Dipartimento di Ingegneria chimica Università di Napoli ... bre… · S. Guido and M. Villone, J....

Dipartimento di Ingegneria chimicaUniversità di Napoli Federico II

Drop deformation and breakup in laminar shear flow

Stefano Guido

COST P21 Student Training School “Physics of droplets: Basic and Advanced topics”Borovets, 13 July 2010

BackgroundBackground

Liquid-liquid biphasic systems

-Examples: emulsions, polymer blends, water-in-waterbiopolymer mixtures

-In many cases system microstructure is characterizedby droplets in a continuousphase

S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

by droplets in a continuousphase

Applications: detergents, personal care, food products, oil recovery

Flow-induced microstructure evolution-Processing (e.g., mixing)-Final product properties and usage

Two main mechanisms governing microstructure dynamics under flow-Droplet collision and coalescence

BackgroundBackground


-Droplet deformation and breakup (this presentation)

x

y z

δ

V

γ = V/δ.

Drop deformation and breakup in simple shear flow

–Isolated droplets–No interfacial agents–Immiscible, Newtonian fluids–Laminar flow. However, results still apply in turbulent flow when

TopicsTopics


δγ V=&

V

δ

Effect of concentration

Non-Newtonian effects

Wall effects

eddies size is larger than droplet diameter (viscous turbulence) Vankova N, Tcholakova S, Denkov ND, Ivanov IB, VulchevVD, Danner Th. J Colloid Interface Sci, 312, 363 (2007)

1) Rallison J M, “The deformation of small viscous drops and bubbles in shear flows”,Ann. Rev. FluidMech.,16, 45-66 (1984).

2) Stone H A, “Dynamics of drop deformation and breakup in viscous fluids”,Ann. Rev. Fluid Mech.,26 ,65-102 (1994) .

3) Tucker III C L and Moldenares P, “ Microstructural evolution in polymer blends”,Ann. Rev. FluidMech., 34, 177-210 (2002).

4) S. Guido and F. Greco,“Dynamics of a liquid drop in a flowing immiscible liquid”, in Rheology

ReviewsReviews


4) S. Guido and F. Greco,“Dynamics of a liquid drop in a flowing immiscible liquid”, in RheologyReviews 2004, Ed. D. M. Binding and K. Walters, The British Society of Rheology, Aberystwyth,pp. 99-142 (2004)

5) Fischer P, Erni P, “Emulsion drops in external flow fields-- The role of liquid interfaces”,CurrentOpinion in Colloid & Interface Science, 12,196-205 (2007).

6) Derkach, S R, “Rheology of emulsions”,Adv Colloid Interface Sci, 151, 1-23 (2009)7) Minale M, “Models for the deformation of a single ellipsoidal drop: a review”,Rheol Acta, online

article8) Frith W J, “Mixed Biopolymer Aqueous Solutions – Phase behaviour and rheology”,Adv Colloid

Interface Sci, advanced online article9) Guido S and Preziosi V,Adv Colloid Interface Sci, advanced online article

Experimental techniquesExperimental techniquesS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

δγ V=&

V

δ

Parallel Band ApparatusG. I. Taylor, Proc. R. Soc. Lond. A, 146, 501-523 (1934)

Couette geometryMighri F and Huneault M A, J. Rheol., 45,783-797 (2001)

Parallel plates (rotational)Levitt L, Macosko C W and Pearson S D, Polymer Eng. Sci., 36, 1647-1655 (1996)

θθθθ

z

r

θθθθr

z

δγ V=&

V

δ

Experimental techniquesExperimental techniquesS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

Parallel plates (translating, high-speed)X. Zhao and J. L. Goveas, Langmuir, 17, 3788-3791 (2001)

Parallel plates (translating)S. Guido and M. Villone, J. Rheology, 42, 395-415 (1998)

2-axes motorized stage

CCD video camera

Tilting and rotary stages Moving plate

Fixed plate

Shear flow workstationShear flow workstationS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

Focusing motor

2-axes motorized stage andfocusing motor controller

CD/DVD recorder

Anti-vibrating table

PC

Microscope translating stage

View along vorticity (video)

Drop shape at small deformationsDrop shape at small deformations


Example: drop of polydimethylsiloxane in polyisobutilene

Newtonian case - Taylor, Chaffey-Brenner, Greco

σγη &

0Car

c=

c

dηηλ =

Capillary number

Viscosity ratio

Nondimensional numbers

ηc viscosity of continuous phaseηd viscosity of drop phase

shear rate r0 drop radius at restσ interfacial tension

Relevant physical quantities

γ. γτστ

&

=Ca

Small deformation theorySmall deformation theoryS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

Predictions

( ) ( ) 221

0

1 CafCafr

rMAX λλ ++=

( ) ( ) 221

0

1 CafCafr

rMIN λλ +−=

( ) 23

0

1 Cafr

rz λ+=Ca

rr

rrD

MINMAX

MINMAX

16161619

++=

+−≡

λλ

σ interfacial tension

D ≡ Deformation parameterG. I. Taylor, Proc. R. Soc. Lond. A, 146, 501-523 (1934)Chaffey CE, Brenner H, J Colloid Interface Sci, 24, 258–269 (1967)Greco F, Phys. Fluids, 14, 946-954 (2002)

Ca)1(80

)32)(1619(

4 λλλπϕ

++++=

Rmax

Rminφ

x

y

z

V

RM

IN,

RM

AX

0.6

0.8

1.0

1.2

1.4

1.6

RZ0.96

0.98

1.00

Stationary droplet shapeStationary droplet shapeS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

Rz

Rp

Drop shape is ellipsoidal up to moderate deformationsS. Guido and M. Villone, J. Rheology, 42, 395-415 (1998)

0.92

0.94

0.96

Ca

0.00 0.05 0.10 0.15 0.20 0.25

ϕϕϕϕ

30

35

40

45

Maffettone-Minale model

Ellipsoidal droplet described by a second order, positive-definite, symmetric tensor S

ΩΩΩΩ = 1/2∇v−∇vT and D = 1/2 ∇v+∇vT, where ∇v is the velocity gradient tensor

g(S) = 3IIIS/IIS, where IIIS and IIS are the third and the second scalar invariant of S (to preserve droplet volume)

Ellipsoidal modelsEllipsoidal modelsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

fi functions chosen to recover Taylor’s small deformation limits

Maffettone PL, Minale M, J Non-Newton Fluid Mech, 78, 227–241 (1998)

Other ellipsoidal modelsWetzel ED, Tucker CL III, J Fluid Mech, 426,199–228 (2001)Yu W, Bousmina M, Grmela M, Palierne J, Zhou C, J Rheol, 46,1381–1399 (2002)Edwards BJ, Dressler M, Rheol Acta, 42,326–337 (2003)

D

0.2

0.3

0.4

0.5

ϕϕϕϕ

25

30

35

40

45

FROM STEADY STATE SHAPE

Interfacial tension measurementInterfacial tension measurementS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

Shear rate, s-1

Carr

rrD

MINMAX

MINMAX

1616

1619

++=

+−≡

λλ

0.0 0.1 0.2 0.3 0.4 0.515

20

25

Shear rate, s-1

Ca)1(80

)32)(1619(4 λ

λλπϕ ++++=

25 µµµµm

FROM DROP RETRACTION

1

2

+++−= τλλ

λ)1619)(32(

)1(40exp0DD

Interfacial tension measurementInterfacial tension measurementS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

tσ/(ησ/(ησ/(ησ/(ηcR0)-6 -4 -2 0 2 4 6 8

ln(D

/D0)

-3

-2

-1

0

1

Overall data

++−= τλλ )1619)(32(

exp0DD

τ = tσ/(ηcR0)

Luciani A, Champagne M F and Utracki L A,,J. Polym. Sci. Phys. Ed., 35, 1393-1403 (1997) .Guido S and Villone M,, J. Colloid Interface Sci., 209, 247-250 (1999).

D

0.0

0.2

0.4

0.6

0.8

1

23 4

5

6

R=19.5 µmShear rate = 0.05 s-1

λ = 0.1

1

20 micron

2

3 4

R = 20 µmλ = 0.1Shear rate = 0.05 s-1

Water-in-water biopolymer mixturesWater-in-water biopolymer mixturesS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

time (s)0 50 100 150 200 250 300

0.0

5 6

time (s)0 50 100 150 200 250

θθθθ

50

60

70

80

90

2

3 4Drop: Na-caseinate rich phaseMatrix: Na-alginate rich phase

Guido, S Simeone M and Alfani, A, Carbohydrate Polymers, 48, 143-152 (2002)

MIN

RMAX

Cohen A and Carriere C J, Rheol. Acta, 28. 223-232 (1989)

)2/ln( 0rL=ε

Droplet retraction after a step strainDroplet retraction after a step strainS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

t, s

RM

IN,

RZ,

RM

AX

RMIN

RZγγγγ = 5

Yamane H, Takahashi M, Hayashi R, Okamoto K, Kashihara H and Masuda T, J. Rheol., 42, 567-580 (1998)

Assighaou S and Benyahia L, Rheol Acta, 49, 677-686 (2010)

FE flat ellipsoid, C spherocylinder, E prolate ellipsoid, S sphere

• Upon increasing Ca, a critical condition (Cacr) is reached where drop shape becomes unstable (video)

λ

Drop breakup in shear flowDrop breakup in shear flow


λ = 1

Ca1>Cacr

λ = 1

Ca2>Ca1

λ<< 1Drop breakup in shear flowDrop breakup in shear flow


Tip streamingDrop: Na-caseinate rich phaseMatrix: Na-alginate rich phase

Cacr

JT modelGrace datade Bruijn data

JT modelGrace datade Bruijn data

Drop breakup in shear flowDrop breakup in shear flow


λλλλGrace H P, Chem. Eng. Commun., 14, 225-277 (1982)de Bruijn R A, PhD thesis, Technische Universiteit Eindhoven (1989)Jackson N E and Tucker III C L, J. Rheol., 47, 659-682 (2003)

Breakup of a liquid threadBreakup of a liquid thread


Tomotika S, Proc. R. Soc. London Ser. A, 150, 322-337 (1935)Elemans P H M, Janssen J M H and Meijer H E H, J. Rheol., 34, 1311-1325 (1990)

Viscous-capillary force balance(inertia is negligible)

rrzvrv d ∂∂≈∂∂≈∂∂ −12222 σηη

ηη

≈ζ d

l

Close to breakup, the external viscous shear stresses associated with thread axial motion become comparable to the internal viscous stresses associated with thread extension

Near critical behaviorNear critical behaviorS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

r

v

∂∂η

z

vd ∂

∂η

ζ characteristic axial length in the pinch-off region

ησ≈

η

w

l ζ characteristic axial length in the pinch-off region

l neck radius

w characteristic velocity

( ) ( )( )ttttw crcr −ησ≈−≡ζ ( )tt cr

d

−ηησ≈l

Lister J R and Stone H A, Phys Fluids, 10, 2758-27 (1998)

Blawzdziewicz J, Cristini V and Loewenberg M, Phys Fluids, 14, 2709-2718 (2002)

λ = 1 Ca = 0.50

50 µµµµm

neck

Time, s0 40 80 120

RMAX /R0

1

2

3

4

5

minimum drop elongation ratemin

dtR/dR

0MAX

Breakup kineticsBreakup kineticsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

daughters satellites

Cristini V, Guido S, Alfani , Blawzdziewicz J and Loewenberg M, J. Rheol., 47, 1283-1298 (2003) t/(1+λλλλ)γγγγ

0 20 40 60

RMAX /R0

1

6

11

16

Ca = 0.433

Ca = 0.505

Ca = 0.595

.

0.00

0.02

0.04

parabolic fit

mindt

R/dR0MAX

Cacr

R'MAX /R0

1

2

30.430.450.470.480.490.50

Ca

Ca

Determination of CacrDetermination of Cacr


Ca

0.44 0.46 0.48 0.50 0.520.00

Time, s

0 40 80 120 1601

Shape evolution for Ca/Cacr = 1.38Shape evolution for Ca/Cacr = 1.38


Cristini V, Guido S, Alfani , Blawzdziewicz J and Loewenberg M, J. Rheol., 47, 1283-1298 (2003)

Daughter drop scalingDaughter drop scaling


Daughter drop sizeindipendent of initial size

scales with critical drop size

Cumulative size distribution

Drop fragment distributionDrop fragment distribution


The distributions for each experiment show two distinct daughter drops and three size classes of satellite drops

Drop fragment distributionDrop fragment distribution


Zhao X, J. Rheol., (2006)

λλλλ = 0.075, Ca = 4.5 Cacr

Liquid-liquid dispersions are alwaysviscoelastic systems, due to interfacial

tension

Our aim is to study the effect of the intrinsic elasticityof the fluid

components on flow-induced morphology

Non-Newtonian effectsNon-Newtonian effects


Model fluids: constant viscosity, highly elastic liquids (Boger fluids)

Outer phase: viscoelastic fluid , innerphase: Newtonian, λλλλ = 1

Ca = 0.4

• Constitutive equation of component liquids: second-order fluid

either the outer or the inner phase is non-Newtonian

(the other being Newtonian)Here

2Ψ=µ γ 2&Ψr

2 additional physical quantities:2

11N γ&Ψ=2

22N γ&Ψ=normal stress differences

Small deformation theory with elastic fluidsSmall deformation theory with elastic fluids


• New nondimensional parameters:1

2ΨΨ=µ

σγ

2

210&Ψ= r

W

• Small deformation conditions Ca<<1 W<<1

12 2

0

12 ≈=≡

ησψ

rCa

WObservable non-Newtonian effects p

Greco F, J. Non-Newt. Fluid Mech., 107, 111-131 (2002)

rMAX /r01.25

1.50

1.75

ϕϕϕϕMAX35

45

Comparison with experimentsComparison with experiments


p = 1.8

Ca0.0 0.1 0.2 0.3

rMIN /r0

0.50

0.75

1.00

Dashed: Newtonian

Continuous: non-Newtonian

λ λ λ λ ==== 1

Ca0.0 0.1 0.2 0.3

25

Ca c

r

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Ca infCa sup

Outer phase: viscoelastic, drop: Newtonian, λλλλ = 0.6

BreakupBreakupS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

Outer phase: Newtonian, drop: viscoelastic, λλλλ = 2.6

Drop breakup is hindered by elasticity of the fluid components

p

0.1 1 100.40

Shape parameters

Nondimensional gap

Nondimensional major axis

d

x

y

z50 µm

R0

Wall effectsWall effectsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

02R

d

gap

x

y

z

L

++

+=λ

λ1

5.21

d

R21DD

3

0T

Shapira M and Haber S, Int. J. Multiphase Flow, 16,305 (1990)

R0

d d >> 2R0


d d ≈ 2R0

L/2a

3

4

5

22

3

45

64 5

150 µµµµm

50 µµµµm 1 1

22

33


t0 50 100 150 200

1

2

0 5 10 151

1

2

1

35

6

Ca = 0.4, λλλλ = 1

d ≈ 2R0 d >> 2R0

4 4

5 5

6 6

Sibillo V, Pasquariello G, Simeone M, Cristini V, Guido S, Phys. Rev. Lett., 97, 054502 (2006)

Comparison with theoretical predictions(left) and numerical simulations (right)

Ca = 0.1


Simulations: Janssen P J A and Anderson P D, Phys Fluids, 19, 043602 (2007)

Theory: Shapira M and Haber S, Int. J. Multiphase Flow, 16,305 (1990)

Wall effects act to stabilize drop shape at λλλλ = 1



Van Puyvelde P, Vananroye A, Cardinaels R, Moldenaers P, Polymer, 49, 5363–5372 (2008)

Even at λ > 4 (no breakup in unbounded shear flow), droplets can still be broken in confined conditions

Vorticity

Flo

w

Wall effects: Shear bandingWall effects: Shear bandingS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

Caserta S, Simeone M, and Guido S, Phys. Rev. Lett., 100, 137801 (2008)

Cone and plate rheometer

Strain0 5000 10000 15000 20000 25000

η/ηη/ηη/ηη/η0000

0.8

0.9

1.0λ = 1

λ = 0.1

λ = 0.04

D

0.16

0.18

0.20

0.22

0.24

(A)

(B)

(C)(D)

(E)

(2)

(3)

(4)

(5)

Shape fluctuations due todrop interactions

Concentrated systemsConcentrated systemsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering

t [s]0 10 20 30 40 50 60

0.14

(A) (B) (C) (D) (E)

(1)

(1)

(2) (3) (4) (5)

Ca = 0.15


The time-averaged value of D depends on Ca only

“Mean field” scaling with blend viscosity


Jansen K M B, Agterof W G M, Mellema J, J Rheol, 45, 227-236 (2001)Caserta S, Reynaud S, Simeone M and Guido S, J Rheol, 51, 585-774 (2007) (data shown here)

Drop shape up to moderate deformations is essentially ellipsoidal and is well represented by small deformation theories and phenomenological models

Numerical simulations are in good agreement with experiments up to breakup

ConclusionsConclusions


Single drop deformation and breakup results can be applied to concentrated systems by using a “mean field” scaling

Drop fragments distribution can be estimated if the original distribution is known

Open issue: effects of surfactants

Wall effects stabilize drop shape and elicit shear bandingphenomena

Dipartimento di Ingegneria chimica Università di Napoli ... bre… · S. Guido and M. Villone, J....

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