Viscoelastic flow of boger fluids in a 4:1 square/square contraction

Abstract

Sudden contraction flows are classical benchmark problems in computational rheology (Hassager, 1988), and a large number of visualization studies in planar and axisymmetric contractions have been published in the literature. In these simple configurations, the flow behaviour of non-Newtonian fluids can be very surprising, and different flow patterns have been observed even for fluids with apparently similar rheological properties. The first visualizations in circular contractions for viscoelastic fluids were carried out by Cable and Boger (1978a, 1978b, 1979) and Nguyen and Boger (1979), who reported a dramatic growth of the salient corner vortex for shear rates higher than a critical value above which the normal stress grew quadratically. These experiments were performed for contraction ratios in the range 7.67:1 to 14.83:1 and at very high Weissenberg numbers the flow became asymmetric and eventually time-dependent. In 1986 Boger et al investigated the behaviour of Boger fluids with similar steady and dynamic shear properties and found different vortex dynamics, thus concluding that a different fluid property had to be taken into account. In his 1987 review paper Boger suggested extensional viscosity as that property, and described in some detail the sequence of flow dynamics in the sudden contraction. For some fluids, only a corner vortex exists, which grows in size as elasticity increases whereas for other fluids the corner vortex extends to the re-entrant corner near which a lip vortex is formed. For high contraction ratios the two vortices are initially separate, as also seen by McKinley et al (1991) in his 4:1 contraction experiments. As the elasticity increased the lip vortex grew at the expense of the corner vortex, while the length of the recirculation remained fairly constant. Eventually, the lip vortex occupied the whole contraction plane region and a further increase in the Weissenberg number lead to an increase of the now concave shaped vortex. At higher Weissenberg numbers a small pulsating lip vortex appeared and lead to unsteady behaviour. The relevance of extensional viscosity was also emphasized in the experimental investigations of White and Baird (1986) in a planar contraction with polystyrene (PS) and low density polyethylene (LDPE): whereas a vortex was found for the LDPE, it was absent from PS and the difference was attributed to their different extensional viscosities. This was further emphasised when they used later (White and Baird, 1988) a constitutive equation that represented correctly the measured extensional viscosity of both fluids and were able to numerically predict the different vortex patterns observed in 1986. In the mid 1980s the experimental work on the 4:1 contraction flow concentrated on assessing the various flow transitions and instabilities and used several experimental techniques, as in McKinley et al (1991).