PHYSICAL REVIEW E 90, 033010 (2014)
Hydrodynamic interaction of two deformable drops in confined shear flow Yongping Chen1,2,* and Chengyao Wang1 1
Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, People’s Republic of China 2 School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou, Jiangsu 225127, People’s Republic of China (Received 5 April 2014; published 16 September 2014) We investigate hydrodynamic interaction between two neutrally buoyant circular drops in a confined shear flow based on a computational fluid dynamics simulation using the volume-of-fluid method. The rheological behaviors of interactive drops and the flow regimes are explored with a focus on elucidation of underlying physical mechanisms. We find that two types of drop behaviors during interaction occur, including passing-over motion and reversing motion, which are governed by the competition between the drag of passing flow and the entrainment of reversing flow in matrix fluid. With the increasing confinement, the drop behavior transits from the passing-over motion to reversing motion, because the entrainment of the reversing-flow matrix fluid turns to play the dominant role. The drag of the ambient passing flow is increased by enlarging the initial lateral separation due to the departure of the drop from the reversing flow in matrix fluid, resulting in the emergence of passing-over motion. In particular, a corresponding phase diagram is plotted to quantitatively illustrate the dependence of drop morphologies during interaction on confinement and initial lateral separation. DOI: 10.1103/PhysRevE.90.033010
PACS number(s): 47.57.jb, 47.85.Np
I. INTRODUCTION
Hydrodynamic interaction of liquid drops is a ubiquitous phenomenon in nature and has attracted particular interest driven by the recent wide applications in microfluidics [1–4], such as emulsification [5–7], polymer blending [8–10], drug delivery [11,12], etc. In particular, the underlying hydrodynamics of this process is essential to active control of relevant technological processes in a manipulated and reproducible manner [13–15]. Since the pioneering work of Taylor [16,17], there have been many attempts to investigate the deformation and breakup of a single drop subjected to the unbounded shear flow. It is well documented that the characteristics of drop deformation and breakup are primarily determined by the magnitude of interfacial tension stresses relative to the magnitude of the flow-generated viscous stresses [18]. Based on the study of the single drop under unbounded shear flow, a lot of effort has been devoted to explore the hydrodynamic interactions between two drops in “unconfined” shear flow. The motion trajectory, deformation, and collision for immiscible drops in free shear flow, as well as the effects of inertia and surface tension on their hydrodynamic interactions were numerically investigated and analyzed [19–21]. In addition, viscosity determines the viscous drag of external fluid on the drop and the drop’s internal viscous resistance for its deformation, which also plays an important role in the hydrodynamic interaction between two drops [22]. It is indicated that the drops roll over each other, and the interactions between two drops lead to an increase in the transverse-flow separation of their centers [23–26]. Loewenberg and Hinch [25] simulated the interaction between a pair of deformable drops in a simple shear flow via a boundary integral formulation, and their results indicated that the interactions do not promote
*
Corresponding author: [email protected]
1539-3755/2014/90(3)/033010(11)
appreciably the breakup of the drops in a certain degree, and the resulting anisotropic self-diffusivities are strongly dependent on viscosity ratio. Recently, Olapade et al. [27] investigated the effects of finite inertia on hydrodynamic interactions between drops in a shear flow. The results illustrate that apart from the usual passing-over trajectory, the finite inertia introduces a new type of reversed trajectory where drops approached each other and then reversed their initial trajectories. In addition to the numerical studies, Guido and Simeon [28] conducted a visualization experiment on binary collision of drops in simple shear flow, and found that the distance between the drop centers along the velocity gradient direction increases irreversibly after collision. For hydrodynamic interaction among multidrops, Beatus et al. [29] discussed phonons in one-dimensional crystals of drops in a microfluidic configuration under the combined flow of a Poiseuille-like response and a potential flow. Furthermore, it is also demonstrated that when the outer shear flow of a single drop is confined, several new effects, such as oscillating transients, drop stabilization against breakup, and the wall-induced distortion, which leads to very elongated drop shape [30], must be considered. The wall effects can elicit the drop migration toward the center plane of a confined shear flow [31]. Being aware of the confinement effect, the morphology development in “simple” confined geometries such as channel flow [32,33] or shear flow [34] is theoretically investigated. Moreover, the experimental studies that focus on the effects of confinement on droplet breakup and droplet deformation have also been reported [30,35,36]. The critical capillary number for droplet breakup is found to be strongly affected by confinement. In the case of low viscosity ratios, the confinement suppresses the breakup, whereas for high viscosity ratios, it promotes the breakup [35]. For drop breakup against an obstacle, there also exists a critical value of the capillary number [37]. In addition, the viscosity contrast is very important for the breakups of confined drops against a linear obstacle [38], while the position of the obstacle controls the degree of breakup and the relative size distribution of the
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resulting dispersion [39]. For drop breakup in junctions of arbitrary angles, the breakup process is controlled by a critical finger length of the drop [40]. In this context, a question arises as to whether confinement also significantly affects the hydrodynamic interactions between drops in confined geometries. However, previous studies on hydrodynamic interaction of isolated droplet pairs are concentrated on the motion trajectories, drop morphologies, and the influence parameters (such as initial position, ratio of viscosity, inertia, and interfacial forces) in free shear flows. Up to now, the role of confinement on hydrodynamic interaction of drops in shear flow is not completely known; especially the regimes of pairwise interaction between drops in confined shear flow are less understood. In addition, the underlying physics of detailed behaviors of binary drops in confined shear flow is still waiting to be explored. For these reasons, the current investigation is undertaken to explore the hydrodynamics of the pairwise interactions between deformable drops in confined shear flows by computational fluid dynamic (CFD) simulations, in an effort to elucidate the effects of confinement and initial lateral separation on the drop morphologies during interaction in confined shear flows.
B. Mathematical model
The volume-of-fluid (VOF) method is utilized to represent the free interface behaviors involved in the interactive drops. In the VOF method, for each additional phase a variable, α, the volume fraction of the phase in the computational cell, is introduced. In each control volume, the volume fractions of all phases sum to unity. Thus the variables and properties in any given cell are either purely representative of one of the phases, or representative of a mixture of the phases, depending upon the volume fraction values. In other words, if the ith fluid’s volume fraction in the cell is denoted as αi , then the following three conditions are possible: αi = 0 The cell is empty (of the ith fluid), 0 < αi < 1 The cell contains the interface between the
(1)
ith fluid and one or more other fluids, αi = 1 The cell is full (of the ith fluid). The average values of density and viscosity are determined by ρ= μ=
II. PAIRWISE INTERACTION OF DROPS IN CONFINED SHEAR FLOW
αi ρi ,
(2)
αi μi ,
(3)
The volume fraction is governed by the transport equation ∂αi + ∇ · (vαi ) = 0, ∂t
A. Computational domain
Here, we develop a two-dimensional mathematical model of the hydrodynamic interaction of two neutrally buoyant and equal-sized circular drops under confined shear flow produced by a Couette geometry as represented in Fig. 1. The upper and bottom plates are set with the confinement of a/H and moved reversely with the velocity of U and −U , respectively, where a is the drop radius and H is the width between two plates. Accordingly, a simple shear flow is produced between the two parallel plates with the shear rate of G = 2U/H [41]. Periodical boundary conditions are imposed except for the top and bottom plates. As indicated in the figure, the drops centers are initially separated symmetrically by x along the flow direction and y along the velocity gradient direction. The distinct phases of drops and matrix liquid are Newtonian fluid and immiscible. In the current study, the confinement of Couette geometry is set as 0.050 ࣘ a/H ࣘ 0.250, which is within the confined shear flow regime for the interaction of two deformable drops [28,42].
(4)
where v is the velocity of the flow. The other governing equations of the VOF formulations on fluid flow can be presented as Continuity equation: ∇ · v = 0.
(5)
Momentum equation: ∂(ρv) + ∇ · (ρvv) = −∇p + ∇ · [μ(∇v + ∇v T )] + F vol . ∂t (6) In Eq. (6), F vol is the source term for calculating the interfacial tension force, F vol = σij
αi ρi κj ∇αj + αj ρj κi ∇αi , 0.5(ρi + ρj )
(7)
U
1
y
y
o x
x
H 2
a
-U 033010-2
FIG. 1. (Color online) Schematic of initial configuration and location of the drops in shear flow.
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where σij is the interfacial tension, and κi and κj are the mean curvature for phase i and phase j , defined as αj αi , κj = . |∇αi | |∇αj |
(8)
In the simple shear flow, the dynamics of interactive drops is governed by the competition between viscous shear stress, interface tension, and inertial force, which can be characterized by the Reynolds number Re = ρm Ga 2 /μm (the ratio of inertia to viscous forces) and capillary number Ca = μm Ga/σ (the ratio of viscous to interfacial forces), where μ is the viscosity, σ is the interfacial tension, Ga is the characteristic velocity which reflects the linear variation of velocity in the shear flow [19,20], and where subscript m represents the matrix fluid. Here, the viscosities of all fluids are assumed to be the same. The deformation of a drop is characterized by the criterion D defined by Taylor [16,17] as D = (L – B)/(L + B), where L and B are the maximum and minimum distances of the drop interface from its center. The governing equations described above are numerically solved by the control volume finite-difference technique. The pressure-velocity coupling is accomplished by the SIMPLE algorithm. The body force weighted scheme is utilized for the pressure interpolation, and a first-order upwind differencing scheme is adopted to discretize the momentum equation. The Geo-Reconstruct scheme based on the piecewise linear interface calculation (PLIC) method is applied to reconstruct the liquid-liquid interface. The under-relaxation factors are used at the following values: 0.2 (pressure), 0.5 (density), 0.5 (body force), 0.2 (momentum). In view of the fact that several previous studies have demonstrated the feasibility of the VOF method on investigating the drop coalescence in the external flow field, the current mathematical model has the ability to numerically predict both the coalescence and the collision between the interacting drops in the shear flow [43–45]. Notably, herein, we concentrate on investigating the hydrodynamics of the collision between two drops which usually plays a considerable role in several practical applications including solvent extraction and emulsification [22]. Therefore, according to the previous studies which indicated that the coalescence between two drops in the shear flow generally occurs when the Ca is smaller than a critical Cac with the magnitude of 10−2 [46], we set the Ca of the current investigated cases as Ca ࣙ 0.1, so as to avoid the drop coalescence. The quality of the mesh plays an important role in the accuracy and reasonability of the numerical results. In particular, during the interaction between two drops involved in the drops’ collision, the flow field information changes sharply in the thin lubrication film region between two drops, which needs to be appropriately solved via the refined meshes. Therefore, a mesh study is conducted to make sure whether the utilized meshes are well resolved to appropriately represent the hydrodynamic interaction between two colliding drops. In the numerical simulation, the computational domain of size Lx × Ly is meshed by the corresponding mesh size of x = y. A mesh independence check is conducted using four mesh sizes of x = y = a/8, a/12, a/16, and a/20. Figure 2 compares the drop deformation during the interaction of two deformable drops for these four kinds of mesh size.
a/8 a/12 a/16 a/20
0.4
D
κi =
0.6
0.2
0.0 -6
-4
-2
0
2 x/ a
4
6
8
10
FIG. 2. (Color online) Drop deformation for four different meshes at Re = 0.2 and Ca = 0.2.
It can be seen that the results converge with the refinement of mesh and little difference in simulation results is seen between the mesh sizes of a/16 and a/20. Based on an overall consideration of computational efficiency and the accuracy of calculation results, the mesh size of x = y = a/16 is adopted. In addition, the current well resolved meshes can also avoid the unreal automatic merging of the interfaces of two approaching drops implicit in the VOF method [47]. C. Validation
In order to validate the present simulation method, a comparison between the simulation results by the present model and the experimental data [28] is plotted in Fig. 3, including the drop deformation, motion trajectory, and the angle φ (inclination of line joining the centroids of the two interacting drops with respect to the y axis). As seen from the figure, the simulations results, including the drop deformation, motion trajectory, and the angle φ, agree well with the experimental data, which validates the reasonability of the present model. III. RESULTS AND DISCUSSION A. Motion behaviors
The motion behaviors intuitively reflect the hydrodynamic interactions of two drops suspended in a shear flow. As shown in Fig. 4(a), the conventional pairwise motion behavior during the drop interaction in an infinite shear flow, i.e., the drops pass over each other [27], is also observed in the confined shear flow regime. Herein, it is defined as passing-over motion. In addition, as illustrated in Fig. 4(b), with the increasing confinement, another type of motion trajectory appears wherein the drops approach each other and then reverse their initial trajectory, which is defined as reversing motion in the current work. Although the drop trajectories of passing-over and reversing motion are inherently different, both the pairwise interaction processes under these two motions could be divided into three stages according to the drop velocity and deformation with respect to time (see Fig. 4): approach, collision, and separation. In the approach stage, drops deform to be ellipsoidal and come close. In the collision stage, the two drops pass over
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Experiment [28] Simulation (a)
0.05 0.00 -6-
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Experiment [28] Simulation (b)
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Experiment [28] Simulation (c)
(1)
d) (d)
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FIG. 3. (Color online) Comparison between simulation results and experiment data (Re = 0.02, Ca = 0.13, y0 /a = 0.43): (a) deformation, (b) relative trajectory, (c) angle φ, (d) drop shape evolution.
each other for passing-over motion but move along the reverse direction under the reversing motion. In the separation stage, drops regain the ellipsoidal shape. In order to provide a detailed analysis of passing-over and reversing motion in confined shear flow, the motion velocities of drops for these two motion behaviors are described in Figs. 4(c) and 4(d), respectively. Note that the dimensionless time tG−1 is denoted by t ∗ , and the motion velocity u is defined as velocity magnitude of the drop centroids. For the passingover motion shown in Fig. 4(c), there is an acceleration for drops at the approaching stage when the shear flow is applied. Subsequently, the drop velocity decreases and then obviously increases during the collision stage. As the hydrodynamic interaction vanishes (i.e., the drops are fully separated), the
droplets separates with a constant velocity. As seen from Fig. 4(d), differing from the variation trend in passing-over motion, the drop velocity is only decreased when two drops interact. As the hydrodynamic interaction is vanishing, u gradually rises and then drops to zero at the final equilibrium location. B. Hydrodynamics of pairwise interaction
The flow fields in the pairwise interaction, including pressure and velocity fields, can provide deep insight into the hydrodynamic behaviors of interacted drops. Figures 5 and 6 present the transient pressure and velocity fields under passingover and reversing motions during the pairwise interaction process.
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(b) t*=0.0
(a) 2 t*=0.0 Δ y/a 0
-2 2 t*=18.2
t*=4.2
Δ y/a 0
-2 2 t*=21.4
t*=8.0
Δ y/a 0
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t*=10.0
Δ y/a 0
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t*=12.0
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t*=26.6
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Δ x/a
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u(s-1) 10
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0 -0.8 -0.4 -3 -2 -1 y 0.0 / a 0.4 0 a x/
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FIG. 4. (Color online) Motion behaviors of two drops in a shear flow (Re = 0.2, Ca = 0.2): (a) passing over (a/H = 0.100), (b) reversing (a/H = 0.200), (c) drop velocity of passing over, (d) drop velocity of reversing. 1. Passing-over motion
As shown in Fig. 5(a) for the passing-over motion, during the approaching stage, the drops deform into an ellipsoidal shape due to shear effect. Meanwhile, owing to the surface tension, a high-pressure area appears at the ends of the
ellipsoidal drops along the major axis and a high-pressure gradient is formed at the interface of the ends [t ∗ = 18.2 in Fig. 5(a)]. As two ellipsoidal drops collide with each other, the pressure at the matrix liquid between the drops rises and drives the matrix liquid towards the low-pressure region, which
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(b)
(a) 2
t*=0.0
y/a 0
p (Pa)
-2 2 t*=18.2 y/a 0 -2 2 t*=21.4
Passing-over flow region Lubrication film
y/a 0 Reversing flow region
-2 2 t*=23.0 y/a 0 -2 2 t*=24.4 y/a 0
High curvature tips
-2 2 t*=27.0 y/a 0 -2 -5
-2.5
0 x/a
2.5
5-5
-2.5
0 x/a
2.5
5
FIG. 5. (Color online) Local transient pressure and velocity contours for passing-over motion (a/H = 0.100, Re = 0.2, Ca = 0.2): (a) pressure contour, (b) velocity contour.
finally results in thinning the lubrication film between the drops. The high pressure in the film causes large interface deformation and the local interfaces of the drops are even flattened [t ∗ = 21.4 in Fig. 5(a)]. When the drops separate, the lubrication film gets thicker with drops recoiling and the film pressure decreases drastically [t ∗ = 24.4 in Fig. 5(a)]. The large pressure gradient in the lubrication film region generates a drag force to suck the interface at the tip of the separating drops, resulting in a high-curvature tip there. Eventually, the pressure of the central flow domain is unvaried after the drops have fully separated. Furthermore, it can be predicted that the thickness of the lubrication film can be decreased by reducing the viscosity of the matrix fluid, because the viscous friction in the matrix fluid suppressing the fluid drainage from the lubrication film is small. It is important to note from Fig. 5(b) that two distinct flow regions are observed in the matrix fluid under both passingover and reversing motions: (1) passing flow represented by the passing-over streamlines in the upper and lower half of the
domain, which generates a drag force to promote the passing over of interacted drops; (2) reversing flow characterized by the reversing streamlines on both sides of the domain, which entrains the drops into the reversing flow of matrix fluid. It can be seen from Fig. 5 that more of the volume of the drop is exposed to the passing-over flow rather than the reversing-flow in matrix fluid during the whole passing-over process. Hence, the drag force caused by passing-flow in matrix fluid plays the dominant role in the moving of the drop, leading to the passing-over motion of the drop during pairwise interaction. 2. Reversing motion
Figure 6 illustrates the detailed flow fields underlying reversing motion of drops during the pairwise interaction. Similar to the passing-over motion, when shear flow is imposed, the circular drops are stretched to be ellipsoidal at the approaching stage. However, when two ellipsoidal
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PHYSICAL REVIEW E 90, 033010 (2014)
2 t*=0.0
p ( Pa)
y/a 0 -2 2 t*=4.2 y/a 0 -2 2 t*=8.0 y/a 0 -2 2 t*=10.0 y/a 0 -2 2 t*=12.0 y/a 0 -2 2 t*=26.6
Additional vortex
y/a 0 -2
-5
-2.5
0 x/a
2.5
5-5
-2.5
0 x/a
2.5
5
FIG. 6. (Color online) Local transient pressure and velocity contours for reversing motion (a/H = 0.200, Ca = 0.2, Re = 0.2): (a) pressure contour, (b) velocity contour.
drops collide, the pressure difference across the approaching interfaces of two ellipsoidal drops in reversing motion [e.g., t ∗ = 8.0 in Fig. 6(a)] is smaller than that in the passing-over motion, resulting in the less flattened approaching interfaces. In addition, for the reversing motion in Fig. 6 under the bigger confinement of a/H = 0.200 with respect to a/H = 0100 in Fig. 5, the larger shear rate in the wall region leads to the greater drag on the ends of deformed drops. Hence, the drop is highly stretched with larger interface curvature at two ends. Therefore, according to the Young-Laplace equation p = 2σ /R which shows the pressure gradient at the interface is proportional to the interface curvature 1/R, the pressure buildup for the reversing motion is a little higher at the drop end region than that of the passing-over motion. Compared with the velocity field under the passing-over motion, the region of reversing flow in matrix fluid under the reversing motion is enlarged due to the confinement, and more of the volume of the drop is squeezed into the reversing flow in the matrix fluid by the confined plates. Therefore, the entrainment induced by reversing flow in matrix fluid drops controls the drops, resulting in the reversing motion of the drops.
C. Influence parameters
The dynamics of the interacted drops in shear flow is essentially governed by the competition among viscous shear stress, interface tension, and inertial force, which is characterized by Re and Ca. Therefore, an investigation of Re and Ca on the hydrodynamic behaviors of the two drops’ collision is performed in the current study, the results of which are respectively plotted in Fig. 7. As shown in Fig. 7(a), the increment in Re leads to the large separation between the drops’ trajectories during the passing-over motion. It is a result of the increasing inertia inducing the large velocities at the drop tips which produces the aerodynamic lift and counterlift on the upper and lower ends of the drop, respectively [48]. These additional lifts towards the wall of the two plates enlarge the separation between the drops’ trajectories. For the effects of the capillary number, Fig. 7(b) indicates that under small Ca (Ca = 0.1), the collided drops follow the reversed motion; as the Ca is increased to the intermediate values (Ca = 0.2, 0.3), drops are able to pass over each other; upon further increase in Ca (Ca = 0.4), drops obey the reversing trajectories again. This variation of the drops’ trajectories with the increasing Ca can be explained by the competition between two effects:
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(a)
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0.5 Re = 0.2 Re = 0.3 Re = 0.4 Re = 0.5
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Ca = 0.1 Ca = 0.2 Ca = 0.3 Ca = 0.4
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FIG. 7. (Color online) Effect of Re and Ca on drop trajectories (x0 /a = 6, y0 /a = 0.8, a/H = 0.075): (a) effect of Re (Ca = 0.2), (b) effect of Ca (Re = 0.2).
(1) a higher Ca corresponds to a higher viscous drag of the matrix fluid, a driving force for the motion of the two drops, which increases the initial momentum of the two drops’ movement towards each other; (2) the big capillary number induces the large deformation of the drops which enlarges the part of the deformed drop subjected into the reversing flow in the matrix fluid. As the Ca just increases moderately, the increment in initial momentum of the two drops’ movement towards each other leads to the big inertia of each drop to continue moving along the original direction, i.e., passing over each other. However, with further increasing Ca, a larger part of the deformed drop is subjected into the reversing flow of the matrix fluid, which plays a dominant role on the motion of the drops, leading to the occurrence of the reversing motion. In view of that our study concentrates on the effect of confinement and initial lateral separation on the hydrodynamic behaviors of the two drops’ collision; the condition of Re = 0.2 and Ca = 0.2 is taken as a representative boundary condition to guarantee single factor analysis on the confinement and initial lateral separation in the current study.
-2 -3
-6
0 Δ x/a
6
3
FIG. 8. (Color online) Effect of confinement on the flow field around two interactive drops under passing-over motions (Re = 0.2, Ca = 0.2): (a) a/H = 0.050, (b) a/H = 0.075, (c) a/H = 0.150.
unconfined shear flow while the reversing-flow region appears in the confined shear flow due to the wall reflection of the perturbation flow produced by the drop. In addition, with the increasing confinement, the reversing-flow region in matrix fluid turns to be expanded in the confined shear flow, leading to the increasing role of reversing flow on the motion of drops under pairwise interaction. Therefore, as indicated in Fig. 9, the motion of drops under pairwise interaction transits from passing-over motion to reversing motion with the increasing confinement under confined shear flow. Furthermore, under the same motion mode of reversing motion, with an increase in confinement, the terminal streamwise separation of the final equilibrium position decreases, but the smallest separation distance is almost unchanged during the collision stage. Figure 10 also depicts the effect of confinement on transient deformation of drops during pairwise interactions under confined shear flow. As shown, differently from the deformation
1. Effect of confinement
1.2 a/H=0.050 (unconfined) a/H=0.100 (confined) a/H=0.150 (confined)
0.6
y/a
Herein, in order to illustrate the hydrodynamic characteristics during the interaction of two deformable drops under confined shear flow, the typical shapes of deformed drops and velocity field at collision stage for the conventional passing-over motions under unconfined shear flow with a/H = 0.050 are illustrated in Fig. 8(a) as a comparison. As shown, even under the same motion patterns, the drops during pairwise interaction are more extended and flattened than those under unconfined shear flow. This phenomenon is attributed to the fact that the ends of the deformed drops are much closer to the wall region of Couette geometry under confined shear flow. Therefore, the large shear rate in the wall region leads to the great drag on the ends of the deformed drops, resulting in highly stretched drops. Moreover, it is important to note that no reversing flow in matrix fluid is observed under
a/H=0.200 (confined) a/H=0.250 (confined)
0.0
smallest separation
terminal separation
-0.6 -1.2 -8
-4
0 x/a
4
8
FIG. 9. (Color online) Effect of confinement on drop trajectories (Re = 0.2, Ca = 0.2, x0 /a = 6, y0 /a = 1.2).
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0.2 a/H=0.050 (unconfined) a/H=0.100 (confined) a/H=0.150 (confined) a/H=0.200 (confined) a/H=0.250 (confined)
0.1 0.0 0
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t* FIG. 10. (Color online) Effect of confinement on drop deformation (Re = 0.2, Ca = 0.2, x0 /a = 6, y0 /a = 1.2).
-2 2
(c)
0 Δ y/a
variation under passing-over motion, the profile of transient deformation exhibits two peak points. With increasing confinement under the passing-over motion, the variation of drop deformation during the whole interaction process tends to be transited from overdamping to underdamping oscillation. In addition, large confinement also leads to the highly deformed droplet due to additional drag force induced by increased wall effect. Another confinement effect is accelerating the propagation of viscous shear force imposed on the drop, which shortens the transient response before the steady deformation is reached as shown in Fig. 10. 2. Effect of initial lateral separation distance
Besides the confinement, the initial lateral separation also affects the motion behavior of pairwise interactions between drops in the confined shear flow. Figure 11 represents the effects of initial lateral separation on drop motion behaviors. As seen from Fig. 11, when the initial lateral separation is small (e.g., y0 /a from 0.4 to 1.2), the reversing motion occurs, and the passing-over motion is observed at larger initial lateral separation (e.g., y0 /a from 1.4 to 1.6). This phenomenon can be explained by the fact that, in the case of large initial lateral offset, the drop is far from the zone of reversing flow in matrix fluid and more exposed into the passing-flow region as illustrated in Fig. 12, leading to the passing-over motion of drops during pairwise interactions. It is also indicated in
-2 -3
-6
0 Δ x/a
6
3
FIG. 12. (Color online) Effect of initial lateral separation on the two interactive drops (Re = 0.2, Ca = 0.2): (a) y0 /a = 0.4, (b) y0 /a = 1.0, (c) y0 /a = 1.6.
Fig. 11 that the terminal streamwise separation between drops in the passing-over motion is larger than that in the reversing motion. In addition, although the movement trajectories are different under various initial lateral separations, the terminal position is unvaried, implying the final balance position of imposed force on the drops after the interaction process is not changed with initial lateral separations. D. Diagram of the flow behavior
According to the current numerical results, a diagram of the flow behavior is plotted in Fig. 13 to illustrate the dependence of the drops’ motion patterns during pairwise interaction on the confinement and initial lateral separation. As shown, the passing-over motion occurs at large initial lateral separation and less confined space. Under the conditions
1.6 1.4
1.0
1.2
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y/a
0.5 0.0
1.0 0.8
y0 / a = 0.8
-0.5
0.6
y0 / a = 1.0 y0 / a = 1.2
0.4
y0 / a = 1.4
-1.0 -6
-4
-2
0 x/a
2
4
0.05
0.15 a/H passing-over trajectory
6
FIG. 11. (Color online) Trajectories of drops in a confined shear flow (a/H = 0.150, Re = 0.2, Ca = 0.2, x0 /a = 6).
0.10
0.20
0.25
reversing trajectory
FIG. 13. (Color online) Regimes of pairwise interaction at finite confinements (x0 /a = 6).
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with weak confinement, the reversing motion occurs rarely; especially, the reversing motion cannot be observed under the confinement of 0.05. The motion of the drops transits from passing over to reversing with increasing confinement. For a given confinement, increasing initial lateral separation may change the drops’ motion from reversing motion to passingover motion. Generally, with the increasing confinement, the transition between passing-over and reversing motion appears at great lateral separation. IV. CONCLUSIONS
In this paper, the drops’ morphologies and underlying mechanisms of hydrodynamic interaction between two neutrally buoyant circular drops in a confined shear flow are investigated by CFD simulation with VOF representation of the interface. The rheological behaviors of interactive drops and the flow regimes are explored, and the underlying transient pressure and velocity contours during the interactive process are presented. The effects of confinement and initial lateral separations on drop trajectory and deformation as well as motion behaviors are examined and analyzed. We find that two types of drop behaviors during interaction occur including passing-over motion and reversing motion, which are governed by the competition between the drag of passing flow and
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the entrainment of reversing flow in matrix fluid. With the increasing confinement, the proportion of the reversing-flow region is enlarged, leading to the increasing volume of the drop exposed to the reversing flow. Meanwhile, the drops’ motion transforms from the passing-over motion to reversing motion, because the entrainment of reversing flow turns to play the dominant role. By enlarging the initial lateral separation, the drop turns to be away from the zone of reversing flow, leading to the increasing drag of ambient passing flow in matrix fluid. And thus, the drops’ behaviors transform from the reversing motion to passing-over motion. In addition, a phase diagram is plotted to quantitatively illustrate the correlation between the effects of confinement and initial lateral separation on the transition between passing-over and reversing motions, which indicates that with the increasing confinement, the transition between passing-over and reversing motion occurs at great lateral separation.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support provided by National Natural Science Foundation of China (Grant No. 51222605) and Natural Science Foundation of Jiangsu Province (Grant No. BK20130009).
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PHYSICAL REVIEW E 90, 033010 (2014) [43] F. Krause, X. G. Li, and U. Fritsching, Eng. Appl. Comp. Fluid Mech. 5, 406 (2011). [44] X. Jiang and A. J. James, J. Eng. Math. 59, 99 (2007). [45] M. Mohammadi, S. Shahhosseini, and Mahmoud Bayat, Chem. Eng. Technol. 37, 27 (2014). [46] D. J. Chen, R. Cardinaels, and P. Moldenaers, Langmuir 25, 12885 (2009). [47] S. Hardt, Phys. Fluids 17, 100601 (2005). [48] Y. Renardy and V. Cristini, Phys. Fluids 13, 7 (2001).
033010-11
Hydrodynamic interaction of two deformable drops in confined shear flow Yongping Chen1,2,* and Chengyao Wang1 1
Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, People’s Republic of China 2 School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou, Jiangsu 225127, People’s Republic of China (Received 5 April 2014; published 16 September 2014) We investigate hydrodynamic interaction between two neutrally buoyant circular drops in a confined shear flow based on a computational fluid dynamics simulation using the volume-of-fluid method. The rheological behaviors of interactive drops and the flow regimes are explored with a focus on elucidation of underlying physical mechanisms. We find that two types of drop behaviors during interaction occur, including passing-over motion and reversing motion, which are governed by the competition between the drag of passing flow and the entrainment of reversing flow in matrix fluid. With the increasing confinement, the drop behavior transits from the passing-over motion to reversing motion, because the entrainment of the reversing-flow matrix fluid turns to play the dominant role. The drag of the ambient passing flow is increased by enlarging the initial lateral separation due to the departure of the drop from the reversing flow in matrix fluid, resulting in the emergence of passing-over motion. In particular, a corresponding phase diagram is plotted to quantitatively illustrate the dependence of drop morphologies during interaction on confinement and initial lateral separation. DOI: 10.1103/PhysRevE.90.033010
PACS number(s): 47.57.jb, 47.85.Np
I. INTRODUCTION
Hydrodynamic interaction of liquid drops is a ubiquitous phenomenon in nature and has attracted particular interest driven by the recent wide applications in microfluidics [1–4], such as emulsification [5–7], polymer blending [8–10], drug delivery [11,12], etc. In particular, the underlying hydrodynamics of this process is essential to active control of relevant technological processes in a manipulated and reproducible manner [13–15]. Since the pioneering work of Taylor [16,17], there have been many attempts to investigate the deformation and breakup of a single drop subjected to the unbounded shear flow. It is well documented that the characteristics of drop deformation and breakup are primarily determined by the magnitude of interfacial tension stresses relative to the magnitude of the flow-generated viscous stresses [18]. Based on the study of the single drop under unbounded shear flow, a lot of effort has been devoted to explore the hydrodynamic interactions between two drops in “unconfined” shear flow. The motion trajectory, deformation, and collision for immiscible drops in free shear flow, as well as the effects of inertia and surface tension on their hydrodynamic interactions were numerically investigated and analyzed [19–21]. In addition, viscosity determines the viscous drag of external fluid on the drop and the drop’s internal viscous resistance for its deformation, which also plays an important role in the hydrodynamic interaction between two drops [22]. It is indicated that the drops roll over each other, and the interactions between two drops lead to an increase in the transverse-flow separation of their centers [23–26]. Loewenberg and Hinch [25] simulated the interaction between a pair of deformable drops in a simple shear flow via a boundary integral formulation, and their results indicated that the interactions do not promote
*
Corresponding author: [email protected]
1539-3755/2014/90(3)/033010(11)
appreciably the breakup of the drops in a certain degree, and the resulting anisotropic self-diffusivities are strongly dependent on viscosity ratio. Recently, Olapade et al. [27] investigated the effects of finite inertia on hydrodynamic interactions between drops in a shear flow. The results illustrate that apart from the usual passing-over trajectory, the finite inertia introduces a new type of reversed trajectory where drops approached each other and then reversed their initial trajectories. In addition to the numerical studies, Guido and Simeon [28] conducted a visualization experiment on binary collision of drops in simple shear flow, and found that the distance between the drop centers along the velocity gradient direction increases irreversibly after collision. For hydrodynamic interaction among multidrops, Beatus et al. [29] discussed phonons in one-dimensional crystals of drops in a microfluidic configuration under the combined flow of a Poiseuille-like response and a potential flow. Furthermore, it is also demonstrated that when the outer shear flow of a single drop is confined, several new effects, such as oscillating transients, drop stabilization against breakup, and the wall-induced distortion, which leads to very elongated drop shape [30], must be considered. The wall effects can elicit the drop migration toward the center plane of a confined shear flow [31]. Being aware of the confinement effect, the morphology development in “simple” confined geometries such as channel flow [32,33] or shear flow [34] is theoretically investigated. Moreover, the experimental studies that focus on the effects of confinement on droplet breakup and droplet deformation have also been reported [30,35,36]. The critical capillary number for droplet breakup is found to be strongly affected by confinement. In the case of low viscosity ratios, the confinement suppresses the breakup, whereas for high viscosity ratios, it promotes the breakup [35]. For drop breakup against an obstacle, there also exists a critical value of the capillary number [37]. In addition, the viscosity contrast is very important for the breakups of confined drops against a linear obstacle [38], while the position of the obstacle controls the degree of breakup and the relative size distribution of the
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resulting dispersion [39]. For drop breakup in junctions of arbitrary angles, the breakup process is controlled by a critical finger length of the drop [40]. In this context, a question arises as to whether confinement also significantly affects the hydrodynamic interactions between drops in confined geometries. However, previous studies on hydrodynamic interaction of isolated droplet pairs are concentrated on the motion trajectories, drop morphologies, and the influence parameters (such as initial position, ratio of viscosity, inertia, and interfacial forces) in free shear flows. Up to now, the role of confinement on hydrodynamic interaction of drops in shear flow is not completely known; especially the regimes of pairwise interaction between drops in confined shear flow are less understood. In addition, the underlying physics of detailed behaviors of binary drops in confined shear flow is still waiting to be explored. For these reasons, the current investigation is undertaken to explore the hydrodynamics of the pairwise interactions between deformable drops in confined shear flows by computational fluid dynamic (CFD) simulations, in an effort to elucidate the effects of confinement and initial lateral separation on the drop morphologies during interaction in confined shear flows.
B. Mathematical model
The volume-of-fluid (VOF) method is utilized to represent the free interface behaviors involved in the interactive drops. In the VOF method, for each additional phase a variable, α, the volume fraction of the phase in the computational cell, is introduced. In each control volume, the volume fractions of all phases sum to unity. Thus the variables and properties in any given cell are either purely representative of one of the phases, or representative of a mixture of the phases, depending upon the volume fraction values. In other words, if the ith fluid’s volume fraction in the cell is denoted as αi , then the following three conditions are possible: αi = 0 The cell is empty (of the ith fluid), 0 < αi < 1 The cell contains the interface between the
(1)
ith fluid and one or more other fluids, αi = 1 The cell is full (of the ith fluid). The average values of density and viscosity are determined by ρ= μ=
II. PAIRWISE INTERACTION OF DROPS IN CONFINED SHEAR FLOW
αi ρi ,
(2)
αi μi ,
(3)
The volume fraction is governed by the transport equation ∂αi + ∇ · (vαi ) = 0, ∂t
A. Computational domain
Here, we develop a two-dimensional mathematical model of the hydrodynamic interaction of two neutrally buoyant and equal-sized circular drops under confined shear flow produced by a Couette geometry as represented in Fig. 1. The upper and bottom plates are set with the confinement of a/H and moved reversely with the velocity of U and −U , respectively, where a is the drop radius and H is the width between two plates. Accordingly, a simple shear flow is produced between the two parallel plates with the shear rate of G = 2U/H [41]. Periodical boundary conditions are imposed except for the top and bottom plates. As indicated in the figure, the drops centers are initially separated symmetrically by x along the flow direction and y along the velocity gradient direction. The distinct phases of drops and matrix liquid are Newtonian fluid and immiscible. In the current study, the confinement of Couette geometry is set as 0.050 ࣘ a/H ࣘ 0.250, which is within the confined shear flow regime for the interaction of two deformable drops [28,42].
(4)
where v is the velocity of the flow. The other governing equations of the VOF formulations on fluid flow can be presented as Continuity equation: ∇ · v = 0.
(5)
Momentum equation: ∂(ρv) + ∇ · (ρvv) = −∇p + ∇ · [μ(∇v + ∇v T )] + F vol . ∂t (6) In Eq. (6), F vol is the source term for calculating the interfacial tension force, F vol = σij
αi ρi κj ∇αj + αj ρj κi ∇αi , 0.5(ρi + ρj )
(7)
U
1
y
y
o x
x
H 2
a
-U 033010-2
FIG. 1. (Color online) Schematic of initial configuration and location of the drops in shear flow.
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where σij is the interfacial tension, and κi and κj are the mean curvature for phase i and phase j , defined as αj αi , κj = . |∇αi | |∇αj |
(8)
In the simple shear flow, the dynamics of interactive drops is governed by the competition between viscous shear stress, interface tension, and inertial force, which can be characterized by the Reynolds number Re = ρm Ga 2 /μm (the ratio of inertia to viscous forces) and capillary number Ca = μm Ga/σ (the ratio of viscous to interfacial forces), where μ is the viscosity, σ is the interfacial tension, Ga is the characteristic velocity which reflects the linear variation of velocity in the shear flow [19,20], and where subscript m represents the matrix fluid. Here, the viscosities of all fluids are assumed to be the same. The deformation of a drop is characterized by the criterion D defined by Taylor [16,17] as D = (L – B)/(L + B), where L and B are the maximum and minimum distances of the drop interface from its center. The governing equations described above are numerically solved by the control volume finite-difference technique. The pressure-velocity coupling is accomplished by the SIMPLE algorithm. The body force weighted scheme is utilized for the pressure interpolation, and a first-order upwind differencing scheme is adopted to discretize the momentum equation. The Geo-Reconstruct scheme based on the piecewise linear interface calculation (PLIC) method is applied to reconstruct the liquid-liquid interface. The under-relaxation factors are used at the following values: 0.2 (pressure), 0.5 (density), 0.5 (body force), 0.2 (momentum). In view of the fact that several previous studies have demonstrated the feasibility of the VOF method on investigating the drop coalescence in the external flow field, the current mathematical model has the ability to numerically predict both the coalescence and the collision between the interacting drops in the shear flow [43–45]. Notably, herein, we concentrate on investigating the hydrodynamics of the collision between two drops which usually plays a considerable role in several practical applications including solvent extraction and emulsification [22]. Therefore, according to the previous studies which indicated that the coalescence between two drops in the shear flow generally occurs when the Ca is smaller than a critical Cac with the magnitude of 10−2 [46], we set the Ca of the current investigated cases as Ca ࣙ 0.1, so as to avoid the drop coalescence. The quality of the mesh plays an important role in the accuracy and reasonability of the numerical results. In particular, during the interaction between two drops involved in the drops’ collision, the flow field information changes sharply in the thin lubrication film region between two drops, which needs to be appropriately solved via the refined meshes. Therefore, a mesh study is conducted to make sure whether the utilized meshes are well resolved to appropriately represent the hydrodynamic interaction between two colliding drops. In the numerical simulation, the computational domain of size Lx × Ly is meshed by the corresponding mesh size of x = y. A mesh independence check is conducted using four mesh sizes of x = y = a/8, a/12, a/16, and a/20. Figure 2 compares the drop deformation during the interaction of two deformable drops for these four kinds of mesh size.
a/8 a/12 a/16 a/20
0.4
D
κi =
0.6
0.2
0.0 -6
-4
-2
0
2 x/ a
4
6
8
10
FIG. 2. (Color online) Drop deformation for four different meshes at Re = 0.2 and Ca = 0.2.
It can be seen that the results converge with the refinement of mesh and little difference in simulation results is seen between the mesh sizes of a/16 and a/20. Based on an overall consideration of computational efficiency and the accuracy of calculation results, the mesh size of x = y = a/16 is adopted. In addition, the current well resolved meshes can also avoid the unreal automatic merging of the interfaces of two approaching drops implicit in the VOF method [47]. C. Validation
In order to validate the present simulation method, a comparison between the simulation results by the present model and the experimental data [28] is plotted in Fig. 3, including the drop deformation, motion trajectory, and the angle φ (inclination of line joining the centroids of the two interacting drops with respect to the y axis). As seen from the figure, the simulations results, including the drop deformation, motion trajectory, and the angle φ, agree well with the experimental data, which validates the reasonability of the present model. III. RESULTS AND DISCUSSION A. Motion behaviors
The motion behaviors intuitively reflect the hydrodynamic interactions of two drops suspended in a shear flow. As shown in Fig. 4(a), the conventional pairwise motion behavior during the drop interaction in an infinite shear flow, i.e., the drops pass over each other [27], is also observed in the confined shear flow regime. Herein, it is defined as passing-over motion. In addition, as illustrated in Fig. 4(b), with the increasing confinement, another type of motion trajectory appears wherein the drops approach each other and then reverse their initial trajectory, which is defined as reversing motion in the current work. Although the drop trajectories of passing-over and reversing motion are inherently different, both the pairwise interaction processes under these two motions could be divided into three stages according to the drop velocity and deformation with respect to time (see Fig. 4): approach, collision, and separation. In the approach stage, drops deform to be ellipsoidal and come close. In the collision stage, the two drops pass over
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0.25 (3)
(6)
0.15
D
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Experiment [28] Simulation (a)
0.05 0.00 -6-
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Experiment [28] Simulation (b)
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(3)
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Experiment [28] Simulation (c)
(1)
d) (d)
-90 -6-
4x/a
FIG. 3. (Color online) Comparison between simulation results and experiment data (Re = 0.02, Ca = 0.13, y0 /a = 0.43): (a) deformation, (b) relative trajectory, (c) angle φ, (d) drop shape evolution.
each other for passing-over motion but move along the reverse direction under the reversing motion. In the separation stage, drops regain the ellipsoidal shape. In order to provide a detailed analysis of passing-over and reversing motion in confined shear flow, the motion velocities of drops for these two motion behaviors are described in Figs. 4(c) and 4(d), respectively. Note that the dimensionless time tG−1 is denoted by t ∗ , and the motion velocity u is defined as velocity magnitude of the drop centroids. For the passingover motion shown in Fig. 4(c), there is an acceleration for drops at the approaching stage when the shear flow is applied. Subsequently, the drop velocity decreases and then obviously increases during the collision stage. As the hydrodynamic interaction vanishes (i.e., the drops are fully separated), the
droplets separates with a constant velocity. As seen from Fig. 4(d), differing from the variation trend in passing-over motion, the drop velocity is only decreased when two drops interact. As the hydrodynamic interaction is vanishing, u gradually rises and then drops to zero at the final equilibrium location. B. Hydrodynamics of pairwise interaction
The flow fields in the pairwise interaction, including pressure and velocity fields, can provide deep insight into the hydrodynamic behaviors of interacted drops. Figures 5 and 6 present the transient pressure and velocity fields under passingover and reversing motions during the pairwise interaction process.
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(b) t*=0.0
(a) 2 t*=0.0 Δ y/a 0
-2 2 t*=18.2
t*=4.2
Δ y/a 0
-2 2 t*=21.4
t*=8.0
Δ y/a 0
-2 2 t*=23.0
t*=10.0
Δ y/a 0
-2 2 t*=24.4
t*=12.0
Δ y/a 0
-2 2 t*=27.0
t*=26.6
Δ y/a 0
-2 -5
-2.5
0
Δ x/a
5-5
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0
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(d) u(s-1)
u(s-1) 10
20
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y/
-1.0 -0.5 0.0 0.5 1.0
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0 -0.8 -0.4 -3 -2 -1 y 0.0 / a 0.4 0 a x/
-2 2 0.8
-4 0
0 a x/
4
FIG. 4. (Color online) Motion behaviors of two drops in a shear flow (Re = 0.2, Ca = 0.2): (a) passing over (a/H = 0.100), (b) reversing (a/H = 0.200), (c) drop velocity of passing over, (d) drop velocity of reversing. 1. Passing-over motion
As shown in Fig. 5(a) for the passing-over motion, during the approaching stage, the drops deform into an ellipsoidal shape due to shear effect. Meanwhile, owing to the surface tension, a high-pressure area appears at the ends of the
ellipsoidal drops along the major axis and a high-pressure gradient is formed at the interface of the ends [t ∗ = 18.2 in Fig. 5(a)]. As two ellipsoidal drops collide with each other, the pressure at the matrix liquid between the drops rises and drives the matrix liquid towards the low-pressure region, which
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(b)
(a) 2
t*=0.0
y/a 0
p (Pa)
-2 2 t*=18.2 y/a 0 -2 2 t*=21.4
Passing-over flow region Lubrication film
y/a 0 Reversing flow region
-2 2 t*=23.0 y/a 0 -2 2 t*=24.4 y/a 0
High curvature tips
-2 2 t*=27.0 y/a 0 -2 -5
-2.5
0 x/a
2.5
5-5
-2.5
0 x/a
2.5
5
FIG. 5. (Color online) Local transient pressure and velocity contours for passing-over motion (a/H = 0.100, Re = 0.2, Ca = 0.2): (a) pressure contour, (b) velocity contour.
finally results in thinning the lubrication film between the drops. The high pressure in the film causes large interface deformation and the local interfaces of the drops are even flattened [t ∗ = 21.4 in Fig. 5(a)]. When the drops separate, the lubrication film gets thicker with drops recoiling and the film pressure decreases drastically [t ∗ = 24.4 in Fig. 5(a)]. The large pressure gradient in the lubrication film region generates a drag force to suck the interface at the tip of the separating drops, resulting in a high-curvature tip there. Eventually, the pressure of the central flow domain is unvaried after the drops have fully separated. Furthermore, it can be predicted that the thickness of the lubrication film can be decreased by reducing the viscosity of the matrix fluid, because the viscous friction in the matrix fluid suppressing the fluid drainage from the lubrication film is small. It is important to note from Fig. 5(b) that two distinct flow regions are observed in the matrix fluid under both passingover and reversing motions: (1) passing flow represented by the passing-over streamlines in the upper and lower half of the
domain, which generates a drag force to promote the passing over of interacted drops; (2) reversing flow characterized by the reversing streamlines on both sides of the domain, which entrains the drops into the reversing flow of matrix fluid. It can be seen from Fig. 5 that more of the volume of the drop is exposed to the passing-over flow rather than the reversing-flow in matrix fluid during the whole passing-over process. Hence, the drag force caused by passing-flow in matrix fluid plays the dominant role in the moving of the drop, leading to the passing-over motion of the drop during pairwise interaction. 2. Reversing motion
Figure 6 illustrates the detailed flow fields underlying reversing motion of drops during the pairwise interaction. Similar to the passing-over motion, when shear flow is imposed, the circular drops are stretched to be ellipsoidal at the approaching stage. However, when two ellipsoidal
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2 t*=0.0
p ( Pa)
y/a 0 -2 2 t*=4.2 y/a 0 -2 2 t*=8.0 y/a 0 -2 2 t*=10.0 y/a 0 -2 2 t*=12.0 y/a 0 -2 2 t*=26.6
Additional vortex
y/a 0 -2
-5
-2.5
0 x/a
2.5
5-5
-2.5
0 x/a
2.5
5
FIG. 6. (Color online) Local transient pressure and velocity contours for reversing motion (a/H = 0.200, Ca = 0.2, Re = 0.2): (a) pressure contour, (b) velocity contour.
drops collide, the pressure difference across the approaching interfaces of two ellipsoidal drops in reversing motion [e.g., t ∗ = 8.0 in Fig. 6(a)] is smaller than that in the passing-over motion, resulting in the less flattened approaching interfaces. In addition, for the reversing motion in Fig. 6 under the bigger confinement of a/H = 0.200 with respect to a/H = 0100 in Fig. 5, the larger shear rate in the wall region leads to the greater drag on the ends of deformed drops. Hence, the drop is highly stretched with larger interface curvature at two ends. Therefore, according to the Young-Laplace equation p = 2σ /R which shows the pressure gradient at the interface is proportional to the interface curvature 1/R, the pressure buildup for the reversing motion is a little higher at the drop end region than that of the passing-over motion. Compared with the velocity field under the passing-over motion, the region of reversing flow in matrix fluid under the reversing motion is enlarged due to the confinement, and more of the volume of the drop is squeezed into the reversing flow in the matrix fluid by the confined plates. Therefore, the entrainment induced by reversing flow in matrix fluid drops controls the drops, resulting in the reversing motion of the drops.
C. Influence parameters
The dynamics of the interacted drops in shear flow is essentially governed by the competition among viscous shear stress, interface tension, and inertial force, which is characterized by Re and Ca. Therefore, an investigation of Re and Ca on the hydrodynamic behaviors of the two drops’ collision is performed in the current study, the results of which are respectively plotted in Fig. 7. As shown in Fig. 7(a), the increment in Re leads to the large separation between the drops’ trajectories during the passing-over motion. It is a result of the increasing inertia inducing the large velocities at the drop tips which produces the aerodynamic lift and counterlift on the upper and lower ends of the drop, respectively [48]. These additional lifts towards the wall of the two plates enlarge the separation between the drops’ trajectories. For the effects of the capillary number, Fig. 7(b) indicates that under small Ca (Ca = 0.1), the collided drops follow the reversed motion; as the Ca is increased to the intermediate values (Ca = 0.2, 0.3), drops are able to pass over each other; upon further increase in Ca (Ca = 0.4), drops obey the reversing trajectories again. This variation of the drops’ trajectories with the increasing Ca can be explained by the competition between two effects:
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FIG. 7. (Color online) Effect of Re and Ca on drop trajectories (x0 /a = 6, y0 /a = 0.8, a/H = 0.075): (a) effect of Re (Ca = 0.2), (b) effect of Ca (Re = 0.2).
(1) a higher Ca corresponds to a higher viscous drag of the matrix fluid, a driving force for the motion of the two drops, which increases the initial momentum of the two drops’ movement towards each other; (2) the big capillary number induces the large deformation of the drops which enlarges the part of the deformed drop subjected into the reversing flow in the matrix fluid. As the Ca just increases moderately, the increment in initial momentum of the two drops’ movement towards each other leads to the big inertia of each drop to continue moving along the original direction, i.e., passing over each other. However, with further increasing Ca, a larger part of the deformed drop is subjected into the reversing flow of the matrix fluid, which plays a dominant role on the motion of the drops, leading to the occurrence of the reversing motion. In view of that our study concentrates on the effect of confinement and initial lateral separation on the hydrodynamic behaviors of the two drops’ collision; the condition of Re = 0.2 and Ca = 0.2 is taken as a representative boundary condition to guarantee single factor analysis on the confinement and initial lateral separation in the current study.
-2 -3
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FIG. 8. (Color online) Effect of confinement on the flow field around two interactive drops under passing-over motions (Re = 0.2, Ca = 0.2): (a) a/H = 0.050, (b) a/H = 0.075, (c) a/H = 0.150.
unconfined shear flow while the reversing-flow region appears in the confined shear flow due to the wall reflection of the perturbation flow produced by the drop. In addition, with the increasing confinement, the reversing-flow region in matrix fluid turns to be expanded in the confined shear flow, leading to the increasing role of reversing flow on the motion of drops under pairwise interaction. Therefore, as indicated in Fig. 9, the motion of drops under pairwise interaction transits from passing-over motion to reversing motion with the increasing confinement under confined shear flow. Furthermore, under the same motion mode of reversing motion, with an increase in confinement, the terminal streamwise separation of the final equilibrium position decreases, but the smallest separation distance is almost unchanged during the collision stage. Figure 10 also depicts the effect of confinement on transient deformation of drops during pairwise interactions under confined shear flow. As shown, differently from the deformation
1. Effect of confinement
1.2 a/H=0.050 (unconfined) a/H=0.100 (confined) a/H=0.150 (confined)
0.6
y/a
Herein, in order to illustrate the hydrodynamic characteristics during the interaction of two deformable drops under confined shear flow, the typical shapes of deformed drops and velocity field at collision stage for the conventional passing-over motions under unconfined shear flow with a/H = 0.050 are illustrated in Fig. 8(a) as a comparison. As shown, even under the same motion patterns, the drops during pairwise interaction are more extended and flattened than those under unconfined shear flow. This phenomenon is attributed to the fact that the ends of the deformed drops are much closer to the wall region of Couette geometry under confined shear flow. Therefore, the large shear rate in the wall region leads to the great drag on the ends of the deformed drops, resulting in highly stretched drops. Moreover, it is important to note that no reversing flow in matrix fluid is observed under
a/H=0.200 (confined) a/H=0.250 (confined)
0.0
smallest separation
terminal separation
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FIG. 9. (Color online) Effect of confinement on drop trajectories (Re = 0.2, Ca = 0.2, x0 /a = 6, y0 /a = 1.2).
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t* FIG. 10. (Color online) Effect of confinement on drop deformation (Re = 0.2, Ca = 0.2, x0 /a = 6, y0 /a = 1.2).
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variation under passing-over motion, the profile of transient deformation exhibits two peak points. With increasing confinement under the passing-over motion, the variation of drop deformation during the whole interaction process tends to be transited from overdamping to underdamping oscillation. In addition, large confinement also leads to the highly deformed droplet due to additional drag force induced by increased wall effect. Another confinement effect is accelerating the propagation of viscous shear force imposed on the drop, which shortens the transient response before the steady deformation is reached as shown in Fig. 10. 2. Effect of initial lateral separation distance
Besides the confinement, the initial lateral separation also affects the motion behavior of pairwise interactions between drops in the confined shear flow. Figure 11 represents the effects of initial lateral separation on drop motion behaviors. As seen from Fig. 11, when the initial lateral separation is small (e.g., y0 /a from 0.4 to 1.2), the reversing motion occurs, and the passing-over motion is observed at larger initial lateral separation (e.g., y0 /a from 1.4 to 1.6). This phenomenon can be explained by the fact that, in the case of large initial lateral offset, the drop is far from the zone of reversing flow in matrix fluid and more exposed into the passing-flow region as illustrated in Fig. 12, leading to the passing-over motion of drops during pairwise interactions. It is also indicated in
-2 -3
-6
0 Δ x/a
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3
FIG. 12. (Color online) Effect of initial lateral separation on the two interactive drops (Re = 0.2, Ca = 0.2): (a) y0 /a = 0.4, (b) y0 /a = 1.0, (c) y0 /a = 1.6.
Fig. 11 that the terminal streamwise separation between drops in the passing-over motion is larger than that in the reversing motion. In addition, although the movement trajectories are different under various initial lateral separations, the terminal position is unvaried, implying the final balance position of imposed force on the drops after the interaction process is not changed with initial lateral separations. D. Diagram of the flow behavior
According to the current numerical results, a diagram of the flow behavior is plotted in Fig. 13 to illustrate the dependence of the drops’ motion patterns during pairwise interaction on the confinement and initial lateral separation. As shown, the passing-over motion occurs at large initial lateral separation and less confined space. Under the conditions
1.6 1.4
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0.4
y0 / a = 1.4
-1.0 -6
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-2
0 x/a
2
4
0.05
0.15 a/H passing-over trajectory
6
FIG. 11. (Color online) Trajectories of drops in a confined shear flow (a/H = 0.150, Re = 0.2, Ca = 0.2, x0 /a = 6).
0.10
0.20
0.25
reversing trajectory
FIG. 13. (Color online) Regimes of pairwise interaction at finite confinements (x0 /a = 6).
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with weak confinement, the reversing motion occurs rarely; especially, the reversing motion cannot be observed under the confinement of 0.05. The motion of the drops transits from passing over to reversing with increasing confinement. For a given confinement, increasing initial lateral separation may change the drops’ motion from reversing motion to passingover motion. Generally, with the increasing confinement, the transition between passing-over and reversing motion appears at great lateral separation. IV. CONCLUSIONS
In this paper, the drops’ morphologies and underlying mechanisms of hydrodynamic interaction between two neutrally buoyant circular drops in a confined shear flow are investigated by CFD simulation with VOF representation of the interface. The rheological behaviors of interactive drops and the flow regimes are explored, and the underlying transient pressure and velocity contours during the interactive process are presented. The effects of confinement and initial lateral separations on drop trajectory and deformation as well as motion behaviors are examined and analyzed. We find that two types of drop behaviors during interaction occur including passing-over motion and reversing motion, which are governed by the competition between the drag of passing flow and
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the entrainment of reversing flow in matrix fluid. With the increasing confinement, the proportion of the reversing-flow region is enlarged, leading to the increasing volume of the drop exposed to the reversing flow. Meanwhile, the drops’ motion transforms from the passing-over motion to reversing motion, because the entrainment of reversing flow turns to play the dominant role. By enlarging the initial lateral separation, the drop turns to be away from the zone of reversing flow, leading to the increasing drag of ambient passing flow in matrix fluid. And thus, the drops’ behaviors transform from the reversing motion to passing-over motion. In addition, a phase diagram is plotted to quantitatively illustrate the correlation between the effects of confinement and initial lateral separation on the transition between passing-over and reversing motions, which indicates that with the increasing confinement, the transition between passing-over and reversing motion occurs at great lateral separation.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support provided by National Natural Science Foundation of China (Grant No. 51222605) and Natural Science Foundation of Jiangsu Province (Grant No. BK20130009).
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