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Hydrodynamic problems with stagnation points are of particular importance influid mechanics as they allow study and investigation of elongational flows. Inthis article, the uniaxial elongational flow appearing at the surface of aviscoelastic drop and its role on the deformation of the droplet at lowinertial regimes is studied. In studies related to viscoelastic dropletsfalling/raising in an immiscible Newtonian fluids, it is well known that byincreasing the Deborah number (the ratio of the relaxation time of the interiorfluid to a reference time scale) the droplet might lose its sphericity andobtain a dimple at the rear end. In this work, the drop deformation isinvestigated in detail to study the reason behind this transformation. We willshow that as the contribution of elastic and inertial forces are increased, thestagnation points at the rear and front sides of the droplet are expanded tocreate a region of elongational dominated flows. At this stage, due to acombined effect of the shear thickening behavior of the elongational viscosityin viscoelastic fluids and the contribution of the inertial force, the interiorphase is squeezed and consequently the droplet finds a shape similar to anoblate. As these non-linear forces are increased further, an additionalcircular stagnation line appears on the droplet surface in the external field,pulling the droplet surface outward and therefore creating a dimple shape atthe rear end. Furthermore, the influence of inertia and viscoelastic propertiesare also studied on the motion, the drag coefficient and terminal velocity ofdrops.

Droplets with circular stagnation lines: combined effects of viscoelastic and inertial forces on drop shapes