cri~ phase phase [mNm- 11 [nm] [$-1] [-] [-] [-]
VE.D.9 VE.C.3 30 o. 554 4. 84 0.11 3.4 1. 0
0. 417 6.42 0.11 4 .1 1.0
VE.D.9 VE,C.3 30 0. 343 8.46 0.10 4. 8 1.1
VE.D.9 VE.C.5 30 0.584 1. 74 0.040 l.B 1.3
0.335 3.38 o. 036 2.7 1.4
0.194 11.19 0.030 5.6 2.8
VE.D.9 VE.C.6 30 0.586 1.58 0.036 1. 7 1. 3
0.370 2.63 0.033 2.3 1.4
0.232 8.89 0.027 5.0 3.0
VE.D.10 VE.C.3 29 0.555 10. 39 0.21 10.5 2.3
VE.D.10 VE.C.5 29 0. 781 0. 74 0, 16 2.5 0. 76
0. 563 1.55 0.12 3. 7 1.2
0. 374 2.26 0.11 4. 6 l. l
0.267 2.95 0.10 5,3 1.0
0.163 >11 0.063 10.4 >2.2
VE.D.10 VE.G.6 29 0.686 1. 05 0.13 3.0 l.1
0. 390 2.19 0.096 4. 5 1.3
0.226 5.39 0.069 7. 4 l.8
0.167 >19 0. 044 14.6 >4. 7
VE.D.11 VE.C.l 33 0.464 36. 57 0.87 0 .46
0. 367 48.11 0.87 0. 48
VE.D.11 VE.C.2 33 0. 628 4.67 0.13 0.53
0.506 6.17 0 .13 0.57
0.386 7 .36 0 .13 0.51
0 .303 9.27 0.13 0, 51
0' 235 11.95 0.13 0. 51
0 .185 14. 78 0.13 0.50
VE.D.11 VE.C.2 33 0. 708 2.93 0.069 0.71
0.530 3. 79 0.069 0.69
0.412 4. 97 0.069 0. 70
o. 310 6. 95 0.069 0. 74
0.220 9.18 0.069 0. 70
VE.D.ll VE.C.5 33 0.563 2.19 0.020 1.4
0.366 2.80 0.020 1.2
0.285 3. 87 0.020 1. 3
o. 479 4.28 0.11 0.22 0. 71
0. 347 5.52 0.11 0 .24 0.66
0 .245 8.20 0.11 0.27 0.69
0.178 10 .10 0.11 0.29 0.62
VE.D.11 VE.C,5 33 0.209 5. 02 0.020 1.2
VE.0.11 VE.C.7 33 0. 355 1.45 0,012 1. 5
0.280 1.51 0.012 l. 5
0.175 1.34 0.012 1.3
0.140 1.36 0.012 1.4
259
TAB LE B.1 (Continued)
Drop Cent. q R :., 3R Ocrit phase phaSEt [mNm-1] [nm] [s-1] [-] [-] [-]
VE.D.12 VE.C.2 33 0. 410 7 .88 0.21 0.2 0.59
0.320 11.06 0.22 0.3 0. 64
0.224 31. 57 0.23 0.4 1.3
0.670 4.lt9 0.21 0.2 0,55
0.524 5.53 0,21 0.2 0.52
o. 414 7.45 0.21 0.2 0.56
0.322 9.31 0.22 0.2 0.55
0.248 11.86 0.22 0.3 0.53
0.193 15. 70 0.22 0.3 0.55
0.154 19.86 0.22 0.36 0.56
VE.D.12 VE.C.3 33 0.675 2.90 0.11 0.20 0.67
VE.D.12 VE.C.5 33 0.605 1.96 0.031 0 .17 1.4
0.402 2. 78 0.032 0.19 1.3
0.291 3.57 0.032 0.21 1.2
0.230 4.42 0.033 0.22 1.2
0.191 6.61 0 .033 0,25 1. 5
VE.D.12 VE.C.6 33 0.524 2.36 0.028 0.18 1. 6
0.332 3 .42 0.028 0.21 1. 5
0.239 4.11 0.028 0.22 1. 3
0.184 5.25 0.029 0.24 1. 3
0 .145 6.54 0,029 0.25 1.2
VE.D.12 VE.C.7 33 0 .386 3.52 0.019 0.19 1.9
0.247 3. 37 0.019 0.21 1.6
VE.D.12 VE.C.7 33 0.176 4. 39 0.019 O.Z2 1.5
0.136 6.15 o. 020 0.25 1.6
0.089 7 .86 0.020 0.27 1.4
VE.D.13 VE.C.2 33 0.571 5.18 0.22 0. 71 0.54
0.446 6.64 0.22 0. 78 o. 54
0.352 9.39 0.22 0.88 0.60
0.272 >30 0.24 1.3 >1.5
VE.D.13 VE.C.3 33 0.448 4.50 0,11 0.68 0.69
0.328 6.34 0 .12 0. 76 0.72
0.230 8. 70 0.12 0.85 0.60
0.167 13 .01 0.12 O.S8 0. 75
VE.D.13 VE.C.5 33 0.723 1. 59 0.032 0.47 1.3
0.44S 2.40 0 .033 0.54 1.2
0.2S7 3.S2 0. 034 0.65 1.3
0.187 5.38 0.034 0. 72 1.2
0 .134 6.S2 0.035 0. 79 l, l
VE.D.13 VE.C.6 33 0.678 1. 68 0.028 0.48 1. 5
0.452 2.58 0.029 0.56 1.5
0.297 3.72 0.030 0. 63 1.5
0.196 5.10 0.030 0. 71 1. 3
0.140 6.91 0.031 0. 79 1. 3
260
TAB LE B.l (Continued)
Drop Cont. " R 7 À 8R 0crit. phase pbase [mNm-1 J [Dili] r.- 11 1-1 [-] (-)
VE.D.13 VE.C.7 33 0.554 l.90 0.019 0.50 2. l
o. 301 3.15 0.020 0. 60 l.9
0.169 4.14 0.020 0.66 1.4
0.126 8.49 0.021 0.65 2.1
0.087 8. 44 0.021 0.84 1.4
VE.D.14 VE.C.2 32 0.567 >18. 66 0.32 3.3 >2.0
0.422 >28.81 0.32 4.2 >2.3
VE.D.14 VE.C.3 32 0.506 3.93 0 .17 1.3 0. 71
0.347 7.03 0.17 1.9 0. 87
0.253 9.05 0.17 2.2 0. 81
0.205 17 .69 0.17 3.2 1.3
VE.D.14 VE.C.5 32 0.590 1. 43 0.052 o. 74 1.0
0.325 2.90 0.052 1.1 1.1
0.598 3. 08 0.246 0.60
0.390 4. 37 0.246 0.55
0.265 5.84 0.246 0.50
VE.D.14 VE.C.5 32 0.196 4.29 0.052 1.4 1.0
0.131 6.33 0 .051 1.8 0.99
VE.D.14 VE.C.6 32 0.652 l.42 0 .046 0. 74 l.3
0.443 2.06 0.046 0.91 1.2
0.281 2.84 0.046 l.l 1.1
0.205 3.67 0.046 1.3 1.0
0 .149 5.02 0.046 1. 5 1.0
0.114 8. 79 0 .045 2.1 1.4
VE.D.14 VE.C.7 32 0.494 1.50 0 .031 0. 76 l.5
0.308 1.98 0.031 0.89 1.2
0.215 2.94 0.031 1.1 l. 3
0.161 3.52 0.030 1.2 1.3
VE.D.15 VE.C.5 31 0,417 2.92 0.22 8.4 1.5
VE.D.15 VE.C.6 31 0.69'i 1.36 0.24 5.6 1.3
0.424 2.27 0 .21 7 .3 1.3
0.308 3.53 0 .18 9.3 l.5
0.214 >10. 46 0.12 16.7 >3.1
0.154 >28.16 0.090 28.5 >6.0
VE.D.16 VE.C.2 35 0.617 4. 65 0.47 0. 49
0. 452 5.93 0.47 0. 46
0,352 8.06 0.47 O.'i9
VE.D.16 VE.C.3 35 0. 684 1.96 0.25 0.44
0.506 3.14 0.25 0.52
0.345 4.63 0.25 0.52
0.233 6.21 0.25 0,47
VE.D.16 VE.C.4 35 o. 410 2.17 0.105 0.68
0.265 3.24 0.105 0.65
VE.D.16 VE.C.5 35 0.571 1.59 0.073 0.99
0.299 3.01 0.073 0.98
261
TAB LE B.l (Continued)
Drop Cont. (f R 7 À SR Ocrit phase pbase [mNm-1 J [nm] [s-1] [-) [-] [-)
VE.D.16 VE.C.6 35 0.632 1.29 0.065 1.0
VE.D.16 VE.C.6 35 0,330 1.89 0.065 0. 77
VE.D.16 VE.C.7 35 0. 578 1.30 0. 043 1.4
0.266 l. 81 0.043 0.89
VE.D.17 VE.C.2 35 0.446 6.11 0.66 o. 79 0.47
0.325 >19 0.65 1.6 >1.l
0,383 17 .25 0.65 1. 5 1.1
0.320 .>27 0.64 1. 9 >1.5
0.521 6.58 0.66 0.83 0.59
0.417 13. 50 0.65 1.3 0. 97
0.371 13. 70 0.65 1. 3 0.87
0.297 >25 0. 64 1.8 >l. 3
0.655 11.47 0.65 1.2 1.3
VE.D.17 VE.C.3 35 0.663 2.64 0.35 0.48 0.57
o. 449 4.40 0.35 0.65 0.64
0,355 >20 0.34 1.6 >2.3
VE.D.17 VE.C.3 35 0 .667 2.16 0.36 0.43 0.47
0.497 3.16 0.35 0.54 0. 51
0.383 6.27 0.35 0.81 0. 78
0.253 >17 .13 0.34 1. 5 >l.4
0.542 2.85 0,35 0.50 0 .50
0. 414 4. 07 0.35 0.62 0.55
0.312 >19 0. 34 1. 6 >2,0
0,364 >14 o. 34 1. 3 >l. 7
0. 586 2. 76 0.35 0. 49 0.52
0.451 3.97 0.35 0.61 0.58
0.337 >19 o. 34 1.0 >2.1
VE.D.17 VE.C.4 35 0.439 1. 58 0.15 o. 35 0.53
0.536 1.48 0.15 0.34 0.60
0.188 4 .28 0.15 0.64 0. 61
0 .134 6.11 0 .15 0. 79 0.62
0.111 >16 0 .15 1.4 >l.3
0. 361 2.30 0.15 0. 44 0. 63
0.221 3 .61 0 .15 0. 58 0.61
0.151 6.96 0 .15 0.85 0.80
0.091 >21 0.14 1. 7 >l,6
262
TABLE B.l (Continued)
Drop Cont. R ; À 8R 0crit phase phase [mNm- 1 ] [mml [s-1] (-] [-] (-)
VE.D.17 VE.C.5 35 0. 540 1.23 O.ll 0.30 0. 72
0. 343 2.08 0 .10 0.42 0. 78
0, 196 3.27 0.10 0.55 0.70
0.133 4.85 0.10 0.69 0. 70
0.096 >17 0.10 1.5 >1.8
0.563 1.47 O.ll 0.34 0.90
0.297 2.00 0.10 0.41 0.65
0.206 3.44 0.10 0.56 0.77
0.126 >13 0.10 1.2 >l.8
0,530 1.50 0.11 0 .34 0,87
0.278 2. 78 0.10 0.50 0.84
0.154 3. 77 0.10 0.59 0.63
0 .117 >17 0.10 1.4 >2.1
0 .500 1.59 0.11 0.35 0. 87
0.281 3.08 0.10 0. 53 0. 94
0.142 5.43 0.10 0. 74 0 .84
VE.D.17 VE.C. 6 35 0.648 o.97 0.095 0.26 0. 78
0.436 l. 31 0. 094 0.32 0.70
0.306 1. 88 o. 094 0.39 0.71
0.194 2.29 0.093 0.44 0. 55
0.157 >9 0.091 0.99 >1.8
0. 440 l. 28 0.094 0. 31 0. 70
0. 316 2.31 0 .093 0. 44 0.90
0.160 3.34 0. 093 0.55 0.66
0.128 >8 0.091 0.92 >1.3
0.625 1.13 0.095 0.29 0.87
0.342 1.59 0.094 0,35 0. 67
VE.D.17 VE.C.6 35 0 .241 2.23 0.093 0 .43 0. 66
0.166 3.73 0.092 0.59 0. 76
0.130 >18 0.090 l. 5 >2.8
0.640 1.32 0.094 0.32 1.0
0.328 2. 46 0.093 0.46 1.0
0.175 3.89 0,092 0.61 0 .84
0.137 >14. 28 0.090 1. 3 >2.4
VE.D.17 VE.C.7 35 o. 512 1. 01 0.063 0.27 0.95
0.322 1.37 0.063 0.32 0.81
0.211 2.21 0 .063 0. 43 0.86
0.131 3.29 0.062 0.55 0.80
0.479 1.16 0. 063 0.29 1.0
0.166 Z.68 0.062 0.48 0.82
0. 078 4. 77 0.061 0,69 0.69
0.063 >64 0.058 3. 3 >7 .5
263
TAB LE B.l (Continued)
Drop Cont. 17 R '1 À SR 11cr1t phase phase [mNm-11 [Olll] [s-1] [-] [-] [-]
Vl!.0.18 VE.C.3 34 0.586 l0.06 0.57 4.7 2.0
0.663 9.22 0 .57 4.5 2.0
0.451 4.59 0.62 3.2 0.69
0.345 13 .40 0.55 5.4 1.5
VE.D.18 VE.C.4 34 0.417 2.26 0.28 2.2 o. 74
0.272 >8 0.25 4.1 1. 7
0.542 2.03 0.29 2.l 0.86
0.314 3. 73 0.27 2.8 0.92
0.220 >12 0.24 5.1 2.1
VE.0.18 VE.C.5 34 0.586 l.27 0.21 1.7 0.83
0,369 2.48 0.20 2.3 1.0
0.194 >14 0.16 5.4 >3.0
0.536 1. 50 0.21 1.8 0.90
0.301 >11 0 .17 4. 9 >3.8
0.305 3.82 0.19 2.9 1.3
0.223 >20 0.16 6.5 >4.9
VE.D.18 VE.C.6 34 0.515 1.18 0.19 1.6 0. 77
0.326 2.06 0.18 2.1 0.85
0 .205 >10 0.15 4 .5 >2.5
0.566 1.13 0.19 1.6 0.81
0.332 2.03 0.18 2.1 0.86
0.232 3.30 0.17 2. 7 0.97
0 .154 >15 0.14 5. 7 >2.9
0.527 1.17 0.19 1. 6 o. 78
o. 314 2.24 0.17 2.2 0. 89
0.694 2.95 0.48 l.6 0. 70
0,488 >9 0.47 3.6 >l.4
VE.D.18 VE.C.6 34 0.200 3.87 0.16 2.9 0.98
0.140 >14 0.15 5.4 >2.5
0.571 l.22 0.18 1.6 0.89
0.328 2.29 0.17 2.2 0.95
0. 232 >ll. 36 0.15 5.0 >3.3
0.181 >20.73 0.14 6.7 >4.8
VE.D.18 VE.C. 7 34 0.460 o. 93 0.13 l.4 0.81
0.295 1. 72 0.12 1.9 0.96
0,188 >8 0.10 4.2 >2.9
0.318 l.47 0.12 l.8 0.89
0.190 3. 97 0.11 2.9 1.4
0.448 1.03 0.13 1. 5 0.88
0 .273 >7 0.10 3' 7 >3.4
o. 572 1.01 0.13 1.5 1.1
0.301 1. 64 0.12 1.9 0.94
0.173 >7 0.10 3' 8 >2.2
264
TAB LE B.1 (Continued)
Drop Cont. " R .; À 8R ocrit phase phase [mNm-1J Cmml (s-ll (-] (-] [-]
VE.D.19 VE.C.2 33 0.448 >20 0.88 7 .o >1.8
0.509 >12 0.89 4.8 >1.2
VE.D.19 VE.C.3 33 0.594 2.84 0 .48 1.6 0.58
O.li49 3. 85 0.48 z.o 0 .60
0.343 >19 0.46 6. 7 >2.3
0.509 6.83 0.47 3.1 1.2
0.689 2.52 0.48 1.4 0.60
0.482 4.10 0.48 2.1 0.68
0.369 >11 0.47 4. 3 >l.4
VE.D.19 VE.C.4 33 0.463 1. 85 0.21 1.1 0.69
0.289 2.91 0.21 1.6 0.68
0 .203 >14 0.20 5.3 >2.3
0.366 2.27 0.21 1.3 0.67
0.232 3 .51 0.21 1.8 0.66
VE.D.19 VE.C.5 33 0.414 l.77 O.H 1.1 0.85
0.235 3.47 0.14 1.8 0 .94
0.139 >24 0.14 7.8 >3.8
0.544 1.31 0.15 0.86 0.82
0.312 1.81 0.15 1.1 0 .65
0.226 2.95 0 .14 1.6 0.77
0.173 >14 0 .14 5. 3 >2.8
0.512 1.52 0.15 0.96 0.90
0.303 2.35 0.14 1.3 0.82
0.188 3. 79 0 .14 l. 9 0.82
0.157 >16 0 .14 5.8 >2.9
VE.D.19 VE.C.6 33 0.322 1. 78 0.13 1.1 0. 75
0.203 2.87 0.13 1.6 0. 76
0.137 >8 0.12 3.2 >l.3
o.539 1.24 0.13 0.82 0.87
0 .318 2.27 0.13 1. 3 0.94
0.172 3. 77 0.13 1. 9 0.85
0.133 >12 0.12 4.6 >2.1
0.563 1.19 0.13 0.80 0.88
0.347 2.23 0.13 1.3 1.0
0.178 >11 0.12 4.3 >2.5
VE.D.19 VE.C.7 33 0.601 0.97 0.087 0.52 1.14
0.305 1. 92 0.086 1.2 1.2
0 .494 1.06 0.087 0.73 1.0
0.288 1. 68 0.086 1.0 0.95
0.163 2.58 0.085 1.4 0.82
0.402 1. 47 0.086 0.94 1.2
0.145 2.91 0.085 1.6 0. 83
0.102 >9 0.083 3.5 >L7
265
TABLE B.2 Shear thinning drop break-up
drop
phase
cont.
phase [mN/ml
R
[mm]
' 7
[l/•l Àm,l [-]
\n,2 [-]
ST.D.l ST.C.l 26.5 0.298 63.83 0.65 4.9E-Ol l.4E-01 2.9E-01
ST.D.l Sî.C.l 26.5 0.560 25.68 0.49 9.7E-Ol 2.9E-Ol 7.0E-01
Sî.D.l Sî.C,1 26.5 0.448 32.58 0.50 8.2E-Ol 2.4E·Ol 5.4E-Ol
ST.D.l ST.C.1 26.5 0.364 40.62 0.50 6.9E-Ol 2.0E-01 4.4E-01
Sî.D.l Sî.C.3 26.5 0.347 8.45 0.63 3.6E-Ol l.lE-01 2.0E-01
ST.D.1 ST.C.3 26.5 0.214 14.90 0.68 2.3E-Ol 6.SE-02 1.2E·Ol
ST.D.l ST.C.5 26.5 0.464 2.18 0.61 3.SE-01 1.0E-01 l.9E-Ol
ST.D.1 ST.C.5 26.5 0.371 2.88 0.65 2.SE-01 8.2E-02 1.5E-Ol
ST.D.l ST.C.5 26.5 0.295 3.65 0.65 2.4E-Ol 7.lE-02 l.2E-Ol
ST.D.l ST.C.5 26.5 0.176 7.40 0.79 l.4E-Ol 4.lE-02 7.lE-02
ST.D.l Sî.C.5 26.5 0.139 10.02 0.84 l.lE-01 3.2E-02 5.SE-02
ST.D.l ST.C.5 26,5 0.108 13.67 0,90 8.SE-02 2.SE-02 4.4E-02
ST.D.l Sî.C.6 26.5 0.405 1.15 0.70 2.3E-Ol 6.8E-02 1.2E-Ol
ST.D.2 Sî.C.l 28
ST.D.2 ST.C.l 28
Sî.D.2 ST.C.l 28
ST.D.2 ST.C.l 28
ST.D.2 ST.C.1 28
ST.D.2 ST.C.1 28
ST.D.2 ST.C.l 28
ST .D.2 ST .C.1 28
ST.D.2 ST.C.3 28
ST.D.2 ST.C.3 28
ST.D.2 ST.C.3 28
ST .D.2 ST ,C. 3 28
ST.D.2 ST.C.3 28
ST.D.2 ST.C.5 28
ST.D.2 ST,C.5 28
ST.D.2 ST.C.5 28
ST .D.2 ST .C.5 28
ST.D.2 ST.C.6 28
ST.D.2 ST.C.6 28
ST.D.2 ST.C.6 28
ST.D.2 ST.C.6 28
ST.D.2 ST.C.6 28
ST.0.3 ST.C.l 29
ST.D.3 ST.C.3 29
ST.D.3 ST.C.3 29
ST.D.4 ST.C.l 29
ST.D.4 ST.C.1 29
ST.D.4 ST.C.1 29
266
0.566 26.93 0.49 5.2E-Ol l.6E-Ol 3.1E-Ol
0.460 35.05 0.52 4.3E-Ol l.4E-Ol 2.5E-Ol
0,366 45.63 0.54 3.5E-Ol 1.lE-01 2.0E-01
0. 286 58. 80 0. 54 3. OE-01 9. 4E-02 l. 7E-01
0. 554 28. 54 0. 51 4. BE-01 l. SE-01 3. OE-01
0.442 34.25 0.48 4.2E-01 1.3E-Ol 2.6E-01
0.333 47.07 0.50 3.4E-01 1.lE-01 2.0E-01
0.268 65.88 0.57 2.7E-Ol 8.SE-02 1.5E-Ol
0,536 5.02 0.54 2.7E-Ol 8.SE-02 l.SE-01
0.434 6.60 0.58 2.2E-Ol 6.9E-02 l,2E-Ol
0.347 9.24 0.65 LSE-01 5.7E-02 9.3E-02
0.271 11.86 o.65 1.5E-Ol 4.7E-02 7.7E-02
0.220 14.93 0.66 l.2E-Ol 3.8E-02 6.5E-02
0.500 2.52 0.72 1.4E-Ol 4.4E-02 8,3E-02
0.400 3.56 0.81 1.lE-01 3.SE-02 6.4E-02
0.398 4.34 0.99 9.5E-02 3.0E-02 5.5E-02
0.215 24.81 3.10 2.BE-03 8.SE-03 1.6E-02
0.571 1.05 0.86 l.lE-01 3.5E-02 6.2E-02
0.452 1.48 0.95 8.9E-02 2.8E-02 4.7E-02
0.383 2.06 1.10 7.0E-02 2.2E-02 3.7E-02
0.287 2.85 1.20 5.6E-02 1.8E-02 2.9E-02
0. 360 2. 29 1. 20 6. 7E-02 2. lE-02 3. 4E-02
0.496 34.28 0.53 2.0E-01 6.8E-02 1.2E-Ol
0.582 9.52 1.10 5.5E-02 1.9E-02 4.2E-02
0.533 12.03 1.30 6,5E-02 Z.2E-02 3.6E-02
0.476 31.15 0.50 5.6E-Ol 2.2E-Ol 4.2E-01
0.386 39.45 0.51 4.8E-Ol 1.9E-Ol 3.BE-01
0. 305 49. 90 0. 51 4. 2E-01 1. 6E-Ol 3. lE-01
TABLE B.2 Continued
drop cont q R ,Y Ocrit >-a >.", 1 (-]
\n,2 [-] phase phasa [mN/mJ [mm] [l/sJ [-] (-)
ST.D.4 ST.C.3 29 0,578 5.47 0.62 2.6E-01 l.OE-01 l.6E-Ol
ST.D.4 ST.C.3 29 0,463 7.77 0.70 2.2E-01 8.SE-02 l.3E-Ol
ST.0.4 ST.C.3 29 0.353 11.33 0.78 l.7E-Ol 6.SE-02 l.OE-01
0.297 15.67 0.91 l.4E-Ol 5.4E-02 8.SE-02
0.236 22.23 1.00 1.21!-0l 4.7E-02 6.9E-02
0. 184 34 .18 1. 20 9. lE-02 3. 5E-02 5. 3E-02
0 .152 50. 34 1. 50 7. 3E-02 2. 8E-02 4. 2E-02
ST.D.4 ST.C.3 29
ST.D.4 ST.C.3 29
ST.D.4 ST.C.3 29
ST.D.4 ST.C.3 29
ST.D.4 ST.C.5 29 0.508 3.18 0.98 1.3E-Ol 5.lE-02 7.SE-02
ST.D.4 ST.C.5 29 0.407 4.65 1.20 1.0E-01 3.9E-02 6.0E-02
ST.D.4 ST.C.5 29 0,315 7.05 1.40 8.lE-02 3.11!-0Z 4,7E-02
ST.D.4 ST.C.5 29 0.248 10.56 1.60 6.4E-02 2,5E-02 3.7E-02
0.194 15.61 1.80 5.lE-02 2.0E-02 2.9E-02
0.536 1.69 1.30 7 .4E-02 2.9E-02 4.3E-02
0.402 2.31 1.30 6.2E-02 2.4E-02 3.SE-02
ST.D.4 ST.C.5 29
ST.D.4 ST.C.6 29
ST.D.4 ST.C.6 29
ST.D.4 ST.C.6
ST.D.4 ST.C.6
ST.D.4 ST.C.6
29
29
29
0.322
0.247
0.193
3.56 1.60
5.66 1.9
9.08 2.4
4, SE-02 1. SE-02 2. BE-02
3.6E-02 l.4E-02 2.lE-02
2. 8E-02 l. lE-02 1. 6E-02
ST.D.4 ST.C.6 29 0.155 14.33 3.1 2.lE-02. 6.2E-03 1.2E-02
ST.D.5 ST.C.l
ST.D.5 Sî.C.l
SI.D.5 SI.C.l
ST.D.5 Sî.C.l
ST.D.5 SI.C.l
ST.D.5 ST.C.3
ST.D.5 ST.C.3
ST.D.5 ST.C.3
Sî .D. 5 ST .C. 3
ST.D ST.C.5
ST.D.5 ST.C.5
ST.D.5 ST.C.5
29.5 0.472 33.21 0.48 3,0E-01 1.2E-01 l.9E-Ol
2.9.5 0.376 53.44 0.61 2.3E-Ol 9.2E-02 l,4E-Ol
29.5 0.2.95 63.12 0.57 2.lE-01 B.4E-02 1.3E-Ol
29.5 0.414 41.71 0.57 2.4E-Ol 9,6E-02 1.7E-01
29. 5 o. 330 61.14 0. 67 1. 9E-Ol 7. 6E-02 1. 3E-02
2.9.5 0.504 9.43 0,91 l.OE-01 4.0E-02. 5.7E-OZ
2.9. 5 0. 403 13, 36 1. 00 7. 8E-02 3. lE-02 4. 7E-02
29.5 0.310 18.07 1.10 6.6E-02 2.6E-02 3.91!-02
29.5 0.248 31.41 1.50 4.8E-02 l.9E-02 2..9E-02
29.5 0.523 3.78 1.10 5.6E-02 2.2E-02 3.3E-02
29.5 0,411
29.5 0,285
5. 74
9,05
1,30 4.5E-02 1.BE-02 2.6E-02
1. 40 3. 5E-02 1. 4E-02 2. OE-02
ST.D.5 ST.C.5 29.5 0.226 18.20 2.20 2.3E-02 9.2E-03 l.4E-02
ST.D.5 ST.C.5 29.5 0.205 19.67 2.20 2.2E-02 6.8E-03 l.3E-02
ST.D.5 ST.C 6 29.5 0.485 3.64 2.40 2.3E-02 9.2.E-03 l.3E-02
ST.D.5 ST.C.6 2.9,5 0,328 6,74 3,00 l,6E-02 6.4E-03 9.SE-03
ST.D.5 ST.C.6 29.S 0.330 6,83 3.10 l.6E-02 6.4E-03 9.4E-03
ST.D.5 ST.C.6
ST.D.6 ST.C.l
ST,D.6 ST.C.l
ST.D.6 ST.C.1
ST.D.6 S!.C.3
ST.D.6 Sî.C.3
29.S 0,245 10,14 3.40 l.3E-02 5.2E-03 7.6E-03
30 0.578 35.08 0.61 1.0E-01 4.lE-02 6.lE-02
30 0.472 47.26 0.67 B.SE-02 3.5E-02 5.1E-02
30 0.378 59.62 0.68 7.5E-02 3.lE-02 4.5E-02
30 0. 611 ll. 09 1. 30 3. OE-02 1. ZE-02 l. BE-02
30 0. 496 15. 32 1. 40 2. SE-02 1. OE-02 1. 5E-02
ST.D.6 ST.C.3 30 0. 432 21. 05 1. 70 2. lE-02 8. 6E-03 1. 2E-02
267
TABLE B.2 Continued
drop
phas&
cont.
phase !mN/mJ
ST.D.6 ST.C.3 30
ST.0.6 ST.C.3 30
ST.D.6 ST.C.5 30
ST.D.6 ST.C.5 30
ST.D.6 ST.C.5 30
ST.D.6 ST.C.5 30
ST.0.6 ST.C.5 30
ST.D.6 ST.C.5 30
ST.0,6 ST.C.6 30
ST.D.6 ST.C.6 30
ST.D.6 S'.l'.C.6 30
ST.D.6 ST.C.6 30
ST.D.6 ST.C.6 30
R
[mm]
7
{l/s] Àm,1 [-]
Àm,2 {-]
0.304 37.56 2.20 1.5E-02 6.lE-03 9.0E-03
0.267 48.67 2.50 1.3E-02. 5.3E-03 7.SE-03
0.307 20.55 3.40 7.6E-03 3.lE-03 4.5E-03
o .194 46. 11 5.10 4. n:-03 l. 9E-o3 2. aE-03
0.419 12.58 2.80 9.9E-03 4.0E-03 5.81!-03
0.376 14.60 2.90 9.2E-03 3.SE-03 5.4E-03
0.262 25.56 3.60 6,7E-03 2.7E-03 3.9E-03
0.203 42.23 4.60 5.lE-03 2.lE-03 3.0E-03
0. 303 15 .15 6 .10 3. SE-03 l. 5E-03 2. lE-03
0. 196 23. 12 6. 20 2. SE-03 1. lE-03 l. SE-03
0.381 12.43 6.30 4.0E-03 1.6E-03 2.3E-03
0.310 18.79 7.80 3.2.E-03 1.3E-03 l.9E-03
0.223 28.14 7.60 2.6E-03 l.lE-03 1.5E-03
ST.D.7 ST.C.l 30.5 0.528 54.91 0.85 4.8E-02 2.lE-02 3.0E-02
ST.0.7 ST.C.l 30,5 0.523 54,84 0.94 4.4E-02 1.9E-02 3.0E-02
ST.D.7 ST.C.3 30.5 0.590 24.68 2.70 l.2.E-02 5.lE-03 7.lE-03
ST.D.7 ST.C.3 30.5 0.426 40.40 3.20 9.0E-03 3.9E-03 5.SE-03
ST.D.7 ST.C.3 30.5 0.345 61.55 4.00 7.2E-03 3.lE-03 4.4E-03
ST.0.7 ST.c.5 30.5 0.455 15.04 3.60 5.3E-03 2.3E-03 3,2.E-03
ST.D.7 ST.C.5 30.5 0.449 27.42 6.50 3.9E-03 l.7E-03 2.4E-03
ST.D.7 ST.C.5 30.5 0.461 20.62 5.00 4,SE-03 l.9E-03 2.7E-03
ST.0.7 ST.C.5 30.5 0.355 33.76 6.30 3.5E-03 l.5E-03 2.lE-03
ST.D.7 ST.C.5 30.5 0.288 52,02 7.90 2.BE-03 l.2E-03 l.7E-03
ST.0.7 ST.C.6 30,5 0.442 20.13 11.70 l.SE-03 7.7E-04 1.l!l-03
ST.D.7 ST.C.6 30.5 0.533 13.55 9.50 2.3E-03 9.9E-04 1.4E-03
ST.D.7 ST.C.6 30.5 0,352 42.05 19.40 l.3E-03 5.6E-04 7.6E-04
ST.0.11 ST.c.4 38
ST.0.11 ST.C.4 38
ST.0.11 ST.C.2 38
ST .lL ll ST ,C.2 38
ST .0.11 ST .C.2 38
ST.D.11 ST.C.2 38
ST.D.12 ST.C.4 35
ST.0.12 ST.C.4 35
ST.D.12 ST.C.4 35
ST.D.12 ST.C.4 35
ST.0.13 ST.C.4 35
ST.D.13 ST.C.4 35
ST.D.13 ST.C.2 35
ST.D.13 ST.C.2 35
268
0,355 15.12 0.89 7.5E-Ol 2.9E-Ol 4.7E-Ol
0.401 10.68 0.71 9.lE-01 3.5E-Ol 6.0E-01
0,448 53,50 1.64 8.BE-01 3.liE-01 5.6E-Ol
0. 464 39. 37 1. 25 l, lE+OO 4. lE-01 7. lE-Ol
0.405 38.25 1.06 l.4E+OO 5.6E-Ol 7.3E-Ol
0,467 44.75 1.43 9.8E-Ol 3.8E-Ol 6.4E-Ol
0.447 19.64 l.58 8.8E-01 3.7E-Ol 7.7E-Ol
0.424 18.47 1.41 9.3E-Ol 3.9E-Ol 8.0E-01
0.458 17.35 1.43 9.5E-Ol 4.0E-01 8.4E-Ol
0.331 28.70 1.71 7.2E-Ol 3.0E-01 5.9E-Ol
0.361 22.93 1.49 9.9E-Ol 4.2E-Ol 7.5E-Ol
0.506 20.97 1.91 8.8E-Ol 3.8E-Ol 8.0E-01
0.482 44.41 1.59 l,2E+OO 5.0E-01 l.6E-OO
0.405 64.15 1.93 l.OE+OO 4.4E-Ol l.2E-OO
APPENDIX C
APPENDIX Cl
EVALUATION OF SINGULARITIES IN BOUNDARY INTEGRAL
METHOD
SINGULARITY IN THE J-INTEGRAL
To calculate the J-integrals in Eq. [5.18] numerically, the
integrand should have a finite value in the singularity E = O.
Since this is not the case the J-integral will be linearised in a
triangular surface t:.S, composed by the vectors ~· z and ~ (see
Fig. Cl.1), and the linearised integral will be evaluated
analytically. The linearisation was done as follows:
1 I - g(,::) dS
r
The area of this triangle is:
óS
0
B
" " ' ' ' ' ', I ', I
- - - -'1- - - .llo,:: - - -' ,, ' ' I ', I '
---~'' ,, ___ ::::+--- 11+6µ
' ~~,, ' / II / ', ' I ' - - - -'t, - - - - - - - -'- - -', ,,, 1', ', 1,, .... , ' '
- - - - - - - - ~ - - - :" - - - .::- '!; - - -'- - - -
' I ', ' ',
• ' ' ' ' ' ... '
À
[Cl.1]
[Cl.2]
A
Fig. Cl.l Discretisation of the triangular surface element àS
269
270
For any vector E e àS the function g(E) can be linearised as
follows:
[Cl.3]
in which A and B can be determined by substituting r=x and r-z:
[Cl. 4]
[Cl. 5]
The integral is evaluated by subdividing the triangle àS in
similar subtriangles, using the parameterisation , E = ~ + µ~.
The surface of such a subtriangle, enclosed by À, À+àÀ, µ and
µ+6µ, is given by:
àStr = àS1 + àS2 6À 6µ 6S + 6À 6µ àS " 2 6À 6µ 6S [Cl.6]
Now the integral I can be calculated:
[Cl. 7]
Rearranging the terms with respect to À and µ leaves us two
integrals which can be solved separately:
I
---------- 26S dµdÀ [Cl.8] LL J 2 2 2 2 A x + 2Àll(~·~> + µ y
To evaluate the first integral we first calculate:
2 2 + JJ y
dµ 1 - ln y (
x+y+z )
x-y+z
The first integral can now be evaluated:
2g0 + (A+B)x2
+ B(~·:O ( x+y+z ) I 1 = ln -- llS
y x-y+z
To evaluate the second integral we first calculate:
[Cl. 9]
[Cl.10]
2 2 + JJ y
dµ = ~ ( :.:: - (~·;> ln ( :::.:.:. J ) y y y x-y+z
[Cl.11]
The second integral can now be evaluated:
y ( :.:: - <~· l> ln ( x+y+z ) ) llS
y x-y+z [Cl.12]
The total linearised J-integrai can now be written as:
I (( 2lls
2(A+B)) (:+y+z) 1 [ zZ+X)
g0+ Y2 ln(;_y+z + ~ (B-A)y(z-x)+(B+A)(z-x) -;-
2llS
y
[Cl.13)
271
272
APPENDIX C2 SINGULARITY IN THE K-INTEGRAL
To calculate the K-integral in Eq. [5.18] an approach similar to
the calculation of the J-integral (see Appendix Cl) was adopted.
This approach involves linearisation of the integrand close to the
singular point and calculation of the linearised integral by
an analytica! method. The following linearisation was used:
I'
with
I l h(f) dS LIS
(C2.l]
[C2.2]
[C2.3]
[C2.4]
After applying the same discretisation of the triangular surface
LIS as in Appendix Bl the integral I' can be rewritten:
I' 1 IÀ 2A'(x.r) + 2B'(z.r) 2 2 - - - -2 2 2llS dµdÀ
>-=O µ=0 X x + 2Àµ(~·l) + µ y [C2.5]
Rearranging the terms with respect to À and µ leaves us two
integrals which can be evaluated separately:
I' = I'1 + I'2 = ï fÀ JÀ=O µ=0
---------- 2llS dµdÀ +
~ 2 2 2 2 À x + 2Àµ(~·l) + µ y
r fÀ µA'(~·l) + µB'(~·l) + µB'y 2
----------- 2LIS dµdÀ
>-=o µ=0 Jx2x2 + 2Àµ(~·l> + µ2y2
(C2.6]
To evaluate the first integral we first calculate:
r (/+(x.z::)J ((x.z::)J) larctan
2llS - arctan
2àS
1
2ÀllS
The first integral 1 1
1 can now be evaluated:
To solve the second integral 1 1
2 we first calculate:
(C2.7)
[C2.8]
IÀ 2 2 µdµ 2 2 " 1 2 ln r:) µ"o >- x + 2Àµ(~·z::> + µ y y l;
-- (A'+B')x2+B'(!!;·~) ~·V ( ) 2/às
[C2.9)
The second integral r• 2 can now be evaluated:
( 2àS ~) (~·Z::) ( (y2+(~·Z::)) I' = ln - - -- arctan
2 / 2llS ((x.z::))) arctan ZllS
(C2.10]
The total integral can now be written as:
273
l' "' [ ((A'+B')(~·;t:)+B'l) lnË) +
2(A'+B')6s ( arctanl2
:~·:t:>) - arctan(<~~~>) ) J 2; 1c2.111
274
STELLINGEN
behorende bij het proefschrift
DEFORMATION AND BREAK.UP OF DROPS IN SIMPLE SHEAR FLOWS
van Robert Antonie de Bruijn
1. De opbreekcriteria voor afschuifsnelheidsverdunnende druppels kunnen beneden een viscositeitsverhouding van 0.1 goed beschreven worden door de criteria voor Newtonse druppels, wanneer men een gemodificeerde viscositeitsverhouding gebruikt, die rekening houdt met een afschuifsnelheid in de druppel; die hoger is dan de opgelegde afschuifsnelheid. (Hoofdstuk 4 van dit proefschrift)
2. In tegenstelling tot wat beweert wordt in een aantal publikaties bemoeilijkt elasticiteit van de vloeistof in de druppelfase het opbreken van druppels. Dit effect is sterker voor viscositeitsverhoudingen van orde 1 dan voor kleine viscositeitsverhoudingen. (Hoofdstuk 4 van dit proefschrift)
3. Voor druppelfasen met een sterke mate van elasticiteit in de vloeistof bestaat een kritische druppelgrootte, waar beneden opbreken in enkelvoudige afschuifstr0mingen niet mogelijk is. (Hoofdstuk 4 van dit proefschrift)
4. In tegenstelling tot beweringen van Rallison, blijkt het opbreekcri terium voor druppels met een viscositeitsverhouding van 1 niet afhankelijk te zijn van de snelheid waarmee de kritische afschuifsnelheid wordt bereikt. (Hoofdstuk 5 van dit proefschrift)
5. Vanneer een druppel een driehoeksvormig afschuifsnelheidsprofiel ondergaat met een maximale afschuifsnelheid die hoger is dan de kritische afschuifsnelheid voor quasi-stationaire afschuifstromingen, zal opbreken alleen plaatsvinden wanneer het afschuifsnelheidsprofiel voldoende lang wordt aangehouden. (Hoofdstuk 5 van dit proefschrift)
6. Het verschijnsel "tipstreaming" kan in enkelvoudige afschuifstromingen optreden wanneer oppervlakte-actieve stoffen aanwezig zijn in concentraties waarbij gradiënten in de oppervlaktespanning kunnen ontstaan, resulterend in een lage oppervlaktespanning nabij de uiteinden van de druppel en een hogere oppervlaktespanning elders. (Hoofdstuk 6 van dit proefschrift)
7. De relaxatie tijden die het lineair visco-elastisch gedrag van een dispersie van hydrodynamisch interacterende capsules beschrijven, kunnen sterk beïnvloed worden door de dikte van de schil van de capsules. (R.A. de Bruijn en J. Kellema, Rheol. Acta 24:159-174 (1985)
8. De toepasbaarheid van het model van Thomas et al. voor de beschrijving van warmte- en massatransportproblemen in poreuze media, is sterk beperkt omdat de thermofysische eigenschappen niet afhankelijk mogen zijn van zowel de temperatuur als het vochtgehalte. (B.R. Thomas, K. Morgan en R.W. Lewis, Int. J. Num. Meth. Eng. 15:1381-1393 (1980) )
9. De toepasbaarheid van het numerieke model van Ohlsson en Bengtsson voor de verwarming van levensmiddelen door microgolven is sterk beperkt in toepasbaarheid door de verwaarlozing van het golfkarakter van deze electromagnetische golven. (T. Ohlsson en N.E. Bengtsson, Microw. Energy Appl. Newsletter 4:3-8 (1979) )
10. De conclusie van Davies dat alleen turbulentie de goede emulgerende werking van hoge druk homogenisatoren kan verklaren is onjuist. (J.T. Davies, Chem. Engng. Sci. 40:839-842 (1985) )
11. De variatie van de gemiddelde druppelgrootte met het toerental van een geroerd vat, welke Stamatoudis en Tavlarides hebben waargenomen, is afhankelijk van de viscositeit van de continue fase. Deze afhankelijkheid kan verklaard worden door de overgang van laminaire naar volledig turbulente stroming in het vat. (M. Stamatoudis en L. Tavlarides, Ind. Eng. Chem. Process Des. Dev. 24:1175-1181 (1985) )
12. De hedendaagse teruggang in het aantal diersoorten en plantesoorten ondersteunt eerder een scheppingstheorie dan een evolutietheorie.
13. Acceptatie van kleine criminaliteit is een grotere bedreiging voor de maatschappij dan grote criminaliteit.
Wymington, september 1989