Use of lubricated skin/core flow of a polymer melt and a hyperbolic shaped converging die are shown to result in essentially pure elongational flow at a constant elongational strain rate. This behavior is demonstrated first by modeling a series of skin/core layered flows : power-law rectangular channel shearing flow; Newtonian fluid converging channel elongational flow; and power-law fluid converging channel elongational flow. Experimental measurements using tracer particles and an image analysis system confirm the predicted behavior and demonstrate the ability to achieve a constant elongational strain rate in the core layer. The constant value is controlled by the volumetric flow rate and die geometry in this confined flow. The viscosity ratio of the core polymer must be at least 100 times that of the core and the converging flow channel must have a specific hyperbolic shape. A Dowex low-density polyethylene skin and a Marlex polypropylene core were used for the experimental measurements and for the rheological parameters used in the modeling.