Concentrated phase emulsion with multicore morphology under shear: A numerical study

We numerically study the dynamic behavior under a symmetric shear flow of selected examples of concentrated phase emulsions with multicore morphology confined within a microfluidic channel. A variety of nonequilibrium steady states is reported. Under low shear rates, the emulsion is found to exhibit a solidlike behavior, in which cores display a periodic planetarylike motion with approximately equal angular velocity. At higher shear rates, two steady states emerge, one in which all inner cores align along the flow and become essentially motionless and a further one in which some cores accumulate near the outer interface and produce a dynamical elliptical-shaped ring chain, reminiscent of a treadmillinglike structure, while others occupy the center of the emulsion. A quantitative description in terms of the (i) motion of the cores, (ii) rate of deformation of the emulsion, and (iii) structure of the fluid flow within the channel is also provided.

Figure 1

Equilibrium configurations of (a) a four-core emulsion, $N=4$, and (b) a 19-core one, $N=19$. Only a section of the channel is shown. The colors correspond to the values of $\varphi$, ranging from 0 (black, middle phase) to $\simeq 2$ (yellow, inner and outer phases), while red lines indicate the droplet interface.

Figure 2

(a) Typical nonequilibrium steady state of a four-core emulsion under a symmetric shear flow with $\dot{\gamma }\simeq {10}^{-4}$ ($v_w=0.01$). A clockwise rotation of the internal cores is triggered by a fluid recirculation produced by the shear within the emulsion. Red arrows indicate the direction of rotation. (b) Steady-state velocity field under shear. The snapshots are taken at $t^{*}=72$. Here, $\mathrm{Ca}\simeq 0.2$ and $\mathrm{Re}\simeq 1.2$.

Figure 3

Time evolution of the $y$ (left axis, top plots) and $z$ (right axis, bottom plots) components of the center of mass of the cores at the steady state. They are computed as $y_{\mathrm{cm},i}\left(t\right)=\frac{{\sum }_{y}y\left(t\right){\varphi }_i(y,z,t)}{{\sum }_y{\varphi }_i(y,z,t)}$ and $z_{\mathrm{cm},i}\left(t\right)=\frac{{\sum }_{z}z\left(t\right){\varphi }_i(y,z,t)}{{\sum }_z{\varphi }_i(y,z,t)}$, taken where ${\varphi }_i>0.01$.

Figure 4

Time evolution of the Taylor parameter of the outer droplet $D_O$ and of one of the cores, $D_1$. Time oscillations are due to reciprocal interactions among internal droplets and with the external interface [34].

Figure 5

Top row: Time evolution of a four-core emulsion under a symmetric shear with $\dot{\gamma }\simeq 2.4\times {10}^{-4}$ ($v_w=0.02$). Cores (a) initially rotate clockwise and, afterwards, (b)–(d) gradually align along the shear flow, due to the large stretching of the external interface which prevents further rotation. Bottom row: Velocity field profile under shear. The internal vortex progressively flattens and the resulting steady-state pattern resembles that of a single fluid under shear. Snapshots are taken at time (a) $t^{*}=2$, (b) $t^{*}=7$, (c) $t^{*}=19$, and (d) $t^{*}=31$. Here, $\mathrm{Ca}\simeq 0.4$ and $\mathrm{Re}\simeq 2.4$.

Figure 6

Time evolution of the Taylor parameter of the outer droplet $D_O$ and of the cores $D_1, D_2, D_3$, and $D_4$. The former value is approximately six times higher than the latter ones. Time oscillations at early times progressively weaken and $D$ stabilizes to a roughly constant value.

Figure 7

Top: Time evolution of the $y$ (left axis) and $z$ (right axis) components of the center of mass of the cores. Oscillations progressively flatten at late times since cores stabilize and align along the shear direction. Bottom: Time evolution of the $y$ (left axis) and $z$ (right axis) components of the velocity of the center of mass of the cores. A quick increase of the speed, at early times, is followed by its rapid decay to zero at the steady state.

Figure 8

(a) Typical nonequilibrium steady state of a CPE with 19 cores under a symmetric shear flow with $\dot{\gamma }\simeq 8\times {10}^{-5}$ ($v_w=0.01$). Once again, a clockwise rotation of the internal cores is triggered by a fluid recirculation produced by the shear within the emulsion. Red arrows indicate the direction of rotation. (b) Steady-state velocity field under shear. Snapshots are taken at $t^{*}=11$. Here, $\mathrm{Ca}\simeq 0.2$ and $\mathrm{Re}\simeq 1.6$.

Figure 9

Top: Time evolution of the $y$ (left axis) and $z$ (right axis) components of the center of mass of five cores (those reported in Fig. 8). All cores follow a periodic dynamics, but the peripheral ones travel along trajectories larger than the ones covered by inner cores. Bottom: Time evolution of the $y$ (left axis) and $z$ (right axis) components of the velocity of the center of mass. Inner cores move at a speed lower than the one of the peripheral cores. The central core is essentially motionless.

Figure 10

Typical time evolution of the angular velocity $\omega$ of four cores of the emulsion in Fig. 8, calculated with respect to $|{\omega }_{max}|\simeq 0.006$. The peripheral and internal cores travel along different paths with approximately equal values of $\omega$.

Figure 11

Top row: Time evolution of a 19-core emulsion under a symmetric shear with $\dot{\gamma }\simeq 2.4\times {10}^{-4}$ ($v_w=0.03$). The outer droplet preserves its circular shape (a) only for a short period of time since, due to the shear flow, it gradually stretches and attains, at the steady state, (b)–(d) an elliptical-like geometry. (a) While, initially, all internal cores rotate clockwise, (b)–(d) later, some remain in the center of the emulsion and others accumulate close to the outer interface and rotate clockwise. Bottom row: Velocity field profile under shear. Like in Fig. 5, the internal vortex progressively flattens, although, at the steady state, its structure exhibits weak deviations from a plain elliptical pattern due to the motion of the internal cores which perturb the field. Snapshots are taken at time (a) $t^{*}=1$, (b) $t^{*}=8$, (c) $t^{*}=13$, and (d) $t^{*}=34$. Here, $\mathrm{Ca}\simeq 0.6$ and $\mathrm{Re}\simeq 5$.

Figure 12

Left: Typical steady-state configuration of the multicore emulsion taken at $t^{*}=34$. Right: Time evolution of the deformation parameter $D$ of the outer droplet ($D_O$) and of the cores (from $D_1$ to $D_7$). At the steady state, the former is roughly six times higher than the latter.

Figure 13

Top: Time evolution of the (a) $y$ and (b) $z$ components of the center of mass of the cores numbered in Fig. 12. Bottom: Time evolution of the (c) $y$ and (d) $z$ components of their center-of-mass velocity. At the steady state, three cores very weakly fluctuate in the middle of the emulsion, while the others rotate periodically clockwise, with $v_{y_{cm}}$ twice as large as $v_{z_{cm}}$.

Figure 14

Approximate nonequilibrium steady states of a 19-core emulsion attained for different values of $\dot{\gamma }$. Green arrows indicate the direction of the moving walls. This applies to all snapshots. For increasing values of the shear rate, the shape of the emulsion varies from (a) a weakly eccentric ellipse towards (b)–(d) a mild and (e),(f) a highly stretched one, up to (g) a state characterized by two bulges connected by a thinner layer of fluid. The cores generally move along elliptical trajectories, but, at (c)–(e) high values of shear rates, some accumulate in the middle of the emulsion where they are repeatedly hit by those in the surroundings. At very high $\dot{\gamma }$, the regular circular motion is replaced by (g) a more chaotic one, in which some cores aggregate and rotate within the bulges of the stretched droplet, while others fluctuate in the connecting layer of fluid.

Figure 15

Velocity profiles observed within and in the surrounding of the emulsions shown in Figs. 14 and 14. The highly stretched recirculation structure shown in (a) is replaced, in (b), by two separate clockwise vortices located at the extremities of the emulsion and connected by a thin layer of fluid where a weaker and irregular fluid pattern is found.