Further considerations of axisymmetric contraction flows

Abstract

Two distinct mechanisms are proposed for the flow of non-Newtonian fluids through abrupt contractions. The first is a quasi-radial flow which is argued to reflect at least qualitatively the flow characteristics at low strain rates. Depending on the rheological properties of the fluid, the second mechanism, referred to as funnel flow, may well be preferred at higher rates of strain. The change from the first to the second flow mechanism is often seen to occur experimentally with the appearance of a so-called ‘lip’ vortex. An approximate analytical inelastic treatment of the funnel flow was presented in an earlier paper. A similar treatment is now offered for the quasi-radial flow and, in addition, an attempt is made to include the effect of elasticity in both flows. Particular attention is placed on the influence of shear viscosity, elasticity (through the first normal stress difference in shear) and extensional viscosity. The analyses suggest that the effects of elasticity and extensional viscosity are opposite, the former resulting for example in a decreased Couette correction while the latter causes the Couette correction to increase. The reduction is seen to be a consequence of the stress boundary conditions which result from the way in which the Couette correction is defined. For realistic fluid parameters the combined effect is for the Couette correction to exhibit an initial decrease followed by a rapid increase which becomes less dramatic as the flow mode changes. As far as the generation and growth of vortices are concerned it would appear that elasticity may enhance the size of the salient corner vortex but again the extensional viscosity has the opposite effect. For funnel flow, in which a recirculating zone is assumed to exist up to the lip of the contraction, vortex growth appears to be almost exclusively dominated by extensional viscosity.