The effects of confinement and inertia on the production of droplets

Rheologica Acta manuscript No. (will be inserted by the editor) Y. Renardy The effects of confinement and inertia on the production of droplets Received: date / Accepted: date Abstract Recent experiments of [1] investigate the effect of walls on flowinduced drop deformation for Stokes flow. The drop and the fluid in which it is suspended have the same viscosities. The capillary numbers vary from 0.4 to 0.46. They find that complex start-up transients are observed with overshoots and undershoots in drop deformation. Drop breakup is inhibited by lowering the gap. The ratio of initial drop radius to wall separation is fixed at 0.34. We show that inertia can enhance elongation to break the drop by examining Reynolds numbers in the range 1 to 10. The volumes of the daughter drops can be larger than in the unbounded case, and even result in the production of monodisperse droplets. Keywords Volume-of-fluid methods drop breakup drop deformation PACS 47.11.-j,47.55D-,47.55.df 1 Introduction The effect of walls at close proximity to drops in processing situations is an important issue in microfluidics industries [2 4]. Ref. [5] investigates the influence of confinement on the steady state microstructure of emulsions sheared between parallel plates, in a regime where the average droplet dimension is comparable to the gap width between the confining walls. It is found that droplets can organize into layers, depending on the migration to the centerline due to wall effects and coalescence. In this paper, we focus on a single drop and examine the effect of inertia on drop deformation when the drop dimension is of the order of the gap width. Y. Renardy Department of Mathematics and ICAM, 460 McBryde Hall, Virginia Tech, Blacksburg VA 24061-0123, USA Tel (540) 953 1823 E-mail: renardyy@math.vt.edu

2 z x y Fig. 1 Schematic of initial condition for the computational domain. In dimensionless form, the upper and lower walls are a unit distance apart, the drop radius is denoted a, and the computational box covers 0 x L x, 0 y L y, 0 z 1. The initial condition for our numerical simulation is a spherical drop of viscosity µ d and density ρ, suspended in another liquid of viscosity µ m and the same density. The drop is placed at the center of the channel, which induces flow symmetry in y and antisymmetry in (x,z) and the drop does not migrate. Figure 1 is a schematic of the initial condition. There is an imposed constant shear rate γ. Time is non-dimensionalized with γ. The top wall moves in the x-direction, and the bottom wall in the opposite direction. The initial velocity field is simple shear in both liquids. This adjusts quickly to the flow solution during the simulation. Distances are non-dimensionalized with respect to the plate separation. The initial dimensionless drop radius is denoted a, fixed at 0.34. The computational box has periodic boundary conditions in the x and y directions. The fluids are incompressible and satisfy the Navier-Stokes equation. At the fluid interface, velocity and shear stress are continuous, and the jump in the normal stress is balanced by interfacial tension force [6]. The dimensionless parameters are: the viscosity ratio of the drop to matrix liquids λ = µ d /µ m, (1) the capillary number, the Reynolds number Ca = µ m γa/σ, (2) Re = ρ γa 2 /µ m, (3) the dimensionless spatial periodicities L x and L y in the x and y directions, respectively, and radius a. L x and L y are chosen sufficiently large that dropdrop interactions can be neglected. Wall separation is denoted L z, kept fixed at 1. We focus on the competition between viscous force, capillary force and inertia when the drop size is comparable to the plate separation. There are

3 examples in unbounded flow where inertia induces drop breakup, and in an analogous way, one expects this to occur for confined flows. Two examples from unbounded flows are: 1. Stokes flow at λ = 1 has a critical capillary number Ca c 0.43, below which the drop does not break [7 14]. The addition of inertia, however, leads to breakup as documented in [11,15]. In particular, Sec. III of [15] describes the mechanism for inertia driven breakup: when inertia is important in the matrix liquid, Bernoulli s equation states that p + ρ v 2 /2 is a constant along each streamline and therefore, the large velocities near the tips induce negative pressures. The resulting suction leads to further tilting upward of the drop. This is an aerodynamic lift on one end, together with a counter-lift at the other. The drop is exposed to larger shear with further tilting, which allows the base flow to pull away the ends. Fig. 17 of [11] and fig. 1 of [15] show the dependence of the critical capillary number on Reynolds number for viscosity ratio 1. In contrast, the effect of inertia on drop deformation in elongational flow is small and stabilizing [16]. 2. Stokes flow with a very viscous drop has a critical viscosity ratio λ c 3.1, beyond which the drop aligns with the flow and will not break. It is shown in [17] for 2D and [18] for 3D that the addition of inertia increases the critical viscosity ratio. At any viscosity ratio, we can find a Reynolds number that is sufficiently high to break the drop. The numerical method used in the simulations is the volume-of-fluid method (VOF) with the parabolic reconstruction of the interface shape in the surface tension force (PROST) [19, 6, 20] or alternatively the continuous surface force formulation (CSF) [21 28] with the piecewise linear interface reconstruction scheme (PLIC) and modifications for low Reynolds number flows described in [11], e.g., the semi-implicit time integration scheme. Sec. 2 is a verification of results of the numerical simulations with experimental data of [1] in Stokes flow. These situations are examined with the addition of inertia in sec. 3 and breakup simulations are presented. 2 Retraction in Stokes flow due to confinement Experimental results of [1] are compared with numerical simulations in figs. 2 and 5. The fluids satisfy λ = 1, and conditions of Stokes flow. The initial drop radius is a = 0.34. In fig. 2, Ca = 0.4 and in fig. 5, Ca = 0.46. Both are below criticality here. In the unbounded case, Ca = 0.4 would be below criticality and Ca = 0.46 above. The numerical simulations in fig. 2 are continued much longer than the experimental data to illustrate the novel oscillation in the length evolution. The vertical axis is the dimensionless length of the drop in the x-z cross-section and the time axis is in units of dimensionless capillary time (dimensional time is t γ 1 ). In the unbounded case, the drop length increases to the stationary value of 2.1 without the overshoot and retraction. Figure 3 shows the velocity field in the x-z cross-section through the drop center, and drop shape for the

4 Fig. 2 Ca = 0.4, a = 0.34. A comparison of experimental data of [1] for Stokes flow (o) vs numerical simulation with Re = 0.1 ( ). t/ca = 6, 66.88, 137.61, and 191.47 correspond to 1-4 on the graph. Dimensional time is t γ 1. The VOF-CSF- PLIC algorithm of [11] is used. The computational domain is 0 x 4, 0 y 2, 0 z 1. Spatial and temporal discretizations are x = y = z = 1/64 and t = 0.00025, respectively. unbounded case. Compared with this, the confined drop elongates twice as much. At locations 1-4 of fig. 2, the computed velocity vectors and drop shapes are shown in fig. 4. The cross-sectional cut for the velocity vector plots is through the center of the drop in the x-z plane. Lengths are scaled with the wall separation L z = 1. There is symmetry in the y-direction into the paper, and anti-symmetry in the x-direction. The computational domain is 0 x 4, 0 y 1, 0 z 1. Initially, the drop has radius a = 0.34 and is centered at x = 2, y = 1, z = 0.5. In the last frame of Fig. 4, the drop tip is at x 0.9 where the shear rate is highest. Figure 5 at Ca = 0.46 shows the stabilizing effect of the walls. Again the circles represent experimental data in the top graph, and the line denotes numerical simulations. The drop tips are pushed to the middle of the channel

5 Fig. 3 Ca = 0.4, unbounded, Stokes flow. Numerical simulation of stationary state, t/ca = 22, L/2a = 2.1. Velocity vector field is shown for the x-z cross-section of the drop through its center, near the tip where shear rates are highest. where elongation ceases and is replaced by retraction. The velocity vector plots are shown at the first two data points given in the experimental data. Without the presence of the walls, the drops would break for this capillary number. A feature of the unbounded case is that when the drop is elongated to this extent, end-pinching always occurs. A drop in an unconfined shear flow deforms into a dumbbell shape once it exceeds a critical amount of deformation (see fig. 6. Fluid drains from the neck of the dumbbell into the ends, which become the largest fragments after the drop breaks. Much smaller fragments form from the remnants of the neck. In a confined flow, the gap between the drop and the wall is much smaller near the ends of the drop than it is near the middle. This leads to an increase in the shear rate of the matrix fluid next to the ends of the drop. Let the velocity be denoted (u x, y y, y z ). Fig. 7 shows the shear rates du x /dz vs z at fixed x = 0.77 in fig. 3 (- -) and at x = 0.9 in the last velocity plot of fig. 4 (-). These are the x-values where the shear rates are highest in each flow. Note that higher values are attained for the confined flow and this persists over a larger interval in z than in the unbounded flow. The positions of the peaks occur close to the drop tip, near the lower part of the interface. Fig. 8 shows the elongational flow component du z /dz vs z at fixed x-values where the highest values are achieved; (- -) for the unbounded flow of fig. 3 and (-) for the final stage of the bounded flow of fig. 4. For both flows, the elongational components and shear rates are highest at roughly the same locations with respect to the drop tips. The higher shear rate has two effects. First, the increased drag extends the drop to a greater length. Secondly, the high shear rates near the tips of the drop would tend to elongate the tips rather than allow them to evolve into

6 Fig. 4 Ca = 0.4, a = 0.34. Stokes flow, Re = 0. Numerical results for velocity vector plots and drop shapes at t/ca = 6, 66.88, 137.61, and 191.47 indicated in fig. 2. In each vector plot, the walls are located at z = 0, 1. The velocities are magnified at the drop tips to show the area where they differ the most from simple shear. Shear rates are highest at the lower bottom of the tip interface.

dumbbells. Compared to the unbounded case, the drop assumes a sausage-like shape without evolving into the dumbbell shape that precedes end-pinching. This can be sustained in equilibrium at much greater lengths than is possible in unconfined flows. If the drop does break, the sausage yields fragments of roughly equal size. 7

8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.5 1 1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.5 1 1.5 Fig. 5 Ca = 0.46, a = 0.34. Stokes flow Re = 0. Experimental data of [1] (o) vs numerical simulation with VOF-PROST of [19] ( ) show drop stabilization due to confinement. Velocity vectors in the x-z plane through the center of the drop are averaged and plotted every 2 grid points at t/ca = 77.4, and 98.20. The velocity field is magnified near the tip where shear rates are highest and the drop center is at x = 3, y = 1, z = 0.5.

γ 9 Fig. 6 Numerical simulation with VOF-PROST of fragmentation in unbounded simple shear. Stokes flow (Re = 0.1), Ca = 0.46, a = 0.125, λ = 1, left to right t/ca = 0, 11, 43, 65, 93. 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z Fig. 7 Ca = 0.4, Stokes flow. Shear rates vs z for x fixed at the tip where the highest shear rates are attained. Unbounded flow (-.-) with x = 0.77 in fig.3; confined flow a = 0.34, x = 0.9 in the final plot of fig. 4 ( ). Shear rates are higher in the confined flow.

10 0.5 0.4 0.3 0.2 du z /dz 0.1 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z Fig. 8 Ca = 0.4, Stokes flow. Elongational component du z/dz vs z for x fixed at the value where the highest elongational component of the velocity field are attained. Unbounded flow (-.-) with x = 0.8 in fig.3; confined flow a = 0.34, x = 1 in the final plot of fig. 4 ( ). Elongation rates are higher in the confined flow.

11 θ 45 40 35 30 25 20 15 10 5 L/2a 5.5 5 4.5 4 3.5 3 2.5 2 1.5 (a) 0 0 50 100 150 200 t/ca 1 0 50 100 150 200 t/ca Fig. 9 Elongation due to inertia for the confined drop at Ca = 0.4, a = 0.34. Evolution of (a) angle of inclination and (b) half-length/initial radius for increasing Reynolds number Re = 0.1 ( ), 1 (...), 2 (-.). Computational domain L x = 4, L y = 2, L z = 1. (b) 3 Elongation due to inertia The effect of inertia on drop deformation in shear flow is strongly destabilizing. As discussed in Sec. 1, this is a result of the Bernoulli suction which results from the increase in fluid speed as the matrix fluid rounds the tip of the drop. This suction pulls the drop outward, increasing the angle it forms with the horizontal and exposing it to a larger shear [11,15]. To exemplify this, we take the case Ca = 0.46, a = 0.34 of sec. 2. With sufficient inertia, this drop would elongate to the point where it breaks. Figure 9 shows the evolution of L/2a (half-length/radius of the drop) at Re = 0.1 (Stokes flow), 1, and 2. The drop retracts for Stokes flow and Re = 1. However, the drop breaks at Re = 2 at the point indicated. The angle of inclination of the drop θ is similar for the Reynolds numbers displayed. The oscillations in drop length also occur without inertia, as in fig. 4; the physical mechanism and determination of the period remain open problems. Figure 10 compares the corresponding drop shapes for time up to t/ca 30 for Stokes flow and Re = 10. At the higher Reynolds number, the drop tips stay closer to the walls; for instance, the tips are 0.14 units away from the walls at Re = 10 and 0.2 units for Stokes flow. The suction toward the walls by inertia leaves the drop tips exposed to higher velocities and eventual breakup, while the Stokes flow case leaves the tips closer to the channel centerline where velocities are lower. We next examine the system at the lower capillary number Ca = 0.4 of fig. 9. With the addition of inertia to the level of Re = 2, the drop breaks, as shown in fig. 11. The drop does not break for Re = 1, so there is a critical amount of inertia that is needed to extend the drop sufficiently for pinch-off to ensue. The final plot (d) in the figure shows daughter drops that are settling to stationary shapes, while the background shear flow would continue to pull those daughters away from each other. The volumes of the daughter drops are comparable to the central drop; thus, the three droplets are of comparable sizes. This monodispersity is in contrast to the case without confinement.

12 Fig. 10 Ca = 0.4, a = 0.34. Drop interface shape in the x-z plane through drop center, drawn every t/ca = 2.2 units, for Stokes flow (top), and Re = 10. Fig. 6 in Sec. 2 shows the typical daughter drop distribution resulting from end-pinching in the unbounded case. The mother drop deforms into a dumbbell shape. Fluid drains from the neck into the ends of the dumbbell, which become the largest fragments after breakup. Much smaller fragments form from the neck. In the presence of the walls however, as we find here, the growth of the dumbbells is inhibited, and the drop assumes a sausage-like shape. Fragments formed when this sausage breaks up are roughly equal in size. Fig. 12 for Ca = 0.46 extends the Stokes flow results from fig. 5. The Re = 2 case breaks at the point indicated. The Re = 5, 10 cases were simulated for only a short time interval. The breakup scenario for Re = 2 is depicted in fig. 13. We conclude that for Ca = 0.46 and a = 0.34, the critical Reynolds number lies between 1 and 2.

13 (a) (b) (c) Fig. 11 Addition of inertia causes breakup for the confined drop, Ca = 0.4, Re = 2, a = 0.34. Numerical simulations with VOF-PROST t/ca = (a) 28, (b) 114, (c) 118, (d) 119. Computational domain 0 x 3, 0 y 2, 0 z 1, spatial discretization x = y = z = 1/64, temporal t = 0.001. In comparison, VOF- CSF-PLIC predicts a similar scenario, with maximum drop length 5% longer and breakup time 3% sooner. (d)

14 10 L/2a 8 6 4 2 0 20 40 20 40 t/ca 60 80 100 60 80 100 40 θ 30 20 10 0 0 t/ca Fig. 12 Ca = 0.46, a = 0.34. Evolution of half-length/radius (L/2a) and angle of inclination θ for increasing Reynolds numbers: Re = 0.1 ( Stokes flow), 1 (- -), 2(...), 5 (-.-), 10 (...). Fig. 13 Addition of inertia causes breakup for the confined drop, Ca = 0.46, a = 0.34, Re = 2, t/ca = 6, 31, 41, 44, 49. Numerical simulations are done with VOFCSF-PLIC in a computational domain Lx = 8, Ly = 2, Lz =1, spatial discretization x = y = z = 1/64 and temporal discretization t = 0.0005.

15 4 Conclusion The influence of inertia on three-dimensional drop deformation and breakup in a confined channel shear flow is investigated with numerical simulations. One viscosity ratio, namely unity, and one value of confinement are studied. The numerical results for Stokes flow are compared with recent experimental data for the purpose of validation and good agreement is found. For unbounded flow, the critical value of the capillary number for viscosity ratio 1 is 0.43. This critical value changes with confinement and Reynolds number. At the fixed confinement of initial drop radius to wall separation ratio 0.34, and capillary number 0.4, the critical Reynolds number is between 1 and 2. There is a slowing down of drop dynamics and breakup inhibition due to wall effects, and a new breakup mode with monodisperse fragment distribution. These issues are relevant for microfluidics applications.

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