Drop deformation in planar extensional flows: Numerical study on viscoplastic drops and viscoplastic surrounding media

The deformation of a drop in an extensional flow of a second immiscible fluid has importance in various industrial processes. In most of the industrial applications, fluids exhibit non-Newtonian behavior. Despite their importance, the mechanism governing the deformation of non-Newtonian drops subjected to an extensional flow remains poorly understood. The same assertion applies to Newtonian drops immersed in non-Newtonian fluids. This work numerically investigates the effects of plasticity on the deformation of a drop immersed in a second fluid undergoing planar extensional flow. We start our analysis with both fluids being Newtonian. The deformation of a viscoplastic drop in a planar extensional flow of a Newtonian fluid is analyzed and qualitatively compared with a work that explored axisymmetric extensional motions. Next, the deformation of a Newtonian drop immersed in a viscoplastic material under planar extensional flow is analyzed. When both liquids are Newtonian, the deformation increases with capillary number for a fixed viscosity ratio. Similarly, the deformation is higher at higher viscosity ratios when the capillary number is kept fixed. For a viscoplastic droplet immersed in a Newtonian fluid, we find that increasing the droplet's plasticity increases its deformation. If the suspending medium is a viscoplastic material and the drop is Newtonian, our results show that stronger plastic effects reduce droplet deformation and increase the critical capillary number. To interpret the results, we recall that the Newtonian–Newtonian system was shown to exhibit an Oldroyd-B-like behavior with the elasticity coming from the interface. Hence, the viscoplastic-Newtonian system can be considered elasto-viscoplastic. We also compared some tendencies obtained in this work with the problem of long drops immersed in a second fluid in a tube under a pressure gradient.