A streamline element scheme for solving viscoelastic flowproblems part II: Integral constitutive models

Abstract

A finite element method for using integral constitutive models in viscoelastic flow simulation is presented. This method is based on a streamline element scheme (S.E.S) which was reported in Part I of this paper. The technique of particle tracking and strain history calculation is discussed in detail. In calculating the infinite memory integral, either Gaussian or Laguerre numerical quadrature formulae are used in our scheme. Some simple and complex flow problems involving the upper-convected Maxwell integral model are solved as test problems and afterwards the effort is concentrated on the K.B.K.Z. model. Much numerical work is devoted to simulating the axial extrusion swell experiments with LDPE sample A of the IUPAC Working Party on Structure and Properties of Commercial Polymers, using a specific version of the K.B.K.Z. model with multiple relaxation times in the memory function designed by Papanastasiou, Scriven and Macosko. It is shown by the numerical results that if the shear viscosity function of the model is kept unchanged, the calculated swelling ratio is very sensitive to the elongational behaviour of the model; it increases (or decreases) monotonically with the increase (or decrease) of the elongational viscosity in the corresponding stretching rate region. When both the shear and elongational response of this model agree well with experiments, the numerical predictions of the swelling ratio also agree well with experimental data at low and high apparent shear rates, while in the medium region the numerical calculation underestimates the swelling ratio. It is also seen that, using this model in our method, the extrusion calculation is surprisingly stable, even at very high Weissenberg numbers or very high extrusion swelling ratios, thus showing the very promising potential of integral models in the field of viscoelastic flow computation.