Flow birefringence provides a means to map stress distributions in complex flows of polymer solutions and melts, which may be used as a basis for evaluating performance of constitutive models in viscoelastic flow simulations. Application of this technique to axisymmetric geometries allows consideration of flows with greater kinematic complexity than is possible in the planar flow fields more traditionally studied using birefringence. This paper reports an extension of this technique to a time-dependent, axially symmetric stagnation flow, in which fluid is periodically forced back and forth against the end of a cylinder with a hemispherical tip. The geometry allows independent variation of Deborah and Weissenberg numbers (through independent control of the amplitude and frequency of the motion), and also imposes periodically reversing kinematics, in which fluid in the stagnation region alternately experiences uniaxial and equibiaxial extension. This new flow can thus provide tests of viscoelastic fluid models under severe conditions. The experimental procedures are validated by experiments on a concentrated but low MW polystyrene solution, with essentially Newtonian theology, to demonstrate the principles and capabilities of the experiment. We further present preliminary data on a shear thinning and viscoelastic solution of higher molecular weight polystyrene, to illustrate the effects of nonlinear viscoelasticity in the stagnation region.