Interfacial viscosity-dictated morpho-dynamics of a compound drop in linear flows

Compound droplets are excellent analogs of complex biological entities such as vesicles or cells. Despite significant advancements toward understanding the morphological evolution of a compound droplet in an incipient flow, the specific role of interfacial rheology toward dictating the same remains unaddressed. Here, we bring out non-trivial implications of interfacial rheology on the deformation of a compound drop subject to an imposed flow. The interfacial viscosity, in effect, interacts with the flow-induced non-uniform surfactant distribution to alter the droplet morpho-dynamics in a rather engaging manner. We employ a closed-form analytical approach to delineate the relative roles of advective and diffusive transport. In the paradigm of diffusion-dominated interfacial transport, viscous interfacial stress arrests the droplet deformation, thus enhancing its stability. However, for large values of the interfacial dilatational viscosity, the drop deformation increases with the interfacial shear viscosity. On the contrary, in the paradigm of surface convection-dominated surfactant transport, the interfacial rheology does not have any significant effect on either the shape deformation or the emulsion rheology. These results may pave a way toward explaining several unique features of complex fluid-fluid interfaces encountered in nature and biology.