Independent confirmation that delayed die swell is a hyperbolic transition

Abstract

We measured shear wave speeds in the same aqueous Xathan solutions used to study delayed die swell by Allain et al. (Eur. J. Mech. A/Fluids, 12 (1993) 175-186). They reported delayed die swell for solutions of 500, 1000, 2000 and 4000 ppm Xanthan in water when the shear rate, 8 U/d, was above a critical value 8 U c/ d, where U is the mean velocity through a round pipe of diameter d = 0.6 mm and U c is a critical value. Their photographs of delayed die swell are exactly like those reported by Joseph et al. (J. Non-Newtonian Fluid Mech., 24 (1987) 31-65) for 17 different polymer solutions, by Joseph (Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990) for 18 and by Hu et al. (J. Non-Newtonian Fluid Mech., 35 (1990) 287-307) for Ml. Allain et al. sent us samples of the four solutions, prepared by them, and we measured wave speeds on our meter. As in all the earlier experiments, the delay occurs when the centerline velocity in the pipe U = 2 U is greater than the wave speed C, and after a distance L the swell is complete and the mean velocity is less than C. This adds four more cases, 23 in all, to those for which the delay can be correlated with the wave speed measured on our meter. So far 23, and only 23, cases have been studied and they are all consistent with calculations from theory giving rise to hyperbolicity and change of type.