Drag on a Cylinder in a Viscoelastic Stokes Flow · The methodology was verified by testing...
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Drag on a Cylinder in a Viscoelastic Stokes Flow by Terence Shiau A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Department of Mechanical and Industrial Engineering University of Toronto ©Copyright by Terence Shiau 2014
Transcript of Drag on a Cylinder in a Viscoelastic Stokes Flow · The methodology was verified by testing...
Drag on a Cylinder in a Viscoelastic Stokes Flow
by
Terence Shiau
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Department of Mechanical and Industrial Engineering
University of Toronto
©Copyright by Terence Shiau 2014
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ABSTRACT
Drag on a Cylinder in a Viscoelastic Stokes Flow
Terence Shiau
Master of Applied Science
Department of Mechanical and Industrial Engineering
University of Toronto
2014
This thesis reports on measurements of drag on an unbounded cylinder in a viscoelastic Stokes
flow, and compares these values with a Newtonian equivalent. Cylinders of diameter 0.5 to 3.34 mm
were submerged 10 to 36 mm into slowly rotating annular tanks with channel widths between 133 to 152
mm. Theoretical formulas and computer simulations were used to correct for the effects of ends and
walls, yielding estimates of the unbounded drag. The methodology was verified by testing Newtonian
fluids and comparing the results to Kaplun’s (1957) prediction for unbounded drag.
The test fluids used were a silicone oil, a polybutene, and two Boger fluids. By comparing the
Boger fluid results to equally viscous Newtonian results, the contributions of elasticity to the drag were
determined. The Deborah number (De) was used to represent the magnitude of flow elasticity, and an
onset of elastic effects was measured between 0.5 and 0.7.
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ACKNOWLEDGEMENTS
My sincere thanks are extended to my supervisor Professor David F. James for his continued
guidance and support throughout my degree at the University of Toronto. His patience and dedication to
education for not just research, but presentation and personal growth have always been qualities I admire.
The opportunity to work with him has been an honor for which I am truly grateful. I would also like to
thank Professor Iain G. Currie for the use of his laboratory facilities in which the majority of my
experimental work was done.
My appreciation is given to the staff at the University of Toronto Mechanical and Industrial
Engineering department. Particularly to Ryan Mendell, Oscar del Rio, Tomas Bernreiter, and Professor
Jason Bazylak for their advice, support, and use of facilities in the design and construction of my
experimental equipment.
I would also like to thank the members of the DFJ research group, past and present, who freely
gave their time, camaraderie, and advice. Thanks especially to Edwin Wang for his training in the use of
the laboratory equipment, Dr. Ronnie Yip for the use of his test fluids T1 and T3, and both of their
willingness to help whenever I had questions.
Finally, I would like to thank my friends and family, who have supported my endeavours both at
and away from the University of Toronto. They have kept me grounded and provided any assistance they
could to lessen my burden and make life more enjoyable while I complete this program.
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TABLE OF CONTENTS
CHAPTER 1: INTRODUCTION .............................................................................................................. 1
1.2 Motivation ............................................................................................................................................... 3
CHAPTER 2: BACKGROUND ................................................................................................................ 5
2.1 Rheological Concepts ............................................................................................................................. 5
2.1.1 Shear flow – Newtonian and non-Newtonian fluids ............................................................... 5
2.1.2 Viscoelastic Fluids .................................................................................................................. 6
2.1.3 Boger Fluids and the Oldroyd-B Constitutive Model ............................................................. 9
2.1.4 Rheometers ........................................................................................................................... 11
2.2 Historical Background .......................................................................................................................... 13
2.2.1 Prior Newtonian Research .................................................................................................... 13
2.2.2 Prior Viscoelastic Research .................................................................................................. 17
2.3 Research Objectives .............................................................................................................................. 22
CHAPTER 3: EXPERIMENT DESIGN ................................................................................................ 23
3.1 Test Fluids ............................................................................................................................................. 23
3.1.1 Newtonian Fluid Characterizations ....................................................................................... 23
3.1.2 Boger Fluid Characterizations .............................................................................................. 26
3.1.3 Selection of Relaxation Time ................................................................................................ 34
3.2 Experimental Apparatus ........................................................................................................................ 35
3.2.1 Newtonian Fluid Characterizations ....................................................................................... 36
3.2.2 Boger Fluid Characterizations .............................................................................................. 37
3.2.3 Force Transducer .................................................................................................................. 38
3.2.4 Second Generation Apparatus ............................................................................................... 43
3.2.5 Final Experimental Setup ...................................................................................................... 45
3.3 Experimental Procedure ........................................................................................................................ 48
3.3.1 Transducer Calibration .......................................................................................................... 48
3.3.2 Data Acquisition – High Velocity Range ............................................................................. 53
3.3.3 Data Acquisition – Low Velocity Range .............................................................................. 54
3.3.4 End Effects ............................................................................................................................ 54
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3.3.5 Effect of the Channel Walls .................................................................................................. 60
3.3.6 Experimental Methodology - Summary ................................................................................ 65
CHAPTER 4: RESULTS AND DISCUSSION ...................................................................................... 67
4.1 Newtonian Results ................................................................................................................................ 67
4.2 Boger Fluids .......................................................................................................................................... 73
4.2.1 Experimental Results ............................................................................................................ 73
4.2.2 Drag Increase ........................................................................................................................ 80
4.2.3 Onset and Discrepancy at Higher De .................................................................................... 81
4.2.4 Comparison to Chilcott and Rallison .................................................................................... 82
3.3 Error Analysis ....................................................................................................................................... 84
CHAPTER 5: CONCLUDING REMARKS ........................................................................................... 87
REFERENCES .......................................................................................................................................... 89
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LIST OF TABLES
Table 3.1: Relevant rheological information for Newtonian test fluids .................................................... 26
Table 3.2: Original Boger fluid composition in wt% and molecular weight ��, from Yip (2011) .... 26 Table 3.3: Relations for relevant fluid characteristics vs. temperature for T1 and T3 ............................... 34
Table 3.4: Annular tank dimensions ......................................................................................................... 36
Table 3.5: Available test cylinder dimensions and material ..................................................................... 39
Table 3.6: Aluminium cantilever beam engineering properties and dimensions ...................................... 41
Table 3.7: Low speed DC motor specifications ........................................................................................ 43
Table 3.8: Polycarbonate cantilever beam engineering properties and dimensions .................................. 45
Table 4.1: Relevant parameters for the silicone oil and the polybutene experimental trials ..................... 67
Table 4.2: Relevant parameters for T1 and T3 experimental trials ........................................................... 73
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LIST OF FIGURES
Figure 1.1: Uniform flow of velocity U around a two-dimensional cylinder of diameter d ....................... 1
Figure 1.2: Flagellum propulsion mechanism ............................................................................................. 3
Figure 2.1: Steady shear flow between parallel plates ................................................................................ 5
Figure 2.2: Comparison of shear stress behavior for Newtonian and non-Newtonian fluid types ............. 6
Figure 2.3: Normal stresses developed as a result of shearing a polymeric fluid ....................................... 7
Figure 2.4: Cone-and-plate rheometer fixture .......................................................................................... 11
Figure 2.5: Comparison of various analytical solutions for the dimensionless drag on a cylinder at low
Re. ............................................................................................................................................................... 15
Figure 2.6: Comparison of Faxen’s formula and Whites solution for specific H/d ratios with Kaplun’s
unbounded solution .................................................................................................................................... 16
Figure 2.7: Jayaweera and Mason’s 1965 experimental data corrected for wall effects by using Faxen’s
formula compared to Kaplun’s solution ..................................................................................................... 17
Figure 2.8: Plan and side views of a confined cylinder experiment where channel Length >> Width (H) >
cylinder diameter (d) ................................................................................................................................... 19
Figure 2.9: Plan and side view of upstream velocity profiles for confined cylinder flow ........................ 20
Figure 3.1: Silicone oil characterization made on AR2000 with 40mm2° cone. Plot of viscosity vs. shear
rate at 21°C, and the viscosity vs. temperature relation ............................................................................. 24
Figure 3.2: Poybutene characterization made on AR2000 with 40mm2° cone. Plot of viscosity vs. shear
rate at 21°C, and the viscosity vs. temperature relation ............................................................................. 25
Figure 3.3: T1 and T3 characterization made on AR2000 with 40mm2° cone at 16°C and 22°C
respectively for shear and dynamic viscosity vs. shear rate and angular frequency, vertical dashed lines
indicate experimental shear rate range ........................................................................................................ 28
Figure 3.4: T1 and T3 shear and polymer viscosity relations vs. temperature as determined from
measurements made on the AR2000 with the 40mm2° cone fit with smooth curves ................................ 29
Figure 3.5: First normal stress coefficient vs. shear rate for T1 at 16°C and for T3 at 22°C, selected value
indicated by horizontal dashed line ............................................................................................................ 30
Figure 3.6: �� relaxation time vs. temperature relation for T1 and T3 ..................................................... 31 Figure 3.7:��/ vs. angular frequency for T1 and T3 ........................................................................... 32 Figure 3.8:�� relaxation time vs. temperature relation for T1 and T3 ...................................................... 33 Figure 3.9: Velocity profile as seen by a cylinder in the rotating annular tank ........................................ 35
Figure 3.10: Annular tank views ............................................................................................................... 36
Figure 3.11: Velocity flow field in a rotating annular tank ...................................................................... 37
Figure 3.12: 3D schematic view of the turntable with the annular tank ................................................... 38
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Figure 3.13: Schematic plan view of the transducer setup including the laser and scale ......................... 40
Figure 3.14: Side view of the first-generation force transducer, including the cantilever beam, cylinder,
slider mechanism, and mirror ..................................................................................................................... 40
Figure 3.15: Simple bending of a cantilever beam ................................................................................... 42
Figure 3.16: Input Voltage vs. Turntable RPM for low speed motor ....................................................... 44
Figure 3.17: Container module with alignment pattern overlay, channel centre line used to position
cylinder, transducer line used to align transducer perpendicular to tank ................................................... 46
Figure 3.18: Side and 3D views of full experimental assembly without the laser and scale positioned
away from the turntable ............................................................................................................................. 47
Figure 3.19: Schematic diagram of calibration system including bending beam, cylinder, pulley, and
calibration weight ....................................................................................................................................... 48
Figure 3.20: Cantilever torsion caused by off centre force ....................................................................... 49
Figure 3.21: Sample relation describing the high force transducers behavior at different immersion depths
for the same applied force .......................................................................................................................... 50
Figure 3.22: Sample high force transducer calibration with three different spline regions ...................... 51
Figure 3.23: Sample calibration data for low force transducer illustrating amount of spread involved in
determining calibration value ..................................................................................................................... 52
Figure 3.24: Variable immersion depths used to extrapolate zero length force ........................................ 55
Figure 3.25: Schematic view of surface disruptions, including fluid climb in front of cylinder and cavity
formation behind ........................................................................................................................................ 56
Figure 3.26: Sample end correction graphs for the polybutene at 21°C, 1.19 mm cylinder, Re = . � ×����, and T3 at 22.5°C, 1.19 mm cylinder, Re = �. � × ���� with linear curve fits ............................... 56 Figure 3.27: Sample % total drag accounted for by end effects plot for PB 1.19 mm cylinder, 18 mm
immersion depth, and T3 1.19 mm cylinder, 36 mm immersion depth ..................................................... 58
Figure 3.28: Sample % total drag graph vs. immersion depth for PB 1.19 mm cylinder, and T3 1.19 mm
cylinder with smooth curve fits and error bars representing standard error ............................................... 59
Figure 3.29: Comparison of Faxen’s formula, Fluent simulations, and Kaplun’s solution for �� =�. ���� at increasing H/d with experimental range marked by vertical lines .......................................... 61
Figure 3.30: Comparison of linear and annular channel Fluent meshes enhanced around cylinder region
.................................................................................................................................................................... 62
Figure 3.31: Fluent simulations run for various Re for the same H/d fit with constant linear curves ...... 63
Figure 3.32: Comparison of Fluent simulations and Faxen’s prediction for various H/d ratios and the
calculated percent difference between the two solutions ........................................................................... 64
Figure 4.1: Silicone oil raw, end corrected, and fully corrected drag coefficient data vs. Re for 0.97 mm
cylinder, dashed line indicates Fluent predictions and solid line indicates Kaplun’s solution .................. 68
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Figure 4.2: Silicone oil and PB end corrected and fully corrected drag coefficient values vs. Re for all
three cylinder diameters, dashed lines indicates Fluent predictions and solid line indicates Kaplun’s
solution ........................................................................................................................................................ 70
Figure 4.3: T1 raw, end corrected, and fully corrected drag coefficient data vs. Re for 2.17 mm cylinder,
dashed line indicates Fluent predictions and solid line indicates Kaplun’s solution ................................. 74
Figure 4.4: T1 and T3 fully corrected data vs. De for all cylinders displayed as ��/������′����� !�� implying that any graphical value of 1 indicates Newtonian behavior. Newtonian equivalent is indicated
by a solid line. Enlarged view of the lower De region shows onset of elastic effects occur between a De
of 0.5 to 0.7 ................................................................................................................................................. 76
Figure 4.5: Comparison of T1 and T3 data to Chilcott and Rallison’s numerical work ........................... 83
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CHAPTER 1: INTRODUCTION
This thesis deals with the drag on an isolated cylinder, at low Reynold’s numbers (Re), with
Newtonian and viscoelastic fluids.
Flow past a cylinder is commonly studied in fluid mechanics, especially for Re > 1 where the
drag coefficient has been extensively investigated. The current study examines two-dimensional Stokes
flow, also dubbed creeping or inertialess flow, which is defined by Re << 1. The boundary conditions
under investigation are for uniform, unbounded flow, as illustrated in figure 1.1, where an infinitely long
cylinder of diameter d is subjected to a uniform flow velocity U.
Figure 1.1: Uniform flow of velocity U around a two-dimensional cylinder of diameter d
The quantity of interest in this thesis is the cylinder drag developed by the flow of a viscoelastic
fluid. This particular flow situation has no theoretical solutions describing what effect the fluid elasticity
has the drag. While approximate solutions for the drag in a Newtonian flow exist, which are further
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explored in section 2.2.1, a direct analytical solution cannot be derived from the Navier-Stokes equations.
Additionally, very little experimentation for Stokes flow around a cylinder has been performed. The lack
of research can be attributed primarily to the difficulty in experimentally replicating the flow and
boundary conditions, and in accurately measuring small drag forces.
In this thesis, for both Newtonian and viscoelastic fluids, the Reynolds number is
�� ="#$� (�. �)
where ρ is the density, η is the viscosity, and d is the diameter. Creating a Stokes flow is not difficult
with a small cylinder, but the resultant drag force is small, making direct measurement difficult. The
situation is complicated further by the conditions of uniform and unbounded flow. With Stokes flow, it is
almost impossible to move walls and boundaries far enough away from an object for their effects not to
be felt. Walls can strongly influence drag measurements and velocity profiles even if they are positioned
over 1000 diameters away as will be detailed in sections 2.2.1 and 3.3.5. Lastly, two-dimensional flow is
impossible to simulate experimentally because the cylinder and fluid container must have finite
dimensions.
These challenges, along with accepted solutions for Newtonian flow, have meant that little effort
has been made to explore this flow situation experimentally. However, there has been a recent increase in
interest for this information from the biomedical field, where microscopic length scales give rise to Stokes
flow conditions. Hence, a fresh experimental look at this situation is presented in this thesis to examine
the drag on a cylinder over a range of fluid and flow conditions.
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1.2 MOTIVATION
There are two primary motivations for this research: an increased interest from the biomedical
field, and a need for experimental data against which theoretical solutions can be compared. The
motivation from the biomedical field comes from the propulsion of microorganisms.
Investigations indicate that microorganism swimming occurs in a Re range of order 10-4 to 10
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based on flagella diameter (Taylor, 1951) (Purcell, 1977). The primary mode of swimming is
reciprocating motion, but in a Stokes flow, where viscous forces dominate, this method is ineffective
because forces are reversible. Instead, microorganisms utilise their flagella in a planar sinusoidal waving
motion or in a helical motion to produce normal and tangential forces, propelling themselves forward
(Lauga, 2007), as shown in figure 1.2.
Figure 1.2: Flagellum propulsion mechanism
By applying a force to the surrounding fluid, an action-reaction pair is formed allowing forward
propulsion to be generated by normal forces. This implies that, if efficiency is ignored, a microorganism
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using an identical propulsive motion between two fluids is expected to swim faster in the more viscous
fluid because a greater drag force is generated.
Microorganisms are commonly found in biological media, such as spermatozoa in cervical
mucus. These media often contain biopolymers, which makes the fluid viscoelastic. As the name
implies, these fluids exhibit both viscous and elastic properties. Many previous investigations about
propulsion, such as that by Taylor, do not account for non-Newtonian effects. In order to understand how
a microorganism propels itself through its environment, the viscoelastic nature of its surrounding media
must be taken into account.
A common measure of the strength of elasticity in a fluid is the dimensionless Deborah number
(De) which will be fully described in section 2.1.2. It is sufficient here to say that as De increases above
1, the effects of elasticity increase. Previous works modelling microorganism swimming analytically or
numerically have indicated that the De ranges between O(10) to O(103) (Fu et al. 2008; Lauga 2007,
2009). The modelling, however, often suffers from oversimplification, such as modelling flagella as two-
dimensional waving sheets or discounting shear thinning in the constitutive fluid model. Depending on
which assumptions were made, different, and sometimes contradictory, results from analytical analyses
have been obtained. For example, in work by Fulford et al.(1998), microorganisms are predicted to swim
faster in biopolymer fluids than in equivalently viscous Newtonian fluids. On the other hand, the analysis
performed by Fu et al.(2009) indicates that swimming speeds are slower in such fluids. Because of the
lack of measurements of drag force, which result from flagella motion, it is uncertain which analysis
better represents reality. To help resolve the disagreement, experimental work under similar flow
conditions becomes essential.
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CHAPTER 2: BACKGROUND
2.1 Rheological Concepts
Because viscoelasticity is important in this thesis, several concepts beyond Newtonian fluid
mechanics are necessary.
2.1.1 Shear flow – Newtonian and non-Newtonian Fluids
Shown in figure 2.1 are two dimensional plates separated by a distance h, while the top plate
moves with constant velocity U.
Figure 2.1: Simple shear between parallel plates.
For simple shear flow in the x direction, the shear rate is
'( )* = $�())$) (. �)
where ) is the transverse and �()) is the velocity. Shear viscosity η is then defined as:
� = +)*'( )* (. )
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where +)* is the shear stress in the ,-direction. For a Newtonian fluid, this shear viscosity, generally referred to simply as the viscosity, is a constant, whereas for a non-Newtonian fluid it can depend on the
shear rate or shearing time.
The three primary categories of variable viscosity fluids are shear-thinning, shear-thickening, and
yield stress behaviour, with shear-thinning fluids, such as paint and ketchup, being the most commonly
encountered. The stress responses of these three types of fluids are illustrated in figure 2.2.
Figure 2.2: Comparison of shear stress behaviour for Newtonian and non-Newtonian fluid types
2.1.2 Viscoelastic Fluids
In this thesis, the fluids of interest are viscoelastic, having both viscous and elastic components.
Typical examples of such fluids include latex paint and molten polymer. In simple shear, in addition to
the shear stress, viscoelastic fluids experience normal stresses -.. and -// illustrated in figure 2.3.
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Figure 2.3: Normal stresses developed as a result of shearing a polymeric fluid
A common measure of fluid elasticity is the first normal stress difference ��, which is defined as:
�� = 0** − 0))(. 2)
This difference increases with shear rate and is approximately proportional to '( *). The first normal
stress coefficient is then defined as:
3� = ��'( *) ,(. �)
which is approximately constant.
Another method of characterising the elasticity of a fluid is to measure its response under
oscillatory shear. By shearing the fluid at a sinusoidal frequency with a small amplitude '�, the stress
response is also sinusoidal but out-of-phase. If the applied strain is defined as:
'*)( ) = '� 567( )(. �)
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then the stress response can be expressed as the decomposition of the in-phase and out-of-phase
components as:
+)*( )'� = �′() 567( ) + �′′():;5( )(. �)
where +)* is the output shear stress and �′ and �′′ are termed the dynamic storage and dynamic loss
moduli respectively. �′ represents the elastic or solid-like response, while �′′ represents the viscous
response. For a Newtonian fluid, shear stress response is always </2 radians out of phase with the applied strain, making �′ = � and �′′ = �. A viscoelastic fluid, due to its partially solid nature, can have a phase shift between the applied strain and its stress response.
Elasticity can also be represented by a relaxation time, which is the time required for a sheared
fluid to relax after the stress has been removed. In a viscoelastic fluid, this time is related to how long it
takes the stress developed in the polymer chains to decay. The fluid relaxation time is a measured
quantity and can be determined in several ways depending on methodology.
As mentioned in section 1.2, the strength of elasticity in a flow can be represented by the non-
dimensional Deborah number (De), which is defined as:
>� = ? (. @)
where ? is the fluid’s relaxation time and is the characteristic time of the flow. A high value of De
implies the fluid exhibits strong elastic effects while a low De value represents a more Newtonian-like
behaviour. For this thesis, De is defined as
>� =?#$ (. A)
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2.1.3 Boger Fluids and the Oldroyd-B Constitutive Model
While most viscoelastic fluids are shear thinning, the degree to which viscosity varies with shear
rate depends on fluid composition. For this thesis, the fluids known as Boger fluids were employed.
Boger fluids, originally developed by Boger (1977), are dilute polymer solutions which are strongly
elastic but only weakly shear thinning, to the point where their viscosities can be considered constant.
This constant viscosity allows for the elastic and viscous responses of the fluid to be separated by
comparing the response of a Newtonian fluid of equivalent viscosity (James, 2009). The response of a
Boger fluid is thus the viscous response of an equivalent Newtonian fluid plus the effects of elasticity.
This eliminates the confusion typically associated with viscoelastic fluids when the effects of shear
thinning and elasticity are intertwined.
According to James (2009), an appropriate constitutive model to represent Boger fluids with a
single relaxation time is the Oldroyd-B model. The constitutive equation for this fluid model is given as
(Bird et al. 1987):
(. B)
where + is the stress tensor, '( is the rate of strain tensor, � is the shear viscosity, ?� is the fluid relaxation
time, and ? is the fluid retardation time. and are the upper convected time derivatives of the stress and strain rate tensors, respectively. The model describes a fluid which has a Newtonian solvent
embedded with a dilute concentration of polymer chains which are modelled as dumbbells connected by
Hookean springs (Larson, 1998, p. 142).
For this thesis, the Oldroyd-B model was used to determine the Boger fluid properties.
According to the model, the stress tensor + in Eq. 2.9 can be written as:
+ = +� + +�(. ��)
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where +� and +� are the stress contributions from the polymer and solvent, respectively. It follows from
mathematical substitution that the viscosity of the fluid can be expressed as:
� = �� + ��(. ��)
where �� and �� are the contributions of the polymer and solvent, respectively.
For the case of steady shearing, the Oldroyd-B model predicts the first normal stress difference to
be (Bird, 1987):
�� = ?���'( (. �)
where ?� is the polymer relaxation time, or simply the fluid relaxation time. This implies that the first
normal stress coefficient 3� should be independent of shear rate. For small amplitude oscillatory shear,
the dynamic response is predicted as:
�′
= ��?�� + C?�D (. �2)
�′′
= �� + ��� + (?�) (. ��)
The quantity �"/ is also known as the dynamic viscosity, and is important because it is the solvent contribution to viscosity as becomes large.
In practice, experimental data do not fit these equations perfectly, as will be shown in section
3.1.2, but reasonable approximations for the relaxation time can still be made by taking appropriate limits.
Equations 2.12 and 2.13 can thus be rearranged to determine the relaxation time as:
?� = � F6G'(→� I ��'( ��J(. ��)
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?� = F6G→�I�′
��J(. ��)
The relaxation times given by these two expressions are generally different, as will be
demonstrated in section 3.1.2. The appropriate relaxation time for the present experiment is the former,
as will be described further in section 3.1.3.
2.1.4 Rheometers
A shear rheometer is designed to characterise fluid properties, such as viscosity, under various
controlled shear flow conditions. For this thesis, a TAInstruments AR2000 rheometer, with a cone-and-
plate geometry fixture, was used to characterise the fluids rheology. A cone-and-plate fixture, shown in
figure 2.4, consists of a flat plate and a shallow-angled cone, generally 2 to 4 degrees, with a fluid sample
placed between them.
Figure 2.4: Cone-and-plate rheometer fixture
For this geometry, as the cone rotates about its shaft axis, the shear rate is the same everywhere
and is given by:
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'( = KL (. �@)
where K is the angular velocity and L is the cone angle. The torque and normal force, each measured by a separate sensor, are the directly-measured outputs from the rheometer. These measurements can then be
used to determine fluid properties. The relation between shear stress + and torque �, for a cone of radius M is defined as (Barrnes et al., 1989):
+ = 2�NM2 (. �A)
which can then be related to viscosity by Eq 2.2 as:
� = 2�LNKM2 (. �B)
allowing viscosity to be measured as a function of shear rate. The first normal stress difference is
determined from the measured total force O on the plate as (Barrnes et al., 1989):
�� = ONM (. �)
The dynamic responses �′ and �′′ of a fluid can be determined by relating Eq. 2.6 to the
measured amplitude of torque response +( ) and its phase shift with applied oscillatory strain.
Rheometric testing on the AR2000 for steady shear can be done by controlling either the shear
stress or the shear rate. For all testing done in this thesis, the input variable was the shear rate because the
experimental process, described in section 3.3, was done for different flow rates.
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2.2 HISTORICAL BACKGROUND
2.2.1 Prior Newtonian Research
The prediction of drag on a cylinder in a creeping flow started in 1851, after Stokes derived an
expression for the drag Fd on a sphere of radius � falling through a viscous fluid to be:
O$ = �N��#(. �)
This formula was derived by solving the Navier-Stokes equations for Re approaching zero, and
has been termed Stokes’ Law or Stokes’ drag. Subsequently, Stokes attempted to solve the flow around a
cylinder. However, he could not find a solution which satisfied both the near and far field boundary
conditions. Stokes regarded this as an indication that no steady flow situation existed, and the problem,
referred to as Stokes’ Paradox, was not resolved until a century later.
Oseen (1910) was the first to give an explanation for the paradox’s existence and provide a partial
resolution (Van Dyke, 1964). He noted that the paradox arises from the assumption that inertial forces
are negligible everywhere. Even when Re << 1, this assumption breaks down and inertial and viscous
stresses have equal orders-of-magnitude in the far field. To solve this issue, Oseen developed an
approximate solution by linearizing the inertial terms in the Navier-Stokes equations. Using a similar
methodology, Lamb (1911) also provided a formula for the planar case (Van Dyke, 1964). Both
formulations describe the far field well, but they break down near the object surface.
The most accepted resolution of Stokes Paradox was presented by Kaplun (1957), and separately
by Proudman and Pearson (1957), through a mathematical technique known as matched asymptotic
expansions (Van Dyke, 1964). The details of this technique are complicated, but a simple description is
to take the solution which is valid near the surface and the solution which is valid in the far field, called
the inner and outer solutions respectively, and matching them where they overlap to generate an
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asymptotic series to form a continuous solution. Accounting for both inertial and viscous stresses, the
resulting formula for the viscous drag coefficient �� for a cylinder of length P, Kaplun’s solution is:
�� = O$�N�#P = �� − �. A@�2 (. )
where� = � − ' − ��U��
A V
and' = �. �@@�.
Although Eq. 2.22 is an asymptotic solution, the formula is highly convergent as Re decreases.
Ironically, while the formula accurately approximates the drag for Re << 1, it is technically not a Stokes’
flow solution because it depends on inertia.
Attempts have been made to improve upon Kaplun’s solution, such as that by Skinner (1975),
but, as can be seen in figure 2.5, all of the proposed solutions differ by less than 10% for Re << 1.
Because Kaplun’s solution provides the simplest, most accurate mathematical relation for ��, it will be
the theoretical basis for the present experimental work.
Very few experimental investigations of drag for Newtonian Stokes flow have been recorded, the
first and best known of which was performed originally by White (1946). He allowed small cylinders to
fall through different viscous fluids contained in jars and tanks, and measured their terminal velocities to
determine drag. To correct for the finite length of cylinders, White used cylinders of the same diameter
but varying length and extrapolated the drag to zero length. This zero length drag was then subtracted
from the total drag. From his data, White proposed the following formula for the drag coefficient ��of a cylinder moving between parallel walls separated by a distance Z which is independent of Re:
�� = �. ��N F;[ \Z$].(. 2)
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Figure 2.5: Comparison of various analytical solutions for the dimensionless drag on a cylinder at low Re
When White performed his experiment, the mathematics that describe wall effects did not yet
exist. Also, although his H/d ratio was as large as 600, later work showed that wall effects are still
significant. This strong dependence on H/d ratio makes a comparison to Kaplun’s unbounded solution
inappropriate.
Faxen (1946) was the first to give an analytical formula for the drag on a cylinder between
moving parallel walls in Stokes’ flow as:
�� = �F7\Z$] − �. B@��@ + �. @��\$Z] − �. @2� \$Z]� (. �)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00001 0.0001 0.001 0.01 0.1
Vis
cou
s D
rag
Co
eff
icie
nt
Re
Lamb's Solution - 1911
Kaplun - 1 Term Solution - 1957
Kaplun - 2 Term Solution - 1957
Skinner's Solution - 1975
16
Shown in figure 2.6, is a comparison of White’s and Faxen’s formulas, at various arbitrary H/d
values.
Figure 2.6: Comparison of equations 2.23 and 2.24 to Kaplun’s unbounded solution
While the formulas provide identical results for sufficiently high H/d ratios, the differences from
Kaplun’s solution illustrate the difficulty in creating an unbounded flow simply by moving the walls away
from the cylinder.
Because creating a completely unbounded flow is physically impossible, to make comparisons to
Kaplun’s unbounded solution, the effect of walls must be corrected for by using theoretical formulas,
such as Faxen’s formula. One such set of corrected data, displayed in figure 2.7, is from Jayaweera and
Mason (1965), who took drag measurements for cylinders falling through viscous fluids.
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.001 0.01
En
d C
orr
ect
ed
Dra
g C
oe
ffic
ien
t
Re
Faxen H/d = 50
White H/d = 50
Faxen H/d = 167
White H/d = 167
Faxen H/d = 500
White H/d = 500
Kaplun
17
Figure 2.7: Jayaweera and Mason’s 1965 experimental data corrected for wall effects by using Faxen’s
formula compared to Kaplun’s solution
With reasonable agreement between analytical solutions and experimental work, no further investigation
of drag on a cylinder has been conducted for the past 50 years, at least for Newtonian fluids.
2.2.2 Prior Viscoelastic Research
For Stokes flow of a viscoelastic fluid around a cylinder, very few published investigations exist,
and the papers which were found did not provide any definite conclusions about drag. For the case of an
unbounded cylinder, no experimental investigations were found.
The numerical work of Chilcott and Rallison (1988), who simulated flow of an Oldroyd-B fluid
around a sphere and cylinder, is the only significant study of unbounded viscoelastic Stokes flow around a
cylinder. For the far field boundary conditions, they defined a circular boundary, between 10 to 20 times
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.01 0.1 1
Cη
= F
/4π
ηU
L
Re
Kaplun
Jayaweera and Mason
18
the cylinder diameter, on which velocity was assumed to be uniform, and meshed the region between
these concentric circles in a uniform pattern with increasing resolution towards the object. This mesh was
further refined to examine the region directly downstream of the rear stagnation point. While this setup
does not provide a true simulation of unbounded flow because their numerical scheme is for an inner
solution bounded by their mesh, their paper serves as an approximate exploration.
In their numerical simulations, Chilcott and Rallison allowed for different maximum polymer
concentrations and different polymer extensibilities over time by using the Finitely Extensible Non-linear
Elastic (FENE) model. They used a time marching solution where the polymers were unstretched and the
outer boundary velocity was made uniform at t > 0. By using formulas which described the stream
function, polymer extension, spring force, and drag force, they ran the simulation until convergence was
achieved. Their results are given in terms of De, as defined earlier. For both a sphere and cylinder, they
predict that as De approaches zero, the viscoelastic flow is equivalent to its Newtonian counterpart,
implying that below a critical value of De, elastic effects are negligible.
For a cylinder, Chilcott and Rallison predict that, above onset, the drag increases with increasing
De, to about 1.2 to 2.5 times the Newtonian equivalent in the De range of 10 to 15. This increase in drag
is attributed to elastic effects. At De values beyond their maximum tested range, Chilcott and Rallison
posited that the viscoelastic drag reaches an asymptote. Their results are further examined and compared
to the findings of this thesis in section 4.2.4.
In the case of spheres, onset behaviour was observed experimentally by several researchers, as
reviewed by James (2009). Experiments were conducted by dropping spheres through viscous fluids and
using their terminal velocities to determine the drag coefficient, similar to White’s experiment.
Experimental results agree that, as De increases from 0, an initial drag reduction occurs, producing values
lower than the Newtonian equivalent; as De increases further, the drag is enhanced by elasticity above the
Newtonian equivalent. Chillcot and Rallison also predict this behaviour in their numerical work for a
19
sphere. However, the De value at which the change from reduction to enhancement occurs is not agreed
upon, nor is the total amount of reduction or enhancement. The reason for such disagreement between
results is currently unknown. When comparing the drag behaviour between spheres and cylinders,
Chilcott and Rallison predict that while drag enhancement eventually occurs as De increases, no region of
drag reduction for a cylinder will occur. Once again, the reason for such behaviour is not fully
understood.
Chilcott and Rallison’s work was restricted by computational capabilities of their day, which
severely limited the mesh size, mesh refinement, De range, time stepping, and convergence criteria.
Nonetheless, their results were still a significant advancement towards understanding viscoelastic flow
behaviour around submerged bodies. Another significant shortcoming is that they did not provide a
comparison of their methodology to accepted theoretical solutions using a purely Newtonian simulation,
which brings into question their non-Newtonian results. Others, such as Huang and Feng (1995), have
run similar numerical simulations, but no significant improvement in understanding has arisen, and the
lack of experimental validation makes these simulations questionable as well.
While no experimental investigations of viscoelastic flow around an isolated cylinder have been
found, a small number of works have dealt with a cylinder confined between parallel channel walls. The
geometry of choice has been a width to diameter ratio (H/d) of 2:1 where the channel and cylinder are
fixed and fluid is pumped through the channel. The experimental works of McKinley et al (1993), and
Verhelst and Nieuwstadt (2004), and the numerical work of Xiong et al (2013), are all recent papers
which use this ratio.
The typical setup for such an experiment, displayed in plan and side views in figures 2.8a and
2.8b respectively, has been an optically clear horizontal channel with a length is of order 100 times the
diameter, and a vertical height (Z) of order 10 times the diameter. A steady parabolic velocity profile is
usually assumed upstream and a roughly flat velocity profile is assumed along the cylinder axis, as shown
20
in figure 2.9. This geometry is designed to create a region of approximately two-dimensional flow around
a cylinder. However, the parabolic upstream velocity across the channel width deviates significantly from
the uniform profile expected of an unbounded flow. Additionally, the region between the channel walls
and the cylinder is one of mixed flow, where shearing and extension are both significant instead of being
shear dominated as expected from an unbounded flow.
Figure 2.8a: Plan view of a confined cylinder experiment where channel width (H) > diameter (d)
Figure 2.8b: Side view of a confined cylinder
Figure 2.9a: Plan view of upstream velocity profiles for confined cylinder flow
21
Figure 2.9b: Side view of upstream velocity profile for confined cylinder flow
Verhelst and Nieuwstadt measured both drag force and velocity profiles for Re in the range of
0.05 to 0.50 and De between 0.1 and 4. They used particle image velocimetry (PIV) to observe the
changes in velocity profile at multiple locations along their channel, both from plan and side perspectives.
Their results indicate that above a critical value, the drag increases with De above the Newtonian
equivalent, similar to Chilcott and Rallison’s prediction. However, their data had discrepancies of 15 to
20 percent compared with their own two-dimensional numerical predictions for this geometry. Upon
further investigation, they discovered that the flow around their cylinder was strongly three-dimensional,
with a flow component in the axial direction of the cylinder. Their discovery implies that it is difficult to
make comparisons between this flow geometry and a two dimensional unbounded flow.
Francois et al. (2008) measured the drag on confined cylinder for H/d up to 25 using a
microfluidics setup. Their work, however, suffered from three-dimensional issues as well, as their
channel height to cylinder length ratio was only 2:1. With this geometry, the upstream flow field was
parabolic along both axes, which forced them to determine the average velocity experienced by the
cylinder by integrating along such a profile. Their data show that an enhancement of drag occurs above
some critical De value; however, not much else can be determined from their results.
A recent Master’s thesis by Wang (2012) explored a new technique for measuring drag on a
cylinder in a Stokes flow. This work aimed at providing methods to compensate for cylinder ends and
channel walls. While the experiment provided results for Newtonian flow that could be related to
22
Kaplun’s unbounded solution, the precision of the equipment and the De range investigated did not allow
conclusions about onset or the amount of drag enhancement to be made. Since no other published
experimental work was found for unbounded viscoelastic flow around a cylinder, and since Wang’s work
suggested a possibly useful technique, the present thesis aimed to develop the technique further because
of the need for reliable drag data, as described earlier.
2.3 Research Objectives
For this thesis, the primary objective was to design and conduct an experiment representative of
unbounded creeping viscoelastic flow past a cylinder. The focus was to examine the effect of elasticity
on the drag.
The specific experimental region of interest was for De between 0.1 and 100. This range was
chosen because previous numerical predictions, such as those by Chilcott and Rallison’s, indicate that an
onset value for elastic effects should exist around De = 1, while microorganism swimming should occur
for De between 10 and 100. The goals of this research were therefore:
• To simulate two-dimensional, uniform, unbounded Stokes flow around a cylinder experimentally.
• To measure drag with Newtonian and viscoelastic fluids, specifically Boger fluids.
• To measure the drag force on a cylinder for De ranging from 0.1 to 100.
• To identify the onset De value, above which elasticity enhances drag.
• To determine differences in drag between Newtonian and viscoelastic fluids.
23
CHAPTER 3: EXPERIMENT DESIGN
3.1 Test Fluids
To perform experiments simulating unbounded flow, large quantities of stable, well-characterized
fluids were necessary. Several types of fluids of sufficient quantity and appropriate properties were
already available in the Flow Measurements Lab at the University of Toronto. They were used as is, and
experimental equipment was designed around them. For consistency, all rheological testing was done on
the TA Instruments AR2000 using a 40 mm diameter, 2° angle cone. Fluid characterizations were
performed throughout the experimental process to ensure accurate fluid properties.
It was discovered that several fluid properties depended significantly on the temperature of the
environment; therefore, fluid characterisations were performed over a range of temperatures ^ [°C], and equations describing the behaviour of their properties were created. The only exception was the
measurement of fluid density, since it was expected to vary less than 1% over the range of possible
laboratory temperatures.
3.1.1 Newtonian Fluid Characterizations
A silicone oil and a polybutene were available in sufficient quantity to be used as Newtonian test
fluids. Because the experimental range was Re < 0.1, the only fluid properties of importance were the
viscosity and density, which were measured using the AR2000 and a precision scale, respectively.
The first fluid to be characterised was the silicone oil. Figure 3.1a shows viscosity versus shear
rate measured at 21°C, and figure 3.1b shows the relationship between viscosity and temperature for the
most recent fluid characterization.
24
Figure 3.1a: Viscosity of the silicone oil at 21°C
Figure 3.1b: Viscosity-temperature relation for the silicone oil, with a linear curve fit
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.1 1 10 100
Vis
cosi
ty [
Pa
.s]
Shear rate [1/s]
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
17 18 19 20 21 22 23 24 25
Vis
cosi
ty [
Pa
.s]
Temperature [ ̊C]
25
Figure 3.2 shows the equivalent graphs for the polybutene.
Figure 3.2a: Viscosity of the polybutene at 21°C
Figure 3.2b: Viscosity-temperature relation for the polybutene, with a polynomial curve fit
0
5
10
15
20
25
30
35
40
0.001 0.01 0.1 1 10
Vis
cosi
ty [
Pa
.s]
Shear Rate [1/s]
0
5
10
15
20
25
30
35
40
45
50
17 18 19 20 21 22 23
Vis
cosi
ty [
Pa
.s]
Temperature [ ̊C]
26
As can be seen from figure 3.2a, the polybutene viscosity exhibits a region of non-Newtonian
behaviour at shear rates from 0.001 to 0.03. This region was ignored because the estimated shear rates for
this fluid during the experiment were above 0.1; therefore, the viscosity values used to create the
viscosity-temperature relation were the constant values above a shear rate of 0.1.
A summary of measured fluid properties and temperature T [°C] relations for the silicone oil and
the polybutene are provided in table 3.1.
Silicone Oil Polybutene
� vs. ^Relation _ = −0.023^ + 1.53 _ = 0.20^d − 11.29^ + 179.93 Density [g[/G2] 1009 957
Table 3.1: Relevant rheological information for the Newtonian test fluids
3.1.2 Boger Fluid Characterizations
Two Boger fluids were available, referred to as T1 and T3, which were originally developed by
Dr. Ronnie Yip at the University of Toronto (2011). Both fluids were dilute polymer solutions of long-
chain polyisobutelene in a solvent mixture of polybutene (PB) and kerosene. The original composition of
these fluids is listed in Table 3.2.
Fluid Composition
T1 0.2 wt% PIB
ij = 4.7 × 10lg/mol 92.8 wt% PB
ij = 635g/mol 7 wt% Kerosene
T3 0.2 wt% PIB
ij = 4.7 × 10lg/mol 92.8 wt% PB
ij = 910g/mol 7 wt% Kerosene
Table 3.2: Original Boger fluid composition in wt% and molecular weight ��, from Yip (2011)
27
Because these fluids were created several years prior, the kerosene may have evaporated to some
extent, and small amounts of mixing between the T1 and T3 fluids may have occurred. These factors may
have led to slightly different viscous responses because the two fluids had different PB solvents.
Since multiple rheological characterizations were performed, the most recent data are presented
here. For T1 and T3, the first property to be characterized was viscosity. Sample characterization plots
of shear and dynamic viscosity are shown in figures 3.3a for T1 at 16°C and 3.3b for T3 at 22°C.
The vertical dashed lines in figure 3.3 indicate the range of shear rates over which the
experiments were conducted, as determined by computer simulation. The viscosity does vary somewhat
over the experimental range; however, for consistency, the value of � used to calculate flow properties was the maximum shear viscosity value determined for each temperature, which is expected to be equal to
the dynamic viscosity at low frequencies. As described in section 2.1.3, the dynamic viscosity at high can be used to estimate the solvent viscosity ��. The polymer viscosity �� can then be estimated as:
�� = � − ��.(2. �)
The shear and dynamic viscosities responses at various shear rates and angular frequencies were
similar at all temperatures for both T1 and T3; thus, temperature was treated as a scaling factor. The
variations of viscosity with temperature for T1 and T3 are presented in figure 3.4
28
Figure 3.3a: Viscous properties for the T1 fluid at 16°C
Figure 3.3b: Viscous properties for the T3 fluid at 22°C
0
2
4
6
8
10
12
14
0.01 0.1 1 10 100 1000
Sh
ea
r V
isco
sity
, D
yn
am
ic V
isco
sity
[P
a.s
]
Shear Rate, Angular Frequency [1/s]
Shear Viscosity
Dynamic Viscosity
Low Re Experimental Range
0
5
10
15
20
25
30
35
0.001 0.01 0.1 1 10 100 1000
Sh
ea
r V
isco
sity
, D
yn
am
ic V
isco
sity
[P
a.s
]
Shear Rate, Angular Frequency [1/s]
Shear ViscosityDynamic ViscosityLow Re Experimental Range
29
Figure 3.4a: Curve fit relations for viscosity and temperature for the T1 fluid
Figure 3.4b: Curve fit relations for viscosity and temperature for the T3 fluid
0
2
4
6
8
10
12
14
16 17 18 19 20 21 22
Ma
xim
um
Sh
ea
r V
isco
sity
, P
oly
me
r V
isco
sity
[P
a.s
]
Temperature [°C]
Shear Viscosity
Polymer Viscosity
0
10
20
30
40
50
60
18 19 20 21 22 23 24
Ma
xim
um
Sh
ea
r V
isco
sity
, P
oly
me
r V
isco
sity
[P
a.s
]
Temperature [°C]
Shear Viscosity
Polymer Viscosity
30
Measurements of �� were also taken for T1 and T3. Sample plots of �� data, corresponding to
the figure 3.3 temperatures, presented as 3� versus '( , are shown in figure 3.5.
Figure 3.5a: First normal stress coefficient for the T1 fluid at 16°C
Figure 3.5b: First normal stress coefficient for the T3 fluid at 22°C
0
2
4
6
8
10
12
1 10
Fir
st N
orm
al
Str
ess
Co
eff
icie
nt
Shear Rate [1/s]
0
10
20
30
40
50
60
70
80
1 10
Fir
st N
orm
al
Str
ess
Co
eff
icie
nt
Shear Rate [1/s]
31
Section 2.1.3 indicated that, for the Oldroyd-B model, 3� should be constant versus shear rate.
As seen in figure 3.5 however, 3� varies with shear rate. Since 3� should ideally be constant, the value selected was an average, calculated from measurements which varied less than 10% from neighbouring
data points as indicated by the horizontal dashed line. By using Eq. 2.14, the measured value of ��, and
the selected values of 3�, the relaxation time ?�� was determined from ��, with values presented in
figure 3.6 versus temperature.
Figure 3.6a: Curve fit relations for the�� relaxation time with temperature, for the T1 fluid
Figure 3.6b: Curve fit relations for the�� relaxation time with temperature, for the T3 fluid
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
16 17 18 19 20 21 22
N1
Re
lax
ati
on
Tim
e [
s]
Temperature [°C]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
18 19 20 21 22 23 24
N1
Re
lax
ati
on
Tim
e [
s]
Temperature [°C]
32
As mentioned in section 2.1.3, �′/ should become constant as → � for an Oldroyd-B fluid. Therefore, �′ data for T1 and T3 are shown in figure 3.7, as �′/versus angular frequency.
Figure 3.7a: �′/ data for the T1 fluid at 16°C
Figure 3.7b: �′/ data for the T3 fluid at 22°C
0.01
0.1
1
10
100
0.01 0.1 1 10 100 1000
G'/
ω2
Angular Frequency [1/s]
0.01
0.1
1
10
100
1000
0.1 1 10 100
G'/
ω2
Angulary frequency [1/s]
33
Figure 3.7 demonstrates asymptotic behaviour beginning to develop below = 0.1, and the actual asymptote is reached below the tested range. Unfortunately, as → 0, the time to make �′
measurements increased exponentially along with the noise associated with the AR2000 measurements;
therefore, the value of �′/ used to determine �′ relaxation time from Eq. (2.16), was an estimated
asymptotic value, shown in figure 3.7 as a dashed line. From the values of ��, and the asymptotic values
of �′/, the relaxation time based on �′, are presented in figure 3.8 versus temperature.
Figure 3.8a: �′ relaxation time vs. temperature for the T1 fluid
Figure 3.8a: �′ relaxation time vs. temperature for the T3 fluid
0
1
2
3
4
5
6
7
8
16 17 18 19 20 21 22
G' r
ela
xa
tio
n t
ime
[s]
Temperature [°C]
0
2
4
6
8
10
12
18 19 20 21 22 23 24
G' R
ela
xa
tio
n T
ime
[s]
Temperature [°C]
34
Because of the difficulty in acquiring accurate measurements of �′ at low enough angular
frequencies to determine an asymptote, the estimations of �′ relaxation time (?�′) were considered less
reliable than those for �� relaxation time (?��).
A summary of equations for the fluid properties of T1 and T3 are presented in table 3.4.
Relation T1 T3
� vs. Temperature _ = 1860 × ^�s.tu _ = 10940 × ^�s.tu ��vs. Temperature _v = 722 × ^�s.td _v = 6610 × ^�d.ws ?�� vs. Temperature xys = 39.9 × ^�s.dl xys = 179 × ^�s.zu ?�′ vs. Temperature x{′ = 920 × |�d.us x{′ = 395 × ^�s.dz
Table 3.3: Relations for relevant fluid characteristics vs. temperature for the T1 and T3 fluids
3.1.3 Selection of Relaxation Time
The relaxation time based on �� was selected for this work because it could be more accurately
measured. The selection was also based on the fact that the relaxation time based on �′ was determined
by an oscillatory process which poorly reflects the flow situation occurring during the experiment.
Furthermore, in the work of James et al. (2012), drag enhancement for cylinders was related to normal
stresses. This work described how an asymmetry in shear rates around a cylinder caused a pressure
gradient between the upstream and downstream sides of a cylinder in a viscoelastic flow due to polymer
dynamics which can be associated with ��.
35
3.2 Experimental Apparatus
The basic requirements of the experiment were to provide a uniform Stokes flow and to make
accurate measurements of drag on a micro-Newton scale.
The apparatus consisted of three components: a container to hold the test fluids, a system to
control the flow rate, and a transducer to measure drag. Based on the design by Wang (2012), an annular
tank mounted on a slowly rotating turntable was used to achieve a steady known flow field which was
nearly uniform on the scale of the cylinder as shown in figure 3.9.
Figure 3.9: Velocity profile as seen by a cylinder in the rotating annular tank
The measurement of drag was achieved by partially immersing a small vertical cylinder, attached to a
flexible cantilever beam, and relating the resulting beam deflection to the applied force. This
experimental setup had an advantage over White’s 1946 experiment in that the flow velocity could be
varied over a wide range, allowing a single fluid and cylinder to give results over broad ranges of Re and
De. The existing equipment provided a first-generation system which was used to verify and refine the
experimental process. A second-generation system was later designed to investigate the onset of elastic
effects. A detailed description of the apparatus is given in the following sections.
36
3.2.1 Annular Tank
The annular tank is shown schematically in figure 3.10. Two tank sizes were available with
dimensions given in table 3.4. Each tank was created by joining two tubes of different diameters to a flat
circular plate. These tubes and the circular plate were made of optically clear acrylic plastic in order to
allow PIV measurements to be taken, though such measurements were not made for this thesis. While not
a uniform flow as illustrated in figure 3.9, the solid body rotation of the annular tank served as the known
flow field as shown in figure 3.11.
Figure 3.10: Annular tank views
Inner Diameter [mm] Outer Diameter [mm] Depth [mm]
Tank 1 190.5 495.3 114.3
Tank 2 177.8 444.5 127.0
Table 3.4: Annular tank dimensions
37
Figure 3.11: Velocity flow field in a steadily rotating annular tank
3.2.2 Drive System
A large industrial strength turntable, shown schematically with the annular tank mounted on it in
figure 3.12, was available for use and provided the first-generation drive system. The turntable consisted
of a clockwise-rotating ½ inch thick metal disk supported by low friction bearings, driven by a heavy duty
2-hp DC motor, and controlled by an analog voltage controller. At its maximum rated 220V, the angular
velocity was 4.19 rad/s. By supplying less voltage, the angular velocity could be slowed to as low as 0.11
rad/s. The turntable easily supported the weight of a full tank, showing no decrease in angular velocity
compared to being unloaded. Unfortunately, due to voltage fluctuation within the building’s electrical
system, the speed controller setting could not be related to the turntable rotational speed.
38
Figure 3.12: A 3D schematic view of the turntable and tank
This system was used to make the first measurements, and the experimental results it produced
helped design a second-generation drive system, for measurements at lower speeds and specifically to
find the onset of elastic effects.
3.2.3 Force Transducer
Several rods were available in the lab as test cylinders, ranging from 0.50 mm to 3.18 mm.
Table 3.6 lists dimensions and materials of the rods. Their lengths were selected to allow the immersion
depth to be up to a third of the tank depth, and the maximum diameter was less than 1/40 of the annular
width to keep the walls as far from the cylinder as possible in either tank.
39
Cylinder Diameter (mm) Cylinder Length (mm) Cylinder Material
0.50 40 Steel
0.97 50 Steel
1.19 60 Brass
1.58 65 Steel
2.17 65 Brass
3.18 45 Acrylic
3.34 65 Acrylic
Table 3.5: Test cylinder dimensions and materials
No standard force transducer was available for this experiment. Commercial force transducers,
such as strain or spring gauges, were incapable of measuring the small forces induced by the slow flows,
and microscopic force transducers would have required isolated mounting. Custom instruments could
have been acquired but were prohibitively expensive.
Accordingly, transducers were designed and built from available materials. The first-generation
design consisted of a test cylinder attached to a flexible cantilever beam with a mirror fixed at the end of
the beam and a scale positioned across the room, as illustrated schematically in figure 3.13. The mirror
and scale were used to determine the cantilever beam deflection more accurately than by directly
measuring so with a ruler, and will be explained shortly. The cylinder was fixed to a sliding mechanism
that was fitted onto the beam, allowing it to be positioned between the mirror and fixed end of the
cantilever. This mechanism also included a vertical slider, which allowed the cylinder to be raised and
lowered while millimeter ruler markings were used to determine the immersion depth. Figure 3.14 shows
the free end of the transducer system along with the mirror, cylinder, and slider.
40
Figure 3.13: Schematic plan view of the transducer setup, including the laser and scale.
Figure 3.14: Side view of the first-generation high force transducer with the slider mechanism.
41
Because excessive deflection of the cantilever would lead to warping of the beam, only small
deflections were allowed, which were challenging to measure accurately. Wang (2012) measured the
deflection directly, using a millimetre ruler and a camera to enhance the beam deflection digitally by
enlarging the photographs on a computer. This methodology was time consuming and did not allow for
the vibrations of the beam to be taken into account, which were discovered to be significant. To provide
an enhanced reading of the cantilever deflection, which could be read quickly during the experimental
process, a laser beam was reflected off the attached mirror. The laser would then strike a scale positioned
between 2 to 4 meters away from the cantilever system, as shown in figure 3.13. When the cantilever
deflected, the laser beam travelled in the horizontal direction on the scale. The further the scale was
placed away, the more the cantilever deflection was enhanced, resulting in a magnification of one to two
orders of magnitude.
The aluminum beam used by Wang (2012) served as the first-generation transducer cantilever.
Aluminium was selected for to its light weight, low elastic modulus, and high yield stress, allowing large
deflections of the beam without plastic deformation. The aluminium beam dimensions and relevant
engineering properties are listed in table 3.6.
Elastic Modulus (E) 69 GPa
Width (b) 1.59 mm
Height (h) 12.7 mm
Area moment of inertia (I) 4.23 × 10-12 m4
Table 3.6: Aluminium cantilever beam properties and dimensions.
According to Gere (2004), a cantilever beam exposed to a point force P through its centroid, has
an end deflection L of:
L = }�~�,(2. )
where l is the beam length from the fixed end to the point force,
moment of inertia. For a rectangular cantilever beam,
� = �2�� ,
where b is the beam thickness parallel to the applied force, and
to the applied force, as illustrated in figure 3.15.
Figure 3.15:
In order to avoid excessive bending, the maximum beam deflection angle was
radians. For the cantilever beam, the maximum force which could be applied
}��* = ~�� L = ~�2�
��N�
The advantage of this transducer system was the wide range of forces
could measure. As fluid viscosities ranged from 1 to 58 Pa.s,
and the mirror-to-scale distance meant that a
42
is the beam length from the fixed end to the point force, E is the elastic modulus, and
moment of inertia. For a rectangular cantilever beam, I is equal to:
is the beam thickness parallel to the applied force, and h is the width of the beam perpendicular
as illustrated in figure 3.15.
Figure 3.15: Simple bending of a cantilever beam
In order to avoid excessive bending, the maximum beam deflection angle was less than
or the cantilever beam, the maximum force which could be applied is then:
N�A.
The advantage of this transducer system was the wide range of forces with
. As fluid viscosities ranged from 1 to 58 Pa.s, the ability to vary both the cantilever length
meant that a single transducer could be used for a wide range of tests.
is the elastic modulus, and I is the area
(2. 2)
of the beam perpendicular
less than 10° or </18
(2. �)
equal resolution it
the cantilever length
be used for a wide range of tests.
43
The first-generation system was used to take measurements of drag for De ranging from 10 to
100. Because the onset of elastic effects was expected to occur around De = 1, both the motor and
transducer system were re-designed.
3.2.4 Second-Generation Apparatus
After disengaging the original motor system, a lower speed DC motor was installed to rotate the
existing turntable at smaller velocities. By doing so, much of the same equipment used for the high Re
system could still be used, such as the fluid tanks and the support structure to which the force transducer
was attached.
In order to achieve a De range between 0.1 and 10 using the same fluids, a roughly 100 times
reduction in turntable angular velocity was required. By driving the turntable with a small friction wheel
attached to the second DC motor, the large turntable surface acted as a natural gear reduction with a ratio
of 20:1. Thus, a motor with an angular velocity 5 times lower than the original motor system was
selected as the second DC motor.
A new digital voltage speed controller, attached to the new motor, greatly decreased the time
required to make experimental measurements because it displayed the exact input voltages �. Subsequently, an equation relating these voltages to turntable RPM was created. The relationship of
voltage input versus turntable RPM is shown in figure 3.16.
44
Figure 3.16: Speed relation for low speed motor
Horsepower 1/100
Max Motor RPM 4
Maximum Turntable [rad/s] 0.022
Minimum Turntable [rad/s] 0.0016
Table 3.7: Low speed DC motor specifications
The relation between turntable RPM and input voltage for the low speed DC motor was then:
�}� = �. ����� × ��.��, �M = �. ����� × ��.��(2. �)
By using angular velocity, the linear velocity U at the location of the immersed rod was calculated as:
# = ��(2. �)
where �� is the distance from the annular tank centre to the immersed rod’s vertical axis, located at the
centre of the channel.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
4 6 8 10 12 14 16
Tu
rnta
ble
RP
M
Input Voltage [V]
45
With fluid velocities 100 times lower, a new cantilever beam had to be created. Equation 3.4
shows that a beam with smaller dimensions and lower elastic modulus was required. However, too long
or too flexible a beam would result in it deflecting under its own weight or the weight of the attached
cylinder and mirror. After a trial and error process with available materials, a polycarbonate beam, about
45 times more flexible than the aluminum beam, was chosen to replace the first beam. Though this
second beam was only about half as flexible as desired, by adjusting the cantilever length and distance to
the scale, it was possible to achieve an equal level of resolution as the first-generation system. The
dimensions and relevant engineering properties of the polycarbonate beam are listed in table 3.8.
Elastic Modulus (E) 2.2 Gpa
Width (b) 1.5 mm
Height (h) 12 mm
Area moment of inertia (I) 3.38 × 10-12 m4
Table 3.8: Polycarbonate cantilever beam properties and dimensions
A major difference between the two transducer systems was that the polycarbonate beam could
not support a fixture to vary cylinder immersion depth. Instead, a cylinder was attached directly to the
cantilever, which was then attached to a digital calliper which could raise and lower the entire transducer
assembly. The calliper also increased the accuracy of measuring the immersion depth by 2 orders of
magnitude. This required change solved a non-linear issue, which will be explained in section 3.3.1.
3.2.5 Final Equipment Setup
To prepare for drag measurements, an annular tank was filled to within 10 mm of its top edge to
achieve sufficient fluid depth for cylinder immersion, while preventing a spill from sudden acceleration.
Markings drawn on both the annular tank and the turntable surface were used to ensure proper concentric
46
alignment. The marking pattern, shown in figure 3.17, was also used to align the cantilever along a line
perpendicular to the flow, and enabled for the cylinder to be placed mid-channel.
Figure 3.17: Container module with alignment pattern overlay
The transducer module was attached to a rigid support above the turntable. This support could be
positioned in three dimensions, allowing for a multitude of transducer positions. Finally, the laser was
positioned so that it would reflect off the transducer mirror to the scale.
Shown in figure 3.18 are side and 3D views of the final experimental setup for the first-
generation system, excluding the laser and scale positioned away from the turntable.
47
Figure 3.18a: Side view of the first-generation full experimental assembly
Figure 3.18b: 3D view of above assembly
48
3.3 Experimental Procedure
3.3.1 Transducer Calibration
After positioning the tank and transducer, the transducer was calibrated with a set of known
weights. Because the transducer length and scale position were both varied to achieve appropriate
measurement resolution, a single relation describing the transducer’s response to applied force was
impractical and a calibration was required for every experimental run.
To calibrate the transducer, a zero depth position for the cylinder was marked by lowering the
cylinder to just touch the fluid surface, and a zero force position was marked on the scale. The cylinder
was then raised, the annular tank moved aside, and a calibration system was attached to the rigid support.
The system is shown schematically in figure 3.19.
Figure 3.19: Schematic side view of the calibration system indicating direction of cantilever deflection
49
The calibration system consisted of a pulley or low friction curved surface aligned directly behind
the cylinder. Fluid flow was assumed to be evenly distributed along the cylinder, which was simulated as
a point force at the centre of the immersion depth. A horizontal thread loop was attached at the
immersion centre on the cylinder and draped over the pulley. Three locations on the cylinder were
marked, representing centres of immersion between the smallest and largest immersion depths. Then by
positioning the calibration pulley accordingly, the response of the transducer was mapped by applying
known weights and recording the location of the reflected laser beam on the scale.
The translation of the laser beam on the scale was not perfectly horizontal and had a vertical
component. This component occurred because the force applied to the transducer was not through the
cantilever beam’s centroid but rather some distance below it. This off-centre force caused the cantilever
to twist, as shown in figure 3.20.
Figure 3.20: Cantilever torsion caused by off centre force
50
It was found that mapping only the horizontal translation of the laser was necessary to accurately describe
the force applied to the cylinder.
To select appropriate calibration weights, Faxen’s formula, Eq. (2.24), was used to estimate the
expected drag. To then estimate the drag with a Boger fluid, the equivalent Newtonian prediction was
multiplied by a value up to 3, depending on the De range.
For the high-force transducer, different immersion depths resulted in different horizontal
displacement of the laser beam when the same calibration force was applied. To relate displacement to
applied force at immersion depths not tested during the calibration, a relation describing the laser
displacement versus immersion depths for the same applied force was created, as shown in figure 3.21.
A similar relation for each applied weight was created, and force versus displacement calibrations were
then created by interpolating the displacement at each immersion depth used in the experiment.
Figure 3.21: Sample relation describing high force transducer’s laser displacement vs. immersion depth
for the same applied force
39.80
40.00
40.20
40.40
40.60
40.80
41.00
41.20
41.40
41.60
41.80
0.00 5.00 10.00 15.00 20.00 25.00 30.00
Lase
r x
-dis
pla
cem
en
t [m
m]
Immersion Depth [mm]
51
The calibration of a transducer for a specific immersion depth was unexpectedly non-linear at times, so a
spline technique was used. A sample calibration of the high force transducer for one immersion depth is
shown in figure 3.22.
Figure 3.22: Sample high force transducer calibration with three different spline regions
By using various weights and the zero point, the transducer calibration consisted of up to three
equations for a single immersion depth. The different displacements in figure 3.21, and the non-linearity
in figure 3.22, were likely caused by the weight of the slider mechanism. That is, while the slider allowed
movement along the cantilever beam and vertical positioning of the cylinder, the shifting weight resulted
in different torques being applied to the cantilever. This meant that the cantilever would have a different
amount of torsion for each vertical position of the slider without any applied force, which would change
its bending behaviour slightly when force was applied.
0.0
0.5
1.0
1.5
2.0
2.5
0 20 40 60 80 100 120 140 160
Ca
lib
rati
on
Fro
ce [
N]
Laser Horizontal Translation [mm]
Low Force Spline
Mid Force Spline
High Force Spline
52
To eliminate this non-linear behaviour for the low-force transducer, a rod was attached directly to
the cantilever beam and the entire beam was moved vertically to change immersion depth. While force
measurements were still taken at three simulated immersion depths, this allowed a single linear equation
to describe the relation between the applied force and laser translation. Because the expected force range
was often between 0.1 to 10 milli-Newtons, even the friction between the polyester thread and the surface
of the calibration equipment could give rise to inaccurate positions of the laser. Consequently, when a
known weight below 0.7 grams was applied, highly scattered results would be obtained compared to
values based on heavier weights. In order to produce a calibration in the appropriate force range, several
measurements of heavier weights were used to find an average value for the calibration equation. Figure
3.23 shows a sample calibration with raw and adjusted data for the lower force transducer where
approximately 20 data points are displayed. As shown, while most points fall close together, 6 data
points lay far from the statistical average and were eliminated as outliers before an average value for the
calibration was determined.
Figure 3.23: Sample calibration data for the low force transducer
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 20 40 60 80 100 120
Ca
lib
rati
on
Fo
rce
[N
]
Laser Horizontal Translation [mm]
Raw Data
Adjusted Average
53
3.3.2 Data Acquisition – High Velocity Range
For the high Re range, the available silicone oil was used as the Newtonian test fluid while both
T1 and T3 were used for the viscoelastic measurements. Three cylinder diameters were used for each
fluid over a range of angular velocities, as will be described fully in sections 4.1 and 4.2 with the
corresponding experimental results. The velocity ranges were selected so that measurements could be
taken over the maximum range of Re and De values without excessive cantilever or cylinder deflection
occurring.
At a fixed immersion depth, the velocity was held constant, the laser translation was recorded,
and then the velocity then changed. Initially, an experiment started at the minimum velocity and was then
repeated starting at the maximum velocity. After several trials, it was determined that the direction of the
speed ramp was irrelevant as long as enough time was given for steady state to be reached.
A stopwatch was used to time a single rotation to determine the turntable velocity and when
steady state had been reached. When at least three consecutive time measurements varying less than 5%
while the cylinder was immersed were noted, steady state was assumed and an average rotational speed
was calculated before recording the laser displacement. Accurate measurements of turntable velocity
were important for calculating the velocities for Re and De, and to make end corrections, as will be
discussed further in section 3.3.4.
Measurements at up to four immersion depths were taken for each angular velocity, and each
measurement was repeated two to three times depending on variation between measurements. Two
measurements varying less than 5% was the minimum required.
As mentioned in section 3.2.2, the high-speed motor system’s voltage controller had a great deal
of uncertainty associated with its settings, varying up to 10% from one measurement to the next.
Additionally, voltage noise and turntable vibration would also make the laser beam position oscillate over
54
a small range at higher speeds. This oscillation was a significant source of error in taking measurements,
as will be discussed in section 4.3.
During testing, the cylinder was bent by the flow to some extent, but the force measurement was
assumed to be perpendicular to the cylinder. The bending was monitored by using a protractor to ensure
that the cylinder end deflection was not greater than 10° or </18 radians.
By using the associated beam calibration, the force experienced by the cylinder was then recorded
as the raw experimental drag.
3.3.3 Data Acquisition – Low Velocity Range
The re-designed equipment used in the low velocity range eliminated many of the time-
consuming problems associated with the high velocity system because the digital speed controller enabled
repeated measurements at almost exactly the same velocity. With the added precision and stability of the
low speed system, steady state was determined by observing a stationary position of the laser for more
than 15 seconds instead of measuring variation in turntable velocity.
Aside from the improvements made to save time and increase reproducibility, the collection of
raw experimental drag with the lower-velocity system was identical to the high-velocity system.
3.3.4 End Effects
In order to convert the raw experimental data to two-dimensional unbounded results, corrections
for end and wall effects were required. First, end effects were accounted for by varying the immersion
depth of the cylinder, as shown in figure 3.24, and extrapolating how much force would remain if the
cylinder were of zero length.
55
Figure 3.24: Variable immersion depths used to extrapolate zero length force
This correction methodology was based on the assumption that at a fixed velocity, the drag
caused by the free end, and by surface effects should be independent of depth. For the high-force
transducer, taking measurements at exactly the same velocity was difficult because of the mentioned
speed variation. During experimentation, surface disruption, such as cavity formation and fluid climb,
occurred for T1 and T3 over the entire high Re range, as shown in figure 3.25. However, the length of the
cavity behind the cylinder and the height of the fluid climb remained constant as long as the velocity
remained constant, implying an independence from immersion depth. To support this hypothesis, during
his investigation of the cavity, Wang (2012) noted that no significant effect on drag could be found.
Therefore, these effects should be compensated for by the variable-length methodology.
56
Figure 3.25: Schematic view of surface disruptions, including fluid climb in front of cylinder and cavity
formation behind
Two sample plots of measured drag versus cylinder immersion, given as P/$, are shown in figure 3.26, with one graph for the polybutene and one for T3.
Figure 3.26a: The dependence of drag on immersion depth to determine end drag for the polybutene at
Re = . � × ����
Extrapolated End Drag = 0.000127 N0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
0.0016
0.0018
0.0020
0 5 10 15 20 25 30 35
Ra
w F
orc
e [
N]
L/d ratio
57
Figure 3.26b: The dependence of drag on immersion depth to determine end drag for the T3 fluid at Re = �. � × ����
As noted, the zero length, or end drag, can be taken as the intercept of a linear fit to the raw data.
This end correction method was used for all conditions and was consistently well fitted by a straight line.
However, the relation between Re and the percentage of drag caused by end effects was extremely
irregular. Two sample plots of end percentage of total drag versus Re for specific immersion depths are
shown in figure 3.27. These two plots are from the same data sets as the end correction plots shown in
figure 3.26 and have two distinctly different scatterings. Because no fixed pattern appeared in the data,
the end drag was taken as the average percentage of total measured drag predicted by the correction
methodology at all Re tested for a specific cylinder at a single immersion depth for each fluid.
Extrapolated End Drag = 0.000610 N0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0 5 10 15 20 25 30 35
Ra
w F
orc
e [
N]
L/d ratio
58
Figure 3.27a: % total drag accounted for by end effects for the polybutene for the 1.19 mm cylinder at 18
mm immersion depth
Figure 3.27b: % total drag accounted for by end effects for the T3 fluid for the 1.19 mm cylinder at 36
mm immersion depth
While this method may seem crude, but inevitable, when calculated this way and graphed as total drag
versus cylinder immersion depth, a smooth curve can be fitted as shown in figure 3.28 for the same data
sets as figure 3.26 and 3.27.
0
2
4
6
8
10
12
14
16
18
0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009 0.0001
% T
ota
l D
rag
Re
0
1
2
3
4
5
6
7
8
0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007
% T
ota
l D
rag
Re
59
Figure 3.28a: End effects as % total drag vs. immersion depth for the polybutene for the1.19 mm
cylinder
Figure 3.28b: End effects as % total drag vs. immersion depth for the T3 fluid for the 1.19 mm cylinder
0
2
4
6
8
10
12
0 5 10 15 20 25 30 35
% T
ota
l D
rag
L/d ratio
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
% T
ota
l D
rag
L/d ratio
60
3.3.5 Effect of the Channel Walls
Even at 100 diameters away, walls have a significant effect on drag value, as was seen in figure
2.6. In order to account for this effect, the first thought was to make use of Faxen’s formula, as given in
section 2.2.1. This formula predicts the drag on a cylinder in a Stokes flow between moving parallel
walls. It was assumed that the flow geometry would have a weak effect on the drag, and an
approximation of the annular tank as a straight channel would be sufficient. However, this was found to
be untrue.
Faxen’s formula consistently overestimated the drag by roughly 10%, obtained after end
corrections were applied to raw data. This consistent discrepancy was eventually attributed to the
difference between circular flow in an annulus rather than straight flow between parallel walls. As no
theoretical solution describing the drag on a cylinder in an annular channel could be found, two-
dimensional computer simulations using the software Fluent were used to predict the drag value for
specific geometries and Re.
The first set of simulations using Fluent was for flow around a cylinder between moving parallel
walls. By using a sufficiently refined mesh and convergence criterion, Fluent predictions of the drag
differed less than 2% in the expected experimental range when compared with Faxen’s formula, as shown
in figure 3.29.
Because Faxen’s formula is an O(104) approximation, its predictions for drag cannot be assumed
accurate for an H/d ratio greater than 1000. The formula also asymptotes to 0; thus at some H/d value it
will eventually predict drag values below Kaplun’s solution as shown in figure 3.29 for an H/d of
100,000. While this is a concern with the formula itself, the experimental range as shown above is well
below 1000.
61
Figure 3.29: Comparison of Faxen’s formula, Fluent simulations and Kaplun’s solution for �� =�. ���� with experimental range marked by vertical dotted lines
Because of the good agreement between Fluent predictions and Faxen’s formula, simulations for
an annular geometry with the experimental dimensions were used to determine the drag on a cylinder.
Figure 3.30 shows a comparison of meshing between linear and annular channels including enhanced
views of the mesh refinement near the cylinder.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1 10 100 1000 10000 100000 1000000
Vis
cou
s D
rag
Co
eff
icie
nt
H/d Ratio
Faxen's Formula
Fluent Prediction
Kaplun's Solution
Experimental Range
62
Figure 3.30a: Linear channel Fluent mesh for H/d = 300, zoomed in to the cylinder region
Figure 3.30b: Annular channel Fluent mesh for H/d = 40, zoomed in to the cylinder region
With these mesh resolutions, Fluent simulations took between 30 to 90 minutes to complete,
making it impractical to run a simulation for every flow scenario. To avoid this, because Faxen’s formula
is independent of Re for Stokes flow, it was surmised that a similar Re independence may apply for the
Fluent simulations. To test this theory, Fluent simulations were carried out with H/d kept constant, and
with viscosity, density, and velocity varied to change the Re. The results show that over a broad range of
63
Re, the drag coefficient predicted by Fluent for the same H/d ratio varies less than 0.1%, essentially
remaining constant, as displayed in figure 3.31.
Figure 3.31: Fluent simulations run for various Re for the same H/d ratio with constant linear fits
By running annular Fluent simulations over the full H/d range used in this experiment and comparing
them to Faxen’s formula predictions for straight channel flow, an equation describing the percentage
difference between the two was created as shown in figure 3.32.
0.20
0.22
0.24
0.26
0.28
0.30
0 0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 0.00014
Dra
g C
oe
ffic
ien
t
Re
H/d = 112
H/d = 64.7
64
Figure 3.32a: Comparison of Fluent predictions for an annular channel and Faxen’s formula for a straight
channel, various H/d ratios
Figure 3.32b: % difference of above data.
0.15
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250 300
Vis
cou
s D
rag
Co
eff
icie
nt
H/d Ratio
Fluent Prediction
Faxen's Formula
0
2
4
6
8
10
12
14
0 50 100 150 200 250 300
% D
iffe
ren
ce
H/d Ratio
65
This comparison between Fluent and Faxen was done to take advantage of the numerical
simplicity of Faxen’s formula. Faxen’s formula was used to calculate the equivalent straight channel drag
and was then scaled to the Fluent prediction for the annular case by multiplying by the appropriate
percentage.
To determine if annular flow can be corrected to become unbounded flow, as H/d increases the
influence of walls must disappear and Kaplun’s solution must be achieved. The Fluent simulations used
to create figure 3.29 and 3.32 support this requirement. The simulations for figure 3.29 indicate that as
H/d grows, Fluent predictions for a straight channel flow around a cylinder approach Kaplun’s solution.
The simulations for annular channel flow in figure 3.32 indicate that, as H/d increases, the percentage
difference between the linear and annular case decreases. Thus, it is reasonable to assume that, at a large
enough H/d value, Kaplun’s solution will also be achieved for annular flow.
Using these annular Fluent simulations, Faxen’s formula, and Kaplun’s solution, the drag
measurements taken in the experiment, corrected for end effects, can be converted to the drag on an
unbounded cylinder with the following formula:
O����)��MM�� �$ = O��$��MM�� �$O������OO�*�� × (� − %$!���M�����! �OO���� ) =
O������OO���� (2. B)
3.3.6 Experimental Methodology – Summary
Because both the end and wall corrections were based on Newtonian fluid behaviour, their
applicability to Boger fluids can be questioned. The end effect corrections should be valid because any
differences in end drag caused by elasticity are taken into account by the variable-depth measurements.
However, the effect of walls is difficult to determine. With no additional research available, the current
wall correction methodology was deemed the best available solution. For the objective of finding onset,
66
the interaction between walls and elasticity should not be a factor because, below onset, the Boger fluids
behave as Newtonian equivalents.
The step by step procedure for the experimental methodology following fluid characterization and the
experimental variations, were as follows:
1. Mount test equipment
2. Calibrate transducer
3. Collect raw drag data for a test cylinder
a. 3 cylinders per fluid
b. 2 to 8 angular velocities for each cylinder
c. 2 to 3 repeated measurements at each velocity
4. Apply end corrections to averaged raw data
5. Apply wall corrections to end corrected data
67
CHAPTER 4: RESULTS AND DISCUSSION
4.1 Newtonian Results
The testing of Newtonian fluids was carried out to validate the experimental method. As such,
the first fluids were the silicone oil and the polybutene, for the high and low Re regions respectively.
Table 4.1 shows the relevant experimental parameters for both fluids.
Silicone oil Polybutene
Annular Tank Size Smaller Smaller
Cylinder Diameters [mm] 0.50, 0.97, 3.18 1.19, 2.17, 3.34
Temperature Range [°C] 22 21 – 22
Viscosity Range [Pa.s] 1.03 30.0 – 30.8
Reynolds Number Range 0.020 – 0.680 0.00002 – 0.0002
Table 4.1: Relevant parameters for the silicone oil and the polybutene experimental trials.
As outlined at the end of section 3.3, the experimental data can be presented in three forms; the
raw data as calculated from the force transducer calibration outlined in section 3.3.1, the data corrected
for end effects as outlined in section 3.3.4, and the data corrected for end effects and for walls as outlined
in section 3.3.5, otherwise referred to as fully-corrected drag. The Newtonian data are presented in terms
of the viscous drag coefficient, defined in section 2.2.1 as:
�� = O$�N�#P(. )
Additionally, Kaplun’s solution has been included for comparison, being the expected value after
correcting for ends and walls. A plot of the raw data for the silicone oil for the mid-size cylinder with
0.97 mm diameter, displayed as �� versus Re at various immersion depths, is shown in figure 4.1a.
68
Figure 4.1a: Silicone oil raw drag data for the 0.97 mm cylinder.
In raw format, the data in figure 4.1a shows significant spread between the various immersion depths and
significant disagreement with Kaplun’s solution.
Figure 4.1b: Silicone oil data corrected for end effects for the 0.97 mm cylinder.
0.10
0.15
0.20
0.25
0.30
0.35
0.01 0.1 1
Ra
w V
isco
us
Dra
g C
oe
ffic
ien
t
Re
15 mm immersion
20 mm immersion
25 mm immersion
30 mm immersion
Kaplun's Solution
0.10
0.15
0.20
0.25
0.30
0.35
0.01 0.1 1
En
d C
orr
ect
ed
Vis
cou
s D
rag
Co
eff
icie
nt
Re
15 mm immersion
20 mm immersion
25 mm immersion
30 mm immersion
Kaplun's Solution
Fluent Prediction
69
After applying end corrections, the data at various immersion depths condenses and agrees well with the
Fluent prediction; however, slight disagreement can be seen above Re = 0.1
Figure 4.1c: Fully corrected silicone oil data for the 0.97 mm cylinder.
Once the data was fully corrected for both ends and walls, the data matched to within 5% of Kaplun’s
solution for Re ≤ 0.1. Figures 4.1b and 4.1c both indicated that above Re = 0.1, the measured and
predicted drag coefficients increasingly diverged. Evidently, inertial effects began to affect the drag, and
the reliability of Kaplun’s solution decreased as it is an inertialess flow solution.
A similarly good agreement was found for all cylinders with both the silicone oil and the
polybutene. Figure 4.2a and 4.2b are plots of silicone oil and the polybutene data corrected for end
effects, respectively, and figure 4.2c shows a plot of all Newtonian data corrected for both ends and walls
displayed as the average value for all depths for each point. Figures 4.2a and 4.2b are presented
separately to allow for clearer comparisons to Fluent predictions for similar cylinder diameters, while
figure 4.2c is given as average values to allow error bars to be displayed.
0.10
0.15
0.20
0.25
0.30
0.35
0.01 0.1 1
Fu
lly
Co
rre
ct V
isco
us
Dra
g C
oe
ffic
ien
t
Re
15 mm immersion20 mm immersion25 mm immersion30 mm immersionKaplun's Solution
70
Figure 4.2a: Silicone oil data, corrected for end effects, for all cylinder diameters, along with Fluent predictions for equivalent H/d ratios
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.01 0.1 1
En
d C
orr
ect
ed
Vis
cou
s D
rag
Co
eff
icie
nt
Re
Silicone oil - 0.50 mm cylinder, H/d = 267
Silicone oil - 0.97 mm cylinder, H/d = 137
Silicone oil - 3.18 mm cylinder, H/d = 42
Kaplun's Solution
Fluent Prediction - H/d = 267
Fluent Prediction - H/d = 137
Fluent Prediction - H/d = 42
71
Figure 4.2b: Polybutene data, corrected for end effects, for all cylinder diameters, along with Fluent predictions for equivalent H/d ratios
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
1E-05 0.0001 0.001
En
d C
orr
ect
ed
Vis
cou
s D
rag
Co
eff
icie
nt
Re
PB - 1.19 mm cylinder, H/d = 112
PB - 2.17 mm cylinder, H/d = 61
PB - 3.34 mm cylinder, H/d = 40
Kaplun's Solution
Fluent Prediction - H/d = 112
Fluent Prediction - H/d = 61
Fluent Prediction - H/d = 40
72
Figure 4.2c: Fully corrected drag for all cylinders and fluids with standard-error bars
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
1E-05 0.0001 0.001 0.01 0.1 1
Fu
lly
Co
rre
cte
d V
isco
us
Dra
g C
oe
ffic
ien
t
Re
Silicone oil - 0.50 mm cylinder
Silicone oil - 0.97 mm cylinder
Silicone oil - 3.18 mm cylinder
PB - 1.19 mm cylinder
PB - 2.17 mm cylinder
PB - 3.34 mm cylinder
Kaplun's Solution
73
The data in figure 4.2b displays an apparent curvature for each cylinder, but because the
differences were within 5% of the corresponding Fluent predictions and the cause could not be identified,
this effect was deemed negligible. Figure 4.2c further emphasises that the difference between the
measured data and Kaplun’s solution increases for Re ≥ 0.1. Because only minor variations from Kaplun’s solution and Fluent predictions occurred in these Newtonian tests, the results displayed in figure
4.2c give confidence in the experimental methodology.
4.2 Boger Fluids
4.2.1 Experimental Results
Testing of the T1 and T3 fluids was done for both high and low De ranges, analogous to high and
low Re ranges for the Newtonian fluids. Presented in table 4.2 are the relevant experimental parameters.
T1 – High De T1 – Low De T3 – High De T3 – Low De
Annular Tank Size Larger Smaller Smaller Smaller
Cylinder Size[mm] 0.5, 0.97, 3.18 1.19, 2.17, 3.34 0.97, 1.58, 3.18 1.19, 2.17, 3.34
T Range [°C] 22 15.2 – 18.5 21 21.9 – 22.4
� Range [Pa.s] 7.1 9.6 – 13.3 38 32.1 – 33.8
Re Range 0.0013 – 0.10 0.00010 –
0.00062 0.00043 – 0.0038
0.000012 –
0.00017
?Range [s] 0.74 1 – 1.2 2.6 2.5 – 2.6
De Range 17 - 184 0.25 – 2.7 21 - 136 0.28 – 4.0
Table 4.2: Relevant parameters for the T1 and T3 experimental trials
Plots for T1 with the 2.17 mm cylinder at low speeds are displayed as �� versus Re in figure 4.3
in raw, end corrected, and fully corrected form similar to the format from Newtonian fluids.
74
Figure 4.3a: T1 raw data for the 2.17 mm cylinder
Figure 4.3b: T1 data for the 2.17 mm cylinder corrected for end effects
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
1E-05 0.0001 0.001
Ra
w V
isco
us
Dra
g C
oe
ffic
ien
t
Re
18 mm immersion
24 mm immersion
30 mm immersion
36 mm immersion
Kaplun's Solution
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
1E-05 0.0001 0.001
En
d C
orr
ect
ed
Vis
cou
s D
rag
Co
eff
icie
nt
Re
18 mm immersion
24 mm immersion
30 mm immersion
36 mm immersion
Kaplun's Solution
Fluent Prediction
75
Figure 4.3c: T1 data for the 2.17 mm cylinder corrected for end and wall effects.
Figure 4.3b and 4.3c show that for Re ≥ 0.0002 there is a deviation from the Fluent prediction for drag. These plots of �� versus Re are informative but cannot be used to compare cylinders of different
diameter. To make such a comparison, and to simplify the identification of the effects of elasticity, all
subsequent data for the Boger fluids are presented as:
��������′����� !��(�. �)
plotted against De.
Figure 4.4a shows data for T1 in both the high and low De ranges for all cylinder diameters and
immersion depths, corrected for both end and wall effects, while figure 4.4b gives the analogous plot for
T3. Figure 4.4c shows both the data given in figures 4.4a and 4.4b, but averages the immersion depths to
give a single data point so that error bars can be included. As one of the primary objectives of this thesis
is to identify elastic onset, figure 4.4d is a plot of the low De region for both the T1 and T3 fluids.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.00001 0.00010 0.00100
Fu
lly
Co
rre
ct V
isco
us
Dra
g C
oe
ffic
ien
t
Re
18 mm immersion
24 mm immersion
30 mm immersion
36 mm immersion
Kaplun's Solution
76
Figure 4.4a: Fully corrected data for all cylinders for the T1 fluid
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.1 1 10 100 1000
Fu
lly
Co
rre
cte
d D
rag
/ K
ap
lun
's S
olu
tio
n
De
Low T1 - 1.19 mm cylinder
Low T1 - 2.17 mm cylinder
Low T1 - 3.34 mm cylinder
High T1 - 0.50 mm cylinder
High T1 - 0.97 mm cylinder
High T1 - 3.18 mm cylinder
Newtonian Equivalent
77
Figure 4.4b: Fully corrected data for all cylinders for the T3 fluid
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.1 1 10 100 1000
Fu
lly
Co
rre
ct D
rag
/ K
ap
lun
's S
olu
tio
n
De
Low T3 - 1.19mm cylinder
Low T3 - 2.17 mm cylinder
Low T3 - 3.34 mm cylinder
High T3 - 0.97 mm cylinder
High T3 - 1.58 mm cylinder
High T3 - 3.18 mm cylinder
Newtonian Equivalent
78
Figure 4.4c: Fully corrected data for the T1 and T3 fluids with standard-error bars
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.1 1 10 100 1000
Fu
lly
Co
rre
cte
d D
rag
/ K
ap
lun
's S
olu
tio
n
De
T3 - 1.19mm low De
T3 - 2.17mm low De
T3 - 3.34mm low De
T3 - 0.97mm high De
T3 - 1.58mm high De
T3 - 3.18mm high De
T1 - 1.19mm low De
T1 - 2.17mm low De
T1 - 3.34mm low De
T1 - 0.50mm high De
T1 - 0.97mm high De
T1 - 3.18mm high De
Newtonian Equivalent
79
Figure 4.4d: T1 and T3 fully corrected drag data vs. De – enlarged low De image
0.0
0.5
1.0
1.5
2.0
2.5
0.1 1 10
Fu
lly
Co
rre
cte
d D
rag
/ K
ap
lun
's S
olu
tio
n
De
T3 - 1.19mm low De
T3 - 2.17mm low De
T3 - 3.34mm low De
T1 - 1.19mm low De
T1 - 2.17mm low De
T1 - 3.34mm low De
Newtonian Equivalent
80
Both figures 4.4a and 4.4b indicate a monotonic increase in drag coefficient for T1 and T3 with
increasing De. While making measurements at lower De was more technically challenging, because the
second-generation equipment eliminated many of the noise factors associated with the first-generation
system, more consistent data was obtained in the low De region compared to the high De region. Despite
the monotonic growth of drag for individual fluids, figure 4.4c shows an unexpected difference in results
between T1 and T3, which will be further discussed in section 4.2.3.
By examining the region around De = 1 as noted in figure 4.4d, the elastic onset point appears to
be a value between 0.5 and 0.7. While the data indicates that the drag force is less than the Newtonian
equivalent below De of 0.7, the actual difference is at most 10%. Below onset, the measurements give a
constant value for drag coefficient rather than a dependence on De, indicating that this phenomenon is
likely an inaccuracy in the system calibration or an unresolved systemic error rather than drag reduction,
which has been observed for the spherical case. So far, the error bars included in all graphs have been for
the purely statistical error associated with data processing. A further error analysis considering machine
calibration is provided in section 4.3.
4.2.2 Drag Increase
An explanation for the drag enhancement comes from the work of Chilcott and Rallison, who
described the formation of a wake region behind a cylinder in a viscoelastic Stokes flow. Wake formation
is a common inertial effect when Re > 1 and can be easily visualised as flow separation for Re > 100. For
Stokes flow where inertial effects should be negligible, wake formation should not occur. However, as
was noted in section 3.3.4, the appearance of a wake for Re < 1 was observed for both T1 and T3 during
this experiment in the high De range and is attributed to elastic effects. In the low De range, asymmetry
between the upstream and downstream sides of the cylinder could only observed at the maximum tested
values of De while near and below onset no visual difference was noted.
81
For a Stokes flow, time reversibility in flow is expected, so that if a snapshot of the streamlines
around an object is observed, the direction of flow should not be identifiable. Chilcott and Rallison
attributed their formation of a wake to polymer chain extension. Extension occurs only when flow
acceleration is present; thus, for the flow field around a cylinder the maximum regions of acceleration
occur upstream and downstream of the stagnation points. A region of high polymer extension occurs
downstream of the rear stagnation point, resulting in a region of low flow velocity. While different in
nature, this low velocity region acts similarly to flow separation for high Re and causes a pressure
differential between the upstream and downstream regions, resulting in an increase in drag. In addition to
observations of cavity formation, flow visualizations made by Wang (2012) using fluorescing seeded
particles in the T1 fluid revealed a low velocity region behind the test cylinder in the high De range.
4.2.3 Onset and Discrepancy at Higher De
The onset value above which elastic effects were present was a De between 0.5 and 0.7. This De
value depends on the selection of the relaxation time. The decision to calculate relaxation time based on
�� measurements was justified in section 3.1.3, but calculating the relaxation time from measurements of
�′ and other methods not discussed in this work is possible.
While onset could be pinpointed, other results are less consistent. As was seen in figure 4.4c, for
the less viscous T1, the rate of drag increase with De was smaller compared to the more viscous T3. A
nearly 40% difference between the drag coefficient value for T1 and T3 is observed at De = 100.
However, T1 and T3’s agreement to the same elastic onset point seems to indicate that more was
occurring than could be accounted for by the methods or models used rather than simply experimental
error. Statistical error analysis of the data does not indicate that this difference is solely the result of
measurement error, and because the same equipment was used for both T1 and T3, machine calibration is
82
also unlikely the cause. The reason for such a discrepancy remains unknown as elastic effects should
have been accounted for by comparing experimental results to De.
The work done by Wang (2012), who used T1 and T3 as the tested Boger fluids, illustrated a drag
enhancement of roughly four times at De = 100, which is similar, though somewhat larger, than the effect
observed in the current work. Unfortunately, the data for T3 had a 40% spread in drag value at all tested
De and was deemed too inaccurate to be used for comparative purposes.
In the work of Yip (2011), who used the T1 and T3 fluids in an experiment to determine the drag
force in a porous media, an onset point at De = 0.5 and an increase in drag of roughly 50% at De = 2 were
recorded. While Yip defined De by using the centre to centre distance between parallel rods in his porous
array, and calculated relaxation time using �′, it is encouraging that there is at least agreement in the
order-of-magnitude with the onset point and amount of drag enhancement found in this work. However,
while there was some difference in drag enhancement between different solidity models, Yip’s data
indicated that the amount of drag enhancement for both the T1 and T3 fluids was approximately equal for
De above onset, which contrasts with the findings of the current work.
The answer to why this discrepancy occurs may lie in a more complete rheological
characterization of the test fluids which is outside the scope of this thesis.
4.2.4 Comparison to Chilcott and Rallison
A comparison of Chilcott and Rallison’s numerical work to the present data is in order, and is
made in figure 4.5. Both sets of results are normalized with Newtonian values, and so the ordinate values
should be 1 at low De. In the figure, two sets of numerical results from Chilcott and Rallison are
presented, where L and c are defined as:
83
P = ��*!���������$}��)��MP��� �}��)��MP��� �� ��� (�. )
� = ��� − �(�. 2)
Figure 4.5: Comparison of T1 and T3 data to Chilcott and Rallison’s work
The experimental and numerical data do not agree as to where onset occurs. Such a large
difference between onset points could not be accounted for by experimental error, determined in section
4.3. However, having an onset value below 1 is logical because a De value of 1 does not imply elastic
effects do not exist. Similar to how inertial effects made Kaplun’s solution less reliable above Re = 0.1 as
illustrated in figure 4.2c, elastic effects likely persist below De = 1.
The large difference between Chilcott and Rallison and the current work may be the result of
differing models used to describe the viscoelastic fluids. While De is defined the same way for both
works, the fluid model used in this thesis allows for infinite polymer extensibility whereas Chilcott and
Rallison set a maximum extensibility. The T1 and T3 fluids also have a higher c than the fluids tested by
0.0
0.5
1.0
1.5
2.0
2.5
0.1 1 10 100
Dra
g C
oe
ffic
ien
t
De
C&R data L = 10 c = 0.5
C&R data L = 10 c = 0.1
Representative T3 Data
Representative T1 Data
Newtonian Expectation
84
Chilcott and Rallison, implying a higher polymer viscosity contribution in the fluids. Furthermore, while
there is a drag reduction below Newtonian expectation for the experimental data, no such reduction was
predicted by Chilcott and Rallison.
While there is disagreement about the onset value, the rate of drag increase above unity is similar
between the experimental and numerical work. For example, the T1 data and the L = 10, c = 0.1 data
both show a growth of 50% above the Newtonian expectation for a 1 decade increase in De. These
differences in data emphasize that further consideration of fluid composition is required to fully describe
the drag behaviour.
4.3 Error Analysis
Experimental uncertainty can be divided into two categories, bias and precision (Coleman and
Steele, 1995). Bias errors are associated with equipment setup while precision errors are determined from
repeatability of measurements. Minimizing bias and precision uncertainty is accomplished by calibrating
equipment and taking multiple measurements, respectively.
The error bars shown in the preceding figures were for the precision uncertainty, and are
calculated as standard error given by:
�~ = ���∑ (*! − ��)�!��√� (�. �)
where � is the population size, *! is the ith value, and �� is the population mean. For all the error bars
associated with drag measurements, a population size of at least 6 was used. For the high speed data,
where greater spread occurred, an error between 8 to 12% was calculated. Given the amount of difficulty
in repeating measurements, this amount of error is considered reasonable. With the more precise second-
85
generation equipment, a standard error of between 1 and 2% was calculated. Such a small error seems
questionable but may be the result of taking many repeated measurements. As equation 4.1 shows, the
standard error is inversely proportional to the sample size, and the low speed final data values were all
calculated from no less than 8 repeated measurements. While taking these low speed measurements was
more technically difficult than their high speed counterparts, the process was more precise. Coupled with
the large sample size, small standard error values were generated.
In order to examine bias, the uncertainty associated with equipment must be considered. Since
much of the equipment was custom built, the bias must be estimated from the associated calibrations.
The transducer system provides the largest amount of uncertainty as both the force calibration and
vertical positioning of the cylinder can add error. By examining the transducer calibrations, it can be
determined that the maximum uncertainties associated with the high and low force transducers are 2%
and 4% respectively. The maximum uncertainty associated with the cylinder immersion depth for the
high force transducer is calculated as 6%, and the analogous uncertainty for the low force transducer is
calculated as 1%. Finally, the uncertainty associated with reading the high force transducer at high
velocities where oscillation of the laser beam on the measurement scale occurs can be estimated as up to
6%.
Because these uncertainties, termed �!, are independent of each other, they can be amalgamated
to give a single uncertainty � � �� by the formula:
� � �� = �(��) + (�)+. . . ��(�. )
Thus the maximum total error associated with bias for the high force transducer is 9% while the
maximum total error for the low force transducer is 4%.
Even if both precision and bias uncertainty are combined, it is not sufficient to explain the nearly
40% difference between T1 and T3 values in the high De range.
86
While the measurement of drag was the primary concern for this thesis, it is still important to
determine the accuracy of both Re and De. To determine the uncertainty associated with Re and De, the
fluid characterizations, measurement of turntable velocity, and measurement of cylinder diameter must be
considered. From the fluid characterizations, the standard error associated with fluid properties
determined by the AR2000 rheometer can be estimated as 2%, while uncertainty associated with density
can is less than 1% and is thus considered negligible. The maximum uncertainty in the measurement of
velocity for the high-speed system was 3% while the more precise lower-speed system had an uncertainty
of 1%. Cylinder diameter was measured with two different digital calipers multiple times, giving an
uncertainty of less than 1% and is thus considered negligible. Using equation 4.2, the maximum
uncertainty associated with Re is 4% and the maximum uncertainty associated with De is 5%. With such
small maximum uncertainties associated with Re and De, horizontal error bars were not added to any of
the preceding plots as they would have impeded readability.
87
CHAPTER 5: CONCLUDING REMARKS
In the present work, the effects of elasticity on a Stokes flow around an isolated cylinder were
experimentally investigated. A set of Newtonian experiments, compared against theoretical solutions,
served as a proof of concept while the use of two Boger fluids allowed for analysis involving elasticity.
Measurements of drag were taken by attaching a small diameter cylinder to a flexible cantilever
beam and immersing the cylinder into a slowly rotating annular tank which contained the test fluids.
Cantilever beam deflection, read as reflected laser translation, was then related to total force on the
cylinder. Multiple tanks, cylinder diameters, and flow velocities were used to take measurements over a
large range of Re from O(10��) to O(10w), and De from O(10�s) to O(10d).
An elastic onset value was identified at a De value between 0.5 and 0.7, above which drag
enhancement was observed. This onset region was identified by both Boger fluids for three different
cylinder diameters and is of importance for future simulations as it provides a reference point against
which the inaccuracy can be judged.
Above elastic onset, drag enhancement for the Boger fluids was observed between 3 to 4 times
the Newtonian expectation at De = 100. However, a difference in the amount of drag enhancement
between T1 and T3 was observed. The more viscous T3 fluid’s measurements indicated that a nearly
40% larger increase in drag occurred compared to the less viscous T1 at De = 100. When analysed
separately, the data for each fluid can be fit with smooth curves, and since both fluids indicated roughly
the same onset point, the discrepancy between their higher De results is thought to be an indication of
more complex effects than could be accounted for by a single relaxation time constitutive model. Error
analysis indicates that the statistical and equipment uncertainties cannot account for this discrepancy, but
it is possible that an unknown systemic error was not accounted for; more experimental investigation
88
would be required to confirm either hypothesis. This discrepancy meant that conclusions about the drag a
microorganism experiences in biological fluids could not be precisely identified by this work alone.
Further investigations of this problem should aim to decrease the number of visual measurements
required, as well as to increase the number and variation of Boger fluids. Only two Boger fluids were
used in this work and both were of similar composition, using identical long chain polymers; as such, an
investigation to determine if chemical composition of the test fluids played a more significant role than
expected is warranted. Additionally, numerical simulations, compared to the findings of this work,
should be performed to investigate the mechanics of drag enhancement, such as the contribution of
normal stresses.
89
REFERENCES
Armstrong RC, Brown RA, McKingley GH (1993). The Wake Instability in Viscoelastic Flow Past
Confined Circular Cylinders. Philosophical Transactions of the Royal Society of London: Physical
Science and Engineering. 344:265-304.
Barnes HA, Hutton JF, Walters K, (1989). An Introduction to Rheology. Elsevier, San Diego.
Bird RB, Armstrong RC, Hassager O (1987). Dynamics of Polymer Liquids, Vol. 1, Fluid Dynamics.
Wiley, New York.
Boger DV (1977). Highly Elastic Constant-Viscosity Fluid. J. Non-Newt. Fluid Mech. 3:87-91.
Bruneau CH, Kellay H, Xiong YL (2013). A Numerical Study of Two Dimensional Flows Past a Bluff
Body for Dilute Polymer Solutions. J. Non-Newtonian Fluid Mech. 196:8-26.
Chilcott MD, Rallison JM (1988). Creeping Flow of Dilute Polymer Solutions Past Cylinders and
Spheres. Journal of Non-Newtonian Fluid Mechanics 29:381-432.
Coleman HW, Steele WG (1995). Engineering Application of Experimental Uncertainty Analysis. AIAA
J. 33:1888-1896.
Faxen OH (1946). Forces Exerted on a Rigid Cylinder in a Viscous Fluid Between Two Parallel Fixed
Plates. Proc. R. Swed. Acad. Eng. Sci. 187:1-13.
Fulford GR, Katz DF, Powell RL (1998). Swimming of Spermatozoa in a Linear Viscoelastic Fluid.
Biorheology. 35:295-309
Francois N, Lanse D, Amarouchene Y, Lounis B, Kellay H (2008). Drag Enhancement with Polymers.
Physical Review Letters. 100:1-4.
Fu HC, Wolgemuth CW, Powers TR (2008). Beating Pattern of Filaments in Viscoelastic Fluids. Physical
Review E. 78:041913:1-12
Fu HC, Wolgemuth CW, Powers TR (2009). Swimming Speeds of Filaments in Nonlinearly Viscoelastic
Fluids. Physics of Fluids. 21:033102:1-12
Gere JM (2004). Mechanics of Materials ed.6. Brooks/Cole, Toronto.
Huang PY, Feng J (1995). Wall Effects on the Flow of Viscoelastic Fluids Around a Circular Cylinder.
J. Non-Newtonian Fluid Mech. 60:179-198.
James DF (2009). Boger Fluids. Annual Review of Fluid Mechanics. 41:129-142.
James DF, Yip R, Currie IG (2012). Slow Flow of Boger Fluids Through Model Fibrous Porous Media.
Journal of Rheology. 56:1249-1277
Jayaweera KOLF, Mason BJ (1965). The Behaviour of Freely Falling Cylinders and Cones in a Viscous
Fluid. Journal of Fluid Mechanics. 22:709-720.
Larson RG (1998). The Structure and Rheology of Complex Fluids. Oxford University Press, New York.
Lauga E (2007). Propulsion in a Viscoelastic Fluid. Physics of Fluids 19:1-13.
90
Lauga E (2009). Life at High Deborah Number. Europhysics Letter 88:64001:1-6.
Purcell EM (1977). Life at Low Reynolds numbers. American Journal of Physics. 45:3-11.
Skinner LA (1975). Generalized Expansions for Slow Flow Past a Cylinder. The Quarterly Journal of
Mechanics and Applied Mathematics 28:333-340.
Taylor G (1951), Analysis of the Swimming of Microscopic Organisms. Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences. 209:447-461
Van Dyke MD (1946). Perturbation Methods in Fluid Mechanics. The Academic Press, Stanford,
California.
Verhelst JM, Nieuwstadt FTM (2004). Visco-elastic Flow Past Circular Cylinders Mounted in a Channel:
Experimental Measurements of Velocity and Drag. J. Non-Newtonian Fluid Mech. 116:301-328
Wang SWE (2012). Slow flow of a Viscoelastic Fluid Past a Circular Cyliner. University of Toronto,
Thesis.
Yip R (2011). Slow Flow of Viscoelastic Fluids through Fibrous Porus Media. Univesrity of Toronto,
Thesis.
White CM (1946). The Drag of Cylinder if Fluids at Slow Speeds. Proceedings of the Royal Society of
London, Series A, Mathematical and Physical Sciences. 186:472-479.
