Determining the elongation and shear viscosity for a number ......Determining the elongation and shear viscosity for a number of complex uids Bart os,V 5887631 erslagV anv Bachelorproject

Determining the elongation and shear viscosity for a number

of complex �uids

Bart Vos, 5887631

Verslag van Bachelorproject Natuur- en SterrenkundeOmvang 12 EC, uitgevoerd in de periode 13-05-2011 tot 05-07-2011

Van der Waals-Zeeman InstituutFaculteit der Natuurwetenschappen, Universiteit van Amsterdam

Ingeleverd op 6 juli 2011Begeleiders: Mounir Aytouna en Daniel Bonn

Tweede beoordelaar: Peter Schall

1 Abstract

In this report the results are presented of measurements on the shear viscosity and elongation viscosityof di�erent concentrations of water-diluted hair gel (Albert Heijn Styling Gel, Mega Strong), di�erentconcentrations of Ceroge 80 xanthane dissolved in water and di�erent concentrations of 1.5 mm �berssuspended in Triton. Using a fast camera, the minimum neck diameter before the breakup of drops isanalysed and �tted to a function of the form Rmin = A |t− t0|P . The value of P goes from 2

3 to 1 forgel and is approximately 1 for the �ber suspension. This means these �uids are in the viscous-capillaryregime, except for the lowest concentrations of gel. The power is approximately constant at 2

3 forxanthane, indicating that the xanthane is in the inertia-capillary regime. The elongation viscosity ofthe diluted gel and the �ber suspension increases with concentration, while it is approximately constantfor xanthane. The shear viscosity of all three �uids is a function of the applied shear rate, going fromlow viscosity at high shear rates to high viscosity at low shear rates. This is in contrast to Newtonian�uids, for which the shear viscosity is independent of the shear rate.

The power n in the Herschel-Bulkley model, τ = τy + k γ̇n, is smaller than 1 and decreasing for geland xanthane, while n is approximately constant at 0.9 for the �ber suspension. Hence, all three �uidsare shear thinning. The yield stress of the gel dilutions goes from 0 Pa at 5% gel to 111 Pa at 90% gel.A small yield stress is found for the �ber suspensions, while the xanthane solution has no yield stress.

2

2 Populair-wetenschappelijke samenvatting

Bart Vos, 5887631

De Aarde: vanuit de ruimte gezien een bol met groene vlakjes, omringd door het blauw van oceanendie tweederde van onze wereld bedekken. Vloeisto�en spelen dan ook een belangrijke rol in ons leven.Van water, bloed en speeksel tot benzine, inkt en bestrijdingsmiddelen, we zijn er de hele dag vanafhankelijk. De enorme toepasbaarheid en relevantie is een belangrijke reden om veel onderzoek tedoen naar het gedrag van vloeisto�en. De zogenaamde �complexe vloeisto�en" krijgen veel aandacht,hoewel (en ook omdat) er nog steeds grote vraagtekens zijn rond deze groep. De complexe vloeisto�enhebben als bijzondere eigenschap dat ze niet-lineair zijn. Dit betekent dat bijvoorbeeld een twee keerhogere druk niet een twee keer zo grote verplaatsing tot gevolg heeft. Een bekend voorbeeld is maizenain water: je kan er op lopen, maar wie stilstaat, zakt langzaam weg!

In dit onderzoek wordt gekeken naar de verhouding tussen de de viscositeit van vloeisto�en dieuitgerekt worden (de rekviscositeit) en de viscositeit van vloeisto�en die tussen twee parallele bewe-gende platen zitten (de schuifviscositeit). Voor Newtoniaanse vloeisto�en (lineaire vloeisto�en) is dezeverhouding altijd 3, maar voor complexe vloeisto�en niet. Van verschillende concentraties haargelverdund met water, xanthaangom opgelost in water en 1.5 mm lange vezels opgelost in Triton is derekviscositeit gemeten. Dit is gedaan door met een hogesnelheidscamera te kijken naar hoe snel hetdunste plekje van een bijna-vallende druppel dunner wordt (zie afbeelding). Ook de schuifviscositeitvan deze vloeisto�en is bepaald. Dit kan door een beetje van de vloeistof tussen een stilstaande eneen draaiende plaat aan te brengen Zo wordt de weerstand gemeten die de vloeistof biedt tegen hetdraaien.

Uit dit onderzoek blijkt dat de rekviscositeit ongeveer constant is, terwijl de schuifviscositeit vande 3 vloeisto�en veel kleiner wordt naarmate ze harder worden gedraaid. De voorspelling dat deverhouding tussen de schuif- en rekviscositeit 3 is, geldt dus niet voor deze complexe vloeisto�en.

40% gel, verdund met water. De tijd voordat de druppel valt staat linksonder in ieder plaatje.

3

Contents

1 Abstract 2

2 Populair-wetenschappelijke samenvatting 3

3 Introduction 5

4 Theory 5

4.1 De�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.1.1 Shear and elongation �ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.1.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.1.3 Strain rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.1.4 Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.1.5 Yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.1.6 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.1.7 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.1.8 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2 Self-similar solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2.1 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.3 Herschel-Bulkley model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Rigid dumbbell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Measurements & results 9

5.1 Gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.1 Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.2 Elongation viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.1.3 Surface tension of yield stress materials . . . . . . . . . . . . . . . . . . . . . . . 125.1.4 Power law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.5 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.2 Xanthane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2.1 Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2.2 Elongation viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2.3 Surface tension of materials without yield stress . . . . . . . . . . . . . . . . . . 175.2.4 Rigid dumbbell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2.5 Power law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2.6 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3.1 Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3.2 Elongation viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3.3 Surface tension of �ber suspension . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3.4 Power law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Acknowledgments 25

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3 Introduction

The behaviour of �uids is a subject in physics that is hundreds of years old, and it still contains manymysteries. While Newtonian �uids are quite well understood nowadays, there is no grand theory fornon-Newtonian �uids. These �uids are of great importance in our daily life, and a theory describingtheir behaviour will be a big step forward in the applicability of non-Newtonian �uids.

In this report a study on 3 non-Newtonian (complex) �uids is presented. The central questionis: �What is the ratio between the elongation viscosity and the shear viscosity of di�erent complex�uids?�. For Newtonian �uids, this ratio is ηE = 3ηS , but this does not hold for non-Newtonian �uids.To answer this question, the elongation and shear viscosity of di�erent �uids need to be determined.

This report starts with a treatment of theoretical concepts necessary to understand the experiments,followed by the measurements and results. To keep this report conveniently arranged, the results havebeen split in three parts, one for each �uid.

4 Theory

4.1 De�nitions

4.1.1 Shear and elongation �ow

Any movement of a �uid can be split up in two components: a shear component and an elongationcomponent. The shear �ow is the �ow where the (parallel) boundaries of the �uid move parallel toeach other, see �gure 1.

Apart from the shear �ow there is also the elongation �ow. In this case a �uid stretches in onedirection, and therefore has to shrink in the other two directions, see �gure 2.

4.1.2 Strain

The strain γ is de�ned as the movement of two parallel plates (with a liquid between them) dividedby the distance between the plates. As length is divided by length, strain is a dimensionless quantity.

4.1.3 Strain rates

The shear rate γ̇ is de�ned as the strain divided by the time it takes the plates to move parallel toeach other; in other words, the velocity divided by the distance between the plates. The dimension ofshear rate is [s−1].

Figure 1: A stationary and a moving plate with a �uid between them. The �uid can be split in layers,each moving with a velocity lower than its upper layer, but higher than the layer below.

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Figure 2: Elongation �ow: the �uid is stretched in one direction, so it becomes thinner in the othertwo directions.

The elongation rate ε̇ is de�ned as the velocity of two parallel plates, moving in the direction normalto their surface, divided by the distance between the plates. The dimension of the elongation rate is[s−1].

4.1.4 Shear stress

When two parallel plates move with respect to each other with a medium between them (see �gure 1),there will be some friction between the upper plate and the adjacent `layer' of the medium. In the caseof a non-solid medium, the layer will start to move with the plate, giving rise to a force on the second`layer' of the �uid. This goes on until the other plate is reached. The force in the medium is calledthe shear stress, with units of force per area. In the case of a solid, moderate shear stresses cause nodeformation or �ow, while a true �uid will always �ow when a shear stress is applied.

4.1.5 Yield stress

Some complex �uids have the property that a small shear stress causes no �ow until a certain minimumshear stress is reached. The yield stress is this threshold that needs to be overcome to start the �owingof a �uid. Newtonian �uids cannot have a yield stress.

4.1.6 Viscosity

The viscosity of a material consists of two components: a shear viscosity (ηS) and an elongationviscosity (ηE). The shear viscosity comes into play when two parallel plates move with respect to eachother, see �gure 1 [1]. Assuming that the friction between the layers of �uid is the same for all layers,the velocity of a certain layer is vx(y) = y V0

a where V0 is the velocity of the upper plate and a theseparation between the plates. The friction between the layers is the shear stress, the moving plate hasto provide an equal but opposite force to keep moving. This force is the di�erence in velocity between

di�erent layers multiplied by a constant called the shear viscosity. In a formula: FS = −η d vx(y)d y , so

viscosity has units of [kgm−1 s−2][s−1] = [kgm−1 s−1], which is often written as [Pa s].

The elongation viscosity plays a role when the �uid is being stretched [2], see �gure 2. In the caseof drop formation, the gravitational force is needed to stretch a �lament of the �uid. The Laplacepressure ∆p = σ

R = 2σh [3] is equal to the elongation stress, which is the elongation viscosity times the

elongation rate. This can be rewritten to obtain the formula

ηE =2σ

hminε̇0= − σ

d hd t

(1)

where ε̇0 = − 2hmin

d hd t is the elongation rate [4].

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The relation between the elongation viscosity and the shear viscosity is ηE = 3ηS for Newtonian�uids [5].

4.1.7 Surface tension

It is clear from everyday life that a �uid tries to minimise its surface. Bubbles of soap are spherical,oil clusters together in a pan and the surface of a cup of milk is �at. This is all caused by the factthat the particles of which a �uid consists are attracted by each other. Any particle on the surface hasonly half of the attractive interactions, which means it is energetically unfavourable for a �uid to havemany particles at the surface. In order to lower its energy, a �uid will decrease its surface as muchas possible, and resist any attempt to change (and therefore increase) its surface. This is called thesurface tension (σ), with units energy per area, [ Jm2 ] = [kg s2].

4.1.8 Elasticity

Elasticity is a property of solid materials like bricks, although clay for example also has some elasticity.When an elastic (also called rigid) material is subjected to a shear stress, it will deform slightly, butwhen the stress is removed, it will go back to its original shape.

4.2 Self-similar solutions

Here quoted is the simpli�ed Navier-Stokes equation presented in [6] for drop breakup

∂tv + v∂zv = −σρ∂z(

1

R1+

1

R2) + 3ν

∂z(∂zvh2)

h2(2)

where v is the velocity in the radial direction of the neck of the drop, σ is the surface tension, ρ is thedensity of the �uid, 1

R1+ 1R2

is the mean curvature which is measured in a similar way as the diameterin �gure 9 [3], ν = ηS

ρ is the kinematic viscosity and h is the local diameter of the �uid neck. Fluids

that can be described by the (full) Navier-Stokes equations are Newtonian �uids. The left hand side ofthe equation represents inertial forces, the �rst part of the right hand side of the equation representsthe surface tension, while the second part comes from viscosity.

In [6] intrinsic units are introduced. These units for length and time are obtained by combiningthe �uid parameters σ, ρ and ν. The results are

li =ν2ρ

σand ti =

ν3ρ2

σ2

In the drop breakup experiment, the minimum diameter of the drop needs to be determined. Whenthis diameter is much larger than the intrinsic length (hmin � li ∝ ν2) the viscous part in equation2 can be neglected, so there is a balance between inertial forces and surface tension. The minimumdiameter, as stated in [6], is hmin = 0.7(σρ )

13 (t− t0)

23 , where t0 is the time of the breakup. Physically,

the viscosity is too low to prevent any �ow, although the mass of the �uid gives an inertial force. Onthe other hand, when hmin � li ∝ ρ

σ , the inertial part is (much) smaller than the surface tension andviscous terms, so inertial forces can be neglected and there is a balance between surface tension andviscosity. The minimum neck diameter in this case is hmin = 0.06 σ

νρ (t− t0). When all three forces are

present, the minimum diameter is given by hmin = 0.03 σνρ (t0 − t).

Hence, when the minimum neck diameter of a �uid behaves as a power of 23 in time, the �uid is in

the inertia-surface tension regime. If the minimum neck diameter behaves as a power of unity in time,the �uid is in the viscous-surface tension regime or in the viscous-inertia-surface tension regime. Another way to distinguish the di�erent regimes is to look at the symmetry of the drop and the remaining�uid at the syringe [7]. An overview is given in table 1.

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Regime ... play no role hmin ∝ tn Symmetric Reynolds numberInertia-surface tension Viscosity n = 2

3 No Re � 1Viscosity-surface tension Inertial forces n = 1 Yes Re � 1

Inertia-viscosity-surface tension - n = 1 No Re ≈ 1

Table 1: Behaviour of di�erent regimes. Instead of surface tension, capillary is often used to indicateregimes (for example, the inertia-capillary regime instead of inertia-surface tension).

Figure 3: Di�erent possibilities for the Herschel-Bulkley model.

4.2.1 Reynolds number

As a check to see in what regime a �uid is, Reynolds number can be calculated [1]. This number givesthe ratio between the inertial forces and the viscous forces. It is de�ned as Re = LU ρ

ηS, where ρ

ηSis

the kinematic viscosity, L is a characteristic length of the �ow and U is a characteristic velocity. Fora low value of Reynolds number viscous forces dominate, while for a high value of Reynolds numberinertial forces dominate.

4.3 Herschel-Bulkley model

The Herschel-Bulkley model [8, 9] describes the stress experienced by a (non-Newtonian) �uid. Thismodel states that

τ = τy + k γ̇n (3)

where τ is the shear stress, τy is the yield stress, k is a constant and γ̇ is the shear rate. If there is alinear relation between shear rate and shear stress and there is no yield stress we have a Newtonian�uid. If the relation goes through the origin but �attens at higher shear rates, which happens forn < 1, the material is called shear thinning. When the material has a linear relation between shearrate and shear stress, but also has a yield stress it is called a Bingham material. A material thatis both non-linear and does not go through the origin is a yield stress, shear-thinning material whenn < 1, or a yield stress, shear-thickening material for n > 1, see �gure 3.

4.4 Rigid dumbbell model

The rigid dumbbell model [10] assumes that a polymer suspended in water behaves like a rigid rod,and is oriented in the direction of the �ow. According to this model, the shear viscosity for a �uid forhigh shear rates is given by

ηS(γ̇) = 3ηsolvent + 0.678nK T λ23 γ̇−

13 (4)

8

Figure 4: Shear stress versus shear rate for gel. The Herschel-Bulkley model has been �tted to theresults.

where ηsolvent is the viscosity of the solvent, in the case of water η = 8.9 ∗ 10−4 Pa s, n is the numberdensity of polymers, K is the Bolzmann constant, T is the temperature, λ is the relaxation time ofthe polymers and γ̇ is the shear rate. For high shear rates, the elongation viscosity is given by

ηE = 3ηsolvent + 6nK T λ (5)

In this equation the elongation viscosity is no longer a function of the shear rate. The rigid dumbbellmodel can only be used for materials with no yield stress.

5 Measurements & results

In this experiment, 3 di�erent �uids were used: di�erent concentrations of water-diluted hair gel(Albert Heijn Styling Gel, Mega Strong), di�erent concentrations of Ceroge 80 xanthane dissolved inwater and di�erent concentrations of 1.5 mm �bers suspended in Triton. The concentrations of thexanthane and �bers are denoted in ppm, parts per million.

5.1 Gel

5.1.1 Shear viscosity

The shear viscosity is obtained using the Rheometric Scienti�c ARES rheometer. This machine consistsof a �at plate and a �attened cone with a �uid between them. By rotating the plate, a shear rate isapplied. To increase the contact and friction between the �uid and the machine, the surface of theplate has been roughened by attaching sand paper on it, the cone has been roughened by gluing sandon it. Before every measurement, a shear rate of 100 s−1 is applied for 60 seconds to remove anystructures created by placing the �uid. By varying the shear rate between 103 s−1 to 10−4 s−1 with10 steps per decade and 10 seconds between every measurement to let the �uid relax, the shear stressand the shear viscosity are measured as a function of the shear rate. The yield stress can be obtainedby �tting equation 3 to the shear stress versus shear rate graphs, see �gure 4. The shear viscosity asa function of shear rate is shown in �gure 5.

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Figure 5: Shear viscosity as a function of shear rate for di�erent concentrations of gel.

Figure 6: A selection of images of the drop breakup for a 40% gel solution. The time before breakupis written in the left bottom corner of every image.

5.1.2 Elongation viscosity

The elongation viscosity can be measured when a �uid is stretched. In this experiment, a drop fallso� the tip of a syringe with a diameter of 1.95 mm. A Phantom V7 video camera is used to record thedrop breakup at 14,000 frames per second. After every measurement a snapshot of a ruler is made forcalibration purposes. The movie is converted to a series of images by Virtual Dub 1.4, a small numberis shown in �gure 6. Using the image processing software Scion Image, the minimum diameter of the�uid neck is determined from the images. In �gure 7 the minimum diameter versus time is plotted fordi�erent concentrations of gel. The time development of the �uid neck, d hd t , is obtained by dividing thechange in neck diameter by the time between two images, 1

14000 s. In the case of 5% and 10% gel, when

P in the formula Rmin = A |t− t0|P is not equal to 1, d hd t is taken 1 ms before breakup. When thechange in neck diameter is very small, the change in neck diameter between 7 images is taken insteadand divided by the time between them, 7

14000 s. The elongation viscosity is calculated by �lling in theparameters in equation 1, ηE = − σ

d hd t

. The determination of the surface tension is explained in the

next section. The elongation viscosity for di�erent concentrations of gel is plotted in �gure 8.

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Figure 7: The minimum drop diameter versus time of gel, a function of the form Rmin = A |t− t0|Phas been �tted to the results.

Figure 8: The elongation viscosity of diluted hair gel, determined by the drop breakup method. Alsoplotted is the shear viscosity at a shear rate of 100 s−1.

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Figure 9: The maximum diameter of a circle in a drop of 40% gel.

5.1.3 Surface tension of yield stress materials

To measure the surface tension of a yield stress material like gel, the formula σ/R = γy is used, whereR is the diameter of the largest possible circle that can be �tted in the drop (see �gure 9). This formulastates that the surface tension tries to minimise the surface, until it cannot overcome the yield stressanymore.

To determine the yield stress, the shear stress as a function of the shear rate, obtained from therheometer experiment, was �tted to the Herschel-Bulkley model (equation 3). To measure R, an imageof the remaining �uid at the syringe was taken after a drop had fallen. Immediately after breakup, thedrop and the �uid at the syringe form a sharp cone, see �gure 10. The drop at the syringe �attens,until surface tension and yield stress are in balance. In a plot of yield stress versus 1

R , the inverse ofthe slope is the surface tension. In �gure 11 the results are plotted for gel. The surface tension is25.3± 1.1mN/m.

5.1.4 Power law

The minimum drop diameter as function of time (see section 5.1.2) was �tted by Origin to a function

of the form Rmin = A |t− t0|P . The value of P for di�erent concentrations of gel is plotted in �gure12. Also plotted is the power in the Herschel-Bulkley model, obtained from the rheology measurement(see �gure 4).

5.1.5 Reynolds number

To calculate Reynolds number for gel, the estimation Re = LU ρηS

= ε̇ L2 ρηS≈ L2 ρ

ηS tis made, where L2

is the minimum neck diameter, ε̇ is the elongation rate and t is the time before breakup. The shearviscosity at a shear rate of 100 s−1 is used, and a density of 1000 kgm−3. Reynolds number for di�erentconcentrations of gel, 1 ms before breakup, is plotted in �gure 13.

5.1.6 Discussion

The ratio between the shear viscosity and the elongation viscosity is not the constant value of 3 aspredicted for Newtonian �uids, see �gure 8. In this �gure the elongation viscosity and the highestmeasured shear rate, 100 s−1, are plotted. 100 s−1 is chosen, because the elongation rate is also high,1200 s−1 for 5% gel 1 ms before breakup and 1500 s−1 for 90% gel.

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Figure 10: The breakup of a 15% gel solution. The satellite drop is a remain of the �uid neck. Notethat the two cones are symmetric, this indicates that the �uid is in the viscous-capillary regime.

Figure 11: 1/R versus yield stress for gel. A straight line is �tted, the surface tension is the inverse ofthe slope of the slope.

13

Figure 12: The power law and the power in the Herschel-Bulkley model for gel.

Figure 13: Reynolds number for di�erent concentrations of gel.

14

Figure 14: Di�erent concentrations of gel diluted in water. The 5% and 10% gel dilutions clearly havean asymmetric breakup point, the 15% gel dilutions is symmetric.

The power in the Herschel-Bulkley is less than 1 for low concentrations of gel and decreasingto a value of about 0.4, which is the case for a shear thinning material. The power in the dropformation of the gel solution (�gure 12) goes from approximately 2

3 at low concentrations to 1 at highconcentrations. For 5% and 10% gel, the breakup is asymmetric, see �gure 14. The 5% gel dilutionalso has no yield stress, contrary to the other concentrations of gel. The breakup of the 15% gel thebreakup is symmetric (see �gure 10), which agrees with the power law being close to 1. This change inbehaviour is a transition from the capillary-inertia regime to the capillary-viscous regime [6]. This issupported by Reynolds number, starting at a high value for low concentrations of gel and decreasingwhen the gel concentration goes up. Because the shear viscosity (and therefore also the kinematicviscosity) is a function of the shear rate, the scaling of �gure 13 can be changed by choosing a di�erentshear rate.

5.1.7 Conclusion

The transition from the capillary-inertia to the capillary-viscous regime takes places in a gel concentra-tion between 5% and 15%. Reynolds number shows a sharp decrease around these concentrations. Theyield stress is 0 Pa for 5% gel, and increases to 111 Pa for 90% gel. The power in the Herschel-Bulkleymodel is smaller than 1, so the diluted hair gel is a yield stress, shear tinning �uid. The shear viscosityat a shear rate of 100 s−1 is 4 to 5 times the elongation viscosity for high concentrations of gel.

Further investigations could investigate the transition of the capillary-inertia to the capillary-viscousregime more closely by studying di�erent concentrations of gel in the range 5% to 15% gel.

5.2 Xanthane

5.2.1 Shear viscosity

The shear viscosity of xanthane is determined in the same way as the shear viscosity of gel. The shearstress versus shear rate for xanthane is plotted in �gure 15. The value of k is smaller than the error,so it is written as 0. The shear viscosity versus shear rate is plotted in �gure 16. A very small yieldstress is found, which is probably a consequence of �tting.

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Figure 15: Shear stress versus shear rate for xanthane. The Herschel-Bulkley model has been �tted tothe results.

Figure 16: Shear viscosity as a function of shear rate for di�erent concentrations of xanthane.

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Figure 17: The minimum drop diameter versus time of xanthane, a function of the form Rmin =A |t− t0|P has been �tted to the results.

5.2.2 Elongation viscosity

The elongation viscosity for xanthane solution is measured in the same way as the elongation viscosityof the diluted gel, see section 5.1.2. d h

d t is taken 1 ms before breakup. The minimum diameter versustime is shown in �gure 17. In this �gure, the �t of the 1500ppm xanthane does not go to the point ofbreakup because shortly before breakup a very thin thread forms, which does not �t Rmin = A |t− t0|P .The elongation viscosity of xanthane is plotted in �gure 18.

5.2.3 Surface tension of materials without yield stress

As the xanthane solution is mostly water, and water is a Newtonian �uid , there is no yield stress.An other method to determine the surface tension is to use the fact that during drop formation, thedownward force is balanced by the surface tension [11]. Right before and immediately after breakupthe surface tension is equal to the gravitational pull, so 2πr f σ = mg, where r is the radius of thetip where the drop forms, f is a numerical correction factor, m is the mass of the drop and g is thegravitational constant. To measure m, a scale was placed beneath the syringe. The surface tensionof di�erent concentrations of xanthane is approximately constant and equal to the surface tension ofwater, see �gure 19.

5.2.4 Rigid dumbbell model

To test the dumbbell method, the elongation viscosity from the drop experiment can be comparedwith the predictions of the rigid dumbbell model. To do this, nK T is treated as one parameter andλ is treated as another parameter in equation 4. By �tting these to the shear viscosity, obtained fromthe rheometer experiment, an estimate can be made of nK T and λ (see �gure 16). Now, nK T andλ can be �lled in into equation 5. The results are plotted in �gure 18.

5.2.5 Power law

The power in the time development of the minimum drop diameter is plotted in �gure 20, togetherwith the power in the Herschel-Bulkley model, obtained from the rheology experiment (see �gure 15).

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Figure 18: The elongation viscosity of the xanthane solution, determined by the drop formation methodand the dumbbell model. Also plotted is the shear viscosity, measured at a shear rate of 100 s−1.

Figure 19: The surface tension of di�erent concentrations of xanthane. The relative large error barsare caused by uncertainty in the radius of the tip.

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Figure 20: The power law and the power in the Herschel-Bulkley model for xanthane.

In �gure 21 is an example of drops of xanthane, which are both asymmetric.

5.2.6 Reynolds number

Reynolds number for the xanthane solution is determined in the same way as for the gel: using

Re ≈ L2 ρηS t

, with ηS at a shear rate of 100 s−1 and t 1 ms before breakup, or 1 ms before the end of the

diameter versus time �t (which is the case for the 1500ppm solution). The result is plotted in �gure22.

5.2.7 Discussion

The prediction that for Newtonian �uids ηE = 3ηS is not true for xanthane, as the shear viscosityis a function of the shear rate. However, for high shear rates the shear viscosity and the elongationviscosity are of the same order of magnitude (see �gure 18) and the shear viscosity versus shear rateseems to stabilise at high shear rates, although this has not been measured.

The yield stress found in the Herschel-Bulkley model is almost zero. It is likely that there is no yieldstress at all, but that the �t with one extra parameter was slightly better. This is in agreement withearlier studies on xanthane. The value of n is smaller than 1 and decreasing for high concentrationsof xanthane, which is the case for a shear thinning material.

The power law in the drop formation of the xanthane solution is constant at 23 , indicating that the

�uid is in the capillary-viscous regime. This is also illustrated by the fact that the drops of xanthaneare asymmetric around the breakup, see �gure 21. Reynolds number is large for low concentrationsof xanthane but decreasing, although the scaling is arbitrary due to the �random� choice of the shearviscosity at a shear rate of 100 s−1.

The results of the dumbbell model are in reasonable agreement for low concentrations of xanthane,but the model seems to be incorrect at higher concentrations of xanthane. A possible explanation isthat the xanthane used in this experiment does not behave as a rigid rod, but has some elasticity. Themodel ignores interactions between the xanthane molecules, which also may cause the model to fail athigher concentrations.

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Figure 21: An image of di�erent concentrations of xanthane solution. The breakup is clearly notsymmetric.

Figure 22: Reynolds number for di�erent concentrations of xanthane.

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Figure 23: Shear viscosity as a function of shear rate for di�erent concentrations of �bers.

5.2.8 Conclusion

We have seen that the power law in the drop formation of the xanthane solution is constant at 23 , which,

combined with the asymmetric breakup, leads to the conclusion that the xanthane solution is in thecapillary-viscous regime. As n in the Herschel-Bulkley model, found in the rheology measurement, issmaller than 1, xanthane is a shear thinning �uid. A zero yield stress is found, which is in agreementwith predictions for xanthane. The shear viscosity at a shear rate of 100 s−1 is an order of magnitudesmaller than the elongation viscosity, so ηE = 3ηS is not true for xanthane.

5.3 Fibers

5.3.1 Shear viscosity

The shear viscosity of the �ber suspended in Triton is determined in the same way as the shearviscosity of gel. The shear stress versus shear rate for the �ber suspension is plotted in �gure 24 andthe shear viscosity versus shear rate is plotted in �gure 23. It was not possible to measure the 5000ppmsuspension at a shear rate of 1000 s−1, or higher concentrations of �bers at low shear rates, as the�bers start to cluster.

5.3.2 Elongation viscosity

The elongation viscosity of the �ber suspension is measured in the same way as the diluted gel. d hd t is

again taken 1 ms before breakup. The minimum diameter versus time is plotted in �gure 25, wheremultiple regimes are �tted for the 500ppm and 2500ppm suspension. For these concentrations, P wasobtained from the �t closest to the breakup. The elongation viscosity of the �ber suspensions is plottedin �gure 26.

5.3.3 Surface tension of �ber suspension

In the same way as for xanthane, the surface tension of Triton was determined. However, the �bersare longer than the radius of the tip, so any �ber that is not oriented exactly in the direction of �ow

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Figure 24: Shear stress versus shear rate for �bers and pure Triton. The Herschel-Bulkley model hasbeen �tted to the results.

Figure 25: The minimum drop diameter versus time of the �ber suspension, a function of the formRmin = A |t− t0|P has been �tted to the results.

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Figure 26: The elongation viscosity of the �ber suspension, determined by the drop formation method.Also plotted is the shear viscosity at a shear rate of 100S−1.

blocks the opening to the tip. Hence, only the surface tension of pure Triton was measured to be43 ± 4mN/m. As the surface tension of the xanthane solution is roughly constant and equal to thesurface tension of its solvent, it is assumed that this also applies to the �ber suspension.

5.3.4 Power law

The power in the time development of the �ber suspensions is plotted in �gure 27 together with thepower in the Herschel-Bulkley model, obtained from the rheology experiment (see �gure 24). A seriesof images of the 500ppm �ber suspension taken before the breakup is shown in �gure 28.

5.3.5 Discussion

The power in the drop formation of the �ber suspension is approximately constant at 1. It is hardto say if the breakup of the �ber suspension is symmetric or not because the thickness of the threadbefore breakup is smaller than the resolution of the camera, although the power law suggests that the�uid is in the viscous-capillary regime. The power in the Herschel-Bulkley model of pure Triton is 1,so pure Triton is a Newtonian �uid. The �ber suspensions have a slightly smaller value of n and asmall yield stress, so it is a yield stress, shear thinning material. An attempt to �t the shear stressversus shear rate graph with n = 1 in the Herschel-Bulkley model works well for high shear rates butslightly overestimates the shear stress for low shear rates, see �gure 29.

A test to see if the yield stress in the Herschel-Bulkley model is accidentally non-zero, or becausethere actually is a yield stress is to shake the �uid and check if small bubbles of air move to the surface(no yield stress) or stay at their original position (yield stress). Sadly, this test can not be performedon the �ber suspension because the �bers slowly sink in the Triton. However, when the shear stressversus shear rate is plotted on a logarithmic scale it is clear that the �ber suspension has a yield stress,see �gure 29.

The shear viscosity as a function of the shear rate is strongly in�uenced by the presence of �bers.Again, the prediction for Newtonian �uids that ηE = 3ηS is not true, see �gure 26. The rigid dumbbellmodel only works for �uids without a yield stress, and as a small yield stress is measured, the modelcannot be used.

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Figure 27: The power law and the power in the Herschel-Bulkley model for �bers.

Figure 28: Some images of a 500ppm �ber suspension. The time before breakup is written in thebottom of each image.

Figure 29: A 2500ppm �ber suspension, �tted with the Herschel-Bulkley model with and without yieldstress.

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5.3.6 Conclusion

The rigid dumbbell method does not work for the �ber suspension, as it should not be applied to amaterial with a yield stress. The �ber suspension is a shear thinning and yield stress material, whilethe pure Triton is a Newtonian �uid as expected. The power in the minimum diameter, obtained bythe drop breakup method, suggests that the �ber suspension is in the viscous-capillary regime. Theshear viscosity at a shear rate of 100 s−1 is of the same magnitude as the elongation viscosity, butηE = 3ηS does not hold.

Further investigations could determine the shear and elongation viscosity of more concentrationsof �bers. Suggested is to measure the shear viscosity of low concentrations (< 500 ppm) to observehow the transition from no to strong shear rate dependency in �gure 23 happens. It would also beinteresting to see what happens when the �bers are suspended in an inertia-capillary �uid instead ofa viscous-capillary �uid like Triton.

6 Acknowledgments

Great thanks go to Mounir Aytouna for all his support and advice during all stages of the project.I also want to thank Jose Paredes for his help with the rheometer, and Daniel Bonn for hosting theproject, his advice during the project and for giving great feedback on this report in multiple stagesof its progress.

References

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[5] H.A. Barnes, J.F. Hutton and K. Walters F. R. S., An Introduction to Rheology, Elsevier,Amsterdam, 1989

[6] J. Eggers, Breakup and coalescence of free surface �ows, book chapter contributed to S.Yip, Handbook of materials modeling, Springer, Berlijn, 2005

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