An approximate solution to flow through a contraction for high Trouton ratio fluids

An approximate analytical solution is presented to predict the flow, vortex and pressure drop behaviour in abrupt axisymmetric contractions. This solution is developed to identify the effect of fluids which exhibit large Trouton ratios at large extensional rates despite having a constant shear viscosity. This behaviour approximates that shown in dilute polymeric solutions, although any transient effects are neglected by removing any strain dependence of the extensional viscosity and by assuming that the steady-state extensional viscosity undergoes a step change from a Trouton ratio of 3 to a large Trouton ratio at a critical rate. The predictions compare well to available experimental results for dilute to semi-dilute solutions of a flexible polymer, polystyrene, prior to the onset of elastic instabilities and for dilute solutions of a semi-rigid polymer, xanthan. The results show that vortex behaviour is determined by the fluid's characteristic time (through the magnitude of the Weissenberg number of the flow), while the steady-state extensional viscosity is of far more importance than transient effects in determining the steady-state pressure drop.