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^{1,2,3}and Rajesh Kumar Singh

^{2,3}

### Abstract

Pair-collision between viscous drops in a confined shear is simulated to show that the confinement alters the trajectories of the drops spatially ordering them at a finite separation in the center of the domain. In contrast to free shear where drops eventually adopt free streamlines with a finite cross-stream separation, here they move towards the centerline achieving zero cross-stream separation but a net stream-wise separation. The latter varies as inverse of capillary number and cube of the confinement (distance between the walls). The final stream-wise separation does not depend on the initial positions of the drops when the drops are in the same shear plane. The separation decreases approximately linearly with the initial separation in the vorticity direction. An analytical theory explaining the phenomenon is presented. Effects of the ratio of drop to matrix viscosity are briefly investigated.

### Key Topics

- Stokes flows
- 8.0
- Hydrodynamics
- 5.0
- Viscosity
- 5.0
- Vortex dynamics
- 5.0
- Microscale flows
- 4.0

## Figures

Hydrodynamic interaction of a pair of drops in a confined shear for Ca = 0.2, L y = 5a, Δx 0/a = 2.5, and Δy 0/a = 0.25. Drops travel towards the center of the domain.

Hydrodynamic interaction of a pair of drops in a confined shear for Ca = 0.2, L y = 5a, Δx 0/a = 2.5, and Δy 0/a = 0.25. Drops travel towards the center of the domain.

(a) Relative trajectory of the drops at Ca = 0.2, Δx 0/a = 2.5, and Δy 0/a = 0.25 for different L y values. (b) The actual trajectories in the domain L y = 4.5a.

(a) Relative trajectory of the drops at Ca = 0.2, Δx 0/a = 2.5, and Δy 0/a = 0.25 for different L y values. (b) The actual trajectories in the domain L y = 4.5a.

Effects of the initial positions on the relative trajectory: (a) Variation of initial separation in the flow direction Δx 0/a for Δy 0/a = 0.50, Ca = 0.20, and L y = 5a. (b) Effects of initial separation in the gradient direction Δy 0/a for Δx 0/a = 2.50 in the same domain and for the same capillary number.

Effects of the initial positions on the relative trajectory: (a) Variation of initial separation in the flow direction Δx 0/a for Δy 0/a = 0.50, Ca = 0.20, and L y = 5a. (b) Effects of initial separation in the gradient direction Δy 0/a for Δx 0/a = 2.50 in the same domain and for the same capillary number.

(a) Effects of the initial separation in vorticity direction on the relative trajectory of a pair of drops at Δx 0/a = 2.5, Δy 0/a = 0.25, Ca = 0.20, and L y = 5a. (b) Effects of Δz 0/a on Δx final /a and y max /a.

(a) Effects of the initial separation in vorticity direction on the relative trajectory of a pair of drops at Δx 0/a = 2.5, Δy 0/a = 0.25, Ca = 0.20, and L y = 5a. (b) Effects of Δz 0/a on Δx final /a and y max /a.

(a) Effects of confinement in the gradient direction (L y ) on Δx final at Ca = 0.20. (b) Δx final increases linearly with 1/Ca.

(a) Effects of confinement in the gradient direction (L y ) on Δx final at Ca = 0.20. (b) Δx final increases linearly with 1/Ca.

(a) Variation of lateral velocity of the drops with y after collision with increasing confinement from the top at Ca = 0.20 along with analytical results (straight lines) of Chan and Leal. 31 (b) The scaling of velocity with L y .

(a) Variation of lateral velocity of the drops with y after collision with increasing confinement from the top at Ca = 0.20 along with analytical results (straight lines) of Chan and Leal. 31 (b) The scaling of velocity with L y .

(a) Variation of the post-collision lateral velocity of the drops with y at different Ca in the confined domain L y = 6a. (b) The scaling of velocity with capillary numbers.

(a) Variation of the post-collision lateral velocity of the drops with y at different Ca in the confined domain L y = 6a. (b) The scaling of velocity with capillary numbers.

(a) Composite scaling of Δx final /a with L y and Ca for many different values of L y and Ca. (b) The relative trajectory of the drops with appropriate scaling.

(a) Composite scaling of Δx final /a with L y and Ca for many different values of L y and Ca. (b) The relative trajectory of the drops with appropriate scaling.

(a) Effects of λ on the relative trajectory of a pair of drops at Δx 0/a = 2.5, Δy 0/a = 0.25, Δz 0/a = 0, Ca = 0.20, and L y = 5a. (b) The effect of λ on Δx final /a.

(a) Effects of λ on the relative trajectory of a pair of drops at Δx 0/a = 2.5, Δy 0/a = 0.25, Δz 0/a = 0, Ca = 0.20, and L y = 5a. (b) The effect of λ on Δx final /a.

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