Journal of Non-Newtonian Fluid Mechanics, 31 (1989) 301-323 zyxwvutsrqponmlkjihgfedcbaZ Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands zyxwvutsrqponmlkji 301 zyxwvut SMIL4RKY SOLUTIONS THAT GIVE RISE TO HYPERBOLICITY AND CHANGE OF TYPE IN STEADY FLOW OF A YISCOELASTIC FLUID CLAUDE V ERDIER and DANIEL D. JOSEPH Department of Aerospace Enginering & Mechanics, University of Minnesota, Minneapolis MN 55455 (U.S.A.) (Received May 20, 1988; in revised form September 13,1988) zyxwvutsrqponmlkjihgfedcbaZYXW Similarity solutions have proved to be a very useful tool for the study of flows of viscoelastic fluids since they allow us to check numerical computa- tions against them. We compute here hyperbolic regions of the vorticity for an upper convected Maxwell model using the similarity solution of Phan- Thien for flow between rotating disks and using the similarity solution of Menon for an accelerated flow in cylindrical coordinates. We show that the extra tension becomes enormous at the edge of disks of ordinary rheometers under operating conditions. 1. Introduction Phan-Thien [1,2] has found a similarity solution of an Oldroyd B fluid between parallel plates which rotate at different speeds around a common axis perpendicular to the plates. The solution is modeled according to a famous one by Von Karman in which the fluid variables are resolved into a quadratic polynomial in r with coefficients to be determined as functions of z. This problem is important because it may model the flow away from edges of the finite disks used in rheometers. A different similarity solution for an upper convected Maxwell fluid has been given by Menon et al. [3]. Menon’s problem is to determine the flow when the fluid is driven by an axisymmetric accelerated surface whose axial velocity w is proportional to the axial coordinate z. This kind of boundary condition is not a common one for devices which could be constructed but it is of interest in a subject where solutions so close to exact ones are so few. 0377-0257/89/$03.50 0 1989 Elsevier Science Publishers B.V.