A semi-hyperbolic converging geometry finds application as an inexpensive elongation rheometer under certain flow conditions. We provide a matched asymptotic solution for the flow of a Newtonian fluid under no-slip boundary conditions. The predicted velocity and pressure profiles agree nearly quantitatively with CFD simulated values. Our theoretical approach has certain advantages over the known similarity solution proposed by James (1991. A.I.Ch.E. Journal 37, 59-64). (c) 2005 Elsevier Ltd. All rights reserved.