The steady shape of a drop of dilute polymer solution falling through a quiescent viscous
Newtonian fluid is considered. Experimentally, we find that an immiscible drop of 0.16% xanthan
gum in 80:20 glycerol/water falling through 9.8 P polydimethylsiloxane oil may exhibit a stable dimple at
its trailing edge. At higher volumes the dimple extends far into the interior of the drop, and
pinches off via a Rayleigh-type instability, injecting oil droplets into the polymer drop. At even
larger volumes, a toroidal shape develops. We show that the dimpled shape can be reproduced
mathematically with axisymmetric solutions for Stokes flow past a non-Newtonian drop, using the
constitutive equation for a Simple Fluid of Order Three.