  Interior stagnation point flows of viscoelastic liquids arise in a wide
variety of applications including extensional viscometry, polymer processing
and microfluidics. Experimentally, these flows have long been known to exhibit
instabilities, but the mechanisms underlying them have not previously been
elucidated. We computationally demonstrate the existence of a supercritical
oscillatory instability of low-Reynolds number viscoelastic flow in a
two-dimensional cross-slot geometry. The fluctuations are closely associated
with the "birefringent strand" of highly stretched polymer chains associated
with the outflow from the stagnation point at high Weissenberg number.
Additionally, we describe the mechanism of instability, which arises from the
coupling of flow with extensional stresses and their steep gradients in the
stagnation point region.
