A slender-body theory for low-viscosity drops in shear-flow
between parallel walls
Citation for published version (APA):
Janssen, P. J. A., Anderson, P. D., & Loewenberg, M. (2010). A slender-body theory for low-viscosity drops in shear-flow between parallel walls. Physics of Fluids, 22(4), 042002-1/10. [042002].
https://doi.org/10.1063/1.3379624
DOI:
10.1063/1.3379624 Document status and date: Published: 01/01/2010 Document Version:
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A slender-body theory for low-viscosity drops in shear flow
between parallel walls
P. J. A. Janssen,1,a兲 P. D. Anderson,1,b兲 and M. Loewenberg2,c兲
1
Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
2
Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286, USA 共Received 20 May 2009; accepted 11 January 2010; published online 29 April 2010兲
A slender-body analysis is presented for the deformation and break-up of a highly confined and highly elongated low-viscosity drop in shear flow between two parallel walls that are separated by a distance less than the drop length. The analysis is simplified by the assumption that the drop has a circular cross section. The results show that confinement enhances the alignment of a low-viscosity drop with the imposed flow, thereby reducing its deformation and increasing the critical flow strength required for breakup. In the intermediate limit, where the wall separation is small compared with the drop length but large compared with its width, the dynamics can be related to that of an unconfined drop at a shear rate reduced by a factor of
冑
3. Under these corresponding conditions, the drop length and cross-section profile are the same for both cases, whereas the centerline deflection of the confined drop is reduced relative to the unconfined case by冑
3. In the intermediate limit of wall separations, the critical flow strength for a confined drop is冑
3 times larger than that for an unconfined drop. © 2010 American Institute of Physics.关doi:10.1063/1.3379624兴I. INTRODUCTION
Predicting the morphology of a polymer blend is impor-tant because the macroscopic properties are governed to a large extent by the microstructure.1The two parameters most studied are the capillary number Ca=Ea/␥and the viscos-ity ratio=i/, where andi are the viscosities of the matrix and drop-phase fluids, respectively, E is the imposed shear rate, ␥ is the coefficient of interfacial tension, and
4
3a3is the drop volume. Recently, there has been focus on
morphology development in confined geometries as found in microfluidic devices.2,3An interesting observation relates to the breakup behavior of drops in shear flow when confined between two parallel walls. Experiments show that confine-ment promotes the breakup of drops with high-viscosity ra-tios 共i.e., lower critical capillary number兲 but hinders the breakup of drops with low-viscosity ratios.4 The former ob-servation may be explained by the fact that high-viscosity drops in unbounded shear flow are stabilized by the rota-tional component of the flow.5 The presence of confining walls hinders the rotational component of the flow inside of the drops; thus drops maintain more alignment with the straining component of flow which promotes breakup. An explanation for the suppressed breakup of the low-viscosity drops is, however, lacking.
Boundary-integral methods have been developed to model the dynamics of drops confined between parallel walls under creeping flow conditions;6–8 an example is shown in Fig. 1, where the length and orientation of a confined and
unconfined low-viscosity drop in shear flow are depicted. The results indicate that confinement reduces drop deforma-tion and enhances their alignment with the imposed flow. Boundary-integral methods are perhaps the best available nu-merical method for simulations of deformable drops in Stokes flows, but even they have difficulty describing the breakup of low-viscosity drops due to the characteristically long and slender drop shapes and pointed tips that form un-der conditions close to breakup. Slenun-der-body theories have been developed to describe the dynamics and breakup of drops in the asymptotic limit →0.9–13 In this paper, we develop a slender-body theory for a low-viscosity drop con-fined between two planar boundaries in the shear flow gen-erated by the tangential motion of these boundaries.
II. PROBLEM FORMULATION A. Slender-body approximation
We consider a drop with a surfactant-free interface con-fined between two parallel walls separated by a distance 2W for the case Ⰶ1. Creeping flow conditions are assumed. The walls undergo a relative tangential motion which gener-ates the imposed shear flow,
u⬁=共Ey,0,0兲, 共1兲
as shown in Fig. 2, where the coordinates and drop-shape parameters are defined. Deformable drops will migrate to the center of the gap between two parallel walls;14,15herein, we assume that the drop is initially centered, as shown in the figure. The drop is assumed to have a slender shape with half-length l and centerline deflection共x,t兲Ⰶl, as sketched in Fig. 2. By symmetry, is an odd function of x with
⬎0 for x⬎0. Following Hinch and Acrivos,12
we invoke the ad hoc assumption that the drop has a circular cross
a兲Present address: Department of Chemical and Biological Engineering,
Uni-versity of Wisconsin, Madison, Wisconsin 53703-1691. Electronic mail: pjjanssen@wisc.edu.
b兲Electronic mail: pda@tue.nl.
c兲Electronic mail: michael.loewenberg@yale.edu.
PHYSICS OF FLUIDS 22, 042002共2010兲
section with local radius R共x,t兲, in order to simplify the analysis. By the assumption that the drop is slender, RⰆl, by symmetry, R is an even function of x, and R = 0 at the ends of the drop x =⫾l. Herein, we assume highly confined condi-tions, where the wall separation is small compared with the drop length,
WⰆ l. 共2兲
We define fixed Cartesian coordinates 共x,y,z兲 with corre-sponding unit vectors ex, ey, ez; the center of the channel is at y = 0. Following Hinch and Acrivos,12we also use a slightly skew local coordinate system共s,yⴱ, zⴱ兲 centered on the drop axis, where yⴱ= y −共x兲, zⴱ= z, and s⬇x is distance along the centerline which is not quite aligned with the x-axis. We define the local cylindrical coordinates共x,r,兲,
yⴱ= y −共x兲 = r sin, zⴱ= z = r cos, 共3兲 where r = 0 coincides with the centerline, and= 0 coincides with the z-axis.
B. Scaling analysis
A scaling analysis reveals certain important features of the problem, supports the use of the slender-body approxi-mation, and provides the relevant length and time scales for nondimensionalization. The scaling arguments presented be-low show that a be-low-viscosity drop in shear fbe-low becomes highly elongated at high shear rates, and nearly aligned with the imposed velocity. Thus, the drop orientation has only a small projection onto the straining component of the im-posed flow. However, this small projection of the drop ori-entation onto the straining component of the flow, or equiva-lently, its slight misalignment with respect to the imposed velocity field,共x兲, is an essential feature of the drop dynam-ics. A slender low-viscosity drop in shear flow generates neg-ligible normal stresses on its interface if it is completely aligned with the imposed velocity, i.e.,⬅0. High elonga-tion relies on the normal stresses generated by the drop mis-alignment to stretch the drop.
Pressure p generated in the continuous phase fluid by the presence of the elongated and nearly horizontal drop scales
asE, where⬃/l is the small inclination angle of the drop with respect to the flow direction, or equivalently, its projection onto the straining axis of the flow. Under the assumption that the centerline deflection scales with the cross-sectional radius,⬃R, we have p⬃ER/l. Balancing the pressure and the O共␥/R兲 capillary stresses, we have alR−2⬃Ca. By conservation of drop volume, the length and cross section of a drop are related by
lR2⬃ a3, 共4兲
thus, we obtain
l/a ⬃ Ca1/2, 共5兲
which indicates that low-viscosity drops 共or bubbles兲 in shear flow are much shorter than in straining flow where l/a⬃Ca2 under strong flow conditions, CaⰇ1.10,12
The flow inside of the slender drop is approximately unidirectional and quasisteady, thus
冋
1 r r冉
r uˆx r冊
+ 1 r2 2uˆ x 2册
= pˆ⬘
, 共6兲where uˆ = uˆx共x,r,兲exand pˆ = pˆ共x兲 are the velocity and pres-sure inside of the drop, and the prime denotes an x-derivative. By the continuity of the velocity at the drop interface, we have uˆx⬃E, and by the assumption that
⬃R, we have ⇑ ⇓ ⇑ ⇓ y = W y =−W x y z R2= z2+ (y− η)2 y = η (x, t) r
FIG. 2. Slender drop in shear flow generated by tangential motion of equi-distant parallel confining walls. Drop has a circular cross section with local radius R共x,t兲 and centerline deflection共x,t兲. Far-field wall-correction ve-locity,. 0 5 10 15 20 0 1 2 3 4 5 6 0 5 10 15 20 0 0.5 1 1.5 2
l
a
a
η(l)
E
t
E
t
(a)
(b)
FIG. 1. Boundary-integral simulations共Ref.8兲 for evolution of initially spherical drops with radius a in shear flow =0.01 and Ca=0.8; 共a兲 half-length l of
drops and stationary shapes共trace of drop interface in shear plane兲 including boundary walls for confined case, 共b兲 alignment, a/共l兲 共see Fig.2兲; isolated drop
uˆx⬃ ER. 共7兲 In flows sufficiently strong to induce breakup, variations of the internal pressure ␦pˆ become comparable to capillary stresses␥/R. Inserting these estimates into Eq.共6兲, and scal-ing x by l and r by R, we obtain Ca共l/a兲⬃1. Combining this result with Eq.共5兲, we obtain an estimate for the critical capillary number
Cacr⬃ −2/3, 共8兲
and accordingly introduce the flow strength parameter
G = Ca2/3. 共9兲
Henceforth, G = O共1兲 is assumed. Thus, from Eqs. 共4兲 and
共5兲, we obtain
R/a ⬃ 1/6, l/a ⬃ −1/3. 共10兲
The result indicates that l/R=O共−1/2兲 which supports the
use of the slender-body approximation forⰆ1.16According to the scaling estimates 共10兲, we shall nondimensionalize lengths in the flow direction by a−1/3 and lengths in the
cross-flow directions by a1/6. The time for convection over
the length of the drop, l/uˆx, sets the time scale for the drop dynamics, thus we shall nondimensionalize time by E−1−1/2, according to Eqs.共7兲and共10兲. We thus introduce the dimen-sionless variables, x ¯ = x a−1/3, ¯ =r r a1/6, t¯ = t E−1−1/2, 共11兲
and the dimensionless drop-shape and confinement param-eters R ¯ = R a1/6, ¯ = a1/6, l¯ = l a−1/3, W¯ = W a1/6. 共12兲 Given the scales for the drop-shape and confinement param-eters, the assumption of highly confined conditions 共2兲 is equivalent to
W¯ Ⰶ −1/2. 共13兲
From the variables共11兲, it follows that the components of the dimensionless velocity and stress are defined
u ¯x= ux Ea1/6, u¯r= ur Ea2/3, 共14兲 u ¯= u Ea2/3, ¯ =p p E1/2, ¯rr= rr E1/2, ¯= E1/2, ¯xx= xx E1/2, 共15兲 ¯r= r E1/2, ¯xr= xr E, ¯x= x E. 共16兲 By the symmetry of the stress tensor, we have
¯rx=¯xr, ¯x=¯x, ¯r=¯r. 共17兲 By continuity of the velocity and the normal stress balance, the velocity and pressure in the drop-phase fluid have the same characteristic scales as the corresponding quantities in the external phase fluid, thus we define
u ¯ˆx= uˆx Ea1/6, ¯ˆ =p pˆ E1/2. 共18兲 However, the components of the viscous stress tensor in the drop-phase fluid are O共兲 weaker than the corresponding quantities in the external phase fluid. Thus, the stress tensor in the drop-phase fluid is dominated by the isotropic contri-bution of pressure and to O共兲 can be approximated as
ˆ = − p¯ˆI. 共19兲
C. External flow
Following Hinch and Acrivos,12the disturbance velocity associated with the slender low-viscosity drop is approxi-mated by distribution of low-order Stokes flow singularities along the centerline of the drop. Thus, the total velocity field at a point x outside of the drop is expressed in unscaled variables as u共x兲 = u⬁共x兲 +
冕
−l +l冋
3 2f共s兲Txy共xˆ;W兲 + 1 2g共s兲S共xˆ;W兲 +1 2h共s兲Dy共xˆ;W兲册
ds, 共20兲where u⬁ is the undisturbed flow 共1兲, x0=共x0, y0, z0兲 is a
singular point, and xˆ = x − x0. Given that the singularities are
located on the drop centerline, we have x0=共s,共s兲,0兲,
where −lⱕsⱕ+l is the distance along the centerline. The singular velocities Txy, S, and Dy are, respectively, the xy-column of the stresslet, a point source, and the y-column of the source-dipole. These solutions decay away from the drop and satisfy no-slip boundary conditions at y =⫾W. The stresslet, source, and source-dipole distributions on the drop centerline are 4f, 2g, and 2h. Note that unscaled vari-ables are used in Eq.共20兲and in the remainder of this sub-section; the different scales for the velocity components given by Eq. 共14兲 make the use of dimensionless variables inconvenient.
The singular velocities are conveniently expressed as
v共xˆ;W兲 = v共0兲共xˆ兲 + v2W共xˆ;W兲, 共21兲
where v is any one of the singular velocities Txy, S, or Dy. Here, v共0兲共xˆ兲 is the free-space singular velocity field for an unbounded fluid and v2W共xˆ;W兲 is the nonsingular two-wall velocity field correction such that v共0兲共xˆ兲+v2W共xˆ;W兲 satisfies no-slip boundary conditions at y =⫾W. Herein, we employ the superposition approximation of the two-wall velocity field correction
v2W共xˆ;W兲 ⬇ vW共xˆ;+ W兲 + vW共xˆ;− W兲, 共22兲
where vW共xˆ; ⫾W兲 is the one-wall velocity field correction such that v共0兲共xˆ兲+vW共xˆ; ⫾W兲 satisfies the no-slip boundary
condition at y =⫾W. The superposition approximation is rig-orous for wall separations large compared with the width of the drop, WⰇR, which is compatible with the assumption that the drop is long compared with the wall separation共2兲 given that the drop is slender. Often, however, approximation
共22兲 is accurate even when this assumption is not satisfied.6,8,17,18 In general, the superposition approximation is reliable provided that the wall correction velocity
v2W共xˆ;W兲 is dominated by the primary velocity v共0兲共xˆ兲.
The one-wall correction field corresponding to a given velocity field v共0兲共xˆ兲 is given by17,19
vW共xˆ; ⫾ W兲 = P共R0+ HR1+ H2R2兲 · v共0兲共xˆ兲. 共23兲
Here, the operators are defined
R0= − Iy− 2yˆⵜ ey+ yˆ2ⵜ2I,
共24兲
R1= − 2ⵜ ey+ 2yˆⵜ2I, R2=ⵜ2I,
where I is the identity tensor, Iy= I − 2eyey, H = y0⫿W, and
yˆ = y − y0. The mirror operator P is defined
P共w兲共x,y,z兲 = Iy· w共x, ⫾ 2W − y,z兲. 共25兲 The requisite singularity solutions v共0兲共xˆ兲 are
Txy共0兲共xˆ兲 = 共x − x0兲共y − y0兲xˆ 兩xˆ兩5 , S 共0兲共xˆ兲 = xˆ 兩xˆ兩3, 共26兲 Dy共0兲共xˆ兲 = ey 兩xˆ兩3− 3 共y − y0兲xˆ 兩xˆ兩5 .
Based on the superposition approximation共22兲, we can ex-press external velocity共20兲as
u共x兲 = u⬁共x兲 + u共0兲共x兲 + uW共x;+ W兲 + uW共x;− W兲, 共27兲
where we separately define the contributions to the distur-bance velocity from the distributions of free-space singulari-ties and the corresponding one-wall correction velocity
u共0兲共x兲 =
冕
−l +l冋
3 2f共s兲Txy 共0兲共xˆ兲 +1 2g共s兲S 共0兲共xˆ兲 +1 2h共s兲Dy 共0兲共xˆ兲册
ds, 共28兲 uW共x; ⫾ W兲 =冕
−l +l冋
3 2f共s兲Txy W共xˆ; ⫾ W兲 +1 2g共s兲S W共xˆ; ⫾ W兲 +1 2h共s兲Dy W共xˆ; ⫾ W兲册
ds. 共29兲The kernel functions Txy共0兲, S共0兲, Dy共0兲, Txy W , SW, and Dy W are defined by Eqs.共23兲–共26兲. D. Internal flow
By the disparity of the length scales in the flow and cross-flow directions, a lubrication approximation is appro-priate for the flow inside of the slender drop; accordingly, the quasisteady unidirectional approximation 共6兲 applies.12 The internal velocity field is obtained by integrating Eq.共6兲and enforcing continuity of the tangential component of the ve-locity at the drop interface. It is convenient to decompose the internal velocity field as
u ¯ˆx= u¯ˆp+ u¯ˆt, 共30兲 where u ¯ˆp= − 1 4共R¯ 2− r¯2兲p¯ˆ
⬘
, 共31兲is the pressure-driven component of the flow that obeys no-slip boundary conditions at the drop interface. The prime denotes a derivative with respect to x¯. The tangentially driven component of the velocity, u¯ˆt共r¯,兲, is the homoge-neous solution of Eq. 共6兲 共Laplace’s equation兲 that satisfies the boundary condition
u
¯ˆt共R¯,兲 = u¯x共R¯,兲, 共32兲
where u¯x共R¯,兲 is the x-component of the external velocity field evaluated on drop interface. The cross-sectional area average of the pressure-driven velocity component is
具u¯ˆp典 = − R ¯2
8¯ˆp
⬘
. 共33兲Using Eq.共A4兲, derived in Appendix A, we obtain the cross-sectional area average of the tangentially driven velocity as the perimeter average of the x-component of the external velocity, 具u¯ˆt典 = 具u¯x典R= 1 2
冕
0 2 u ¯x共R¯,兲d. 共34兲The axial volume flux inside the drop, q¯ =R¯2具u¯ˆ
x典, is gov-erned by the conservation relation
2R¯ R¯˙ + q¯
⬘
= 0, 共35兲where the dot denotes a derivative with respect to t¯. Integrat-ing this relation with symmetry condition q¯共0兲=0 yields
具u¯ˆx典 = − 2 R ¯2
冕
0 x ¯ R ¯ R¯˙ dx¯. 共36兲Combining this result with Eqs. 共30兲, 共33兲, and 共34兲, and integrating yields the internal pressure,
p ¯ˆ = p¯ˆ0共t¯兲 − 8
冕
0 x ¯ R ¯−2关具u¯ˆ x典 − 具u¯x典R兴dx¯, 共37兲where the uniform pressure p¯ˆ0共t兲 is determined from the con-straint imposed by the conservation of drop volume,
3 4a3
冕
−l +l R2dx =3 4冕
−l¯ +l¯ R ¯2dx¯ = 1. 共38兲E. Boundary conditions on interface
Under the assumption that surface tension gradients are absent, the stress balance at the interface is given by
关−ˆ兴 · n = 2␥n. 共39兲
Here,andˆ are the stress tensors in the continuous- and drop-phase fluids; n andare the outward normal vector and mean curvature on the drop interface which are given by
n =关− 共R
冑
⬘
+⬘
sin兲,1,0兴1 +共R
⬘
+⬘
sin兲2 , 2=1
R
冑
1 + R⬘
2− R⬙
. 共40兲Using the dimensionless variables共11兲 and 共12兲, we obtain the approximations
n =关− 1/2共R¯
⬘
+¯⬘
sin兲,1,0兴 + O共兲,共41兲
2¯ = R ¯−1+ O共兲,
where¯ = a1/6is the dimensionless curvature.
The leading-order stress balance at the drop interface is obtained by inserting the pressure-dominated approximation of the drop-phase stress tensor 共19兲 and the geometric ap-proximations 共41兲 into Eq. 共39兲. Expressed in component form and in dimensionless variables共11兲–共18兲, we obtain,
¯xr= O共兲, ¯rr=共GR¯兲−1− p¯ˆ + O共兲,
共42兲
¯r−共R¯
⬘
+¯⬘
sin兲¯x= O共兲, where G is the flow strength parameter共9兲.Mass conservation at the drop interface requires the ki-nematic condition,
R˙ +˙ sin= u · n. 共43兲
In dimensionless variables with n given by Eq. 共41兲, this becomes
R
¯˙ +¯˙ sin= −共R¯
⬘
+¯⬘
sin兲u¯x+ u¯r, 共44兲 where dot denotes a derivative with respect to t¯ and prime denotes a derivative with respect to x¯.For a given drop shape, Eqs. 共27兲–共29兲,共37兲, and 共42兲 provide a closed set of equations for the velocity; the evolu-tion of the drop shape is governed by the kinematic condievolu-tion
共44兲subject to the volume constraint共38兲.
III. SOLUTION OF THE HYDRODYNAMIC PROBLEM
In this section, we evaluate the velocity and associated stress and present the results in terms of dimensionless vari-ables共11兲–共18兲and the dimensionless singularity distribution functions, f¯= f Ea21/3, g¯ = g Ea25/6, h¯ = h Ea3. 共45兲
The singularity distribution functions are then determined from the stress balance equations共42兲 and are presented at the end of this section.
A. External flow
It is convenient to express the velocity in terms of local coordinates共3兲relative to the drop centerline, noting that in these coordinates, the boundary walls are located at
yⴱ= W¯⫾ⴱ = ⫾ W¯ −¯共x¯兲. 共46兲
Accordingly, we express the external velocity field共27兲as
u共xⴱ兲 = u⬁共xⴱ兲 + u共0兲共xⴱ兲 + uW共xⴱ;W¯−ⴱ兲 + uW共xⴱ;W¯+ⴱ兲,
共47兲
and similarly express the pressure and stress, p ¯共xⴱ兲 = p¯⬁共xⴱ兲 + p¯共0兲共xⴱ兲 + p¯W共xⴱ;W¯ − ⴱ兲 + p¯W共xⴱ;W¯ + ⴱ兲, 共48兲 共xⴱ兲 =⬁共xⴱ兲 +共0兲共xⴱ兲 +W共xⴱ ;W¯−ⴱ兲 +W共xⴱ;W¯+ⴱ兲. 共49兲
In presenting the results below, we exploit the symmetry relation共17兲. In local cylindrical coordinates and dimension-less variables, the imposed shear flow共1兲and its associated stress are given by
u ¯x⬁=¯共x兲 + r¯ sin, u¯r⬁= u¯⬁= 0, 共50兲 ¯xr⬁= sin, ¯x⬁= cos, 共51兲 p ¯⬁=¯xx⬁=¯rr⬁=¯⬁ =¯r⬁= 0.
The contributions to the disturbance flow from the distribu-tions of free-space singularities, obtained from Eq.共28兲, are given by u ¯x共0兲= f¯ r ¯sin+ O共 log 兲, 共52兲 u ¯r共0兲= f¯¯
⬘
2r¯ + g ¯ r ¯ +冉
h ¯ r ¯2− f¯⬘
冊
sin+ f¯¯⬘
2r¯ cos 2 + O共 log 兲, 共53兲 u ¯共0兲= − h ¯ r ¯2cos+ f¯¯⬘
2r¯ sin 2+ O共 log 兲, 共54兲 where the neglected O共 log 兲 terms result from higher-order derivatives of the dimensionless singularity distribu-tions, f¯⬙
, g¯⬘
, h¯⬘
; finite-length effects enter at O共兲.Henceforth, second- and higher-order harmonic varia-tions inwill be omitted from the disturbance velocity field because the corresponding tractions with this angular depen-dence are incompatible with the assumed circular cross sec-tion of the drop.12The pressure and stress associated with the remaining terms in the velocity field共52兲–共54兲are
p ¯共0兲= −2f¯
⬘
r ¯ sin, ¯xr 共0兲= − f¯ r ¯2sin, 共55兲 ¯x共0兲 = f¯ r ¯2cos, ¯r 共0兲=再
4h¯ r ¯3 − f¯⬘
r ¯冎
cos, ¯xx共0兲= 4f¯⬘
r ¯ sin, ¯rr共0兲= − 2g¯ r ¯2 − f¯¯⬘
r ¯2 +再
2f¯⬘
r ¯ − 4h¯ r ¯3冎
sin, 共56兲 ¯共0兲=2g¯ r ¯2 + f¯¯⬘
r ¯2 + 4h¯ r ¯3sin,which are compatible with the ad hoc assumption that the cross section of the drop is circular. The pressure generated by the free-space singularities is entirely associated with the stresslet distribution because the free-space source and source-dipole are potential flows.
The leading-order contribution to the x-component of the disturbance flow from the wall-correction velocity, obtained from Eq.共29兲, is u ¯x W = f¯ ¯ sinr − 2W ¯ ⫾ ⴱ 4W¯⫾ⴱ2− 4r¯W¯⫾ⴱ sin+ r¯2+ O共W ¯ ⫾ ⴱ兲, 共57兲 where O共W¯ ⫾
ⴱ兲 terms are negligible by assumption 共13兲.
The result satisfies the no-slip boundary condition, i.e., u
¯x共0兲+ u¯xW⬅0 at y¯ⴱ
= r¯ sin= W¯⫾ⴱ; however, it is incompatible with the assumed circular cross section of the drop because it contains higher-order harmonic variation in. Higher-order harmonics are purged from u¯xW
by expanding formula共57兲in powers of r¯/W¯⫾ⴱ and truncating the series for terms beyond the first harmonic to obtain
u ¯xW= − f¯ 2W¯⫾ⴱ
冋
1 +1 2共r¯/W ¯ ⫾ ⴱ兲sin+ O共共r¯/W¯ ⫾ ⴱ兲2cos 2兲册
. 共58兲This approximation does not satisfy the no-slip boundary condition at y¯ⴱ= W¯⫾ⴱ but it is a rigorous approximation of the wall-correction velocity 共57兲 in the near field 共r¯⬇1兲, pro-vided that W¯ Ⰷ1. In any case, approximation共58兲is consis-tent with and required by the ad hoc assumption of a circular drop cross section that was used to eliminate the higher-order harmonics from the velocity field associated with the distri-bution of free-space singularities共52兲–共54兲.
We similarly obtain the leading-order contributions to the r- and -components of the disturbance flow from the wall-correction velocity, expanding the results for r¯/W¯⫾ⴱⰆ1, and keeping terms only up to the first harmonic variation in
, u ¯rW= f¯
⬘
¯rW 4 + f¯¯⬘
¯rW 8W¯⫾ⴱ +再
f¯⬘
2 + f¯¯⬘
2W¯⫾ⴱ冉
r ¯W2 8 − 1冊
+ ¯g W¯⫾ⴱ冉
r ¯W2 8 − 1冊
+ 3 4 h ¯ W¯⫾ⴱ2冉
r ¯W2 4 − 1冊
冎
sin, 共59兲 u ¯W =再
f¯⬘
2冉
1 − r ¯W2 2冊
− f¯¯⬘
2W¯⫾ⴱ冉
1 +¯rW 2 8冊
+ g ¯ W¯⫾ⴱ冉
3r¯W2 8 − 1冊
+3 4 h ¯ W¯⫾ⴱ2冉
3r¯W2 4 − 1冊
冎
cos, 共60兲where we define r¯W= r¯/W¯⫾ⴱ. In obtaining these results, O共W¯⫾ⴱ2兲 terms have been neglected which is permissible by assumption共13兲. The pressure and stress contributions asso-ciated with the wall-correction velocity共58兲–共60兲are
p ¯W =
冉
¯ +g f¯¯⬘
2冊
1 W¯⫾ⴱ2 + h ¯ W¯⫾ⴱ3 +再
− f¯⬘
2W¯⫾ⴱ2 +共g¯ + f¯¯⬘
兲 1 W¯⫾ⴱ3 + 3h ¯ 2W¯⫾ⴱ4冎
r ¯ sin, 共61兲 ¯xrW= − f¯ 4W¯⫾ⴱ2 sin, ¯xW= − f¯ 4W¯⫾ⴱ2 cos, 共62兲 ¯rW=再
− f¯⬘
4W¯⫾ⴱ2 + ¯g 2W¯⫾ⴱ3 + 3h ¯ 4W¯⫾ⴱ4冎
r ¯ cos, ¯xx W = − f¯⬘
W¯⫾ⴱ −共g¯ + f¯¯⬘
兲 1 W¯⫾ⴱ2 − h ¯ W¯⫾ⴱ3 −再
冉
g¯ +3f¯¯⬘
2冊
1 W¯⫾ⴱ3 + 3h ¯ 2W¯⫾ⴱ4冎
r ¯ sin, 共63兲 ¯rrW= f¯⬘
2W¯⫾ⴱ −冉
g¯ +f¯¯⬘
4冊
1 W¯⫾ⴱ2 − h ¯ W¯⫾ⴱ3 +再
f¯⬘
2W¯⫾ⴱ2 −冉
¯g 2+ 3f¯¯⬘
4冊
1 W¯⫾ⴱ3 − 3h ¯ 4W¯⫾ⴱ4冎
r ¯ sin, 共64兲 ¯W= f¯⬘
2W¯⫾ⴱ −冉
¯ +g f¯¯⬘
4冊
1 W¯⫾ⴱ2 − h ¯ W¯⫾ⴱ3 +再
f¯⬘
W¯⫾ⴱ2 −冉
3g¯ 2 + 3f¯¯⬘
4冊
1 W¯⫾ⴱ3 − 9h ¯ 4W¯⫾ⴱ4冎
r ¯ sin. 共65兲The wall-correction velocities for the source and source-dipole are not potential flows thus, all of the singularity dis-tributions contribute to the pressure associated with the wall-correction velocity共61兲.
1. Large wall separations
According to Eqs. 共58兲–共60兲, the wall correction pro-duces an O共1兲 velocity in the y-direction at large wall sepa-rations, W¯ Ⰷ1,
uW= u¯ y W⬁
ey+ O共r¯/W¯ 兲. 共66兲
This far-field flow is produced by the derivative of stresslet distribution on the drop centerline,
u ¯yW⬁= f¯
⬘
共x¯兲冕
−⬁ +⬁ 共s − x¯兲Txyy W 共x¯ − s;W¯ 兲ds = f¯⬘
2, 共67兲 where Txyy W 共q;W兲 =48qW4− 3q3W2 共q2+ 4W2兲7/2 . 共68兲The nondecaying contribution of the wall-correction velocity is a consequence of the highly confined conditions共13兲 as-sumed in our analysis. The far-field form of the wall-correction velocity 共66兲–共68兲 applies to wall separations in the intermediate limit,
1Ⰶ W¯ Ⰶ −1/2. 共69兲
At separations large compared with the drop length, W¯ Ⰷ−1/2, the wall-correction velocity vanishes as u¯W⬃W¯−2,
according to Eqs. 共23兲–共26兲. Equations 共61兲–共65兲 indicate that stresses associated with the wall-correction velocity de-cay asW⬃W¯−1 for W¯ Ⰷ1.
B. Internal flow
Collecting the contributions to the x-component of the external velocity from Eqs.共50兲,共52兲, and共58兲and combin-ing them accordcombin-ing to the decomposition共47兲yields
u ¯x=¯ − f¯w1 2 +
冋
¯ +r f¯ r ¯冉
1 − r ¯2w 2 4冊
册
sin, 共70兲 where we define wk=冉
1 W¯ −¯冊
k +冉
1 − W¯ −¯冊
k . 共71兲Evaluating the perimeter average共34兲of the external veloc-ity共70兲and inserting the result into Eqs.共36兲and共37兲yields the internal pressure field,
p ¯ˆ = p¯ˆ0共t¯兲 + 8
冕
0 x ¯冋
R¯−2冉
¯ −f¯w1 2冊
+ 2R ¯−4冕
0 x ¯ R ¯ R¯˙ dx¯册
dx¯ , 共72兲where the value of p¯ˆ0共t¯兲 is determined by the volume con-straint共38兲.
The unidirectional internal velocity field is not needed for obtaining the evolution equations because pressure domi-nates viscous stresses inside the drop. For completeness, we present the solution for the internal velocity field in Appendix B.
C. Singularity distributions
The singularity distributions f¯, g¯, and h¯ are determined from the stress balance at the drop interface. Combining the stress contributions given in Eqs.共51兲,共55兲,共56兲, and 共62兲–
共65兲according to the decomposition共49兲and inserting them into the stress balance equations共42兲and solving the linear system yields f¯= R¯2D +1 −1, f¯
⬘
=关2R¯⬘
− 4¯⬘
R w3兴R¯D+1−2, 共73兲 g ¯ =冋
R¯ 2 2 共p¯ˆ − 共GR¯兲 −1−¯⬘
兲 − 2f¯R¯⬘
Rw3 D2 + f¯⬘
R¯冉
Rw1 2 − D+1Rw3 D2冊
册
D3−1, 共74兲 h ¯ = 关f¯共R¯⬘
−¯⬘
Rw3兲 − g¯Rw3兴R¯D2−1, 共75兲 where the internal pressure is given by Eq. 共72兲. Here, we have definedRwk= 2−kR¯kwk, 共76兲
with wkdefined by Eq. 共71兲, and D⫾1= 1⫾ Rw2, D2= 1 + 3Rw4,
共77兲
D3= 1 + 2Rw2− 4D2 −1
Rw32 .
1. Large wall separations
Expanding Eqs.共72兲–共77兲for R¯ /W¯ Ⰶ1, we obtain p ¯ˆ = p¯ˆ共0兲+ p¯ˆ共2兲共R¯/W¯ 兲2+ O共R¯/W¯ 兲4, 共78兲 f¯= f¯共0兲
冋
1 −1 2共R¯/W¯ 兲 2册
+ O共R¯/W¯ 兲4, 共79兲 f¯⬘
= f¯共0兲⬘关1 − 共R¯/W¯ 兲2兴 + O共R¯/W¯ 兲4, g ¯ = g¯共0兲关1 − 共R¯/W¯ 兲2兴 +冋
R¯ 2¯ˆp共2兲 2 +¯ R¯ R ¯⬘
册
共R¯/W¯ 兲2 + O共R¯/W¯ 兲4, 共80兲 h ¯ = h¯共0兲冋
1 −1 2共R¯/W¯ 兲 2册
+ O共R¯/W¯ 兲4. 共81兲Here, f¯共0兲, g¯共0兲, h¯共0兲are the leading-order singularity distribu-tions for R¯ /W¯ Ⰶ1,
f¯共0兲= R¯2, f¯共0兲⬘= 2R¯ R¯
⬘
, 共82兲 g ¯共0兲=R ¯2 2 共p¯ˆ 共0兲−共GR¯兲−1−¯⬘
兲, h¯共0兲= R¯3R¯⬘
,and p¯ˆ共0兲, p¯ˆ共2兲are the first two terms in the expansion for the internal pressure, p ¯ˆ共0兲= p¯ˆ0共t¯兲 + 8
冕
0 x ¯冋
¯ R¯−2+ 2R¯−4冕
0 x ¯ R ¯ R¯˙ dx¯册
dx¯ , 共83兲 and p ¯ˆ共2兲= − 8 R ¯2冕
0 x ¯ ¯ dx¯ . 共84兲As discussed below Eq.共69兲, the stresses associated with the wall-correction velocity vanish for W¯ Ⰷ1, thus the leading-order singularity distributions共82兲and internal pressure共83兲 correspond to those for an isolated drop共i.e., R¯/W¯ =0兲 and agree with the analysis of Hinch and Acrivos.12
By the symmetry depicted in Fig.2and discussed in Sec. II A, we have p¯ˆ共2兲⬍0 and ¯ R¯
⬘
⬍0. Accordingly, Eqs.共78兲–共81兲indicate that for a given drop shape, the presence of the boundaries reduces the strength of all singularity dis-tributions and the internal pressure induced by the flow.
IV. EVOLUTION EQUATIONS
Collecting the contributions to the velocity, according to the decomposition共47兲and inserting the result into the kine-matic condition 共44兲, we obtain the evolution equations for the drop-shape parameters,
R ¯˙ = −¯ R¯
⬘
−R ¯¯⬘
2 + f¯ R ¯共R¯⬘
Rw1+⬘
Rw2兲 + f¯⬘
Rw1 2 + g ¯ R ¯, 共85兲 ¯˙ = −¯¯⬘
− R¯ R¯⬘
+共2u¯y W⬁ − f¯⬘
兲 − f¯ R ¯冉
R ¯⬘
D −1−¯⬘
Rw3 2冊
−¯g R ¯共2Rw1− Rw3兲 + h ¯D4 R ¯2 , 共86兲where dot denotes a derivative with respect to t¯ and prime denotes a derivative with respect to x¯. The singularity distri-butions f¯, g¯ and h¯ are given by Eqs.共73兲–共75兲, Rwkis defined by Eq.共76兲, D⫾1is defined by Eq.共77兲, and
D4= 1 − 3Rw2+ 3Rw4. 共87兲
The nonvanishing component of the wall-correction velocity, u
¯yW⬁ is given by Eq.共67兲. Equations 共85兲 and 共86兲 enforce conservation of drop volume, given that the uniform pressure p
¯ˆ0共t¯兲 in Eq.共72兲is determined by the volume constraint共38兲. Equations 共85兲 and 共86兲 were integrated using the nu-merical method described by Hinch and Acrivos12in Sec. IV of their paper. At a given wall separation, the flow strength parameter G was increased by small steps, integrating the
evolution equations in time until a stationary state was achieved. The stationary state was used as the initial condi-tion for the next larger value of G. The flow strength param-eter was increased until the time required to reach stationary state began to diverge, signaling G⬇Gcr; we did not attempt
to accurately determine the critical flow strength parameter by extrapolation.12,20 We found it convenient to expand the coefficient functions in the evolution equations in powers of 1/W¯ . Only even powers are generated because the drop is centered between the two walls. Numerically converged re-sults for W¯ ⱖ4 are obtained by retaining terms up to O共1/W¯4兲. The results are discussed in Sec. IV B.
A. Intermediate limit of wall separations
In the limit of large wall separations, R¯ /W¯ →0, evolu-tion equaevolu-tions共85兲and共86兲reduce to
R ¯˙ = −¯ R¯
⬘
− R¯¯⬘
−共2G兲−1+ R ¯ 2¯ˆp 共0兲, 共88兲 ¯˙ = −¯¯⬘
− R¯ R¯⬘
, 共89兲where p¯ˆ共0兲is given by Eq.共83兲with p¯ˆ0共t¯兲 determined by the volume constraint共38兲. These equations are obtained by sub-stituting the singularity distributions 共82兲and internal pres-sure共83兲for an isolated drop and using the limiting results wk→0, D⫾→1, D4→1. Given the highly confined condi-tions共13兲assumed in our analysis, Eqs.共88兲and共89兲 corre-spond to the intermediate limit of wall separations共69兲.
For an isolated drop, the evolution equations, derived by Hinch and Acrivos,12 are
R ¯˙ = −¯ R¯
⬘
− R¯¯⬘
−共2G兲−1+ R ¯ 2¯ˆp 共0兲, 共90兲 ¯˙ = −¯¯⬘
− 3R¯ R¯⬘
, 共91兲where p¯ˆ共0兲 given by Eq. 共83兲. These equations cannot be recovered from Eqs.共85兲and共86兲because of the nonvanish-ing component of the wall-correction velocity, u¯Wy⬁, that ap-pears in Eq. 共86兲. The evolution equations for an isolated drop共90兲and共91兲, corresponding to W¯ →⬁, differ from the evolution equations共88兲 and 共89兲, corresponding to the in-termediate limit of wall separations共69兲. The dynamics of an isolated drop are only recovered at separations that are large compared with the drop length, W¯ Ⰷ−1/2.
There is, however, a simple rescaling transformation by which the evolution equations for intermediate wall separa-tions共69兲become identical to the evolution equations for an isolated drop. The appropriate transformation is
¯ = 3−1/2˜ , t¯ = 31/2t˜, G = 31/2G˜ , p¯ˆ = 3−1/2˜ˆ ,p 共92兲 with x¯ and R¯ unchanged. Transforming Eqs.共83兲,共88兲, and
共89兲accordingly yields R ¯˙ = −˜ R¯
⬘
− R¯˜⬘
−共2G˜ 兲−1+ R ¯ 2˜ˆp 共0兲, 共93兲 ˜˙ = −˜˜
⬘
− 3R¯ R¯⬘
, 共94兲 p ˜ˆ共0兲= p˜ˆ0共t¯兲 + 8冕
0 x ¯冋
˜ R¯−2+ 2R¯−4冕
0 x ¯ R ¯ R¯˙ dx¯册
dx¯ , 共95兲 where dot denotes a derivative with respect to t˜ and prime denotes a derivative with respect to x¯.1. Discussion
The transformed evolution Eqs. 共93兲–共95兲 are identical to the evolution Eqs.共83兲,共90兲, and共91兲for an isolated drop in shear flow. According to Eq.共92兲, the dynamics of a con-fined drop with intermediate wall separation共69兲 and flow strength parameter G =
冑
3G˜ corresponds to that of an isolated drop in a weaker flow, with flow strength parameter G = G˜ . Under these “corresponding conditions,” the drop cross-section profile R¯ 共x¯兲 and length are the same for the two cases. However, the centerline deflection profile¯共x¯兲 for the confined drop is less than that of the isolated drop by a factor of冑
3. Moreover, the confined drop evolves more slowly than the isolated drop by the same factor under these correspond-ing conditions.Beyond a critical flow strength parameter, Gcr, stable
stationary states cease to exist. According to the definition of corresponding conditions given above, the critical flow strength parameter for a confined drop with intermediate wall separation共69兲 is given by
Gcr=
冑
3Gcr共0兲, 共96兲where Gcr共0兲⬇0.0541 is the critical flow strength parameter for an isolated drop in an unbounded shear flow, as computed by Hinch and Acrivos.12 The critical drop length 共length at the critical flow strength兲 for confined drops with intermedi-ate wall separation is the same as the critical length for iso-lated drops.
For given flow strength, a confined drop is shorter and more aligned with the velocity than an unconfined drop at steady-state. Here, alignment is characterized by¯共l¯兲−1. This
statement follows from the above results for corresponding conditions and the assumption that the stationary drop length and alignment increase monotonically with flow strength.
For wall separations in the intermediate range共69兲, the leading-order effect of confinement results from the convec-tion by the nondecaying component of the wall-correcconvec-tion velocity, u¯yW⬁
. According to Eqs. 共67兲, 共79兲, and 共82兲, we have
u
¯yW⬁= R¯ R¯
⬘
. 共97兲Accordingly, u¯yW⬁
is an odd function of x¯ with u¯yW⬁⬍0 for x
¯⬎0, as shown schematically in Fig.2. We conclude that the far-field wall-correction velocity rotates a drop, enhancing its alignment, and thereby reducing its projection onto the straining component of the imposed flow and reducing its length at a given flow strength.
B. Numerical results
The results that we obtained by numerical integration of Eqs. 共85兲 and 共86兲 are depicted in Fig. 3 in terms of the transformed variables 共92兲. As explained above, in these variables the limiting results for R¯ /W¯ →0 correspond to in-termediate wall separations 共69兲 with flow strength param-eter given by G =
冑
3G˜ and to an isolated drop with G=G˜ . The results shown in Fig.3indicate that the evolution equa-tions for intermediate wall separaequa-tions共93兲–共95兲are accurate for W¯ ⱖ10. The terminal points of curves shown in Fig. 3correspond approximately to G⬇Gcr, as discussed in Sec.
IV. Accordingly, the results indicate that the critical flow strength parameter exceeds the limiting result共96兲for finite values of wall separation.
The boundary-integral simulations depicted in Fig. 1
correspond to conditions characterized by G˜ =0.0214. Re-sults are shown for an unconfined drop and a confined drop with W¯ =2.4. The shape profiles depicted in Fig.1共a兲indicate that the drops are not slender, therefore our slender-body description does not apply; a quantitative comparison would require smaller values of, with G=O共1兲 and thus Ca⬃−2/3, but this is computationally impractical. The
re-sults of our boundary-integral simulations are nevertheless qualitatively consistent with the prediction that, for a given flow strength, confined drops are more aligned with the im-posed flow and therefore shorter than isolated drops.
0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.2 0.4 0.6 0.8 1 1.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
¯l
η
˜
¯
l
−1˜
G
G
˜
(a)
(b)
FIG. 3. Steady state drop length l¯ and alignment˜共l¯兲−1as functions of flow strength parameter G˜ for wall spacing W¯ =4 共dotted curves兲, W¯ =10 共dashed
curves兲, and intermediate limit, Eq.共69兲共solid curves兲.
ACKNOWLEDGMENTS
P.J.A.J. and P.D.A. acknowledge the support of Grant No. 446 from the Dutch Polymer Institute; M.L. acknowl-edges the support of grant CTS Grant No. 0553551 from the National Science Foundation. The authors thank Professor Jerzy Bławzdziewicz for help in developing the wall-correction velocities.
APPENDIX A: TANGENTIALLY DRIVEN COMPONENT OF INTERNAL VELOCITY FIELD
Here, we present the solution for the tangentially driven component of the unidirectional internal velocity field, u
¯ˆt共r¯,兲, defined in Sec. II D. Accordingly, we seek a nonsin-gular solution of Laplace’s equation on a circular disk r
¯ⱕR¯, 0ⱕ⬍2with boundary condition共32兲prescribed on the boundary r¯ = R¯ . Under these conditions, we have
u ¯ˆt共r¯,兲 = a0+
兺
n=1 ⬁ r ¯n关ancos n+ bnsin n兴, 共A1兲 where the constants an 共n=0,1,2,¯兲 and bn 共n=1,2,¯兲 are given by integrals of u¯xon the perimeter.
a0= 1 2
冕
0 2 u ¯x共R¯,兲d, 共A2兲 an= 1 冕
0 2 u ¯x共R¯,兲cos nd, 共A3兲 bn= 1 冕
0 2 u ¯x共R¯,兲sin nd; n = 1,2,¯ ,⬁.A useful result that follows from Eqs.共A1兲and共A2兲 is the equality of the area-averaged value of u¯ˆt共r,兲 over the drop cross section and its perimeter-averaged value,
1 R¯2
冕
0 2冕
0 R ¯ u ¯ˆt共r¯,兲r¯dr¯d= a0= 1 2冕
0 2 u ¯x共R¯,兲d. 共A4兲APPENDIX B: SOLUTION FOR INTERNAL VELOCITY FIELD
Here, we present the solution for the unidirectional in-ternal velocity field defined by decomposition共30兲. Inserting the pressure field共72兲into Eq.共31兲, we obtain the pressure-driven component of the velocity,
u ¯ˆp= − 2关1 − 共r¯/R¯兲2兴
冉
¯ − f¯w1 2 + 2R ¯−2冕
0 x ¯ R ¯ R¯˙ dx¯冊
. 共B1兲 The tangentially driven component of the internal velocity field is obtained by inserting the x¯-component of the externalvelocity共70兲into Eqs.共A2兲and共A3兲. Accordingly, Eq.共A1兲 yields u ¯ˆt= a0+ b1¯ sinr , 共B2兲 where a0=¯ − f¯w1 2 , b1=
冋
1 + f¯ R ¯2冉
1 − R ¯2w 2 4冊
册
. 共B3兲Inserting the stresslet distribution function 共73兲 into Eqs.
共B1兲–共B3兲yields a solution for the internal velocity in terms of the drop-shape and confinement parameters.
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