  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.
