Deformation and burst of a liquid droplet with viscous surface moduli in a linear flow field

Suspensions of fluid particles with complex interfacial architecture (for instance, capsules, vesicles, polymersomes, lipid bilayers, and emulsions embedded with certain surface-active agents and surfactants) find an immense number of applications in the field of engineering and bioscience. Interfacial rheology plays an important role in the dynamics of many of these systems, yet little is understood about how these effects alter droplet deformation and breakup. In this study, we develop a theoretical model that explores the deformation and breakup of a single droplet with a viscous surface modulus suspended in an unbounded immiscible fluid under a general linear flow field. The viscous interface is modeled as a two-dimensional surface having a surface shear viscosity ${\eta }_{\mu }$, surface compression/dilational viscosity ${\eta }_{\kappa }$, and a constant surface tension over the interface, using a Boussinesq-Scriven constitutive relationship. We present the droplet breakup analysis in Stokes flow in the limit of small droplet deformation using a turning point analysis similar to that of Barthes-Biesel and Acrivos [J. Fluid Mech. 61, 1 (1973)]. In particular, we examine how the critical capillary number for breakup depends on the interfacial viscosity for different viscosity contrasts between the inner and outer fluid and different flow types. For all the flows considered, we observe that ${\eta }_{\kappa }$ is found to have a destabilizing impact on droplet breakup, whereas ${\eta }_{\mu }$ has a stabilizing effect. We explore the physical picture behind these observations in this work.

Figure 1

Problem setup: dimensional quantities.

Figure 2

Shape of a deformed droplet placed under uniaxial extensional flow.

Figure 3

Droplet deformation versus Ca in uniaxial flow, from both the $O\left(\text{Ca}\right)$ (blue line) and $O\left({\text{Ca}}^2\right)$ (red line) analyses ($\lambda =1$, ${\text{Bq}}_{\mu }=0$ and ${\text{Bq}}_{\kappa }=0$). The $y$ axis is $-D/3$, where $D$ is the $x_1$ component of the shape tensor $D_{ij}$. Positive values of Ca correspond to extension along the $x_3$ axis (prolate shape), and negative values of Ca correspond to compression along the $x_3$ axis (oblate shape). For this particular set of parameters, a droplet with $D<-2.2$ forms a prolate shape, while a droplet with $D>-2.2$ forms an oblate shape. The stable and unstable prolate droplet shapes at $\text{Ca}=0.05$ are shown in the $x_1\text{−}{x}_3$ plane.

Figure 4

${\text{Ca}}_C$ versus $\lambda$ for a clean droplet in uniaxial extensional flow. The snapshots are deformed droplet shapes at ${\text{Ca}}_C$. Note: for $\lambda =0.02$, the shape from the perturbation analysis is not valid because the droplet is highly deformed. Slender body droplet theories like in Refs. [12, 51, 52] are more valid in this regime ($\lambda \ll 1$).

Figure 5

(a) ${\text{Ca}}_C$ versus $\lambda$ for different dilational Boussinesq numbers ${\text{Bq}}_{\kappa }$ in uniaxial extensional flow. The shear Boussinesq number ${\text{Bq}}_{\mu }=0$. The snapshots are at ${\text{Ca}}_C$. (b) Schematic of the mechanism behind reduced stability. Black lines represent the existing circulation pattern, and purple lines are the induced counterflow from interfacial effects.

Figure 6

(a) ${\text{Ca}}_C$ versus $\lambda$ for different shear Boussinesq numbers ${\text{Bq}}_{\mu }$ in uniaxial extensional flow. The dilational Boussinesq number ${\text{Bq}}_{\kappa }=0$. The snapshots are at ${\text{Ca}}_C$. (b) Schematic of the mechanism behind enhanced stability. Black lines represent the existing circulation pattern, and purple lines are the induced flow from interfacial effects.

Figure 7

${\text{Ca}}_C$ versus $\lambda$ at different values of ${\text{Bq}}_{\mu }={\text{Bq}}_{\kappa }=\text{Bq}$ under uniaxial flow. $\text{Bq}=0$ corresponds to a clean droplet.

Figure 8

Combination of Boussinesq numbers that have a ${\text{Ca}}_C$ identical to that of a clean droplet under uniaxial flow at $\lambda =5$. “Less stable” (“more stable”) implies that the droplet has its critical point at a lower (higher) ${\text{Ca}}_C$ than a clean droplet at the same viscosity ratio.

Figure 9

Deformed droplet shape in planar hyperbolic flow.

Figure 10

$D_1$ ($x_1$ component of shape tensor) versus Ca from both $O\left(\text{Ca}\right)$ (blue lines) and $O\left({\text{Ca}}^2\right)$ (red lines) analyses for planar hyperbolic flow. Lower branches (bold lines) represent stable solutions, and the upper branches represent unstable states.

Figure 11

Comparison between theoretical and experimental values of ${\text{Ca}}_C$, for a clean droplet (${\text{Bq}}_{\mu }={\text{Bq}}_{\kappa }=0$) in planar hyperbolic flow. Experimental results are from Bentley and Leal [13]. The theory does not provide accurate predictions for $\lambda \ll 1$ (demarcated by vertical dotted line), as deformations are large.

Figure 12

${\text{Ca}}_C$ versus $\lambda$ at different values of ${\text{Bq}}_{\mu }$ in planar hyperbolic flow. The dilational Boussinesq number is ${\text{Bq}}_{\kappa }=0$. The snapshots are at ${\text{Ca}}_C$.

Figure 13

${\text{Ca}}_C$ versus $\lambda$ at different values of ${\text{Bq}}_{\kappa }$ in planar hyperbolic flow. The shear Boussinesq number is ${\text{Bq}}_{\mu }=0$. The snapshots are at ${\text{Ca}}_C$.

Figure 14

Droplet placed under simple shear flow.

Figure 15

${\text{Ca}}_C$ versus $\lambda$ for a clean droplet (i.e., ${\text{Bq}}_{\mu }={\text{Bq}}_{\kappa }=0$) in shear flow along with experimental results from Torza et al. [57]. Experimental images on the right are from Rumscheidt and Mason [58]. These images represent how a droplet's shape will change as Ca increases for a low-viscosity ratio droplet ($\lambda =2\times {10}^{-4}$) and a high-viscosity ratio droplet ($\lambda =6$). At large Ca, a low-viscosity ratio droplet forms pointed ends from which small droplets eject (as depicted by the dashed lines emanating from the droplet tips).

Figure 16

${\text{Ca}}_C$ versus $\lambda$ at different values of ${\text{Bq}}_{\mu }$ in shear flow. The dilational Boussinesq number is ${\text{Bq}}_{\kappa }=0$. The snapshots are shown at ${\text{Ca}}_C$.

Figure 17

${\text{Ca}}_C$ versus $\lambda$ at different values of ${\text{Bq}}_{\kappa }$ in shear flow. The shear Boussinesq number is ${\text{Bq}}_{\mu }=0$. The snapshots are shown at ${\text{Ca}}_C$.

Figure 18

Combination of Boussinesq numbers that have a ${\text{Ca}}_C$ identical to that of a clean droplet under shear flow at $\lambda =1$. “Less stable” (“more stable”) implies that the droplet has its critical point at a lower (higher) ${\text{Ca}}_C$ than a clean droplet at the same viscosity ratio.