Morphology of multicore compound drops in oscillatory shear flows

Abstract

Dynamics and morphology of two-dimensional multicore (dual and triple core) compound drops, subjected to oscillating shear flows have been addressed in this work. We use the binary-phase-field method to deduce numerical solutions for the flow field and the droplet deformation characteristics. Our results reveal that depending on the geometric configuration, the inner cores as well as the outer drop may exhibit irregular temporal variations in the deformation, because of a relatively larger number of modes contributing to their shapes. We establish that displacement of the center of masses of the drops generally facilitates this irregular deformation pattern, while for stationary drops, smooth and periodic temporal variations in the shape are observed. As a result, the inner cores in multicore drops are far more prone to exhibit irregular shape variations, while the outer drops tend to follow similar trends when the inner cores are not symmetrically distributed with respect to the imposed flow. The time period of oscillation does not fundamentally alter the morphological characteristics, although it does increase the extent of deformation and impacts the phase lag between the deformation and the imposed flow. Our results may have important implications in characterizing the rheology of double emulsions, when subject to unsteady flows.

Appendices

Appendix A Cahn number and grid independence

Fig. 10

Cahn number independence study for the dual core compound drop in oscillating shear, for the top symmetric configuration. a Outer drop deformation $(D_o)$ vs time (t) for various Cahn numbers: $Cn = 0.03, 0.015$ and 0.01. b Droplet shapes at the time of maximum outer droplet deformation, for those same choices of Cn. Other relevant parameters are: $T_{\text {osc}} = 1, Re = 1, Ca = 0.25$

Figure 10 depicts the temporal variations in the deformation parameter ($D_o$) of the outer drop in panel (a) and the droplet shapes at the time of maximum deformation of the outer drop in panel (b), for the top symmetric configuration of a dual core drop, shown in Fig. 1b, for various choices of $Cn = 0.03,\;0.015$ and 0.01. It is evident that as the Cahn number falls below 0.015, the results for the droplet shape and deformation parameter become independent of its value, thus indicating Cahn-number independence of the numerical results reported herein.

At the same time, since the grid size is chosen based on Cn (see Table 2), the results of Fig. 10 are also sufficient to establish grid independence.

Appendix B Stress variations across the interfaces

Fig. 11

Contours of stress components for the normal stress components $T_{xx}$ in a, $T_{yy}$ in b and the Shear stress component $T_{xy}$ in c, for the dual-core drop in top symmetric configuration with $T_{\text {osc}} = 1$, at the time of maximum outer drop deformation. d–f plot $T_{xx}$, $T_{yy}$ and $T_{xy}$ for the dual-core drop with asymmetric configuration and $T_{\text {osc}} = 3$, at the time of maximum outer drop deformation; the major axes of the drops are represented by the dashed lines, showing the positions of maximum deformation. Other relevant parameters are: $Re = 1, Ca = 0.25, R_1 = 0.3$

In an effort to gain further insights into the local deformation of drop surfaces, we analyze the stress variations across their interfaces. The normal stress balance at the drop interface between fluids i ( = 1, 2) and j (= 2, 3), which governs the local curvature of the drops, reads:

$$\begin{aligned}{}[[\textbf{T}_i - \textbf{T}_j]\varvec{\cdot }\hat{\textbf{n}}_{ij}]\varvec{\cdot }\hat{\textbf{n}}_{ij} = \frac{\varvec{\nabla }\varvec{\cdot }\hat{\textbf{n}}_{ij}}{Ca}. \end{aligned}$$

(B1)

Where, $\hat{\textbf{n}}_{ij}$ is the surface normal at the interface between the i-th and the j-th fluid and $\textbf{T}_i$ is the viscous stress tensor in the i-th fluid ($i = 1,2$), defined as: $\textbf{T}_i = -p_i \varvec{\delta } + [\varvec{\nabla }\textbf{u}_i + (\varvec{\nabla }\textbf{u}_i)^T]$. Since the jump in the normal stresses is balanced by the surface tension acting along the curvature, it follows that larger stress jumps will result in larger curvatures. However, since the phase field formalism inherently possesses a diffuse interface, defining a stress jump across the interface may be non-trivial. Nevertheless, one may still get valuable insights on droplet deformation by simply exploring the normal and the shear stress components in the entire fluid domain.

As such, Fig. 11 exhibits the contours of the normal $(T_{xx},T_{yy})$ and the shear stress $(T_{xy})$ components, for two distinct configurations: (i) top symmetric dual core drop for $T_{\text {osc}} = 1$ in panels (a)–(c) and (ii) asymmetric dual core drop for $T_{\text {osc}} = 3$ in panels (d)–(f). The contour plots have been depicted at the time of maximum deformation in the outer drop, after the periodic state is reached. The major axes of all the drops have been represented as dashed lines (showing the positions of maximum deformation). Other relevant parameters are mentioned in the caption.

It is evident from Fig. 11 (refer to panels (a), (b), (d) and (e)) that for all the drops, the jumps in the normal stresses $(T_{xx},T_{yy})$ across the surface are indeed maximum at the points of maximum curvature (point of intersection of major axes and the droplet interface). On the other hand, the stress jumps are relatively smaller at the points lying close to intersection between the perpendicular to the major axis and the interface, resulting in relatively flatter shapes. The shear stress jump is however, smaller as compared the normal stress differences, because of the pressure contributing to the variations in $T_{xx}$ and $T_{yy}$.