Nonlinear electrohydrodynamics of a viscous droplet

5.7 MB·English

by  Paul F. Salipante

Nonlinear electrohydrodynamics of a viscous droplet
Paul F. Salipante and Petia M. Vlahovska,
School of Engineering, Brown University, Providence, RI 02912
(Dated: January 24, 2013)
A classic result due to G.I.Taylor is that a drop placed in a uniform electric field becomes a
prolate or oblate spheroid, which is axisymmetrically aligned with the applied field. We report an
instability and symmetry-breaking transition to obliquely oriented, steady and unsteady shapes in
strong fields. Our experiments reveal novel droplet behaviors such as tumbling, shape oscillations,
and chaotic dynamics even under creeping flow conditions. A theoretical model, which includes
anisotropyinthepolarizationrelaxationduetodropasphericityandchargeconvectionduetodrop
fluidity, elucidates the interplay of interfacial flow and charging as the source of the rich nonlinear
dynamics.
3
1
PACSnumbers: 47.15.G-,47.55.D-,47.55.N-,47.57.jd,47.52.+j,47.65.Gx
0
2

n Nonlinear phenomena such as instabilities and turbu-
a
lence naturally occur in fluid dynamics because of the
J
nonlinearityoftheNavier-Stokesduetotheinertialterm.
3
In the absence of inertia, the Stokes equations governing
2
thefluidflowarelinearandevolvingboundaryconditions

] aretheonlysourceofnonlinearity[1]. Forexample,asin-
n
gleparticleexhibitscomplexdynamicsifitisdeformable:
y Ε
a capsule or a red blood cell in shear flow [2–6], a drop
d
- in oscillatory strain-dominated linear flows [7], or a drop A B C
u
sedimenting in an electric field [8].
l FIG. 1: Exposure to a uniform direct current (DC) electric field
f If the particle is rigid, chaotic motions are observed with increasing strength excites a variety of drop responses [12]
.
s with ellipsoids in shear flow [9] or uniform electric fields (see videos [13]). A. A castor oil drop suspended in silicon oil is
c spherical in the absence of electric field. B. Weak fields induce
[10,11]. Inthelattercase,thenonlineardynamicsarises
i axisymmetric oblate deformation. C. In strong fields, the drop is
s
from anisotropy in the polarization relaxation due to as-
y tiltedwithrespecttotheappliedfielddirectionandtheflowhasa
h phericity of the particle shape. A question arises - what rotationalcomponent. Thesketchesillustratethedropshapeand
p if the particle is not rigid, but fluid and its shape is not flowstreamlines.
[ fixed? Wouldparticlefluidityanddeformabilitygiverise

tonew,richerbehaviors? InthisLetter,weexplorethese
1 where σ and (cid:15) denote conductivity and dielectric con-
v questionsontheexampleofaviscousdropsubjectedtoa stant, respectively. The product of R and S compares
2 uniformdirectcurrent(DC)electricfield. Uponincrease
the charge relaxation times of the media [14, 15]
8 in the field strength, this system was found to undergo a
3 τ (cid:15) (cid:15)
.5 seyamr mfloewtryil–lbursetraaktiendg tinranFsiigtuiorne f1r.omBaxainsdymCme[1t2ri]c, twohliicnh- RS = τcc,,einx , where τc,in = σiinn τc,ex = σeexx . (2)
1
hints upon the possibility of more complex physics. We
0 If RS < 1 (τ > τ ), the conduction response of
first discuss nonlinear drop electrohydrodynamics theo- c,in c,ex
3 the exterior fluid is faster than the particle material. As
1 retically, and then we describe its experimental realiza-
a result, the induced dipole is oriented opposite to the
: tion. We find that depending on the fluids viscosity, the
v applied electric field direction. This configuration is un-
droplet may exhibit a range of intriguing dynamics such
Xi astumbling,sustainedshapeoscillations,andevenchaos. favorableandbecomesunstableaboveacriticalstrength
of the electric field [10, 16, 17]. A perturbation in the
r Theoretical model for drop polarization and shape evo-
a dipolealignmentgivesrisetoatorque,whichdrivesphys-
lution. When placed in an electric field, a particle po-
ical rotation of the sphere. The induced surface-charge
larizes because free charges carried by conduction accu-
distribution rotates with the particle, but at the same
mulate at boundaries that separate media with different
time the exterior fluid recharges the interface. The bal-
electric properties. This is illustrated in Figure 2 on the
ance between charge convection by rotation and supply
exampleofasphereinauniformelectricfield. Themag-
by conduction from the bulk results in an oblique dipole
nitude and orientation of the induced dipole depend on
orientation shown in Figure 2.(b). The rotation rate ω
the mismatch of electric properties between the particle
is determined from the balance of electric and viscous
(“in”) and the suspending fluid (“ex”)
torques acting on the particle, P×E = A·ω, where A
σ (cid:15) is the friction matrix.
R= in , S= ex , (1)
σ (cid:15) The spontaneous spinning of a rigid sphere in a uni-
ex in
2
ω y x
a
θ
E
ω
E φ a
P
P ν
FIG. 2: Charge distribution and induced dipole P for a sphere
with RS < 1. Above a critical field strength E0 > EQ , where FIG. 3: Sketch of the model for drop electrorotation. Drop as-
E0 =|E| and EQ is given by Eq. 4, constant rotation around an phericityischaracterizedbytheaspectratioβ=a(cid:107)/a⊥. Therigid
axisperpendiculartotheelectricfieldisinducedbythemisaligned body rotation is related to the tilt angle of the deformed shape,
dipole moment of the particle (right). The rotation can be either θ, byω=dθ/dt. Intheco-rotatingframetheinterfacevelocityis
clock-orcounter-clockwise. ν(−yβ,x/β).
form DC electric field has been known for over a cen- (cid:18)dθ 2β (cid:19)
α + ν =P E −P E +(χ∞−χ∞)E E
tury, first attributed to the work of Quincke in 1896. In ⊥ dt 1+β2 (cid:107) ⊥ ⊥ (cid:107) (cid:107) ⊥ ⊥ (cid:107)
this case, the friction matrix is diagonal, and a straight- (5)
forward calculation [16–18], assuming instantaneous po- where α is the friction coefficient, χ∞ is the high-
⊥
larization, yields three possible solutions: no rotation, frequency susceptibility, and (cid:107), ⊥ denote components
ω =0, and parallel and perpendicular to the axis of symmetry, see
(cid:115) Figure 3. Drop fluidity is accounted by interfacial ve-
1 E2
ω =± 0 −1, (3) locity with frequency ν. The latter is determined from
τmw EQ2 thebalanceofelectricalenergyinputandviscousdissipa-
tion, in a similar fashion to the analysis of drop rotation
where the ± sign reflects the two possible directions of
in a magnetic field [20], or red blood cell tank-treading
rotation and
in shear flow [21]. The polarization relaxation equations
(cid:15) +2(cid:15) 2σ µ (R+2)2 in a coordinate system rotating with ω are
τ = in ex and E2 = ex ex . (4)
mw σ +2σ Q 3(cid:15) (cid:15) (1−RS)
in ex ex in ∂P 1
(cid:107) =−νP β− [P −(χ0−χ∞)E ], (6a)
τ , the Maxwell-Wagner polarization time, is the char- ∂t ⊥ τ (cid:107) (cid:107) (cid:107) (cid:107)
mw (cid:107)
acteristic time-scale for polarization relaxation. The ∂P νP 1
⊥ = (cid:107) − [P −(χ0 −χ∞)E ], (6b)
dipole “tilt” is steady; the angle between the dipole and ∂t β τ ⊥ ⊥ ⊥ ⊥
the electric field is φ = arctan(cid:2)(τ ω)−1(cid:3). Eq. 3 shows ⊥
mw
thatrotationispossibleonlyiftheelectricfieldexceedsa whereτ andχ0 aredirectionalMaxwell-Wagnerre-
(cid:107),⊥ (cid:107),⊥
criticalvaluegivenbyE . Hence,ifE ≤E ,thesphere laxation timescales and low-frequency susceptibility re-
Q 0 Q
andthesuspendingfluidaremotionless. IfE >E ,the spectively. If ν = 0, Eq. 5 and Eq. 6 reduce to the
0 Q
sphererotatesanddragsthefluidinmotion;theresulting equations of motion for a rigid ellipsoid [11], and predict
flow is purely rotational. three types of behavior: alignment of the symmetry axis
If polarization relaxation, i.e., non-instantaneous with the electric field, oscillations around the field direc-
charging of the interface described by ∂ P = ω ×P− tion(“swinging”), andcontinuousflipping(“tumbling”).
t
τ−1 (P−P ), is included in the analysis, the polariza- The additional torque associated with the interface flu-
mw eq
tionevolutionequationandthetorquebalancemaponto idity (characterized by ν) modifies these behaviors to:
theLorenzchaosequations[10,19]. Asecondbifurcation steady axisymmetric orientation (Taylor regime), steady
occurs in stronger fields and the sphere exhibits chaotic tilted orientation, swinging around a non-zero tilt angle
reversal of the rotation direction. with respect to the electric field, and tumbling. More-
Unlike solid particles, drops are fluid and have a free over, chaotic switching between the swinging and tum-
boundary. The electric stress deforms the drop and, as bling states is also found. The phase diagram resulting
a result, the friction matrix and polarization relaxation from the numerical solution of Eq. 5 and Eq. 6 is shown
become anisotropic and dependent on the drop orienta- in Figure 4. In all cases except for the chaotic behavior,
tion relative to the applied electric field. Moreover, the the motion of the dipole mirrors the motion of the el-
interface does not move as a rigid body and its veloc- lipsoid. The model predicts that chaos is observed with
ity differs from that the Quincke rotation. The modified high viscosity drops (drop viscosity at least four times
torque balance (assuming for simplicity that drop shape greater than the suspending medium). In this case, the
remains an axisymmetric oblate spheroid with fixed as- chaotic motion stems from the lack of synchronization
pect ratio once rotation is initiated) is between ellipsoid and dipole orientations.
3
vestigate the effect of drop viscosity, the viscosity ratio
4
Sw λ = µin/µex is varied in the range 1 to 14. The electric
3.5 field is increased in small steps of about 0.1E and the
Q
systemisallowedtoequilibrateateachsteptoavoidspu-
3 Tb Tb Ch Tb
rioustransients. Forexample,itispossibletoexcitedrop
Q
E/E02.5 tumblinguponasuddenincreaseofthefieldstrengthbut
c Field 2 Ti tdheetadilrsopabeovuetnttuhaelleyxpseetrtilmeseninttcoasntebaedyfotuinltdedinst[a1t2e].. More
ectri 1.5 TypicaldropbehaviorsareillustratedinFigure6. The
El Sw
unsteady dynamics of high viscosity drops (λ = 14) is
1
chaotic tumbling, while low viscosity drops (λ = 1) un-
0.5 dergo steady shape oscillations. Intermediate viscosity
Al Al
drops (λ = 4) exhibit both behaviors, namely a cycle
0
1 1.5 2 2.5 consistingofshapeoscillationswithincreasingamplitude
Aspect Ratio
followed by several tumbles. These behaviors are better
FIG.4: Phasediagramforafluidellipsoidwithviscosityratioλ≡ seen in the insets of Figure 6, where the angle between
µin/µex=14. Thelinesrepresenttheboundariesbetweenvarious the drop major axis and the applied field direction is
behaviors computed from the model. The symbols correspond to
plotted.
experimental data: (cid:3) - Taylor (steady axisymmetrically oriented
ellipsoid), (cid:5) - steady tilted ellipsoid, (cid:52) - chaotic tumbling. The For high viscosity drops λ = 14, unsteady behavior is
linescrossingthesymbolsreflectthevariationsintheaspectratio observedataspectratioaboveβ >1.2inboththemodel
ofthedropduringtherotation. and experiment. Experimentally, we observe a perturba-
tion to the oblate drop shape to slowly grow until reach-
ingachaotictumblingstate,inwhichthedirectionofro-
Of course in reality drop shape does not remain an el-
tation reverses chaotically. The transitions between the
lipsoidwithfixedaspectratio,andtheexperimentsshow
Taylor, tilted and tumbling states of a drop are given by
pronouncedshapeoscillationsinthecaseoflow-viscosity
lines in Figure 6.a. The largest stretching occurs while
drops (see Figure 5.a). This is due to the fact that the
the short axis of the oblate drop is aligned with the field
rotation period is comparable with the surface tension
(seeFigure5.bat0.7sand2.37s). Atthispoint,thedrop
relaxation time aµ /γ (γ being the interfacial tension,
ex can either break up or continue to rotate.
µ the suspending fluid medium, and a the initial drop
ex Lowviscositydrops(λ=1)exhibitsteadyoscillations
radius). However, in this case the main axis is found to
between a prolate and a tilted oblate shapes, see Figure
oscillate around a defined tilted angle and hence this be-
5.a. These shape oscillations can be explained by con-
havior qualitatively corresponds to the swinging mode.
sidering the dynamics of the induced dipole and shape.
High-viscositydropsremainnearlyundeformedthrough-
The lower fluid viscosity allows the drop to be easily de-
outoneperiodofrotationandtumblelikerigidellipsoids
formedbytheelectricfieldactingontheinducedsurface
(see Figure 5.b)
charge. As the interface charges, the drop is squeezed
intoanoblateshape. Theaccumulatedsurfacechargero-
a)
tates due to the electric torque until the induced dipole
is nearly aligned with the field. In this dipole configu-
ration, the drop is pulled into a prolate shape until the
b)
surfacechargerelaxes. Thecyclerepeatsbytheinterface
recharging and deforming into an oblate shape. At fields
0.0 s 0.3 s 0.7 s 1.0 s 1.3 s 1.7 s 2.0 s 2.3 s 2.7 s strengthswellabovethetransitiontounsteadybehavior,
the amplitude and frequency of oscillation increase until
FIG. 5: Examples of oscillatory and tumbling drop behavior.
thedropstretchesintoahighlydeformedprolate”dumb-
a) λ=1, E0=9.9kV/cm, a=1.8mm. b) λ=14, E0=9.7kV/cm,
a=3.0mm bell” shape, which then either undergo breakup or col-
lapses back into itself, disrupting its rotational velocity.
Our “fluid ellipsoid” model for drop electrorotation Aftercollapsing,theoscillationswillbegintobuildagain
identifies drop viscosity as a key control parameter for in the same direction as before.
drop behavior. Next we experimentally test the model The different unsteady behaviors are attributed to the
andshowthatitqualitativelycapturesthevarietyofdrop interplay of shape and dipole dynamics. The more vis-
responses. cousthedropfluid,thehigherresistancetofluidmotion.
Experiment: Weaklyconductingfluidsareusedforthe Accordingly, high viscosity drop tend to behave as rigid
drop(siliconeoil)andcontinuousphase(castoroil). This particles and tumble in the electric field. They deform
experimental system is characterized by conductivity ra- very little during the dipole rotation. In contrast, low
tio R = 0.03 and permittivity ratio S = 1.8. To in- viscositydropsundergolargedeformationsseenasshape
4
4 oscillations. The limited deformation of high viscosity
dropsallowsforthefixedshapemodeltoaccuratelypre-
3.5
dict the transition to unsteady behavior.
β π/2
3 3 Br
Q 2 In conclusion, in this Letter we report novel nonlin-
d E/E 2.5−π 1 0θ Un eDarropdlyentatmuimcsbloinfga,dsrhoappleetoisnciullnatifioornms, DanCdeclehcatorticicfireoldtas-.
el
c Fi 2 tions occur under creeping flow conditions, where non-
tri linearphenomenaarerare. Wehaveconstructedaphase
Elec 1.5 −π/2 Ti digram of drop behaviors as a function of viscosity ratio
1 and field strength. The experimental data qualitatively
agrees with a theory which models the drop as a fluid el-
0.5
Ta lipsoid. Thefavorablecomparisonissurprisinggiventhe
many simplifications in the theory and encouraging for
0
1 2 3 4 5 6 futureeffortstobuildamorerefinedandaccuratemodel.
Drop Diamater (mm)
Thenonlineardropphysicscouldinspirenewapproaches
4
in electromanipulation for small-scale fluid and particle
3.5 Br motion in microfluidic technologies.
3 β π/2
Q 3
d E/E2.5 2 1 θ Un
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[13] See supplemental material at link for movies and details
0.5 Ta
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