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

Hydrodynamic problems with stagnation points are of particular importance in fluid mechanics as they allow study and investigation of elongational flows. In this article, the uniaxial elongational flow appearing at the surface of a viscoelastic drop and its role on the deformation of the droplet at low inertial regimes is studied. In studies related to viscoelastic droplets falling/raising in an immiscible Newtonian fluids, it is well known that by increasing the Deborah number (the ratio of the relaxation time of the interior fluid to a reference time scale) the droplet might lose its sphericity and obtain a dimple at the rear end. In this work, the drop deformation is investigated in detail to study the reason behind this transformation. We will show that as the contribution of elastic and inertial forces are increased, the stagnation points at the rear and front sides of the droplet are expanded to create a region of elongational dominated flows. At this stage, due to a combined effect of the shear thickening behavior of the elongational viscosity in viscoelastic fluids and the contribution of the inertial force, the interior phase is squeezed and consequently the droplet finds a shape similar to an oblate. As these non-linear forces are increased further, an additional circular stagnation line appears on the droplet surface in the external field, pulling the droplet surface outward and therefore creating a dimple shape at the rear end. Furthermore, the influence of inertia and viscoelastic properties are also studied on the motion, the drag coefficient and terminal velocity of drops.