Elasto-inertial deformations of viscoelastic compound drops in extensional flows

Compound drops of polymeric liquids have become increasingly prominent in applications related to particle and cell encapsulation, targeted drug delivery, and various other micro- and millifluidic systems. However, comprehensive analysis of their morphology even in simple canonical flows such as shear or extensional flows is still in their infancy. To address this, here, we explore the deformations of such compound polymeric drops subject to externally imposed extensional flows. As such, we treat the polymeric liquids using the nonlinear viscoelastic Giesekus constitutive model, and employ a ternary phase-field formalism to numerically simulate the flow and shape deformations of the shell and the core. The ternary phase-field scheme enables us to consider the properties of all three phases as distinct, while venturing beyond the constraints of low inertia and weak viscoelasticity. Our results reveal that depending on the extent of inertia, viscoelasticity of the constituent phases may induce shape oscillations of the drops during the initial transience. At the same time, when the polymers have finite extensibility, their elasticity inhibits droplet deformation, whereas inertial forces act in exactly the opposite way. However, when the polymers are infinitely extensible (special case of the Oldroyd-B fluid), the excess polymeric stresses may actually enhance the deformation. We show that there exists a critical extensional rate beyond which the drops tend to undergo continuous elongation in extensional flows. The elastic nature of the polymeric fluids, however, tends to push this critical rate to higher values, thus inhibiting continuous elongation and potential breakup in the process.