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Yield Stress Fluids: Summer Reading Group 2009

http://web.mit.edu/nnf

A 50 Cent Rheometer for Yield Stress Measurements

( N. Pashias, D. V. Boger, J. Summers, and D. J. Glenister J. of Rheology, 1996) Cited 57 times.

Measuring yield stress via “slump-test”

A field engineering technique comes to lab

Trush Majmudar

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Slump Test:

Not this one!

Slump test in physiotherapy to

Figure out back injuries!Slump test in civil engineering to determine

“workability” of concrete.

Steps involved in carrying out a slump test:

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Some definitions and the theme of the paper:

Yield Stress:

Workability:

4

The current paper deals with experiments and modeling

the slump test:

Slump test procedure:

1) Fill a conical or cylindrical (current case) container with cement

2) Lift the container

3) Measure the final height of the sample

Factors affecting slumping of the sample:

1) Yield stress of the material (material parameters)2) Height, and width of the sample (geometric parameters)

3) Viscosity, gravity, density…

Features that might shed light on the properties of the material:

1) Slump height2) Shape of the mound as it slumps

3) Rate of slumping

Important dimensionless number: !y

'=

!y

"gH

!y

(dimensionless yield stress)

Current paper:

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Range of values:

Yield stress: (30 - 800 Pa)

Velocity of lifting: (0.1 - 30 m/s)

Apsect ratio R/H was around 1.0

These ranges imply that the dimensionless

yield stress values were between 0.01- 0.5

!y

'= 0.01"0.5[ ]

Materials and Methods:

Samples: Mineral suspensions of:

Zirconia: Zirconia dioxide

Titania: Titanium dioxide

Bauxite residue (thixotropic)

ZrO2 (! = 5800 kg / m3)

TiO2 (! = 4000 kg / m3)

(!= 3200 kg / m3)

Figure exhibits the variation of yield stress with the

pH of the suspension.

The samples used were at pH, which resulted in

maximum yield stress: “isoelectric point”.

The yield stresses measured using vane geometry.

Four cylinders with aspect ratios 0.78, 0.97, 1.17, and

1.28 were used for the slump test.

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⇒Linearly increasing stress distribution with depth: 0 at the apex, maximum at the base

Interfacial layer between yielded and unyielded layer assumed to be flat - slumping only

due to radial flow. The flow stops when the stress in the yielded portion is back to yield

stress value. In the yielded portion, height is divided into elemnets of thickness dz, which

evolve to thicknesss dz1; incompressibility condition gives the relation:

Theory:

Pz= z!g

!z=

1

2z"g

!z

'=

1

2z'

h0: height of theunyielded region

h1: height of the yielded region

h : total final height (h = h0+ h

1)

z : vertical coordinate

! z : shear stress

! y : yield stress

dz1=

rz

rz1

!

"#

$

%&

2

dz

At any given height z, pressure is approximated as:

Maximum shear stress on an ideal elastic solid is: (Poisson ratio 1/2)

Dimensionless form (stress scaled with ):!gH

h1= dz

1

h0

h

!Height given by:

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s'= 1! h

0

' ! h1

'

s'= 1! 2" y

'1! ln 2" y

'( )#$

%&

! rz( )2

"z= r

z1

( )2

"y

h1

'=

! y'

1

2z'dz

'

h0'

1

"

h1

'= #2! y

'ln h

0

'( )

! y'=

1

2h0'

" h1'= #2! y

'ln(2! y

')

Theory (Contd.)

No flow between horizontal planes;

flow occurs until the cross-sectional area

increases so that stress required to support

the weight is reduced to yield stress:

Substituting previous equations into the

height integral gives:

Dimensionless slump value is given by:

Exanding the log term: ln 2!y

'( ) " 2!y

'#1( )

Simplified expression for the dimensionless slump value:

!y

'=

1

2"

1

2s' or s

'= (1! 2"

y

')2

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Results:

Slump height decreases with increasing

yield stress;

(Reduced state: suspension mixed for prolonged

periods of time causing structural decay and reduction

in yield stress)

Red mud (bauxite)

Dimensionless yield stress and slump height

as a function of agitation time, to test if slump height depends on yield stress for suspensions

with different structures; agitation time 140 h.

•Yield stress intially high; slump height low

•As it gets sheared more, yield stress decreases,

slump height increases.

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Results (contd.)

Main result: combined data with theoretical fits

All of the data falls on roughly a common

curve. Solid line is the exact theoretical curve.

Dashed curve is the approximate solution.

•Slump height independent of material

•Slump height is a unique function of the

yield stress.

⇒Slump test can be used to measure/infer

yield stress of a material.

The exact theory fits well upto

Thereafter it underpredicts data.

The approximate curve over-predicts data until

!y

'"0.15

!y

'"0.2

Possible reasons: sample is not an elastic solid as assumed.

horizontal layers may not remain horizontal

frictional effects between sample and walls, and the base.

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Summary of the results:

•Slump test can be modeled by a simple force balance equations.

•The fluid slumping under gravity yields at the base upto some height and

the rest of the fluid is unyielded.

•The flow continues until the stresses within the yielded portion return back to

the yield stress and support the weight of the fluid above.

•Dimensionless slump height is a unique function of the dimensionless yield stress

of the fluid.

•By measuring the slump height we can effectively measure the yield stress.

•The slump height does not depend on the velocity of lifting the cylinder.

•The slump height does not depend on the base surface.

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A more sophisticated theory: “slip-line field theory”

Previous theory can be considered as “uniform stress theory”:

All stresses vertical, and uniform across any horizontal plane; no friction anywhere.

Slip-line field theory simplistically means that when a rigid-plastic material fails,

it does so along certain directions called slip lines, which are curvilinear boundries.

The slip-lines are oriented in the directions of maximum shear stress.

Friction at the base is included and treated as Coulombic friction.

J. Chamberlain, J. Sader, K. Landman, D. Horrobin, L. White, Int. J. of Mech. Sci.,44 (2002)

Slip lines and stresses are

calculated iteratively, starting

from the free surface going

inwards towards the base

and centerline.

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Slip-line field theory:

Predictions of this model:

There is a critical friction value below which

the incipient failure height increases with increasing

friction, and above which it does not change.

The incipient failure height is also a function

of the scaled radius of the cylinder.

Uniaxial yield stress can be deduced from a

plot of inverse scaled height vs. aspect ratio.

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h=

! y

H"g

R0

H

vs.

The idea is to measure slump height H and

radius R0, which gives you X-coordinate,

then using appropriate friction curve (?!),

read off scaled yield stress from Y axis.

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More slip line theory based results:

(JNNFM, 158, 91-100, 2009)

This paper also calculates slip lines, and profiles of the slump, and reports experiments

with Carbopol, but the setup is more suitable for shallow flow or the “dam-break” problem.

( Full profile) (zoom at the edge)

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Force balance:

Haar-Karman hypothesis:

Yield condition:Free surface condition:Force at the base:

Coulomb condition:

Rescaling:New force balance:

Equations:

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References:

• J. Chamberlain et al., Int. J. of Mech. Sci., 43, 793-815, 2001.

• J. Chamberlain et al., Int. J. of Mech. Sci., 44, 1799-1800, 2002.

• J. Chamberlain et al., J. of Rheology, 47 (6), 1317-1329, 2003.

• A. Saak, H. Jennings, S. Shah, Cement and Concrete Research, 34, 363-371, 2004.

• N. Dubash et al., JNNFM, 158, 91-100, 2009.

• W. Schowater and G. Christensen, J. of Rheology, 42 (4), 865-870, 1998.

• J. -M. Piau and K. Debiane, JNNFM, 127, 213-224, 2005.

• J. -M. Piau, J. of Rheology, 49 (6), 1253-1276, 2005.

• J. -M. Piau, JNNFM, 135, 177-178, 2006.

• N. Roussel and P. Coussot, J. of Rheology, 49 (3), 705-718, 2005.