Morphology of compound viscoelastic drops in extensional flows

Compound droplets of complex fluids (such as polymeric liquids) are becoming increasingly prominent for their applications in targeted drug delivery, cell and particle encapsulation, and many other micro- and millifluidic processes. Despite this, the morphological behaviors of such drops are yet to be properly addressed even in simple canonical flows. The main challenge in this regard perhaps originates from the complex and non-linear constitutive relation of the constituent fluids. To address this, here, we analyze the flow field and the deformations of a compound viscoealstic drop, subject to uniaxial extensional flows. All three phases are considered to obey the Giesekus constitutive model, known for its ability to accurately capture the rheological properties of many polymeric liquids. We derive asymptotic solutions for the limiting case of small deformation and weak viscoelasticity, and subsequently validate them against full numerical simulations based on the ternary phase field method. The results, derived mainly from the asymptotic analysis, demonstrate that the elongational elastic stresses help reduce the deformations in both the shell and the core, and this is facilitated by the shear-thinning nature and the finite extensibility of the Giesekus model. We also show that depending on the extent of viscoelasticity of the outermost phase and the core size, the shape of the shell may change from prolate to oblate and vice versa for the core. The viscoelasticity of the core on the other hand has relatively little influence on the deformation of the shell, although it is found to significantly impact the core's evolution.