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.