Viscoelastic Flows in Abrupt Contraction-Expansions II. Relaxation Time and the Deborah Number

II. Relaxation Time and the Deborah Number NB In what follows we use viscometric information from Note I ( Fluid Rheology). To determine the Deborah number for the axisymmetric contraction-expansion we need to select both a characteristic timescale for the flow and a representative measure of the spectrum of relaxation times of the fluid. A characteristic strain rate based on the local flow conditions near the plane of the contraction region is defined by Ú / +vz,2 / R2 where +vz,2 = Q/ R2 2 is the average velocity into the contraction and R2 is the radius of the contraction. A characteristic convective time of the flow can then be taken to be / / R2 / +vz,2 = Ú . The simplest choice for the fluid time scale is of course the longest or Zimm relaxation time z determined from the linear viscoelastic measurements. Even though the viscosity does not have a strong rate dependence, the first normal stress coefficient does. It is therefore important to note that the average relaxation time, which can be calculated from the viscometric properties of the fluid previously determined, is a function of shear rate [1]. Discrepancies between different estimates of the relaxation time have been discussed at length by Keiller et al. [2] and by Boger et al. [3]. In particular, they point out that numerical calculations based on the (constant) longest relaxation time need to achieve a very large values of Deborah number. For scaling purposes, in our present work the characteristic relaxation time of the 0.025% PS/PS solution is reported using the Maxwell relaxation time evaluated in the limit of zero shear rate. After substituting for the asymptotic value of 10 obtained from the Rouse-Zimm model the characteristic relaxation time becomes