considered the flow of the XPP fluid past a cylinder using a spectral element approach

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended PomPom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.

[1]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[2]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[3]  A. A. Amsden,et al.  A simplified MAC technique for incompressible fluid flow calculations , 1970 .

[4]  R. Tanner.  A theory of die‐swell , 1970 .

[5]  R. Bird.  Dynamics of Polymeric Liquids , 1977 .

[6]  S. Okubo,et al.  On the Normal Stress Effect and the Barus Effect of Polymer Melts , 1980 .

[7]  R. Keunings,et al.  Finite-element Analysis of Die Swell of a Highly Elastic Fluid , 1982 .

[8]  R. Tanner,et al.  Numerical Simulation of Non-Newtonian Flow , 1984 .

[9]  R. Keunings,et al.  Numerical-simulation of the Flow of a Viscoelastic Fluid Through An Abrupt Contraction , 1984 .

[10]  Xiaolin Luo,et al.  A streamline element scheme for solving viscoelastic flow problems. Part I. Differential constitutive equations , 1986 .

[11]  J. Song,et al.  Numerical simulation of the planar contraction flow of a giesekus fluid , 1988 .

[12]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[13]  R. Armstrong,et al.  Finite element methdos for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity , 1990 .

[14]  G. Buscaglia.  A finite element analysis of rubber coextrusion using a power‐law model , 1993 .

[15]  M. F. Tomé,et al.  GENSMAC: a computational marker and cell method for free surface flows in general domains , 1994 .

[16]  S. Richardson,et al.  Explicit numerical simulation of time-dependent viscoelastic flow problems by a finite element/finite volume method , 1994 .

[17]  N. Phan-Thien,et al.  Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method , 1995 .

[18]  M. Deville,et al.  Unsteady finite volume simulation of Oldroyd-B fluid through a three-dimensional planar contraction , 1997 .

[19]  F. Pinho,et al.  Numerical simulation of non-linear elastic flows with a general collocated finite-volume method , 1998 .

[20]  D. Joseph,et al.  Delayed-die swell and sedimentation of elongated particles in wormlike micellar solutions , 1998 .

[21]  M. Webster,et al.  A second-order hybrid finite-element/volume method for viscoelastic flows 1 Dedicated to Professor M , 1998 .

[22]  R. Larson,et al.  Molecular constitutive equations for a class of branched polymers: The pom-pom polymer , 1998 .

[23]  T. McLeish,et al.  Predicting low density polyethylene melt rheology in elongational and shear flows with , 1999 .

[24]  T. Phillips,et al.  Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method , 1999 .

[25]  T. McLeish,et al.  Numerical simulation of the transient flow of branched polymer melts through a planar contraction using the `pom–pom' model , 1999 .

[26]  N. Phan-Thien,et al.  Galerkin/least-square finite-element methods for steady viscoelastic flows , 1999 .

[27]  N. Phan-Thien,et al.  Viscoelastic finite volume method , 1999 .

[28]  P. Rubio,et al.  LDPE melt rheology and the pom–pom model , 2000 .

[29]  M. F. Tomé,et al.  Numerical Simulation of Axisymmetric Free Surface Flows , 2000 .

[30]  F. Baaijens,et al.  Differential constitutive equations for polymer melts: The extended Pom–Pom model , 2001 .

[31]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[32]  F. Baaijens,et al.  Viscoelastic analysis of complex polymer melt flows using the extended Pom-Pom model , 2002 .

[33]  M. F. Tomé,et al.  A finite difference technique for simulating unsteady viscoelastic free surface flows , 2002 .

[34]  T N Phillips,et al.  Contemporary Topics in Computational Rheology , 2002 .

[35]  M. F. Webster,et al.  Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows , 2002 .

[36]  M. Pasquali,et al.  Free surface flows of polymer solutions with models based on the conformation tensor , 2002 .

[37]  M. Webster,et al.  The distinctive CFD challenges of computational rheology , 2003 .

[38]  Ramesh Nallapati,et al.  A moving unstructured staggered mesh method for the simulation of incompressible free-surface flows , 2003 .

[39]  F. Pinho,et al.  A convergent and universally bounded interpolation scheme for the treatment of advection , 2003 .

[40]  F. Denaro,et al.  On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions , 2003 .

[41]  R. Rutgers,et al.  On the evaluation of some differential formulations for the pom-pom constitutive model , 2003 .

[42]  M. Aboubacar,et al.  Time-dependent algorithms for viscoelastic flow: Bridge between finite-volume and finite-element methodology , 2003 .

[43]  F. Baaijens,et al.  Numerical simulations of the planar contraction flow for a polyethylene melt using the XPP model , 2004 .

[44]  H. Demir,et al.  Numerical modelling of viscoelastic cavity driven flow using finite difference simulations , 2005, Appl. Math. Comput..

[45]  T. Phillips,et al.  Efficient and stable spectral element methods for predicting the flow of an XPP fluid past a cylinder , 2005 .

[46]  M. Aboubacar,et al.  Modelling pom-pom type models with high-order finite volume schemes , 2005 .

[47]  P. Colella,et al.  A stable and convergent scheme for viscoelastic flow in contraction channels , 2005 .

[48]  K. Ahn,et al.  High-resolution finite element simulation of 4:1 planar contraction flow of viscoelastic fluid , 2005 .

[49]  J. Guillet,et al.  Three-dimensional numerical simulation of viscoelastic contraction flows using the Pom-Pom differential constitutive model , 2005 .

[50]  J. P. Aguayo,et al.  Extensional response of the pom-pom model through planar contraction flows for branched polymer melts , 2006 .

[51]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[52]  R. Tanner.  On the congruence of some network and pom-pom models , 2006 .

[53]  V. G. Ferreira,et al.  A Stable Semi-Implicit Method for Free Surface Flows , 2006 .

[54]  Manuel Laso,et al.  Numerical simulation of 3D viscoelastic flows with free surfaces , 2006, J. Comput. Phys..

[55]  R. G. Owens,et al.  A numerical study of the SPH method for simulating transient viscoelastic free surface flows , 2006 .

[56]  Andrea Prosperetti,et al.  A second-order boundary-fitted projection method for free-surface flow computations , 2006, J. Comput. Phys..

[57]  M. F. Tomé,et al.  A marker-and-cell approach to viscoelastic free surface flows using the PTT model , 2007 .

[58]  J. L. Doricio,et al.  Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach , 2007 .

[59]  B. Gautham,et al.  Simulation of viscoelastic flows of polymer solutions in abrupt contractions using an arbitrary Lagrangian Eulerian (ALE) based finite element method , 2007 .

[60]  P. M. Phillips,et al.  The numerical prediction of planar viscoelastic contraction flows using the pom-pom model and higher-order finite volume schemes , 2007, J. Comput. Phys..

[61]  V. G. Ferreira,et al.  A finite difference technique for solving the Oldroyd-B model for 3D-unsteady free surface flows , 2008 .

[62]  José Alberto Cuminato,et al.  An implicit technique for solving 3D low Reynolds number moving free surface flows , 2008, J. Comput. Phys..

[63]  José Alberto Cuminato,et al.  Stability of numerical schemes on staggered grids , 2008, Numer. Linear Algebra Appl..

[64]  T. Phillips,et al.  Numerical simulation of flow past a cylinder using models of XPP type , 2009 .

[65]  Numerical solution of the upper-convected maxwell model for three-dimensional free surface flows , 2009 .

[66]  M. F. Webster,et al.  The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheology , 2009 .

[67]  Z. Cai,et al.  An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type , 2009 .

[68]  F. Pinho,et al.  The log-conformation tensor approach in the finite-volume method framework , 2009 .

[69]  T. Phillips,et al.  Numerical prediction of extrudate swell of branched polymer melts , 2010 .