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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
-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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
λ = 1
Ca1>Cacr
λ = 1
Ca2>Ca1
λ<< 1Drop breakup in shear flowDrop breakup in shear flow
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
Cristini V, Guido S, Alfani , Blawzdziewicz J and Loewenberg M, J. Rheol., 47, 1283-1298 (2003)
Daughter drop scalingDaughter drop scaling
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
Daughter drop sizeindipendent of initial size
scales with critical drop size
Cumulative size distribution
Drop fragment distributionDrop fragment distribution
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
The distributions for each experiment show two distinct daughter drops and three size classes of satellite drops
Drop fragment distributionDrop fragment distribution
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
• 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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
Wall effectsWall effectsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
d d ≈ 2R0
L/2a
3
4
5
22
3
45
64 5
150 µµµµm
50 µµµµm 1 1
22
33
Wall effectsWall effectsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
Wall effectsWall effectsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
Wall effectsWall effectsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
Wall effectsWall effectsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
Concentrated systemsConcentrated systemsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
The time-averaged value of D depends on Ca only
Concentrated systemsConcentrated systemsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
“Mean field” scaling with blend viscosity
Concentrated systemsConcentrated systemsS. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
S. GuidoUniversity of Naples Federico IIDepartment of Chemical Engineering
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
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