The Axisymmetric Contraction-Expansion: 
The Role of Extensional Rheology on Vortex Growth Dynamics and the Enhanced Pressure Drop

Jonathan P. Rothstein and Gareth H. McKinley
 Dept. of Mechanical Engineering, Massachusetts Institute of Technology
Cambridge, MA 02139; USA


The flow of a polystyrene Boger fluid through axisymmetric contraction-expansions having various contraction ratios (2 b 8) and degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry and digital particle image velocimetry are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of b = 2 a steady elastic lip vortex is observed while for contraction ratios of 4 b 8 no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects. 

Keywords: axisymmetric contraction-expansion, entrance pressure drop, dissipative stress, polystyrene Boger fluid, PIV, LDV, vortex growth dynamics, entrance lip singularity, extensional rheology 

Published  J. Non-Newt. Fluid Mech., 98(1), 2001, p 33-63.