Planar Flow of Dilute Polymer Solutions

Shear & Extensional Effects in Internal Flows of

Dilute Polymer Solutions

by

Shamsur Rahman

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Department of Mechanical & Industrial Engineering

University of Toronto

© Shamsur Rahman 2011

ii

Abstract

Shear & Extensional Effects in Internal Flows of Dilute Polymer Solutions

Shamsur Rahman

Master of Applied Science, 2011

Department of Mechanical & Industrial Engineering

University of Toronto

Shear and extensional flows of dilute polymer solutions were studied experimentally in an

attempt to understand the mechanism of polymer-induced drag reduction. A flowcell capable of

simulating the dynamics of a turbulent boundary layer, involving the motion of counter-rotating

vortices, was designed and fabricated. The pressure drop across the flowcell was measured for

different flow arrangements, first with a Newtonian fluid and then with drag reducing, dilute

polymer solutions. The pressure drop in excess of the Newtonian baseline, after accounting for

viscous effects, was used as a measure of elastic effects.

With the dilute polymer solutions, elastic effects were observed both in shear, extensional, as

well as presheared extensional flows. These effects can be attributed to additional normal

stresses generated by shearing. For extensional flows, the observed effects were independent of

elongation rates, indicating that a conclusion regarding the mechanism of drag reduction cannot

be made from the flowfield investigated.

iii

Table of Contents

Chapter 1: Introduction ................................................................................................................1

1.1 Previous Work with Drag-Reducing Fluids...............................................................................5

1.2 Research Objectives...................................................................................................................8

Chapter 2: Experimental Methodology .....................................................................................10

2.1 Conceptual Design ...................................................................................................................10

2.2 Design Considerations .............................................................................................................15

2.2.1 Main Channel Geometry ................................................................................................16

2.2.2 Side Channel Geometry .................................................................................................19

2.2.3 Channel Lengths.............................................................................................................20

2.2.4 Exit Channel Length.......................................................................................................21

2.2.5 Pressure Tap Locations ..................................................................................................21

2.2.6 Pressure Dropo & Choice of Test Fluids .......................................................................22

2.3 Flowfield with Final Design ....................................................................................................23

2.4 Fabrication of Flowcell ............................................................................................................27

Chapter 3: Test Fluids .................................................................................................................28

3.1 Non-Newtonian Fluids.............................................................................................................28

3.1.1 Boger Fluids ...................................................................................................................29

3.1.2 Rheometry ......................................................................................................................29

3.2 Shear Viscosity of Test Fluids .................................................................................................31

3.3 Critical Concentration..............................................................................................................33

iv

3.4 Relaxation Time & First Normal Stress Difference ................................................................35

3.4.1 First Normal Stress Difference.......................................................................................35

3.4.2 Oldroyd-B Model ...........................................................................................................36

3.5 Elastic Modulus .......................................................................................................................39

3.6 Summary of Fluid Prpoerties ...................................................................................................42

Chapter 4: Experimental Results and Discussions ...................................................................43

4.1 Main Channel and Side Channel Combined Flow...................................................................43

4.1.1 Newtonian Results..........................................................................................................45

4.1.2 Results with PEO Solutions ...........................................................................................47

4.2 Main Channel Flow without Side Flow...................................................................................51

4.3 Analyses of Results..................................................................................................................53

4.3.1 Flow Instability in Shear ................................................................................................53

4.3.2 Hole Pressure Error ........................................................................................................55

4.4 Side Channel Flow without Main Channel Flow.....................................................................58

4.4.1 Elastic Effects in Extension............................................................................................62

4.4.2 Numerical Analysis ........................................................................................................64

4.4.3 N1 Effect .........................................................................................................................68

4.5 Comparison with Prior Work...................................................................................................71

Chapter 5: Concluding Remarks................................................................................................74

5.1 Summary ..................................................................................................................................74

5.2 Conclusions..............................................................................................................................76

v

5.3 Future Work .............................................................................................................................76

Chapter 6: References .................................................................................................................78

Appendix A: Numerical Simulation ...........................................................................................81

Appendix B: Fluid Mechanics ....................................................................................................84

Appendix C: Oldroyd-B Model ..................................................................................................91

Appendix D: Pressure Transducer.............................................................................................95

Appendix E: Engineering Drawings of Flowcell .......................................................................99

vi

List of Figures

Figure 1.1 Setup of turbulent burst experiment (Reproduced from Kim et al, 1971).....................2 Figure 1.2 Photographic plate showing H2 bubble lines (Reproduced from Kim et al, 1971) .......3 Figure 1.3 (a) Formation of a hairpin vortex. (b) A group of hairpin vortices being lifted up from the surface. (c) A pair of counter-rotating vortices exerting upward force on the fluid. (Reproduced from Davidson, 2004) ................................................................................................5 Figure 1.4 Setup of presheared extensional flow experiment. (Reproduced from James et al, 1987) ................................................................................................................................................7 Figure 2.1 Fully developed flow through a rectangular channel. Parabolic velocity profile........11 Figure 2.2 Planar extensional flowfield. The streamlines are typically hyperbolic......................12 Figure 2.3 Cross-slot geometry in three-dimensions (Reproduced from Winter et al., 1979)......12 Figure 2.4 Setup of experimental flowcell....................................................................................14 Figure 2.5 Setup of experimental apparatus..................................................................................14 Figure 2.6 Comparison between velocity profiles for flow in a rectangular channel with different aspect ratios, and flow between two parallel plates. Channel width = 2a, height = 2d .................15 Figure 2.7 Streamwise component of velocity along vertical centreplane ...................................25 Figure 2.8 Enlarged view of streamwise velocity at the main channel entrance ..........................25 Figure 2.9 Transverse component of velocity along vertical centreplane. Color legend shows velocity in metres per second (m/s). ..............................................................................................26 Figure 3.1 Working principle of a cone-and-plate rheometer.......................................................30 Figure 3.2 Steady shear viscosity measurements for test fluids ...................................................32 Figure 3.3 Determination of intrinsic viscosity ............................................................................34 Figure 3.4 a) Step input in strain and the corresponding stress relaxation of b) a Newtonian fluid and c) a viscoelastic fluid and solid (Reproduced from Macosko, 1994)......................................35 Figure 3.5 First normal stress difference measurements in response to steady shearing for the 1200 ppm PEO solution in PEG solvent........................................................................................38

vii

Figure 3.6 Elastic modulus measurements in response to small-amplitude oscillations ..............40 Figure 4.1 Three-dimensional view of streamlines for combined flow from the main and the side channels showing how flow from the side channels is superposed on main channel flow ...........44 Figure 4.2 Measured pressure drop and prediction by COMSOL, at two different shear rates, corresponding to main channel flow rates of 4 ml/s (Re=7) and 5.5 ml/s (Re=8).........................46 Figure 4.3 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent at a wall shear rate of 1100 s-1, corresponding to a main channel flow rate of 4 ml/s and a Reynolds number of 7. ..............................................................................................48 Figure 4.4 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent at a wall shear rate of 1500 s-1, corresponding to a main channel flow rate of 5.5 ml/s and a Reynolds number of 8. ...........................................................................................48 Figure 4.5 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent at a wall shear rate of 2000 s-1, corresponding to a main channel flow rate of 7.5 ml/s and a Reynolds number of 11. .........................................................................................49 Figure 4.6 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent for variation in main channel flow with no flow from the side channels. The Reynolds numbers corresponding to the lowest and highest flow rates are 2 and 11 respectively ........................................................................................................................................................52 Figure 4.7 Pressure measurement in (a) a Newtonian fluid and (b) a viscoelastic fluid (Reproduced from Bird et al., 1987)..............................................................................................55 Figure 4.8 Three-dimensional view of streamlines for side channel flow only............................58 Figure 4.9 Pressure measuring arrangement for the side flow experiment...................................59 Figure 4.10 Normalized pressure drop measurements for two viscoelastic test fluids and the Newtonian solvent for variation in extensional rates with side channel flow with no flow from the main channels. For the 1200 ppm fluid, the Deborah number corresponding to the extension rates are also shown. ......................................................................................................................60 Figure 4.11 Uniaxial Trouton ratio for a Boger fluid (a semidilute solution of 0.31 wt% polyisobutylene in polybutene) stretched over a range of extensional rates and plotted as a function of Hencky strain. (Reproduced from McKinley and Sridhar, 2002) ...............................63 Figure 4.12 Side view of section from flowcell showing intersection of the main, side and the slanted exit channels ......................................................................................................................65 Figure 4.13 Numerical results for velocity in the flow direction along channel centerline, obtained from COMSOL. ..............................................................................................................65

viii

Figure 4.14 Calculated values of Hencky strain in the flowfield plotted as a function of Deborah number ........................................................................................................................67 Figure 4.15 Measurements of First Normal Stress Difference for the 1200 ppm PEO solution in PEG Solvent. Solid line shows 1N from the Oldryod-B model fitted to the first six data points ........................................................................................................................................................70 Figure 4.16 Comparison of elastic pressure drop for the 1200 ppm PEO solution in PEG with extrapolated values of N1 corresponding to the wall shear rate downstream of the side flows. ....70 Figure A1 Tetrahedral mesh in channel geometry........................................................................81 Figure A2 Enlarged section showing mesh at the entrance to the main channel..........................82 Figure A3 Comparison between analytical results for flow in a rectangular channel with aspect ratio a/d=6.7 and channel height 2d, flow between two parallel plates, and numerical results from COMSOL...................................................................................................................83 Figure B1 Turbulent flow over a flat plate ...................................................................................87 Figure C1 Characterization of the original Boger fluid prepared by Boger, 1977. The blue symbols represent the shear stress and the red symbols represent the first normal stress difference. (Reproduced from James, 2009)..................................................................................93 Figure D1 Low pressure calibration curve using a column of water ............................................97 Figure D2 High pressure calibration curve using a column of water ...........................................97 Figure E1 Exterior view of flowcell .............................................................................................99 Figure E2 Interior view showing the channels in the flowcell .....................................................99 Figure E3 Side view of flowcell .................................................................................................100 Figure E4 Front view of flowcell................................................................................................100

ix

List of Tables Table 2.1 Summary of channel dimensions used in the final design of the flowcell ....................25 Table 3.1 Fluid properties of test fluids ........................................................................................43

x

Nomenclature

a Channel half-width [m]

b Parameter determining type of extension

*c Crtitical concentration [ppm]

or xy Strain

O Initial strain

shear rate [s-1]

C Critical shear rate [s-1]

D Downstream channel wall shear rate [s-1]

U Upstream channel wall shear rate [s-1]

xy shear rate in the x-y plane [s-1]

d Channel half-height [m]

D Deformation rate tensor [s-1]

D Upper convected derivative of the deformation rate tensor [Pa.s-1]

De Deborah number

P Pressure drop [Pa]

elasticP Elastic pressure drop [Pa]

NewtonianP Newtonian pressure drop [Pa]

t Change in time [s]

v Velocity gradient tensor [s-1]

Tv Transpose of the vcelocity gradient tensor [s-1]

xi

xV Change in velocity in the flow direction [m/s]

y Displacement in the transverse direction [m]

Hencky strain

Extensional rate [s-1]

max Maximum extensional rate [s-1]

F Force [N]

F Inertial force [N]

F Viscous force [N]

G Dynamic storage modulus (slastic modulus) [Pa]

G Dynamic loss modulus (shear modulus) [Pa]

h Height of main and side channels [mm]

Uh Upstream channel height [mm]

Dh Downstream channel height [mm]

Shear viscosity [Pa.s]

E Extensional viscosity [Pa.s]

P Polymer viscosity [Pa.s]

PEG Viscosity of PEG Solvent [Pa.s]

S Solvent viscosity [Pa.s]

SP Specific viscosity [Pa.s]

Intrinsic viscosity [Pa.s]

Cone angle [rad]

xii

l Stretched length of fluid filament [m]

El Inlet length [mm]

ol Unstretched length of fluid filament [m]

L Main channel length [mm]

xL Length-scale in flow direction [m]

yL Length-scale in transverse direction [m]

SL Side channel length [mm]

eL Slanted exit channel length [mm]

or 1 Relaxation time [s]

2 Retardation time [s]

P Polymer contribution to the relaxation time [s]

M Applied torque [N.m]

1N First normal stress difference [Pa]

2N Second normal stress difference [Pa]

Kinematic viscosity [m2/s]

P Pressure [Pa]

OP Stagnation pressure [Pa]

1P Upstream pressure [Pa]

2P Downstream pressure [Pa]

*P Hole pressure error [Pa]

MQ Flow rate [ml/s]

xiii

SQ Side channel flow rate [ml/s]

r Cone radius [m]

Re Reynolds number

ReI Reynolds number based on fluid acceleration

Density [kg/m3]

t Time [s]

T Observation time [s]

Stress tensor [Pa]

0 Initial stress [Pa]

Upper convected derivative of the stress tensor [Pa.s-1]

S Solvent contribution to the stress tensor [Pa]

P Polymer contribution to the stress tensor [Pa]

P Upper convected derivative of the solvent contribution to the stress tensor [Pa.s-1]

W Wall shear stress [Pa]

xy Shear stress in the x-y plane [Pa]

xz Shear stress in the x-z plane Normal stress in the x-direction [Pa]

yz Shear stress in the y-z plane [Pa]

xx Normal stress in the x-direction [Pa]

yy Normal stress in the y-direction [Pa]

zz Normal stress in the z-direction [Pa]

*u Friction velocity [m/s]

xiv

U Bulk velocity [m/s]

U Free-stream velocity [m/s]

v Velocity [m/s]

Savgv , Side channel average velocity [m/s]

xv x-component of velocity [m/s]

yv y-component of velocity [m/s]

zv z-component of velocity [m/s]

Vol Volume [m3/s]

w Main channel width [mm]

Sw Side channel width [m]

Wi Weissenberg number

CWi Critical Weissenberg number

Frequency of oscillation [s-1]

Angular velocity [rad/s]

*y Spatial parameter for turbulent boundary layer

1

Chapter 1: Introduction

Drag reduction, also known as Toms’ effect, has been a topic of interest in fluid mechanics over

the last 60 years. In 1948, Toms discovered that the addition of a small amount of high-

molecular-weight polymer to a Newtonian turbulent flow reduced the wall shear stress by up to

70%. Considering the extent by which an inertia-driven phenomenon such as turbulence can be

interrupted by a polymer concentration as low as 5 parts per million, this observation was quite

remarkable, more so because the addition of such low quantities of polymer hardly causes any

change in the fluid’s viscosity, implying that the phenomenon is not a viscous effect. Since its

discovery, drag reduction has been extensively used in industrial pipe flows such as those in long

transcontinental oil pipelines, as an effective method to reduce power consumption. In addition,

this phenomenon has applications in many other areas where it is desirable to increase the flow

rate without increasing the pumping costs, such as in the hoses of fire fighting equipment (Sellin

and Ollis, 1980; Khalili et al., 2002).

Despite the widespread application of the technique, the mechanism of drag reduction has not

been fully understood (White and Mungal, 2008). However, it is believed that this mechanism

pertains to the fluid elasticity caused by the long polymer chains. The induced elasticity,

although weak, is believed to be sufficient to provide resistance to fluid stretching during vortex

formation in the turbulent boundary layer, where vortices are generated though a process known

as the turbulent burst.

Turbulent bursts were characterized in an experiment conducted by Kim et al. (1971). Using

hydrogen bubble time-streak markers in a turbulent flow of water over a flat surface,

Chapter 1: Introduction 2

they observed the formation of streamwise vortices in the viscous sublayer, the region closest to

the wall in the turbulent boundary layer. With a Newtonian fluid, the vortex is lifted up from the

wall and grows in the buffer layer where inertial forces compete with viscous forces.

Subsequently, the vortex enters the log region, where inertial stresses completely outweigh

viscous stresses and the flow is overwhelmingly turbulent, and breaks up with the onset of even

more chaotic fluctuations and the cycle starts again. This process, starting with the formation of a

vortex in the viscous sublayer and ending with its break-up in the log region, is termed the

turbulent burst. Figure 1.1 shows the bubble markers and Figure 1.2 is an image from a

photographic plate showing the streamwise vortex just after formation.

Figure 1.1 Setup of turbulent burst experiment (Reproduced from Kim et al, 1971)


Figure 1.2 Photographic plate showing H2 bubble lines (Reproduced from Kim et al, 1971)

Donohue et al (1972) conducted a set of experiments to determine if the degree of drag reduction

can be correlated with the rate of turbulent burst. As turbulent bursts can be characterized as

streaky structures traveling at low speeds, they decided to use the spacing between streaks as a

measure of the amount of bursting. Using a dye-injection method for flow visualization, they

compared the spacing between low speed streaks in a turbulent channel flow between water and

a drag-reducing fluid: a dilute solution of polyethylene oxide in water. They observed an

increase in streak spacing with the polymer solution compared to that with the solvent at the

same flow rate, indicating fewer bursts in the polymer solution. A separate experiment involving

pressure drop measurements in a turbulent pipe flow with water and the same polymer solutions

was conducted in order to determine the percentage of drag reduction. A correlation of the results

from the two experiments showed that an increase in the amount of drag reduction corresponds

to a decrease in the rate of turbulent bursting.


Although results from Donohue et al (1972) confirmed that drag reduction is related to the

turbulent burst, the exact mechanism by which polymer molecules interact and suppress the

bursting process was still not understood. In order to relate the interaction of polymer molecules

with the turbulent boundary layer, it is necessary to closely examine the underlying fluid motion

during vortex formation in a burst. Particular emphasis needs to be given to the first stage of the

bursting process which involves the lifting of fluid away from the wall, requiring a force to be

exerted on the fluid in the perpendicular direction. Prevention of lifting should therefore suppress

turbulent burst. Hence, it is necessary to understand the details of this mechanism.

Fluid lifting in a turbulent burst takes place via formation of counter-rotating vortices known as

Hairpin vortices. When a flow at high Reynolds number is perturbed, the vorticity field

associated with the flow rearranges in such a way that a pair of counter-rotating vortices exerts

an upward force on the fluid causing the fluid to be lifted up from the wall in the shape of

hairpins, and so creates “hairpin vortices”. The formation of hairpin vortices is an extensional

motion, as shown in Figure 1.3, and it is this mechanism that causes lifting of fluid away from

the wall.


Figure 1.3 (a) Formation of a hairpin vortex. (b) A group of hairpin vortices being lifted up from

the surface. (c) A pair of counter-rotating vortices exerting upward force on the fluid. (Reproduced from Davidson, 2004)

1.1 Previous Work with Drag Reducing Fluids

Drag reducing fluids are dilute polymer solutions, i.e, a solution in which the macromolecules

are so few that there is no interaction between them. The only interaction in these fluids is

between the polymer chains and the solvent. Several researchers (Metzner and Metzner 1970,

Chauveteau 1981, James and Saringer 1980) reported that, in laminar flow of dilute polymer

solutions, the onset of elastic effects usually take place when the extensional rate, , exceeds the

inverse of the longest relaxation time, , of the fluid, i.e,

1 ,


corresponding to a Deborah number )( De greater than unity. For a drag-reducing

concentration of an aqueous polymer solution, the longest relaxation time is of the order of 1

millisecond, and thus the critical extensional rate required for elastic effects to take place is

~ 1000 s-1. However, measurements by Muller & Gyr (1986) showed that the extensional rate

in a turbulent burst is only about 50 s-1, a value almost two orders of magnitude lower than the

critical extensional rate required in laminar flow. This discrepancy can be explained by realizing

that in a laminar extensional flow, a macromolecule generally enters the extensional flowfield

from a “strain-free” region where it undergoes zero or very little deformation and hence the

polymer chain remains close to its equilibrium configuration before being extended by the

extensional field. However, in the turbulent boundary layer, the macromolecule is subjected to

considerable shearing near the wall and is already partially extended before reaching the

extensional flowfield. James et al. (1987) conducted an experiment to investigate whether

preshearing has an effect on the laminar extensional flow of dilute polymer solutions.

The apparatus for their experiment, as shown in Figure 1.4, was a rectangular channel with an

axisymmetric orifice channel placed in the bottom wall at a downstream location in the channel.

The fluid was presheared in the rectangular channel before entering the orifice where it was

extended. The pressure drop was measured between the location of the fluid entering the orifice

and that of the fluid leaving the orifice. The shear rate was varied by varying the flow rate in the

rectangular channel while the extensional rate was varied by varying the flow rate through the

orifice by controlling the flow restriction in the orifice.


Figure 1.4 Setup of presheared extensional flow experiment.

(Reproduced from James et al, 1987)

The results from this experiment showed that, even with preshearing, the extensional rates

required for obtaining elastic effects were much higher than the value measured in the turbulent

boundary layer. For an upstream shear rate of 800 s-1, extensional rates greater than 600 s-1 were

required to produce elastic effects with a polymer concentration of 20 ppm. Moreover, this

experiment did not completely resemble the turbulent boundary layer because the motion of

counter-rotating vortices was not simulated in this experiment. Also, in this experiment, in order

to undergo extension, the fluid had to flow into the wall while in the turbulent boundary layer the

fluid undergoes extension while being lifted away from the wall.

Planar vs Uniaxial Extension

Since it is generally accepted that drag reduction is an elastic effect in extension, it is worthwhile

to investigate whether this phenomenon can be correlated to elasticity in uniaxial extension, the

simplest and the most common form of extensional flow. James and Yogachandran (2006)

demonstrated that the breaking length of fluid filaments under uniaxial extension can be a


measure of the fluid’s elasticity. They then attempted to correlate drag reduction to this measure

of elasticity in uniaxial extension. However, no correlation could be established. The reason was

attributed to the fact that, in uniaxial extension, the polymer chains are extended a considerable

amount of their original length; however, in drag reduction, the chains are not extended as much

because shear is the dominant mode of deformation in the wall region where the polymer is

operative. By performing numerical simulation of polymerstretching in the turbulent boundary

layer, Terrapon et al. (2004) showed that the polymer chains are stretched a large fraction of their

full extension but not stretched to their fullest extent. This is important because extensional

stresses are roughly proportional to the cube of the effective length of the polymer chain. Their

simulation results also indicated that the polymer chains are highly extended in regions of planar

extension which is preceded by shearing.

1.2 Research Objectives

Results from the two experimental works described above indicate that neither pure uniaxial

extension nor axisymmetric extension coupled with preshearing was able to accurately model the

mechanism of drag reduction. No attempt has yet been made, however, to simulate this

phenomenon by creating a planar extensional flowfield coupled with preshearing, or to model the

extensional motion involving counter-rotating vortices. As planar extensional motion created by

counter-rotating vortices is a crucial step in the formation of vortices in the turbulent boundary

layer, an accurate simulation of these motions should lead to a representative model of the

underlying dynamics of the turbulent boundary layer. The objective of the present study is

therefore to understand the mechanism of drag reduction by examining the effect of fluid

elasticity in a laminar flowfield created to simulate the first stage of a turbulent burst. With a


laminar flow at low Reynolds number, it should be possible to identify elastic responses in the

flowfield. These responses are not easy to identify in a turbulent flow because in turbulent flows

inertial effects are dominant and hence elastic effects are difficult to separate experimentally

from inertial effects . The goals of the research are then:

To design and fabricate a flowcell capable of simulating a turbulent burst using a laminar

flow.

To minimize the Reynolds number in this flow in order to eliminate inertial effects.

To measure the pressure drop and flow rate in this flowcell first with a Newtonian fluid

and then with dilute polymer solutions, and explain the difference in results.

10

Chapter 2: Experimental Methodology

In order to run a controlled experiment with the desired flowfield, an approach involving the

following steps was adopted:

Design and fabrication of a flowcell capable of simulating with a laminar flow the fluid

mechanics of preshearing and planar extension in a turbulent boundary layer. Previous

work and numerical simulation will be used to determine the optimum geometry of the

flowcell.

A mechanism to vary shear and extensional rates.

Testing Newtonian and non-Newtonian fluids, specifically, dilute polymer solutions.

Comparison of measurements between the fluids to determine the elastic effects.

2.1 Conceptual Design

The first objective of the design problem is to establish a planar extensional flowfield combined

with preshearing. In order that the dynamics correspond to a turbulent burst, this flowfield should

be able to generate a minimum shear rate of 1000 s-1, a minimum extensional rate of 50 s-1, and

resemble the motion of counter-rotating vortices responsible for causing extensional motion in

the turbulent boundary layer. In addition, the Reynolds number needs to be low enough to ensure

that inertial effects can be neglected.

Preshearing can be achieved by flow through a wide rectangular channel built into the flowcell.

If this flow is fully developed, the velocity profile will be parabolic, with the highest shear rate at

the walls. As the wall shear rate is proportional to the flow rate in the channel, the required

Chapter 2: Experimental Methodology 11

shear rates can be achieved by adjusting the flow rates. Figure 2.1 below shows the schematic of

a fully-developed shear flow through a rectangular channel.

Figure 2.1 Fully developed flow through a rectangular channel. Parabolic velocity profile

A planar extensional flowfield, on the otherhand, can be set up in several ways. One common

method is to use a stagnation point flow. This type of flow is particularly preferred because the

shear stresses are identically zero in this flowfield, except at confining walls, causing the flow to

be purely extensional away from the walls. A planar extensional flowfield is shown in Figure

2.2. A particular technique for generating a stagnation point flow involves flow in a cross-slot

device, as shown in Figure 2.3. With opposing inlet and outlet flows, a stagnation point is created

at the centre of the device.

Figure 2.2 Planar extensional flowfield. The streamlines are typically hyperbolic


Figure 2.3 Cross-slot geometry in three-dimensions (Reproduced from Winter et al., 1979)

A modified flowfield consisting of a horizontal bottom plate and the top half of a stagnation

point flowfield can be used to effectively simulate the planar extensional flowfield created in the

first stage of the turbulent burst cycle. In this way, the effects of the counter-rotating vortices of a

turbulent burst can be achieved by the hyperbolic streamlines of stagnation point flow exerting

an upward force and corresponding extensional stresses. This flowfield retains the prime features

of a purely extensional flow as well as the linear dependence of the normal stresses on the spatial

dimensions. Also, extensional rates can be varied by varying the flowrate through the flowfield.

This concept of a presheared planar extensional flowfield led to the design of the flowcell shown

in Figure 2.4.


This flowcell consists of three inlets and one outlet. The flow in the main channel is a gravity-

driven flow from an overhead reservoir and its purpose is to provide preshearing, to generate

shear rates greater than 1000 s-1. The flow from the side channels is generated by a pressurized

container and is meant to establish a planar extensional flowfield with extensional rates higher

than 50 s-1. This flow, superimposed on the shear flow from the main channel, is meant to create

the motion of counter-rotating vortices in the turbulent boundary layer. The exit channel has

been slanted at an angle of 13° to the horizontal in order to resemble the lifting motion of a

streamwise vortex as illustrated in Figure 1.2 from Kim et al (1971). All the channels have

rectangular cross-sections.

The flow rate in the main channel and the combined flow rates in side channels are the quantities

to be controlled in each experiment. The flow rate in the main channel is controlled using a valve

before the entrance of the main channel, as shown in Figure 2.5. The flow rate in the side

channels is controlled by the pressure in the container which generates this flow. The fluid

exiting the flowcell is collected and weighed over time using a digital balance and a timer in

order to determine the total flow rate.

This proposed flowcell should be a good model because it incorporates all of the essential

features present in the first stage of a turbulent burst. The side flows simulate extensional motion

caused by counter-rotating vortices while the main flow ensures that the flow is presheared, high

enough to produce elastic effects. Thus, the flowcell should be a good first attempt to model

turbulent burst using a laminar flow.


Figure 2.4 Setup of experimental flowcell

Figure 2.5 Setup of experimental apparatus


2.2 Design Considerations

Although rectangular channel flow is present in all components of the flowcell, the calculations

to design the flowcell can be greatly simplified if the flow can be modelled as pressure-driven

flow between two parallel plates. For this assumption to be valid, the channel aspect ratio of

width to height has to be high enough for the influence of the side walls to be negligible. In order

to test the validity of this assumption, the analytical solution for flow in a rectangular channel,

with various aspect ratios, was compared with the solution for flow between two parallel plates.

The details of this comparison can be found in Appendix B1. The results are shown in Figure

2.6.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

y/d

Ve

loci

ty (

m/s

)

Rectangular channel with a/d = 6.7




Flow between parallel plates

Figure 2.6 Comparison between vertical velocity profiles midway between the side walls for

flow in a rectangular channel with different aspect ratios, and flow between two parallel plates. Channel width = 2a, height = 2d


As the plot shows, at an aspect ratio greater than 3, the velocity profile calculated assuming 2D

flow between two parallel plates is in agreement with the analytical solution for the velocity

profile in a rectangular channel. Hence, an aspect ratio of 6.7 was chosen as the design aspect

ratio to neglect the influence of the side walls. Moreover, negligible influence of the side walls

means that the assumption of pressure-driven flow between parallel plates can be used to

perform design calculations for the flowcell. This assumption was therefore used in the

subsequent design calculations.

Computational fluid dynamics (CFD) is another method for analyzing the flow in the flowcell

geometry. The advantage of this method is that no simplifying assumptions are required and flow

behaviour, predicted by numerically solving the Navier-Stokes equations, can be visualized in all

sections of the flowcell, including in regions close to the side walls. Hence, simulation results

were used to validate the design of the flowcell and verify that many of the desired flow

parameters are achievable. For this purpose, results from numerical simulations performed using

COMSOL Multiphysics 4.0, are provided in the relevant sections to validate a number of design

decisions. Details of the software package and numerical computations can be found in

Appendix A2.

2.2.1 Main Channel Geometry

In order to eliminate inertial effects, it is desirable to have as low a Reynolds number as possible.

Hence, one of the objectives during the design of the flowcell was to minimize the Reynolds

number. In the present work, the Reynolds number, Re, based on the of the main channel is

given by:


W

QVh M

Re , (2.1)

where is the fluid density, is the fluid viscosity, h is the height, and V is the average

velocity given by:

WhQV M / , (2.2)

where MQ is the flow rate.

Because the Reynolds number is proportional to the flow rate and inversely proportional to the

viscosity, lowering the flow rate and increasing the viscosity of the fluid are two methods to

reduce the Reynolds number. As the primary objective is to model the first stage of a turbulent

burst, one estimate of the experimental Reynolds number is the local Reynolds number in the

viscous sublayer. This Reynold’s number can be estimated from the definition of *y , given in

Appendix B as:

yu

y ** , (2.3)

where *u is the friction velocity, and is the kinematic viscosity of the fluid. Thus the quantity

*y is analogous to the definition of Reynolds number and can be used as an estimate of the

design Reynolds number. In the viscous sublayer, *y ranges from 0 to 5. Therefore, a Reynolds

number of 5 in the flowcell should correspond to the flow in the viscous sublayer.

As discussed in section 1.1, for dilute aqueous polymer solutions, the onset of drag reduction

takes place at a shear rate of 1000 s-1. This critical value is based on the reciprocal of the


relaxation time of aqueous polymer solutions with relaxation times in the order of 1 ms. As most

viscoelastic fluids have relaxation times greater than 1 ms, the value of 1000 s-1 should be an

upper bound for the critical shear rate and should be high enough for the onset of drag reduction

with most viscoelastic fluids. Therefore, the flowcell should be able to generate a shear rate

considerably higher than this value in order to successfully simulate drag reduction effects.

The wall shear rate, W , can be derived from the approximation of flow between two parallel

plates as:

2

6

Wh

QMW . (2.4)

That is, the wall shear rate is inversely proportional to the square of the channel height. Hence,

the wall shear rate can be increased by reducing the channel height or increasing the flow rate.

The above equation should provide a reasonably accurate estimate of the centreline wall shear

rate in the flowcell, half-way from both the sidewalls.

To allow sufficient tolerance for machining of the flowcell, a minimum height of h = 1.5 mm

was chosen as the height of both the main and the side channels. A main channel flow rate range

of 4 ml/s to 7.5 ml/s was chosen because this range falls within the range of flow rates that can

be generated by the overhead reservoir. These values of h and MQ combined with the design

aspect ratio of W/h = 6.7, generates, according to equation 2.4, a wall shear rate range of W

from 1067 s-1 to 2000 s-1, which meets the design objective of having a wall shear rate greater

than 1000 s-1. From equation 2.1, the Reynolds numbers corresponding to this geometry and flow


rates range from 2 to 11. This range is comparable with the Reynolds numbers in the viscous

sublayer.

2.2.2 Side Channel Geometry

As discussed in section 1.1, the extensional rate in turbulent burst is approximately 50 s-1. In the

flowcell, extensional motion is created when the flow from the side channels is superimposed on

the flow from the main channels. This motion takes place in the region where the side channels

meet the main channel, as shown in Figure 2.4. The extensional rate is given by (Coventry &

Mackley, 2008):

,2hW

Q

S

S (2.5)

where SQ is the combined flow rate in both side channels, and h and SW are the side channel

height and width respectively. Thus, the extensional rate is proportional to the side channel flow

rate and inversely proportional to the side channel height and the square of the side channel

width. The obvious way of increasing the extensional rate is to increase the side flow rate, but

the maximum flow rate is limited by the pressure limit of the tank used to drive the side flow.

Therefore, a maximum side flow rate of 1 ml/s, corresponding to the tank pressure limit, was

used in the calculations. To maintain geometric conformity and to allow sufficient machining

tolerance, the side channel height was kept the same as the main channel height, i.e, h = 1.5 mm.

The side channel width therefore needs to be as small as possible in order to maximize the

extensional rate. To allow for sufficient machining tolerance, the side channel width, SW , was

chosen to be 2 mm. Substituting these values of SQ , h and SW into equation 2.5 yields a


maximum extensional rate of = 333 s-1, which is well above the required extensional rate in a

turbulent burst. From equation 2.1, the maximum Reynolds number corresponding to this

geometry and flow rate is 11. This value is comparable to the value of 5 in the viscous sublayer,

and thus ensures that inertial effects can be neglected.

2.2.3 Channel Lengths

To ensure that a parabolic velocity profile is established in the vertical centreplanes and the shear

rate at the wall is maximum, it is necessary for the flow in the main channel to become fully

developed before reaching the flow from the side channels. The minimum length, El , of the main

channel required for the flow to become fully developed can be calculated using the equation for

fully developed laminar flow in a rectangular channel assuming uniform flow at the entrance

(Schilchting, 1960, p. 171):

416.0 2Vh

lE , (2.6)

where is the kinematic viscosity of the fluid.

Using the design channel height, the maximum main channel flow rate, and a kinematic viscosity

of 8.5 x 10-5 m2/s, the maximum inlet length was calculated as El = 18 mm. This value of

viscosity is the kinematic viscosity of a 33.3% solution of polyethyelene glycol in water. The

justification for using this viscosity will be provided in the section describing the choice of

fluids. To ensure that the length is long enough for the flow to become fully-developed, a main

channel length of L = 28 mm was chosen.


Similarly, a side channel length of LS = 10 mm was chosen to ensure that the side channel flow is

also fully-developed.

2.2.4 Exit Channel Length

The exit channel length was chosen as 13 mm. This distance allows enough space to install a

pressure tap in the exit channel wall, leaving a workable machining distance on either side of the

tap.

2.2.5 Pressure Tap Locations

The pressure drop along the channel will be measured at each flow rate. For a Newtonian

laminar flow, this pressure drop arises due to the fluid’s viscosity. For a polymeric liquid,

however, the pressure drop can be caused by both viscosity as well as elasticity. For such a fluid,

the extra pressure drop obtained above values for a Newtonian fluid with the same viscosity is a

measure of the fluid’s elasticity. Therefore, if this experimental design is indeed a model for

turbulent burst, the elastic pressure drop obtained should be a function of the amount of drag

reduction.

As shown in Figures 2.4 and 2.5, the pressure difference between a location upstream in the main

channel flow and a location downstream of the flow from the side channels is measured. These

locations would ensure that the measured pressure drop is between a region where the fluid has

not been extended and a region where it has been subjected to planar extension. The criterion for

choosing the locations was obtaining pressure drop readings across a portion of the channel

where the flow has been subjected to both preshearing and planar extension. To do this, the high


pressure tap was placed at a distance of 11 mm from the main channel entrance while the low

pressure tap was placed 6.5 mm from the slanted channel exit.

2.2.6 Pressure Drop & Choice of Test Fluids

The pressure transducer used in the experiment is a Honeywell differential pressure sensor which

generates an electrical signal based on the deflection of its silicone membrane. This was the

transducer that was available in the laboratory and therefore was used for this exploratory work.

This device was calibrated using a column of water. Detailed specifications of the pressure

transducer as well as the results of the calibration can be found in Appendix D. This pressure

sensor can provide a minimum pressure reading of 0.1 psi in its linear range. Hence, it is

necessary to ensure that the pressure drop in the channel is high enough to be measured by the

transducer. High values would also ensure that the readings are not affected by noise associated

with weak signal. The pressure drop, P , along the flowcell can be estimated using the

following equation based on flow between parallel plates:

3

12

wh

LQP M (2.7)

Although water is usually the fluid of choice in experiments involving channel flows. However,

using water in the flowcell gives pressure drops from 0.005 psi to 0.01 psi, values much below

the minimum 0.1 psi of the pressure transducer. To increase the pressure drop to a measurable

level, it is necessary to increase the viscosity of the fluid to about 20 times that of water. Since

drag reduction is observed with solutions of very low polymer concentration, the addition of the

very small quantities of a drag-reducing polymer would not have much effect in altering the fluid

viscosity. Hence it was decided that a viscous solvent would be used instead of water.


Experiments conducted by Dontula et al (1998) showed that dissolving moderate quantities of

polyethylene glycol (PEG) in water can increase the viscosity of the solution by up to 200 times

that of water while the fluid remains Newtonian. PEG is a low molecular weight polymer and

hence non-Newtonian effects are not observed as long as the concentration is below

approximately 43% by weight. The advantage of using an aqueous PEG solution over other

viscous liquids like glycerol or silicone oils is that a PEG solution is capable of dissolving water-

soluble polymers, a property necessary for the present work.

For the present experiments, polyethylene glycol, with a molecular weight of 8000, was

dissolved in water to prepare a fluid with a concentration of 33.3% by weight. Dontula et al

(1998) reported this fluid has a viscosity 86 times that of water. Hence, this fluid was used as the

solvent for dissolving drag-reducing polymers. The density of PEG is almost identical to that of

water, i.e, = 1000 kg/m3. This fluid is subsequently referred to as the PEG solvent.

For a flow rate of 7.5 ml/s and viscosity of 0.086 Pa.s, numerical simulation using COMSOL

gives a pressure drop of 0.7 psi along the channel between the pressure taps. This value is within

the measurable range of the pressure transducer.

2.3 Flowfield with Final Design

Since the final design of the flowcell is now complete, this design geometry, with dimensions

summarized in Table 2.1, can be used in COMSOL to obtain numerical velocity profiles in

different parts of the flowcell. These velocity profiles are important in understanding the


flowfield and to observe if the flow behaviour is as expected. The PEG solvent’s fluid properties,

namely the values of the viscosity and density, were used to obtain the simulation results. For

running these simulations, a uniform flow velocity of 0.5 m/s, corresponding to a flow rate of 7.5

ml/s, was used as the main channel entrance flow velocity while a uniform flow velocity of 0.2

m/s, corresponding to a flow rate of 0.6 ml/s, was used as the side channel entrance flow velocity

in each side channel. These velocities were chosen to correspond to the upper limits of the design

flow rates in order to achieve maximum shear and extensional rates in the flowcell.

Geometric Parameter Value Unit

Main channel width, W 10 mm

Main channel height, h 1.5 mm

Main channel length, L 28 mm

Side channel width, WS 2 mm

Side channel height, h 1.5 mm

Side channel length, LS 10 mm

Exit channel height, h 1.9 mm

Exit channel length, Le 13 mm

Slant angle 13 degrees

Table 2.1 Summary of channel dimensions used in the final design of the flowcell

Figure 2.7 is a snapshot from COMSOL showing the magnitude of the flow velocity along the

vertical centreplane of the main channel. An enlarged view of the streamwise (x-component)

velocity at the entrance section is shown in Figure 2.8. The velocity magnitudes are shown in the


accompanying color legends, which indicate that a uniform flow velocity of 0.5 m/s at the

entrance becomes fully developed with a maximum of about 0.75 m/s.

Figure 2.7 Streamwise component of velocity along vertical centreplane

Figure 2.8 Enlarged view of streamwise velocity at the main channel entrance


Figure 2.9 shows the streamwise velocity component (z-direction) in the side channel vertical

centreplane. This plot confirms that the flows in the side channels are fully developed and are

equal and in opposite directions.

Figure 2.9 Streamwise component of velocity in the vertical centreplane of the side channels. .

Color legend shows velocity in metres per second (m/s).

The above velocity plots provide some insight into the expected flow behavior in the flowcell.

The flow in both the main and the side channels become fully developed, confirming that the

channel lengths are sufficiently long, even for the highest flow rates to be used in the

experiments. Further numerical simulation using COMSOL will be used in subsequent chapters

to analyze the experimental results, in particular to calculate strain and strain rates, which are

useful in explaining the cause of elastic effects.


2.4 Fabrication of Flowcell

Detailed engineering drawings of the parts and assemblies required to fabricate the flowcell were

prepared using SolidWorks. The detailed drawings for each part can be found in Appendix E.

The material was polycarbonate, chosen for its visual transparency, water-resistance and ability

to form chemical bonds with adhesives. The flowcell was constructed at the University of

Toronto’s MIE machine shop.

28

Chapter 3: Test fluids

As described in section 2.2.6, a 33.3% by weight of polyethylene glycol solution dissolved in

water was chosen as the inelastic fluid for establishing the Newtonian baseline, against which

pressure drop measurements of dilute polymer solutions will be compared. Several viscoelastic

fluids were prepared by dissolving different concentrations of polyethylene oxide in the

polyethylene glycol solvent. This chapter describes the fluid characterization tests conducted on

these fluids and the results. These tests were carried out to measure the relevant viscous and

elastic properties of these viscoelastic fluids. The rheological concepts behind these fluid

properties, as well as the laboratory techniques used to conduct the tests are presented first.

3.1 Non-Newtonian Fluids

The most important fluid property that differentiates a non-Newtonian fluid from a Newtonian

fluid is viscosity. For simple shearing, the viscosity, , is defined as:

xy

xy

, (3.1)

where xy is the shear stress in the x - y plane and xy is the shear rate defined as

y

vxxy

, (3.2)

where xv is the velocity component in the flow (x) direction. Unlike Newtonian fluids, the

viscosity of most non-Newtonian fluids decreases with shear rate, i.e, the fluids are shear-

thinning. There also exist fluids, such as a concentrated suspension of corn starch in water,

whose viscosity increases with shear rate, and as such are termed as shear-thickening fluids.

Chapter 3: Test Fluids 29

Another important property of non-Newtonian fluids is elasticity. Many polymeric liquids

exhibit behaviours such as stringiness which indicate that these fluids possess elasticity in

addition to viscosity.

3.1.1 Boger Fluids

Boger fluids are dilute polymer solutions whose viscosity remains almost constant with respect

to shear rate. This property makes these fluids special because it enables elastic effects to be

clearly separated from viscous effects. Although most polymer solutions and melts are inherently

shear thinning, the polymer concentrations in Boger fluids are low enough that the variation in

viscosity can be ignored. These fluids were first introduced by Boger in 1977 and, since then,

they have been an effective means of studying elastic effects of polymer solutions (Boger, 1977).

In an experiment conducted with two fluids: a Boger fluid and a Newtonian fluid with the same

viscosity, the difference in outcomes at the same flow rate can be attributed to elasticity alone. In

experiments where the viscosities between the fluids are different, the results can still be

compared by making use of appropriate dimensionless groups. Thus, Boger fluids have made it

possible to determine whether an observed non-Newtonian effect is caused by shear thinning, or

elasticity, or both (James, 2009). Therefore, although drag-reducing fluids are usually not Boger

fluids, because drag reduction is caused by elasticity, Boger fluids can be used to understand the

role of elasticity in causing drag reduction.

3.1.2 Rheometry

Shear rheometers are the most common instruments used to characterize non-Newtonian fluids.

A wide range of rheological shear properties including viscosity, first normal stress difference,


and viscous and elastic moduli can be measured by these instruments. In the present study, a

cone-and-plate rheometer was used to characterize the test fluids.

A cone-and-plate rheometer consists of a fixture, as shown in Figure 3.1, mounted over a flat

plate leaving a small gap for the fluid sample to be inserted in the space between the fixture and

the plate. The cone angle is typically between 0.5 to 2 degrees, while the diameter is usually

between 2 cm to 6 cm.

Figure 3.1 Working principle of a cone-and-plate rheometer

As the cone rotates with a constant angular velocity, , it generates a uniform shear rate, ,

throughout the fluid:

, (3.3)

where is the cone angle. Thus a wide range of shear rates can be obtained depending on the

range of angular velocity of the machine’s motor.

The shear stress, , in the fluid can be expressed in terms of the torque, M , according to the

following relationship (Macosko, 1994):


32

3

R

M

, (3.4)

where R is the cone radius. Using equation 3.1, equation 3.4 can be rewritten to obtain an

expression for the fluid viscosity, :

32

3

R

M

(3.5)

Using the dimensions of the cone geometry, the rheometer measures the torque in order to

determine the viscosity for each angular velocity and calculates the shear rate corresponding to

this angular velocity and thus produces viscosity measurements at different shear rates.

3.2 Shear Viscosity of Test Fluids

The drag reducing polymer used in this work was polyethylene oxide (PEO) with a molecular

weight of 4 million. PEO is well-known to cause drag reduction and has been used in many prior

studies (for example James et al. 1987, and Scrivener 1974). Solutions with concentrations of

100 ppm, 500 ppm, 750 ppm, 1000 ppm, and 1200 ppm PEO dissolved in the PEG solvent were

prepared. Shear viscosity measurements of these fluids were made with an AR2000 rheometer

using a 60 mm 0.5 degree cone-and-plate fixture at a temperature of 25° C. The temperature was

chosen to coincide with the laboratory temperature when the flowcell experiments were

conducted. Figure 3.2 shows the viscosity data as a function of shear rate. The test fluids were

sheared up to a shear rate of 2000 s-1, corresponding to the maximum shear rate expected in the

flowcell.

The plot shows that the viscosity of PEG solvent alone is independent of the shear rate,

confirming that the solvent exhibits Newtonian behavior in this range of shear rates. The


constant viscosity of PEG solvent was found to be 85 mPa.s, which is within 1% of the value of

86 mPa.s reported by Dontula et al (1998) for a solution with the same concentration of PEG.

With this value of viscosity, the pressure drop in the main channel should range from 2800 Pa to

5500 Pa, corresponding to a range of 0.4 psi to 0.8 psi. This pressure drop is within the 0.1 psi –

1 psi measurable range of the pressure sensor.

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

10 100 1000 10000Shear Rate (s-1)

Vis

cosi

ty (

Pa.

s)

1200 ppm PEO in PEG Solvent



PEG Solvent

Figure 3.2 Steady shear viscosity measurements for test fluids

Also shown in Figure 3.2 are the viscosity measurements for the 100 ppm, 500 ppm, and the

1200 ppm PEO solutions. These results indicate that the addition of the 100 ppm PEO increases

the viscosity of the PEG solvent by 4%, 500 ppm by 20%, and 1200 ppm by 50%. The solutions’

viscosities were virtually constant indicating that the fluids are Boger fluids.


3.3 Critical concentration

As described earlier, drag reducing fluids are dilute polymer solutions, i.e, a solution in which

the macromolecules are so few that there is no interaction between them. The only interaction in

these fluids is between the polymer chains and the solvent. An experimental measure of the

boundary separating dilute and semi-dilute solutions is defined by the critical concentration, *c .

A widely accepted definition of *c , provided by Graessley (1980), is:

][

77.0*

c , (3.6)

where ][ is the intrinsic viscosity defined as

csp

c

0lim][ (3.7)

where c is the concentration, and sp is the specific viscosity defined as:

s

ssp

, (3.8)

where is the overall viscosity and s is the solvent viscosity. The quantity csp

is known as

the reduced viscosity.

The critical concentration of the PEO/PEG solution was determined by plotting the reduced

viscosity, cSP

, versus the concentration for four different concentrations, as shown in Figure

3.3.


520

522

524

526

528

530

0 200 400 600 800 1000 1200 1400

Concentration (ppm)

Red

uce

d V

isco

sity

(η

sp/c

)

ηsp=(η-ηs))/ηs

[η]

Figure 3.3 Determination of intrinsic viscosity

The intrinsic viscosity was evaluated by extrapolating the plotted data in Figure 3.3 to zero

concentration to obtain the intercept of 521 ppm-1.

The critical concentration, *c , was then obtained, using equation 3.6, as

ppmc 148077.0*

Thus, the test fluids were all within the dilute regime for this polymer/solvent combination, even

though their concentrations are much higher than the O(10) ppm concentration of well-known

aqueous drag-reducing fluids. Because the solutions are dilute, the test fluids can be considered

Boger fluids.


3.4 Relaxation Time & First Normal Stress Difference

When a Newtonian fluid is subjected to a step increase in strain, as shown in Figure 3.4 a), the

stress, , relaxes instantly to zero, as shown in Figure 3.4 b). However, when a viscoelastic fluid

is subjected to the same deformation, the stress decays as shown in Figure 3.4 c) (Macosko,

1994, p.110).

Figure 3.4 a) Step input in strain and the corresponding stress relaxation of b) a Newtonian fluid

and c) a viscoelastic fluid and solid (Reproduced from Macosko 1994, p.110).

A viscoelastic fluid’s relaxation time indicates how quickly the fluid relaxes and is another

measure of the fluid’s elasticity. In this work, relaxation time was a key property used to

characterize fluids and to explain experimental results.

3.4.1 First Normal Stress Difference

The first normal stress difference, N1, of a viscoelastic fluid is defined as the difference between

the normal stress components in the flow direction and in the direction perpendicular to the flow

direction, i.e:

yyxxN 1 , (3.9)


where xx and yy are respectively the normal stress components in the streamwise and

transverse directions. The first normal stress difference is a measure of fluid elasticity in shear

and its value is identically zero for Newtonian fluids.

By measuring the axial force, F , a cone-and-plate rheometer can produce values for 1N

according to the following expression (Macosko, 1994):

21

2

R

FN

, (3.10)

where R is the cone radius. However, this axial force generated by weakly elastic fluids in shear

is usually very small and often lower than the measurable range of the force transducer in the

rheometer. Therefore, if the test fluids are able to generate large enough normal stresses in shear,

it should be possible to obtain 1N measurements for these fluids.

3.4.2 Oldroyd-B model

The Oldroyd-B constitutive equation is a mathematical model commonly used to predict the

behaviour of Boger fluids. It has been derived from the dynamics of a dilute suspension of bead-

spring dumbbells in a viscous fluid, resembling the dynamics of polymer chains dissolved in a

viscous solvent (Prilutski et al, 1983). This model is particularly appropriate for Boger fluids

because the separate contributions of the solvent and the polymer viscosities are included in the

constitutive equation, according to:

SP , (3.11)

where is the fluid viscosity, P is the contribution to the viscosity by the polymer and S is

the solvent viscosity.


For steady shear flow, the Oldroyd-B equation predicts 1N to have a quadratic dependence on

shear rate, i.e:

21 2 PN (3.12)

where is the shear rate, is the relaxation time. Constitutive equations of the Oldroyd – B

model can be found in Appendix C.

Equation 3.12 can be used to determine a fluid`s relaxation time from measurements of 1N .

Therefore, measurements of first normal stress difference, 1N , were made under steady shearing

using an AR2000 cone-and-plate rheometer and attempt was made to obtain a relaxation time.

Reliable 1N measurements could be obtained for only one of the fluids, the 1200 ppm solution,

the data for which are shown in Figure 3.5. The figure is a plot of 21 / N vs shear rate, . If the

fluid is an Oldroyd-B fluid the quantity 21 / N should be independent of the shear rate.


0.001

0.01

100 1000Shear Rate (s-1)

N1/γ

2

Figure 3.5 First normal stress difference measurements in response to steady shearing for the

1200 ppm PEO solution in PEG solvent

As is evident from the data, this fluid follows the Oldroyd-B prediction in the range of shear

rates below 150 s-1, making the quantity 21 / N independent of shear rate in this range. Using

this constant value of 21 / N , together with the polymer viscosity, P , equation 3.12 can be

used to determine a relaxation time, , for this fluid as:

382 2

1

P

Nms


Thus, the relaxation time for the 1200 ppm solution, as determined from 1N measurements is

approximately 38 milliseconds.

3.5 Elastic modulus

One other measure of a fluid`s elasticity is a quantity known as the elastic modulus. This

quantity, defined in oscillatory shear flow, was also used to characterize the test fluids.

When a fluid is sheared sinusoidally with a small amplitude 0 at a frequency , the stress

response is also sinusoidal.

For a strain input: )sin()( 0 ttxy ,

the stress-response, consisting of an in-phase and an out-of-phase component, is given by:

),cos(")sin(')(

0

tGtGtxy

(3.13)

where xy is the output shear-stress, 'G is known as the dynamic storage modulus or the elastic

modulus, and "G is known as the dynamic loss modulus or the shear modulus. For a viscoelastic

fluid, 'G is a measure of the fluid’s elasticity while "G is a measure of its viscosity. For a

Newtonian fluid, xy is always 90° out of phase with xy and therefore, by equation 3.13, 'G is

identically zero for a Newtonian fluid.

Measurements of 'G and "G can be obtained from a cone-and-plate rheometer, which determines

these quantities by applying an oscillatory strain and measuring the amplitude of the torque

response and its phase shift with the applied strain (Collyer and Clegg, 1998).


Figure 3.6 shows measurements of the elastic modulus, 'G , in response to small amplitude

oscillatory shear, performed using the ARES rheometer with a 5 cm 0.5 degree fixture at 25° C.

For all concentrations shown, 'G increases with the frequency of oscillation, , indicating that

the solutions possess elasticity. Moreover, at each frequency, 'G values for the 1200 ppm is

almost 70% higher than that of the 500 ppm indicating that the 1200 ppm solution is

considerably more elastic than the 500 ppm solution. As shown in the plot, 'G values are below 1

Pa for all the fluids in the range of frequencies used. These values agree with measurements

made by Dontula et al. (1998) for PEO/PEG solutions with similar concentrations. Reproducible

'G measurements could not be obtained for the 100 ppm solution indicating that the elasticity of

this solution is below the measurable range of the rheometer. Similarly no reproducible

measurements could be obtained for the Newtonian solvent, 'G readings for which should be

identically zero.

0.01

0.1

1

1 10 100

ω (1/s)

G' (

Pa)

1200 ppm

1000 ppm

750 ppm

500 ppm

Figure 3.6 Elastic modulus measurements in response to small-amplitude oscillations


For small frequencies, i.e, as 0 , the Oldroyd-B model predicts 'G to have a quadratic

dependence on the frequency, ie:

2' PG , as 0 , (3.14)

where and P once again are respectively the fluid’s relaxation time and the polymer

contribution to the viscosity. The derivation of the above equation is given in Appendix C.

According to this equation, a logarithmic plot of 'G versus , as that in Figure 3.6, should have

a slope of 2. But a slope of 2 is not observed in this plot, even though several other properties

confirmed that the fluids are Boger fluids. This discrepancy can be explained by noting that the

above relationship holds only for very small values of . Thus, it is possible that a slope of 2

can be obtained at lower frequencies than were used in these measurements; however, at lower

frequencies, reliable measurements could not be obtained for any of the fluids, as lower values of

'G most likely fall outside the measurable range of the rheometer.


3.6 Summary of Fluid Properties

Table 5.1 is a summary of the relevant fluid properties determined from the fluid characterization

measurements described above for the test fluids.

Fluid Fluid

Viscosity, (Pa.s)

Polymer Viscosity, P

(Pa.s)

Solvent Viscosity, S

(Pa.s)

RelaxationTime,

(s)

Density, (kg/m3)

PEG Solvent 0.085 - - - 1000

100 ppm PEO in PEG 0.088 0.003 0.085 - 1000

500 ppm PEO in PEG 0.103 0.018 0.085 - 1000

1200 ppm PEO in PEG 0.128 0.043 0.085 0.038 1000

Critical Concentration: *c = 1480 ppm

Table 3.1 Fluid properties of test fluids

43

Chapter 4: Experimental Results & Discussions

Results of pressure drop measurements from the flowcell for the Newtonian and viscoealstic

fluids are presented and discussed in this chapter. Results from numerical analyses, performed in

order to explain some of the results, are also discussed where applicable.

4.1 Main Channel & Side Channel Combined Flow

A presheared planar extensional flowfield was established by combining flows in the main

channel and the side channels. Figure 4.1 shows the streamlines in such a flowfield as predicted

by COMSOL for a Newtonian fluid. As shown by this plot, the streamlines from the side

channels bend towards the exit channel as they meet the streamlines from the main channel, and

push the main channel streamlines towards the centre of the channel. Although the streamlines

do not resemble the motion of counter-rotating vortices in the turbulent boundary layer, they

demonstrate that extensional motion is present in the flowfield in the region where the side flow

is superimposed on the main channel flow.

In order to ensure that the flowfield was symmetric, the flow rate from each side channel was

measured individually. Keeping one channel blocked, the flow rate from the other channel was

measured, and vice versa. Equal flow rates in each channel confirmed that the flowfield was

symmetric.

Chapter 4: Experimental Results & Discussions 44

Figure 4.1 Three-dimensional view of streamlines for combined flow from the main and the side

channels showing how flow from the side channels is superposed on main channel flow

The extensional rate, , in this flowfield was estimated as:

,2hW

Q

S

S

where SQ is the combined flow rate in both side channels, SW is the side channel width, and h

is the side channel height.

Drag reduction is thought to be a phenomenon caused by elasticity in extension. Therefore if the

designed flowfield is a model of the beginning of turbulent burst, an elastic effect that depends

on the rate of extension in this flowfield should have a correlation with the amount of drag

reduction. It is therefore necessary to study the effect of extensional rates on the pressure drop, at

different values of preshearing. The wall shear rate, W , that indicates the amount of presheraing,

was calculated using equation 2.4 as:

2

6

Wh

QMW ,


where MQ is the main channel flow rate, W is the main channel width and h is the main channel

height.

For each test, the main channel flow rate was set to a value corresponding to a certain wall shear

rate, while the side channel extensional rate was varied to vary the extensional rates. By

repeating the tests at different main channel flow rates, the effect of varying the shear rate was

studied.

4.1.1 Newtonian Results

The first experiments were conducted with PEG solvent in order to establish a Newtonian

baseline. Figure 4.2 shows pressure drop measurements for the PEG solvent at two main channel

flow rates, at shear rates above the critical value of 1000 s-1 required for the onset of drag

reduction. The pressure drop is plotted versus extensional rate, , and the corresponding side

channel flow rate, SQ . The pressure drop predicted by COMSOL, for a Newtonian fluid with the

same density and viscosity as the PEG solvent, is also shown in the graph.


2500

3000

3500

4000

100 150 200 250 300 350 400 450

Pre

ssu

re D

rop

(P

a)

Wall Shear Rate = 1500/s

Wall Shear Rate = 1100/s

COMSOL

Qs (ml/s) = 0.6 0.9 1.2 1.5 1.8 2.1 2.4

έ (1/s) =

Re = 7

Re = 8

Fluid: 33.3% solution ofpolyethylene glycol in water (PEG solvent)

Figure 4.2 Measured pressure drop and prediction by COMSOL, at two shear rates,

corresponding to main channel flow rates of 4 ml/s (Re=7) and 5.5 ml/s (Re=8), for various side channel flow rates.

The plot shows that the pressure drop increases linearly with side channel flow rate and

extensional rate at both shear rates, although the variation is slight. The pressure drop at a shear

rate of 1500 s-1 is approximately 36% higher than that at 1100 s-1. As expected, this difference

remains almost constant in the range of extensional rates studied, because for a Newtonian fluid,

the pressure drop, P , is proportional to the main channel flow rate, i.e:

3

12

Wh

LQP M ,

where is the fluid viscosity, and L is the length between the pressure tap locations. Because

the main channel flow rate was increased to increase the wall shear rate, the pressure drop

increased proportionately.


The increase in pressure drop with extensional rate is caused by the increase in the side channel

flow rates required to increase the extensional rate. By continuity, this increases the flow rate in

the exit channel and thus increases the pressure drop. Since the side channel flow rate is only a

fraction of the main channel flow rate, the increase in the total flow rate is small and hence the

increase in the pressure drop is also slight. Since PEG solvent is an inelastic fluid, the variation

of pressure drop with shear and extensional rates is caused entirely by the fluid’s viscosity and

hence is linear.

The pressure drop predicted by COMSOL is in agreement with the experimental data, indicating

that the software is capable numerically of solving this flowfield for a Newtonian fluid.

4.1.2 Results with PEO Solutions

The experiments described above were then repeated with two viscoelastic fluids: 100 ppm and

500 ppm PEO dissolved in the PEG solvent. The 100 ppm concentration was chosen because this

is the minimum concentration where the solution viscosity can be discerned from the solvent

viscosity. The 500 ppm solution was chosen because 'G measurements show that this fluid is

considerably elastic. Since these fluids had higher viscosities than the solvent, P was

normalized by multiplying it by /PEG , the ratio of viscosities, to eliminate viscous effects and

thereby reveal any elastic effects. The normalized pressure drop,

PEGPx , therefore represents

the pressure drop of a fluid with the same viscosity as the PEG solvent. Figures 4.3 to 4.5 show

plots of this quantity versus extensional rate for different shear rates, corresponding to three main


channel flow rates. All of these shear rates are higher than the critical value of 1000 s-1, required

for the onset of drag reduction.

2000

2500

3000

3500

4000

0 50 100 150 200 250 300 350 400 450

Extension Rate (s-1)

∆P

x (η

PE

G /

η)

(Pa)

500 ppm

100 ppm

PEG Solvent

Preshearing:

Wall shear rate = 1100 s-1

Figure 4.3 Normalized pressure drop measurements for the viscoelastic test fluids and the

Newtonian solvent at a wall shear rate of 1100 s-1, corresponding to a main channel flow rate of 4 ml/s and a Reynolds number of 7.


3000

3500

4000

4500

5000

0 50 100 150 200 250 300 350 400 450


∆P

x (η

PE

G /

η)

(Pa)

500 ppm

100 ppm

PEG Solvent

Preshearing:



Newtonian solvent at a wall shear rate of 1500 s-1, corresponding to a main channel flow rate of 5.5 ml/s and a Reynolds number of 8.

5000

5500

6000

6500

7000

0 50 100 150 200 250 300 350 400 450


∆P

x (η

PE

G /

η)

(Pa)

500 ppm

100 ppm

PEG Solvent

Preshearing:



Newtonian solvent at a wall shear rate of 2000 s-1, corresponding to a main channel flow rate of 7.5 ml/s and a Reynolds number of 11.


In the above plots, since the pressure drop readings have been normalized to eliminate viscous

effects, any data above the PEG solvent’s baseline signify a non-viscous effect, i.e, an elastic

effect. Thus, elastic effects are observed at all shear rates with the 500 ppm solution and at shear

rates greater than 1500 s-1 with the 100 ppm solution. It should be noted from these results that

the magnitudes of the observed effects are dependent on the shear rates and not on the

extensional rates. If the elastic effect was an extensional one, it would vary non-linearly with

extensional rate, a behavior characteristic of extensional effects but absent in the present results.

The observed elastic effect, however, grows with increasing shear rates. At a shear rate of 1100

s-1,

PEGPx is about 8% higher than the PEG solvent for the 500 ppm solution, whereas the

difference is about 15% at a shear rate of 1500s-1, and almost 20% at a shear rate of 2000s-1. This

observation led to the need for investigating the effect of shearing, independently, on the

measured pressure drop.

For the three shear rates tested, the pressure drop varies linearly with extensional rate in the

range of extensional rates tested. This linear increase in pressure drop with extensional rate,

observed both with the Newtonian PEG solvent as well as the polymer solutions, is a viscous

effect caused by the increase in the side channel flow rate.

In this experiment, the Reynolds number, defined as Vh

Re , has a maximum value of 11. For

channel flows, the transition from laminar to turbulent flows takes place at a Reynolds number of

around 1500 and hence Re = 11 present in the experiment should be low enough to assume that

inertial effects can be neglected. However, since the velocity in the flow direction is changing,


inertial forces may still play a role in this flowfield. To examine the role of inertia in this case, a

more appropriate form of Reynolds number is:

2Re

VhI

, (4.1)

where V is the change in velocity in the flow direction. A derivation of this Reynolds number

can be found in Appendix B6. For the experiment with combined flow in the main and side

channels, the change in velocity can be obtained using the flow rates and the dimensions of the

flowcell. The maximum Reynolds number obtained with V is ReI = 0.7. Thus inertial inertial

effects are neglected in the first instance.

4.2 Main Channel Flow without Side Flow

As the elastic effects observed in the side flow experiment varied with the wall shear rate, a

different experiment involving flow in the main channel only, with no flow in the side channels,

was conducted to study the effect of shearing alone. The results from this experiment are shown

in Figure 4.6. In order to eliminate viscous effects, the quantity

PEGPx is plotted once again;

this time, however, the independent variable is the shear rate in the main channel.


0

1000

2000

3000

4000

5000

6000

7000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Shear Rate (s-1)

∆P

x (η

PE

G /

η)

(Pa)

500 ppm

100 ppm

PEG Solvent

Re = 2

Re = 11

Qside = 0


Newtonian solvent for variation in main channel flow with no flow from the side channels. The Reynolds numbers corresponding to the lowest and highest flow rates are 2 and 11 respectively.

The results show that an elastic effect is observed with the 500 ppm PEO solution in PEG as the

pressure drop, after compensating for the difference in viscosities between the fluids, increases

above the Newtonian baseline beyond a certain shear rate. The above data also shows an onset of

elastic effects, as reported in the literature for dilute polymer solutions. This onset takes place at

a shear rate of approximately 1000 s-1, beyond which the effect appears to grow with increasing

shear rate. No effect is observed with the 100 ppm solution, the data for which falls on the

Newtonian baseline.


Since there is no flow from the side channels, there is no identifiable extensional motion taking

place in this flowfield. The elastic effect observed with the 500 ppm solution therefore has to be

attributed to shearing. Since the highest shearing takes place at the walls, it is necessary to

examine the wall shear rates and the elastic stresses generated by shear.

4.3 Analyses of Results

The results of the main channel experiment show an elastic effect that grows with the wall shear

rate. The next step is therefore to understand and explain the cause of this effect. However, first,

it is necessary to ensure that the observed phenomenon is not caused by a flow instability.

4.3.1 Flow Instability in Shear

The shear flow of a Boger fluid can become unstable when the shear rate exceeds a critical value.

Hence, in order to differentiate between an elastic effect and a flow instability, it is necessary to

know this critical value. The value is given in terms of the Weissenberg number, ,Wi defined as

the product of the fluid’s characteristic time and the rate of shear, i.e:

, Wi (4.2)

Phan-Thien (1985) derived an expression for the critical shear rate of an Oldroyd-B fluid using

the geometry of a cone-and plate rheometer. For a Boger fluid with viscosity and polymer

viscosity P , the critical Weissenberg number, cWi , for the onset of shear instability to take

place is given by:

5

2 PcWi (4.3)


For flow in a cone-and-plate rheometer, since the shear rate is proportional to the angular

velocity, , in this particular context, the Weissenberg number, Wi , can be written as:

, Wi (4.4)

The critical shear rate, c , can then be obtained as:

5

2 Pc , (4.5)

where is the fluid relaxation time and is the cone-angle of a cone-and-plate fixture.

As indicated by equation 4.5, the critical shear rate depends on the fluid’s relaxation time, which,

unfortunately, could not be determined for the 500 ppm PEO solution in PEG solvent. However,

the relaxation time for the 1200 ppm solution can be used to obtain a lower bound for the critical

shear rate because the critical shear rate varies with the inverse of the relaxation time. As the 500

ppm solution is certainly less elastic than the 1200 ppm, it is reasonable to assume that the

relaxation time for the 500 ppm solution is shorter than that of the 1200 ppm solution. Hence, for

a cone angle of 0.5 degrees, and for the fluid properties in Table 4.1, the critical shear rate can be

calculated as:

2505103.05

018.02

5.0038.0

180

x

x

xc 1s

Thus, the critical shear rate of the 500 ppm solution is at least 2500 1s . This values is higher

than the maximum shear rate in the experiment, and hence the possibility of a shear instability

affecting the results can be ruled out.


4.3.2 Hole Pressure Error

The observed elastic effects may have been caused by a phenomenon known as hole pressure

error. To understand this concept, it is necessary to identify the differences between measuring

the pressure of a Newtonian fluid and that of non-Newtonian fluid using a pressure tap.

When a Newtonian fluid flows between two parallel plates, the pressure, 1P , measured by a

pressure transducer flush mounted on the wall is the same as the pressure, 2P , measured by a

recessed transducer placed at the end of a pressure tap at the same downstream location as the

flush-mounted transducer, i.e, 21 PP , as shown in Figure 4.7 (a).

Figure 4.7 Pressure measurement in (a) a Newtonian fluid and (b) a viscoelastic fluid

(Reproduced from Bird et al., 1987)

However, because normal stresses can develop in a viscoelastic fluid, the reading given by a

pressure transducer is the pressure P plus yy , the component of normal stress acting on the

surface of the transducer. At the transducer surface, yy is generally lower than it is at the surface

of the flush-mounted transducer, making the readings of the two transducers different, i.e,

0)()( 21 yyyy PP , as indicated in Figure 4.7 (b). This difference in pressure readings is


known as the hole pressure error, *P . For pressure taps with circular cross-section, this quantity

is given by (Bird et al., 1987, p. 68):

W

xyxy

dNN

P

0

21*

3

1, (4.6)

where 1N & 2N are respectively the first and the second normal stress differences. Since 1N is

the dominant quantity and 2N is usually taken as one-tenth of 1N , ie, 21 1.0 NN (Bird et al.,

1987) equation 4.6 can be written as

)( 1* NfP (4.7)

where )( 1Nf is the function representing the right-hand side of equation 4.6. )( 1Nf will be

used in the subsequent sections to refer to this function.

An attempt will now be made to apply the principles of hole pressure error to flow in the main

channel. In this setup, the two pressure taps are located in different channel sections, as shown

earlier in Figure 2.4. However, since the channel heights at the two locations are different, the

wall shear rates are also different. More specifically, by equation 2.4, the wall shear rate, ,W

varies with the inverse square of the channel height, h , i.e:

2

1

hW .

Using the subscripts U and D to denote the upstream and the downstream sections respectively,

the ratio of heights from the channel geometry is:

79.09.1

5.1

mm

mm

h

h

D

U .

Hence, in the experiment involving only the main channel flow, the ratio of wall shear rates is


6.179.0

122

U

D

WD

U

h

h

,

Because the upstream wall shear rate is 60% higher than that in the downstream section, the first

normal stress difference, 1N , is higher upstream.

As mentioned, for a viscoelastic fluid, presence of normal stresses affects the measured pressure

at the location of each pressure tap because the normal stress component, yy , contributes to the

pressure reading. As the first normal stress difference 1N is different in the locations of the two

pressure taps, so is the quantity yy . For low shear rates and correspondingly small values of 1N ,

the difference in yy between the upstream and the downstream sections is small and hence does

not appear as an effect in the measured pressure drop. As 1N increases with the wall shear rate,

this difference in yy becomes larger and measurable, causing the onset of elastic effect in Figure

4.6. Also, according to the Oldroyd-B equation, 1N increases as the square of the shear rate and

thus the observed elastic effect also grows non-linearly with shear rate.

It is of interest to see if the observed increase in the pressure drop can be related to the

magnitude of 1N . This, however, could not be done because reliable measurements for 1N could

not be obtained for this particular test fluid, the 500 ppm PEO solution with the available

rheometers.


4.4 Side Channel Flow without Main Channel Flow

Since extensional flow, in combination with shear flow, did not produce any elastic effect, the

next step in the investigation was to examine if extensional flow alone was able to produce an

effect. To do this, an experiment involving side channel flow only was carried out. This

experiment was conducted by blocking the entrance to the main channel and having a pressure-

driven flow in the side channels. Figure 4.8 show the streamlines obtained from COMSOL for

this flowfield. The streamlines pattern is similar to that for stagnation point flow, indicating that

an extensional flowfield has been created. Although this setup was considerably different from

the previous two experiments, the arrangement for measuring the pressure difference was the

same as before, as shown in Figure 4.9.

Figure 4.8 Three-dimensional view of streamlines for side channel flow only


Figure 4.9 Pressure measuring arrangement for the side flow experiment

This time, since there is no flow in the main channel the pressure at location 1 is the stagnation

pressure, 0P , and hence the pressure difference, P , measured by the transducer is:

20 PPP , (4.14)

where 2P is the pressure at location 2. Thus, P is a measure of the pressure downstream of the

flow from the side channels, which is the pressure of interest in this case.

The results from this experiment, conducted with the PEG solvent as well as with two

viscoelastic fluids: the 500 ppm and the 1200 ppm solutions, are shown in Figure 4.10.


0

40

80

120

0 50 100 150 200 250 300έ (s-1) =

∆P

x (η

PE

G /

η)

(Pa

) 1200 ppm

500 ppm

PEG Solvent

QMain = 0

De (for 1200 ppm) = 1.9 3.8 5.7 7.6

Re=1

Re=11

Figure 4.10 Normalized pressure drop measurements for two viscoelastic test fluids and the Newtonian solvent with side channel flow with no flow in the main channel. For the 1200 ppm

fluid, the Deborah numbers corresponding to the extensional rates are also shown.

Like the previous graphs, the normalized pressure drop,

PEGPx is again the ordinate and this

time the abscissa extensional rate corresponding to the side channel flow rate. The results show

that, with side channel flow alone, elastic effects are observed with both solutions, with the 1200

ppm solution producing the larger effect. The onset of elastic effects takes place at a lower

extensional rate with the 1200 ppm solution than it does for the 500 ppm solution. These

observations are in accordance with elastic properties of the two test fluids. That is, the 1200

ppm fluid, being more elastic than the 500 ppm, requires a lower extensional rate to generate

elastic stresses necessary for onset.


The maximum Reynolds number, as shown in the plot, was normally 11, while the Reynolds

number based on the change in velocity as described in Appendix B6 , is ReI = 0.2. Hence

inertial forces are only a fraction of viscous forces and thus inertia does not play a significant

role in this case either.

Since neither inertia nor viscosity has any contribution to the results, it can be deduced that the

observed phenomenon is caused by fluid elasticity. The Deborah number, defined as a ratio of

the characteristic time of the fluid to the characteristic flow time, governs the extent to which

elasticity manifests itself in response to fluid acceleration or non-homogeneous deformation

(Dealy, 2010). Specifically, Deborah number, ,De is:

TDe

, (4.15)

where , the fluid’s relaxation time, is usually taken as the characteristic time, and T is the flow

time. For extensional flows, the observation time can be taken as the reciprocal of the rate of

extension. Thus,

,1

T (4.16)

Since it was possible to determine the relaxation time for one of the fluids, the 1200 ppm

solution, the Deborah number, corresponding to the extensional rates tested for this fluid, is also

plotted on the x-axis. The extensional rates were found using equation 2.5. For this fluid, the

onset is at a Deborah number of about 1.9. Although this result fulfills the condition of

)1(ODe required for extensional effects, the data for both fluids show maxima, at 6.7De

for the 500 ppm solution and at 7.5De for the 1200 ppm. These maxima indicate that the


observed elastic effect may not have been caused by extension because extensional effects are

expected to increase with De .

4.4.1 Elastic Effects in Extension

In order to understand whether the observed elastic effect is caused by extension, it is necessary

to identify the conditions required for an extensional effect to take place.

Boger fluids can produce large elastic effects in extension. A filament stretching rheometer can

produce purely uniaxial extension at constant extensional rates and thereby yields a true value of

extensional viscosity, E (Tirtaatdmadja and Sridhar 1993, Anna et al 2001, McKinley and

Sridhar 2002). Results from such an instrument are shown in Figure 4.11. For this typical Boger

fluid, data are plotted as the Trouton ratio, E , versus the Hencky strain, , defined as:

0

lnl

l , (4.17)

where l is the stretched length of a fluid filament with an initial length, 0l .

The Hencky strain can also be determined as:

dt , (4.18)

where is the extensional rate and t is the time with 0t corresponding to the start of

stretching. For a constant extensional rate, as produced by a filament stretching rheometer,

equation 4.18 can be simplified to give:

t . (4.19)


As shown by Figure 4.11, for small values of the Hencky strain, the Trouton ratio remains nearly

constant at the Newtonian value of 3. When the Hencky strain exceeds about 2, elastic effects

begin and E starts to rise sharply above the Newtonian value.

Figure 4.11 Uniaxial Trouton ratio for a Boger fluid (a semidilute solution of 0.31 wt%

polyisobutylene in polybutene) stretched over a range of extensional rates, plotted as a function of Hencky strain. (Reproduced from McKinley and Sridhar, 2002)

Using 2 as the critical Hencky strain, the amount of extension, c

l

l

0

, required by the

polymer chains to produce elastic effects can then be estimated from equation 4.17:

4.7)2exp(0

cl

l (4.20)


Thus, in order to produce extensional effects, polymer chains need to be extended at least 7 times

their length in equilibrium configuration.

In view of the above criterion, an analysis was performed, with the help of numerical simulation,

in order to determine whether, in the present experimental flowfield, the Hencky strain exceeds

the value of 2 required for producing elastic effects.

4.4.2 Numerical Analysis

Figure 4.12 shows a side view of the flowcell where the side channels meet the main channel.

For this analysis, the z -coordinate is used as the direction of flow, with 0z defined at the

edge of the side channel meeting the main channel, Sw is the width of each side channel, and the

centrelines of the main and the exit channels are as shown. Figure 4.13 shows the velocity,

obtained with COMSOL, along the main and exit channel centerlines, plotted against Swz / , the

normalized displacement in the flow direction. This centerline velocity is shown for three flow

rates covering the range of extensional rates which produced the maximum elastic effect. This

plot indicates that velocity is non-zero in the portion of the main channel approaching the side

channel intersection, even though there is no through flow in the main channel. The reason is

that, as shown by the streamlines in Figures 4.9, the fluid close to the wall of the side channel

passes through a region near the end of the main channel before entering the slanted exit channel.

This causes the velocity magnitude to be non-zero at negative values of Swz / .


Figure 4.12 Side view of section from flowcell showing intersection of the main, side and the

slanted exit channels

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3 4 5 6 7 8

z/ws - along flow direction

Vel

oci

ty (

m/s

)[i

n t

he

flo

w-d

irec

tio

n]

Q = 1 ml/sQ = 0.7 ml/sQ = 0.6 ml/s

ws : side channel width

Figure 4.13 Numerical results for the velocity along the main and exit channel centerlines,

obtained using COMSOL.

From the slope of each curve in Figure 4.13, it is possible to determine how the extensional rate

varies along the centreline for the particular flow rate. The extensional rate, by definition, is the


velocity gradient in the direction of flow. If iV is the velocity in the flow direction at a point i on

the curve corresponding to the location iz , the local extensional rate, i , is given by:

ii

ii

i

ii zz

VV

z

V

1

1 , (4.21)

where 1i is the point adjacent to the i th point. The slope is a measure of the extensional rate,

, which is a maximum where the slope is steepest. From equation 4.21, the maximum

extensional rate at a flow rate of 1 ml/s was found to be 120 s-1. This value is close to the

extensional rate of 150 s-1 estimated using equation 2.5 for the side flow experiment at the same

flow rate.

As mentioned above, the Hencky strain is the primary criterion for determining whether the

observed elastic effect is an extensional effect. For a variable extensional rate, the Hencky strain,

, is defined as:

dt , (4.22)

where t is the time. This integral can be evaluated numerically as:

N

iii tdt

1

, (4.23)

where N is the total number of points along a curve and the time, it , required for a fluid

element to travel from iz to 1iz , is given by:

i

iii V

zzt

1 , (4.24)


Figure 4.14 shows the Hencky strain, calculated using equation 4.23, plotted versus the Deborah

number, De , calculated numerically as:

max De , (4.25)

where is the 38 ms relaxation time for the 1200 ppm PEO solution and max is the maximum

extensional rate calculated using equation 4.21 for each of the three flow rates shown in Figure

4.13.

1.25

1.3

1.35

1.4

1.45

2 2.5 3 3.5 4 4.5 5

Deborah Number

Hen

cky

Str

ain

Figure 4.14 Calculated values of Hencky strain in the flowfield plotted as a function

of Deborah number

The maximum Hencky strain achieved in this flowfield is about 1.38, corresponding to a

Deborah number of 4.7. This value is less than the critical value of 2 required for producing

elastic effects in extension. Moreover, the increase of Hencky strain with Deborah number is also

very slight, an observation which can be explained by the fact that, although the rate of extension

increases with the increase of Deborah number, the time of travel shortens, and hence the


Hencky strain, being a product of the extensional rate and time, does not change much as the

Deborah number increases.

This numerical analysis was based on flow of a Newtonian fluid. If these numerical results are

valid for the viscoelastic test fluids, then the elastic effect observed in the side channel flow

experiment was not likely caused by extension. Therefore, it is necessary to investigate if the

effect was caused by shearing.

4.4.3 N1 Effect

As discussed earlier, the first normal stress difference N1 is a measure of fluid elasticity in shear.

Figure 4.15 a) shows 1N measurements obtained from an ARES rheometer for the 1200 ppm

PEO solution in PEG solvent plotted as 21

N

versus the shear rate . This plot shows that 21

N

approaches a constant value at low shear rates, as predicted by the Oldroyd-B model according to

equation 3.12. Using this constant value of 21

N

, the fluid’s relaxation time of 38 ms was found,

as outlined in section 3.4. Figure 4.15 b) shows 1N data obtained from the rheometer as well as

the Oldroyd-B model using the 38 ms relaxation time. The lowest shear rate at which 1N can be

measured by the rehometer is about 160 s-1. Below this shear rate, reliable 1N measurements

cannot be obtained from the instrument, which is unfortunate because calculations using

equation 2.4 show that, for the side flow experiment, the range of wall shear rates in the

downstream channel is much lower than the measurable range shown in this plot. However, it is

evident that at shear rates below 250 s-1, 1N for this fluid has quadratic dependence on shear

rate. Hence, by assuming that this fluid continues to behave like an Oldroyd-B fluid at shear rates


below the measurable range, the line representing the Oldroyd-B model can be extrapolated to

obtain 1N values at shear rates corresponding to the wall shear rates in the side flow experiment.

These extrapolated values of 1N have been plotted in Figure 4.16 along with the elastic pressure

drop, obtained in the side flow experiment. The elastic pressure drop, elasticP , is calculated by

subtracting the Newtonian pressure drop, NewtonianP , at a particular flow rate from the normalized,

viscosity compensated pressure drop, icViscoelast

PEGPx

, for the polymer solution at the same

flow rate, i.e:

Newtonian

icViscoelast

PEGelastic PPxP

(4.26)

The shear rates in the elastic pressure drop data in Figure 4.16 correspond to the flow rates in the

PEG solvent’s Newtonian baseline in Figure 4.10. For each of these shear rates, the viscoelastic

pressure drop was obtained by fitting a curve through the available data for the 1200 ppm

solution, also shown in Figure 4.10. The elastic pressure drop was then obtained by subtracting

the Newtonian pressure drop from the viscoelastic pressure drop.


Figure 4.15 Measurements of First Normal Stress Difference for the 1200 ppm PEO solution in PEG Solvent plotted a) as 2

1 / N showing a plateau for low shear rates and b) along with a slope of 2 obtained by fitting the Oldryod-B model to the first six data points (solid line)

0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Shear Rate (s-1)

N1

(Pa)

N1 Oldroyd-B

∆P_elastic

Figure 4.16 Comparison of elastic pressure drop for the 1200 ppm PEO solution in PEG with extrapolated values of N1 corresponding to the wall shear rate downstream of the side flows.

0.1

1

10

100

1000

10000

10 100 1000

Shear Rate (s-1)N

1 (P

a)

Data

Oldroyd - B

0.001

0.01

100 1000Shear Rate (s-1)

N1/γ

2

Data


As is evident from Figure 4.16, the elastic pressure drop is in the same order of magnitude and

has a similar trend as the first normal stress difference corresponding to the wall shear rate at the

channel, indicating that the observed elastic effect is possibly an 1N effect. As the wall shear rate

in the exit channel increases, so do the normal stress components. As the transducer measures the

pressure plus the transverse normal stress component, an elastic effect is observed when the

shear rates are high enough to generate normal stresses that are measurable by the pressure

sensor. As 1N is a measure of these stresses, the observed elastic effect has a similar trend as

1N .

4.5 Comparison with Prior Work

Numerous works about elastic effects of dilute polymer solutions have been reported in the

literature. Some of these works relate to drag reduction and so are discussed here. Comparisons

of results are made where appropriate and relevance of the present findings in understanding

prior results is discussed.

Most works with drag-reducing fluids involved experiments with polyethylene oxide (PEO)

dissolved in water. Hence, it is difficult to make a direct comparison with the present experiment

which have been conducted with PEO dissolved in a viscous solvent containing another polymer,

namely polyethylene glycol (PEG). However, comparisons of results reveal that, to produce and

elastic effect, higher concentrations of PEO are needed in the PEG solvent than is required when

PEO is dissolved in water alone. In other words, it is easier for PEO to manifest its elasticity in

water than it is in the presence of another polymer.


In their presheared extensional flow experiments, discussed in section 1.1 and illustrated by

Figure 1.4, James et al. (1987) obtained elastic effects with aqueous polymer solutions with

concentration as low as 20 ppm. For an upstream shear rate of 800 s-1, the pressure drop in excess

of the Newtonian baseline increased with extensional rate for extensional rates greater than 600

s-1 and reached a maximum at an extensional rate of about 1500 s-1 before starting to decrease

towards the Newtonian baseline. The same pattern, of reaching a macimum and then declining, is

shown in Figure 4.10.

For extensional flow without preshearing, i.e with no upstream shear flow, James et al. observed

no elastic effect with the 10 ppm or the 20 ppm PEO solutions and observed an elastic effect

with the 40 ppm solution but only at very high extensional rates, in excess of 3000 s-1. In the side

flow experiment of the present work, conducted in absence of any upstream shear flow, elastic

effects were obtained with both the 500 ppm and the 1200 ppm PEO solutions at much lower

extensional rates between 50 s-1 and 250 s-1. These results indicate that the extensional flowfield

in the side channel setup of the present work is more capable of generating elastic effects than

the axisymmetric contraction flowfield used by James et al. However, in the present experiment

too, the elastic pressure drop approached the Newtonian baseline at high extensional rates after

reaching a maximum, as was observed by James et al. in their presheared extensional flow

experiments.

James and Saringer (1982) observed elastic effects with extensional flow of dilute PEO solutions

in water through axisymmetric converging channels. They observed large effects with polymer

concentration as low as 10 ppm. They also observed onsets in elastic effects, which reached a


maximum before declining towards the Newtonian baseline, as was observed in the present

work.

The above works and other prior works involving extensional flows have attributed non-

Newtonian behaviour to extensional effects. In the present work, the observed elastic effects,

although appearing to be extensional effects at first sight, appear to be caused by normal stresses

generated by shearing. Extension was ruled out because the Hencky strain was not high enough

to produce extensional effects. However, none of the prior works involving extensional channel

flows made any attempt to measure or calculate the Hencky strain in their flowfields. Therefore,

it is possible that many of the observed extensional effects reported in the literature may not

actually have been caused by extension. Instead, shear effects may have been responsible for

causing these elastic responses. For example, in a converging channel, which is a widely-used

method for creating an extensional flowfield, there is also shearing on the channel walls. As the

flow rate through the channel increases, the extensional rate increases but so does the shear rate

at the wall. Hence, it is not immediately clear whether an elastic effect that increases with flow

rate through such a channel is really caused by extension, or by shearing at the channel wall, or

by both.

74

Chapter 5: Concluding Remarks

5.1 Summary

In this work, the effect of fluid elasticity on three flows of dilute polymer solutions through a

complex geometry was studied in an attempt to understand the mechanism of drag reduction. A

flowcell was designed and fabricated as an attempt to simulate the first stage of turbulent burst in

a turbulent boundary layer. While a definitive conclusion regarding the mechanism of drag

reduction cannot be made from the present results, measurable elastic effects were recorded in

each of the flows investigated.

For each flowfield, a Newtonian baseline was established by measuring the pressure drop for a

Newtonian fluid. Pressure drop measurements for dilute polymer solutions, which exhibited

behavior of Boger fluids, were normalized to eliminate viscous effects and compared with the

Newtonian baseline to identify elastic effects.

For the primary flow, a combined flow from the main channel and the two side channels, non-

Newtonian effects were independent of extensional rate in the range of extensional rates

comparable to those in the turbulent boundary layer, indicating that the present flowcell model

does not simulate the turbulent boundary layer. However, an elastic effect which increased with

increasing shear rate was observed with one of the viscoelastic test fluids, the 500 ppm solution

of PEO in PEG solvent.

Chapter 5: Concluding Remarks 75

For main channel flow alone, without any flow from the side channels, an elastic effect that

increases with shear rate was observed with the 500 ppm polymer solution at shear rates greater

than 1000 s-1. This effect was analyzed and is attributed to the difference in the first normal stress

difference, 1N , between the upstream and the downstream sections of the channel. This

difference increases with shear rate and generates an extra normal stress which gives rise to the

extra pressure drop, i.e, one above the Newtonian baseline. Moreover, analysis showed that the

observed effect is not caused by a shear instability since wall shear rates are well below the

critical shear rate required to cause a shear instability.

For side channel flow alone, without any flow in the main channel, elastic effects were observed

with both viscoelastic fluids tested: the 500 ppm and the 1200 ppm polymer solutions. The onset

of the effect took place at a Deborah number of 1.9 for the 1200 ppm fluid, satisfying the

Deborah number condition for elastic effects in extension. However, numerical analysis showed

that the Hencky strain in the flowfield is less than 2, the value needed for extensional effects. For

the 1200 ppm solution, the elastic pressure drop and values of 1N were similar quantitatively and

qualitatively, indicating that the observed elastic effect is possibly an 1N effect.


5.2 Conclusions

The observed elastic effects obtained with preshearing were independent of the

extensional rate, indicating that the extensional flowfield is not a representative model of

the first stage of turbulent burst.

With main channel flow only, an elastic effect is observed at shear rates greater than

1000s-1. This elastic effect can be attributed to a higher first normal stress difference at

the upstream sections of the channel.

With side channel flow only, an elastic effect is observed at extensional rates greater

than 45s-1 for the 1200 ppm solution. Numerical analysis shows that the Hencky strain in

this flowfield is less than 2 and hence the observed effect with side flow alone is not an

extensional effect.

The elastic pressure drop is similar to extrapolated values of 1N , indicating that the

observed effect is possibly an 1N effect.

The Reynolds number based on fluid acceleration was less than 1 in all the experiments,

indicating that inertial effects can be neglected.

5.3 Future Work

The primary objective of this work was to model the first stage of a turbulent burst. However, as

the results showed, this objective was not fulfilled. Some suggestions are made in this section

regarding possible modifications to the flowcell and the experimental setup. By making these

modifications, it may be possible to make a better model of a turbulent burst.


In the present design, the exit channel was slanted to model lifting of fluid from the wall of a

turbulent boundary layer. As a result of this slant, the fluid bent upwards and followed the

geometry of the channel. However, in a turbulent burst, lifting of fluid takes place due to the

force exerted by the motion of the counter-rotating vortices. Therefore, if more vertical height is

provided in the region where the side channels meet the main channel, the fluid coming from the

main channel should have a greater possibility of being lifted up by the flow from the side

channels. Alternatively, the height of the side channels can be shortened to serve the same

purpose. Also, the flow rates from the side channel may need to be much higher than those used

in the present experiment in order to generate a counter-rotating motion. In fact, the required

flow rates may be so high that a pair of opposing jets may need to be created. The slant in the

exit channel made no difference to the results. Therefore, to simplify the design and fabrication,

this channel section can be made flat.

An alternative method could be to utilize flow focusing, a technique used in microfluidics to

form droplets. Having an outer flow converge to a main flow creates an extensional motion in

the direction of the flow. If the outer flow is made to attack the main flow, not only at a

horizontal angle but also at a vertical angle, it may also be possible to cause lifting of fluid.

78

Chapter 6: References

Anna SL, McKinley GH, Nguyen DA, Sridhar T, Muller SJ, Huang J, James DF (2001). An interlaboratory comparison of measurements from filament-stretching rheometers using common test fluids. J. Rheol. 45:83-114. 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. Chauveteau G (1981). Proceedings of the 56th Annual Fall Technical Conference and Exhibition of the SPE of AIME, San Antonio, Texas. p. 14. Collyer AA, Clegg DW (1998). Rheological Measurement. Chapman & Hall, London. Coventry KD, Mackley MR (2008). Cross-slot flow birefringence observations of polymer melts using a multi-pass rheometer. J. Rheol. 52(2):401-405. Currie IJ (2003). Fundamental mechanics of fluids. Marcel Dekker, New York. Davidson PA (2004). Turbulence – An introduction for scientists and engineers. Oxford University Press, New York. Dealy JM (2010). Weissenberg and Deborah numbers – their definitions and use. Rheology Bulletin 79(2):14-18 Donohue GL, Tiederman WG, Reischman MM (1972). Flow visualization of the near-wall region in a drag reducing channel flow. J. Fluid Mech. 56(3):559-575. Dontula P, Macosko CW, Scriven LE (1998). Model elastic liquids with water-soluble polymers. AIChE J. 44:1247-55. Graessley WW (1980). Polymer chain dimensions and the dependence of viscoelasic properties on the concentration, molecular weight and solvent power. Polymer 21:258–262. James DF (2009). Boger fluids. Annu. Rev. Fluid Mech. 41:129-142. James DF, McLean BD, Saringer JH (1987). Presheared extensional flow of dilute polymer solutions. J. Rheol. 31(6):453-481. James DF, Saringer JH (1982). Flow of dilute polymer solutions through converging channels. J. Non-Newt. Fluid Mech. 11:317-339.

Chapter 6: References 79

James DF, Saringer JH (1980). Extensional flow of dilute polymer solutions. J. Fluid Mech. 97: 655-671. James DF, Yogachandran N (2006). Filament-breaking length – a measure of elasticity in extension. Rheol. Acta 46: 171-170. Khalil MF, Kassab SZ, Elmiligui AA, Naoum FA (2002). Applications of drag-reducing polymers in sprinkler irrigation systems: sprinkler head performance. J. Irrig. Drain Eng. 128:147-152 Kim HT, Kline SJ, Reynolds WC (1971). The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50:133-160. Macosko CW (1994). Rheology: Principles, Measurements, and Applications. Wiley, New York. Maxwell JC (1867). On the viscosity and internal friction of air and other gases. Phil. Trans. Roy. Soc., A157:49-88 McKinley GH, Sridhar T (2002). Filament-stretching rheometry of complex fluids. Annu. Rev. Fluid Mech. 34:375-415. Metzner AB, Metzner AP (1970). Stress levels in rapid extensional flows of polymer fluids. Rheol. Acta 9(2):174-181. Muller A, Gyr A (1986). On the vortex formation in the mixing layer behind dunes. Journal of Hydraulic Research 24:358-375 Phan-Thien (1985). Cone-and-plate flow of the Oldroyd-B fluid is unstable. J. Non-Newt. Fluid Mech. 17:37-44. Prilutski G, Gupta RK, Sridhar T, Ryan ME (1983). Model viscoelastic liquids. J. Non-Newt. Fluid Mech. 12:233-241. Schlichting (1960). Boundary layer theory. McGraw-Hill, New York. Scrivener, O (1974). A contribution on modifications of velocity profiles and turbulence structure in a drag reducing solution. Proc. Int. Conf Drag Reduction, St. Johns College, Cambridge, ed. N. G. Coles. C66-70. Sellin RHJ, Ollis M (1980). Polymer drag reduction in large pipes and sewers: results of recent field trials. J. Rheol. 24:667-684 Terrapon VE, Dubief Y, Moin P, Shaqfeh ESG (2004). Simulated polymer stretch in a turbulent flow using Brownian Dynamics. J. Fluid Mech. 504:61–71.

Chapter 6: References 80

Tirtaatmadja V, Sridhar T (1993). A filament stretching device for measurement of extensional viscosity. J. Rheol. 37:1081-1102. Toms B (1948). Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. Proceedings of the 2nd International Rheological Congress. p. 135–141. Ward-Smith AJ (1980). Internal Fluid Flow - the Fluid Dynamics of Pipes & Ducts. Oxford University Press, New York. White CM, Mungal MG (2008). Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40:235–56. Winter HH, Macosko CW, Bennett KE (1979). Orthogonal stagnation flow, a framework for steady extensional flow experiments. Rheol. Acta 18:323-334

81

APPENDIX A: Numerical Simulation

A1 Mesh

For the purpose of running the simulation, a tetrahedral mesh with a mesh refinement setting of

maximum mesh spacing of 0.04 mm and a minimum mesh spacing of 0.02 mm was used. Figure

A1 below shows the mesh in the channel geometry. The mesh at the entrance section of the

channel is shown enlarged in Figure A2.

Figure A1 Tetrahedral mesh in channel geometry

Appendix A: Numerical Simulation 82

Figure A2 Enlarged section showing mesh at the entrance to the main channel

A2 Accuracy of Numerical Results

In order to verify the accuracy of numerical simulations prior to performing computational

analysis for the flowcell geometry, results from COMSOL were obtained for a flowfield which

can be solved analytically. A rectangular channel, with width-to-height aspect ratio of 6.7, same

as that of the main channel of the flowcell, was chosen as the geometry to be tested. Figure A3

shows a comparison of the centreplane velocity obtained from this analytical solution with that

obtained from COMSOL. Agreement between the analytical and the numerical results attest to

the accuracy of the simulations for the given settings of mesh refinement.

Appendix A: Numerical Simulation 83

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

y/d

Ve

loci

ty (

m/s

)

Rectangular channel with a/b = 6.7Flow between parallel platesCOMSOL

Figure A3 Comparison between analytical results for flow in a rectangular channel with aspect

ratio a/d=6.7 and channel height 2d, flow between two parallel plates, and numerical results from COMSOL.

84

APPENDIX B: Fluid Mechanics

B1 Comparison between Rectangular Channel Flow and Flow Between Parallel Plates

The velocity, v , in a rectangular channel of width a2 and height d2 is given by (Ward-Smith

1980):

,coscosh

cosh)1(4

2

1),(

03

22

nn

n

n

n

n

yNaN

zN

Ndyd

dx

dPyzv

(B1)

d

nNn 2

)12( , (B2)

where is the fluid viscosity anddx

dP is the pressure gradient in the direction of flow.

The velocity profile in a pressure driven flow between two parallel plates, on the other hand, can

be obtained by solving the Navier-Stokes equations for two-dimensional flow:

,2

2

dx

dP

dy

vd x (B3)

where is the dynamic viscosity of the fluid, xv is the x-component of fluid velocity, and dx

dP

once again is the pressure gradient in the direction of flow. If h is the gap between the parallel

plates, with no-slip boundary conditions at the walls, equation B3 can be solved to obtain the

velocity profile in terms of the pressure gradient in the flow:

yhydx

dPyvx

21

)( . (B4)

Appendix B: Fluid Mechanics 85

B2 Shear Rate

The shear rate, , is given by:

dy

dvx . (B5)

Substituting equation B4 in equation B5 gives:

yh

dx

dPy

2

1)(

, (B6)

where h is the height of the channel. The volumetric flow rate, MQ , in the channel is given by:

w

z

h

y

xM dydzyvQ0 0

)( . (B7)

Substituting equation B4 into equation B7 and integrating gives:

3

12

1h

dx

dPwQM

, (B8)

where w is the width of the main channel. Then, obtaining dx

dP in terms of MQ from equation

B8 and substituting in equation B6 gives:

y

h

wh

Qy M

2

12)(

3 (B9)

The wall shear rate, W , can then be obtained by setting 0y in equation A9:

,6

2wh

QMW (B10)

Equation A10 was used to calculate the wall shear rates in the experimental flowcell.


B3 Extensional rate

The planar extensional rate, , in a cross-slot stagnation point flow can be estimated as

(Coventry and Mackley, 2008):

S

S

w

V2 , (B11)

where SV is the average velocity in each of the in-flow side channels and Sw is the width of each

side channel. Equation B11 can be written in terms of the side channel flow rate ,SQ as

,2

2hw

Q

S

S (B12)

where SSS hVwQ , and h is the height of both the side and the main channels. Equation B12

was used to estimate the extensional rates in the flowcell.

B4 Pressure Loss

Equation A8 can be rearranged to obtain an expression for the pressure gradient in the flow in

terms of the flow rate as:

3

12

wh

Q

dx

dP M , (B13)

which can be integrated as:

2

1 03

12P

P

LM dx

wh

QdP

(B14)

to obtain the pressure drop, 21 PPP , between points 1 and 2 in the flow separated by a

distance L as below:

3

12

wh

LQP M (B15)


Although equation B15 gives the pressure drop in a straight channel and does not take into

consideration the complex geometry of the flowcell or the extra flow from the side channels, the

pressure drop calculated using this equation should nevertheless be a reasonable estimate of the

actual pressure drop in the actual flowcell geometry, because the side channel flow rate is only a

fraction of the main channel flowrate. Hence, this equation was used to estimate the pressure

drop between the locations of the two pressure transducers.

B5 Turbulent Boundary Layer

Turbulence is perhaps the most significant unsolved phenomenon in fluid mechanics that is

characterized by inertia-driven, unsteady, chaotic fluctuations in flow parameters such as

velocity, momentum, and kinetic energy. A turbulent flow consists of many structures among

which the turbulent boundary layer is an important one which is of particular interest to this

present study.

Figure B1 Turbulent flow over a flat plate


Considering a turbulent flow with free-stream velocity U over a flat surface as shown in Figure

B1, y as the vertical distance above the surface, xv as the x-component of the fluid velocity, and

as the shear viscosity of the fluid, from equation 2.3, the wall shear stress, W , is given by:

0

y

xW y

v . (B16)

Defining a friction velocity, *u , as

Wu * , (B17)

where is the fluid density.

A spatial parameter, *y , can then be defined as

yu

y ** , (B18)

where is the kinematic viscosity of the fluid defined as: / .

The turbulent boundary layer consists of three layers categorized by their distance from the wall

and the fluid dynamics that take place in each one. The first layer, closest to the wall

where 5*0 y , is called the viscous sublayer. The flow here is dominated by viscous forces

rather than inertial forces because, close to the wall, the fluid’s viscosity is the primary cause of

generating stresses. The next layer, in the region 40*5 y , is known as the buffer layer. Here

the inertia driven turbulent stresses are larger than they are in the viscous sublayer and are

comparable to the viscous stresses. Hence, the buffer layer is a transitional layer where the

viscous forces compete with inertial forces. The third and the last layer of the turbulent boundary

layer is known as the log region where 40* y . Here the inertial stresses completely outweigh


the viscous stresses and the flow is overwhelmingly turbulent (Schlichting, 1960., p. 465 & Kim

et al., 1971).

B6 Inertial Effects

Inertial effects are usually indicated by Reynolds number greater than 1. Reynolds number is

defined by

VL

Re , (B19)

where is the fluid density, V is the velocity, L is a length-scale, and is the viscosity. For

flows with no acceleration, eg, pipe flow, pressure loss is independent of , even for Re O(102).

To include acceleration, perhaps the Reynold number should be

VL

Re . (B20)

V represents the change in velocity in flow direction. More fundamentally,

F

F

forceviscous

forceinertial

_

_Re . (B21)

The inertial force, F , can only arise with acceleration, i.e:

xaVolmaF , (B22)

where xa is the acceleration in the flow (x) direction and Vol is the volume.

The viscous force, F , depends on the shear stress and area, i.e:

)()( AreaL

VArea

y

VAreaF

y

xxxy

, (B23)


where y and yL are the distances in the transverse direction over which the velocity changes

by xV .

Then,

Area

Vol

V

aL

AreaL

VaVol

F

F

x

xy

y

x

x

)(

)(Re , (B24)

where Area is a surface area, one on which the shear stress acts to resist the flow,

xLArea

Vol , (B25)

where xL is the length-scale in the flow direction.

Then

xx

yx aV

LL

Re . (B26)

Now, xa is acceleration in the flow (x) direction, which is the change in velocity xV over the

time t . The time is the length in the x-direction divided by the mean velocity, i.e,

./ x

xx

xx

xxx L

VV

VL

V

t

Va

(B27)

Then, xyxx

xxyx VLLV

VVLL

Re (B28)

This definition of Reynolds number, denoted in the text as ReI , could be used to determine whether inertial effects are important:

That is, if ,1

xy VL

inertia is not important

whereas, if ,1

xy VL

inertia likely plays a role.

[Reference: James, 2011, Private Communications]

91

APPENDIX C: Oldroyd-B Model

The Oldroyd-B equation is given by:

DD ˆ2ˆ 21 , (C1)

where is the stress tensor, D is the deformation rate tensor, 1 is the relaxation time, 2 is the

retardation time, is the viscosity, and and D are the upper convected time derivatives of the

stress and deformation rate tensors respectively. This derivative is given by (Bird et al., 1987, p.

296):

vvvt

T

, (C2)

where v is the fluid velocity vector, is the gradient of the stress tensor, v is the fluid

velocity gradient tensor, and Tv is the transpose of that tensor.

Expressing the total stress in the fluid as a sum of the stresses, S and P , generated respectively

by the solvent and the polymer, i.e:

PS . (C3)

Equation B2 can be re-written as two separate equations expressing the contribution of the

polymer and the solvent. The polymer contribution is given by:

DPPPP ˆ , (C4)

where P is the polymer relaxation time, P is the polymer viscosity, and P is the upper

convected time derivative of the polymer contribution to the stress.

Appendix C: Oldroyd – B Model 92

The stress generated by the Newtonian solvent is given by:

DSS , (C5)

where S is the solvent viscosity.

Since the solvent is Newtonian, it has no contribution to the relaxation time, and hence:

P 1 . (C6)

The fluid retardation time is a fraction of the relaxation time and is given by:

PS

S

2 , (C7)

and the overall viscosity of the fluid is given by:

SP (C8)

Response to Shear Flow

For steady shear flow, the Oldroyd-B equation can be simplified to obtain the following

expressions for the stress components (Bird et al., 1987, p347):

xyxy (C9)

22 xyPyyxx (C10)

0 zzyy , (C11)

where xy is the shear rate, xy is the shear stress, is the relaxation time, P is the polymer

viscosity, and xx , yy , and zz are respectively the normal stress components in the streamwise,

transverse, and spanwise directions. The quantity yyxx is known as the first Normal Stress

Difference, 1N , and is a measure of fluid elasticity in shear.

yyxxN 1 (C12)


The quantity zzyy is known as the second Normal Stress Difference, 2N . For a Newtonian

fluid both 1N and 2N are zero because all the normal stress components in a Newtonian fluid

are identically zero.

zzyyN 2 (C13)

The Oldroyd-B model predicts the shear stress and the first Normal Stress Difference to

respectively have linear and quadratic dependence on the shear rate. It also predicts the second

normal stress difference to identically zero at all shear rates.

Figure C1 Characterization of the original Boger fluid prepared by Boger, 1977. The blue

symbols represent the shear stress and the red symbols represent the first normal stress difference. (Reproduced from James, 2009)


Figure C1 shows measurements of xy and 1N under variation of xy for the original Boger fluid

prepared by Boger (Boger 1977). As shown by this plot, 1N of Boger fluids tend to follow the

Oldroyd-B model, i.e a semi-logarithmic plot of 1N vs xy has a slope of 2, in the low shear rate

range up to a certain shear rate beyond which the slope decreases and the variation is no longer

quadratic. It is therefore possible to determine the relaxation time, , of an Oldroyd-B fluid

from 1N measurements in this low-shear rate range as:

21

2 xyP

N

(C14)

For small amplitude oscillatory shear flow, the Oldroyd-B model simplifies to:

2

"

)(1

PS

G (C15)

2)(1

'

PG

(C16)

Equation 2.42 can also be used to determine a relaxation time from 'G measurements of

Oldroyd-B fluids in the range of small frequencies, i.e as 0 . In this case, the equation

reduces to:

P

G

2

' (C17)

For a viscoelastic fluid, 'G is a measure of the fluid’s elasticity while "G is a measure of its

viscosity. For a Newtonian fluid, xy is always 90° out of phase with xy and therefore, 'G is

identically zero for a Newtonian fluid.

95

APPENDIX D: Pressure Transducer

D1 Specifications

From manufacturer

From manufacturer

Appendix D: Pressure Transducer 96

D2 Components & Features

From manufacturer


D3 Calibration

y = 0.0016x - 0.0298

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

1700 1800 1900 2000 2100 2200 2300 2400 2500

Pressure (psi)

Pre

ssu

re S

enso

r V

olt

age

Ou

tpu

t (m

V)

Honeywell Pressure Sensor

Calibration Using Column of Water

Figure D1 Low pressure calibration curve using a column of water

y = 0.0016x + 0.2269

27.8

28

28.2

28.4

28.6

28.8

29

29.2

29.4

29.6

17600 17800 18000 18200 18400 18600 18800

Pressure (Pa)

Pre

ssu

re S

ens

or

Vo

ltag

e O

utp

ut

(mV

)

Honeywell Pressure Sensor

Calibration Using Column of Water

Figure D2 High pressure calibration curve using a column of water


D4 Engineering Drawing

From manufacturer

99

APPENDIX E: Engineering Drawings of Flowcell

E1: Snapshots from SolidWorks

Figure E1 Exterior view of flowcell

Figure E2 Interior view showing the channels inside the flowcell

Appendix E: Engineering Drawings of Flowcell 100

Figure E3 Side view of flowcell

Figure E4 Front view of flowcell


E2: Engineering Drawings of Individual Parts and Assemblies


















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Planar Flow of Dilute Polymer Solutions