Spatial ordering due to hydrodynamic interactions between a pair of
colliding drops in a confined shear
Kausik Sarkar1,2,3 and Rajesh Kumar Singh2,3
Department of Mechanical and Aerospace Engineering
George Washington University, Washington, DC 20052
2
Department of Mechanical Engineering
University of Delaware, Newark, DE 19716, USA
3
Biomechanics and Movement Science Program
University of Delaware, Newark, DE 19716, USA
1
PACS numbers: 47.55.D-, 47.57.-s, 47.57.ef, 47.61.Ne
Abstract
Pair-collision between viscous drops in a confined shear is numerically simulated to show that the
confinement drastically alters the trajectories of the drops. In contrast to free shear, drops here move
towards the centerline giving rise to a zero cross-stream separation and a net stream-wise separation. The
latter varies as inverse of capillary number and the cube of the confinement (distance between the walls).
The stream-wise separation does not depend on the initial positions of the drops. An analytical theory for
the phenomenon is offered.
Hydrodynamic interactions between deformable particles and the bounding walls in a
confined shear is important in microfluidic applications[1-4] and microcirculatory flows[5]. Due
to the small size and velocity, the flow is governed often by the inertia-less Stokes flow. Stokes
flow is linear and therefore reversible. A number of counterintuitive phenomena are observed in
particulate Stokes flows due to the flow reversibility[6]. For instance, in a free shear, a rigid
sphere does not experience any cross-stream motion[7], or a pair of rigid spheres continues in
their original streamlines after collision maintaining the pre-collision cross-stream separation.
However, for drops, the reversibility is broken; a drop migrates away from a bounding wall[8, 9],
and after collision a drop-pair increases their cross-stream separation leading to an enhanced
shear induced particle diffusion in an emulsion.[10, 11] Reversibility is also broken in presence of
finite inertia [12, 13]. Finite inertia induced particle migration to an intermediate position (0.6
radial distance) in a Poiseuille flow first observed by Segre and Silberberg [14] inspired a series
1
of theoretical and experimental efforts targeted at understanding the underlying physics of inertial
migration [15-20].
Recently, we showed that deformation and inertia can work in unison to generate a new
type— reversed (type II)—of trajectories for a pair of drops in free shear not seen in Stokes flow
[12, 13]. Such reversed trajectories are also seen in presence of inertia for a pair of rigid spheres
[21]. The underlying mechanism has been identified as the inertia induced reversed streamlines
around a particle [22, 23]. On the other hand, in the Stokes flows limit, in a confined shear a
similar reversed (called swapping trajectory by the authors) trajectory for a pair of rigid spheres is
discovered due to interaction with the bounding walls [24]. Reversibility leads to a swapping of
pre-collision streamlines between drops. Swapping trajectories have been proposed as a probable
cause for anomalous particle diffusion observed in an experimental study [25]. Here, we show
that in presence of both deformation and confinement, pair interaction gives rise to a specific
spatial positioning of drops in the center of the confined domain.
Figure 1: Pair of drops interacting in a confined shear for Ca=0.2, =5, ∆xo/a = 2.5 and ∆yo/a =
0.25. Drops travel towards the center of the domain.
2
We numerically simulate the collision of a pair of initially spherical drops of radius a in a
confined shear bounded by walls oriented along the x -axis separated by a distance y = Ly using
the front tracking finite difference method [26-29]. The method has been used to study a number
of different problems, including pair interactions in an unbounded shear, where the simulation
showed an excellent match with prior experimental observations[10, 12] The walls are moved
with equal and opposite x -directional velocity to generate a shear γ . The dynamics depends on
the capillary number=
Ca µmγ a Γ , viscosity ratio ( λ = µd µm ) and degree of confinement
1.5
∆y/a
1
1
xb
0.5
ymax
y/a
0.5
Ly= 4.5a
Ly= 5a
Ly= 6a
Ly= 7a
Ly= 10a
Ly= 13.4a
0
-0.5
-1
-20
0
0
-10
0
10
x/a
10
∆x/a
20
20
30
Figure 2: Relative trajectory of the drops at Ca = 0.2, ∆xo/a = 2.5 and ∆yo/a = 0.25 for
different Ly values. Inset shows the trajectory of the drops in the domain Ly =4.5a.
Ly / a . Here µm and µd are matrix and drop phase viscosities, and Γ is the interfacial tension.
Since the code is not fully implicit, we are limited to simulations with small but finite non-zero
inertia. We consider Re ( = ρ mγ a 2 µ m ) =0.02 as a surrogate for Stokes flow simulation. ρ m is the
density of the matrix phase. We use a computational domain Lx = 50a and L z = 5a . Here, for
3
brevity, we concentrate on a viscosity matched system λ =1. In the vorticity direction, they are in
the same central z-plane. We vary Ly to study the effects of confinement on the trajectory of the
drops. In the flow (x) and the vorticity (z) directions periodic boundary conditions are used.
The drops driven by the imposed shear interact, deform—maximum deformation being
when they press against each other along the compression axis of the imposed shear—then
separate and move in opposite directions (a typical case is shown in Figure 1). However, in
contrast to free shear, here after collision drops do not eventually follow any free streamlines [10,
11]. Neither do they achieve a net cross-stream separation. Instead, drops experience a wall
induced lateral migration that moves them to the center line, progressively reducing their crossstream displacement to zero (Figure 2). Finally, they achieve a state of relative equilibrium
separated by an equilibrium distance ∆x final / a at the centerline. This is shown explicitly for the
case of Ly / a = 4.5 in the inset of Figure 2. For much larger Ly / a , i.e. weaker confinement,
1.5
∆x0/a =
∆x0/a =
∆x0/a =
∆x0/a =
1.2
2.5
3
3.5
4
∆y/a
0.9
1.5
∆y0/a = 0.25
∆y0/a = 0.5
∆y0/a = 0.75
∆y0/a = 1.0
1
∆y/a
0.6
0.3
0.5
0
0
-5
0
0
20
5
∆x/a
10
∆x/a
40
60
15
20
25
Figure 3: Effects of the initial positions on the relative trajectory: Variation of initial separation
in the flow direction
for
, Ca=0.20 and
. Inset shows the effect of
the separation in the gradient direction
same capillary number.
for
4
in the same domain and the
∆x final / a becomes larger, and hence requires much longer simulation in far longer (larger Lx a )
computational domain. However, after collision, the drop trajectory eventually becomes a straight
line (as can be seen in Figure 2) and therefore, ∆x final / a can be determined by linear
extrapolation. The validity of this extrapolation procedure has been carefully examined and
established for several Ly / a by using simulations in longer domains. Only in the limit of very
large inter-wall separation ( Ly / a  20 ), wall effects are negligible.
In Figure 3, we investigate the effects of initial separation on the drop trajectory. Changing
initial separation changes trajectory type—increasing initial stream-wise or decreasing initial
cross-stream separation leads to reversed or swapping (type II) trajectory both for rigid spheres
160
∆xfinal /a
120
150
Ly= 4.5a
Ly= 5a
Ly= 6a
80
120
∆xfinal/a
90
60
40
30
0
0
150
250
2
4
(Ly /a)3
350
6
1/Ca
8
450
Figure 4: Effects of confinement in the gradient direction (Ly) on
Inset of the figure shows that
10
550
at Ca=0.20.
decreases with increasing Ca.
and drops [12]. However, here we consider those initial positions which do not change the
trajectory type. With this restriction, Figure 3 shows that ∆x final / a remains independent of
5
initial positions. For the results in this paper, we choose initial separation in the flow and the
2.5 and ∆y0 / a =
gradient directions fixed at ∆x0 / a =
0.25 .
Figure 2 studies trajectories for a number of different Ly / a , for a capillary number of
Ca = 0.2 resulting in ∆x final / a  ( Ly / a )3 shown in Figure 4. By varying capillary number for
three different confinements, we obtain ∆x final / a  1/ Ca shown in the inset of Figure 4. For
Ca > 0.35 drops experience too large a stretching and possible breakup—confinement is known
to delay breakup [30]. They are not considered here.
0
-0.02
uy
0
Ly= 4.5a
Ly= 5a
Ly= 5.5a
Ly= 6a
Ly= 7a
C&L
uy×(Ly/a)
3
-1
-0.04
-2
-3
0
-0.06
0
0.2
0.4
0.6
y/a
0.2
0.4
y/a
0.6
0.8
Figure 5: (Color online) Variation of lateral velocity of the drops with y after collision with
increasing confinement from the top along with analytical results due to Chan and Leal (1979).
Straight lines Inset shows the scaling of velocity with for different domains.
To investigate the reason for the numerically observed scalings, we investigate the drop
velocity. In the flow direction the velocity of the drop post collision is dominated by the imposed
shear and therefore can be approximated as u x = γ y , neglecting the small slip velocity as well as
6
the effect due to the interaction with the other drop, the result becoming more accurate as the drop
approaches the centerline. In Figure 5, we note that the lateral velocity u y ~ − y especially near
the centerline after the effects of collision decay. This explains the straight line trajectory of the
drop after collision: dx
=
/ dy u x / u y ≈ constant . Furthermore, inset of Figure 5 shows
u y ~ − y ( a Ly ) . Chan and Leal [31] performed a perturbative analysis of a drop migrating in a
3
plane shear between two parallel plates to get the following migration velocity:
uy
γ a
2
16 + 19λ 3 ( 54 + 97λ + 54λ )  a
Ca 
2
L
16 + 16λ
70 (1 + λ )
 y



2


8y *
−y*−
,
2 2 

−
1
4
*
y
(
)

(1)
where y * is the cross-flow distance from the centerline nondimensionalized by Ly ( y* = y / Ly ).
For small y * this can be linearized to obtain for λ = 1 :
 a
uy
=
−21.62 × Ca 
L
γa
 y
3
  y
 y
    −α  
a
 a
(2)
Simulated velocity matches very well this relation in Figure 5 for different Ly / a . Noting the
symmetry between the top and the bottom drop, one can integrate to obtain
y
∆y
= 2 max e −α tγ ,
a
a
(3)
∆xb 2  ymax  ∆y
∆x ∆xb 2  ymax 
−α tγ
= + 
 (1 − e ) = + 
− .
a
a α a 
a α a  a
(4)
Here ± ymax is the post-collision vertical positions of the top and the bottom drops (measured
from the centerline) wherefrom they follow a linear trajectory (inset of Figure 2). ∆xb is the flow
wise separation at that instant. From Figure 2 it seems reasonable and therefore we assume that
ymax and ∆xb are almost independent of Ly and Ca . The relation suggested by (4) is verified by
3
the collapse of relative trajectories for different Ly and Ca while scaling ∆x with Ca (a / Ly ) in
the inset of Figure 6. The same relation (4) after putting ∆y final / a =
0 ) gives rise to
7
 Ly
∆x final ∆xb
2
=
+

21.62 × Ca  a
a
a
3
  ymax 
 
,
a



(5)
explaining ∆x final / a  ( Ly / a )3 / Ca , if one neglects ∆xb . Figure 6 shows this scaling for a
number of different Ly / a and Ca .
35
Ca.∆xfinal/a
25
Ca= 0.20 & Ly= 4.5a
Ca= 0.225 & Ly= 4.5a
Ca= 0.125 & Ly= 5a
Ca= 0.20 & Ly= 5a
Ca= 0.20 & Ly= 6a
Ca= 0.25 & Ly= 6a
Ca= 0.20 & Ly= 10a
1.5
∆y/a
15
1
0.5
5
0
50
150
Figure 6: Composite scaling of
0
250
(Ly/a)
with
3
0.02
350
0.04
(Ca.∆x/a)/(Ly/a)
3
450
for different
0.06
0.08
550
. Inset
shows the same for relative trajectory of the drops.
Drops achieving a finite separation ( ∆x final / a ) in the flow direction in a confined shear
is an interesting physical phenomenon analogous to others where particulate system organizes
into specific spatial ordering [32]. However, here it is mediated exclusively by hydrodynamic
interactions. It assumes further importance in view of the independence of their final separation
of the initial drop positions. It indicates that a dilute emulsion of drops in a confined shear has a
tendency to organize into a single file separated by a specific distance that would depend on
intrinsic hydrodynamic parameters, viz., capillary number and degree of confinement. Note that
the parameters studied here are realizable in microfluidic devices. In a 10 micron channel a
velocity of 1 cm/s produces a shear rate γ  103 s −1 ; with µ  1-100 mN/m (water viscosity 1
8
mN/m), Γ  1-100 mN/m/s, for a 2 micron drop ( Ly / a = 5)
capillary number is
Ca  0.00002 − 0.2 also obtained in microfluidic devices [1, 33]. We have investigated the
drop interaction in linear shear instead of in a pressure driven flow, more often used in such
devices, because it separates the shear effects on migration from those due to shear gradient
present in the latter. The present phenomenon of spatial ordering can be interrogated, e.g.
optically, as a means for determining either size or deformability, both parameters affecting
capillary number. Differential migration also offers a way of filtering based on the same
parameters. There has been a recent surge in innovative applications of size-differentiated inertial
migration of rigid particles in pressure driven microfluidic devices for developing sorting,
focusing and flow cytometry [34-37]. Deformation provides an additional parameter to control
migration and in systems with inertia will create additional migratory effects. Linear chain of
droplets separated by a fixed distance have recently seen many novel applications such as
determination of the time evolution of reaction kinetics, protein crystallization and concentration
indexing using specially designed droplet-pairs [38-40].
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