THE TIME-DEPENDENT EXTRUDATE-SWELL PROBLEM OF AN OLDROYD-B FLUID WITHSLIP ALONG THE WALL

We demonstrate that viscoelasticity combined with nonlinear slip acts as a storage of elastic energy generating oscillations of the pressure drop similar to those observed experimentally in extrusion instabilit ies. We consider the time-dependent axisymmetric incompressible Poiseu ille and extrudate-swell flows of an Oldroyd-B fluid. We assume that s lip occurs along the wall of the die following a slip equation which r elates the shear stress to the velocity at the wall and exhibits a max imum and a minimum. We first study the stability of the one-dimensiona l axisymmetric Poiseuille flow by means of a one-dimensional linear st ability analysis and time-dependent calculations. The numerically pred icted instability regimes agree well with the linear stability ones. T he calculations reveal that periodic solutions are obtained when an un stable steady-state is perturbed and that the amplitude and the period of the oscillations are increasing functions of the Weissenberg numbe r. We then continue to numerically solve the time-dependent two-dimens ional axisymmetric Poiseuille and extrudate-swell flows using the elas tic-viscous split stress method for the integration of the constitutiv e equation. Again, oscillations are observed in the unstable regime; c onsequently, the surface of the extrudate is wavy. However, the amplit ude and the period of the pressure drop oscillations are considerably smaller than in the one-dimensional flow. The most important phenomeno n revealed by our two-dimensional calculations is that the flow in the die is periodic in the axial direction. (C) 1998 The Society of Rheol ogy.