Abstract
In this paper, we study the transient flow of branched polymer melts with contrasting shear and elongational properties in planar 4:1 abrupt and rounded-corner contractions. This includes Single and Double Extended forms of the Pom–Pom model (SXPP and DXPP), comparing the transient behaviour for these two different models. With the DXPP version, the evolution of the molecular-chain backbone stretch (λ) is described by a dynamic equation, whilst in the SXPP form, stretch is an instantaneous algebraic function of the stress tensor (τ). Simulations are performed with a hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm. We demonstrate some novel features due to the influence and imposition of realistic transient boundary conditions on evolutionary flow-structure. The different effects of various model parameter choices are also exposed through transient field response in principle stress difference fringe patterns, rates of deformation, first and second normal stress difference, stress and stretch.
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Abbreviations
- Symbol:
-
Name
- U :
-
fluid velocity
- p :
-
hydrodynamic pressure
- τ :
-
extra-stress
- t:
-
time
- d :
-
rate of deformation tensor
- I :
-
unit tensor
- \(\mathop{\boldsymbol{\tau}} \limits^{\nabla}\) :
-
upper-convected material derivative of τ
- Re :
-
Reynolds number
- We :
-
Weissenberg number
- U :
-
characteristic velocity
- L 0 :
-
characteristic length
- ρ :
-
fluid density
- μ :
-
fluid viscosity
- μ p :
-
Polymeric component
- μ s :
-
Newtonian solvent component
- β :
-
solvent fraction ratio
- q :
-
number of side-branch arms
- ε :
-
system entanglement
- λ :
-
stretch of back-bone segment
- α :
-
anisotropy parameter
- λ 0b :
-
backbone-orientation relaxation time
- λ 0s :
-
backbone-stretch relaxation time
- G 0 :
-
linear relaxation modulus
- N 1 :
-
first normal stress difference
- N 2 :
-
second normal stress difference
- \(\dot{\gamma}\) :
-
shear-rate
- \(\dot{\varepsilon}\) :
-
extension-rate
References
Aboubacar, M., Webster, M.F.: A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows. J. Non-Newton. Fluid Mech. 98, 83–106 (2001)
Aboubacar, M., Matallah, H., Webster, M.F.: Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows. J. Non-Newton. Fluid Mech. 103, 65–103 (2002)
Aguayo, J.P., Tamaddon-Jahromi, H.R., Webster, M.F.: Extensional response of the pom-pom model through planar contraction flows for branched polymer melts. J. Non-Newton. Fluid Mech. 134, 105–126 (2006)
Aguayo, J.P., Phillips, P.M., Phillips, T.N., Tamaddon-Jahromi, H.R., Snigerev, B.A., Webster, M.F.: The numerical prediction of planar viscoelastic flows using the pom-pom model and high-order finite volume schemes. J. Comput. Phys. 220, 586–611 (2007)
Baloch, A., Webster, M.F.: A computer simulation of complex flows of fibre suspensions. Comput. Fluids 24, 135–151 (1995)
Baltussen, M.G.H.M., Verbeeten, W.M.H., Bogaerds, A.C.B., Hulsen, M.A., Peters, G.W.M.: Anisotropy parameter restrictions for the extended pom–pom model. J. Non-Newtonian Fluid Mech. 165(19–20), 1047–1055 (2010)
Belblidia, F., Keshtiban, I.J., Webster, M.F.: Stabilised computations for viscoelastic flows under compressible implementations. J. Non-Newton. Fluid Mech. 134, 56–76 (2006)
Bishko, G.B., Harlen, O.G., McLeish, T.C.B., Nicholson, T.M.: Numerical simulation of the transient flow of branched polymer melts through a planar contraction using the ‘pom–pom’ model. J. Non-Newton. Fluid Mech. 82, 255–273 (1999)
Blackwell, R.J., McLeish, T.C.B., Harlen, O.G.: Molecular drag-strain coupling in branched polymer melts. J. Rheol. 44, 121–136 (2000)
Bogaerds, A.C.B., Grillet, A.M., Peters, G.W.M., Baaijens, F.P.T.: Stability analysis of polymer shear flows using the extended pom–pom constitutive equations. J. Non-Newton. Fluid Mech. 108, 187–208 (2002)
Clemeur, N., Rutgers, R.P.G., Debbaut, B.: Numerical simulation of abrupt contraction flows using double convected pom–pom model. J. Non-Newton. Fluid Mech. 123, 105–120 (2004)
Clemeur, N., Rutgers, R.P.G., Debbaut, B.: On the evaluation of some differential formulations for the pom–pom constitutive model. Rheol. Acta 42, 217–231 (2003)
Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford University Press, Oxford (1986)
Donea, J.: Taylor-Galerkin method for convective transport problems. Int. J. Numer. Methods Eng. 20, 101–119 (1984)
Hawken, D.M., Tamaddon-Jahromi, H.R., Townsend, P., Webster, M.F.: A Taylor-Galerkin based algorithm for viscous incompressible flow. Int. J. Numer. Methods Fluids 10, 327–351 (1990)
Inkson, N.J., McLeish, T.C.B., Harlen, O.G., Groves, D.J.: Predicting low density polyethylene melt rheology in elongational and shear flows with “pom–pom” constitutive equations. J. Rheol. 43, 873–896 (1999)
Inkson, N.J., Phillips, T.N.: Unphysical phenomena associated with the extended pom–pom model in steady flow. J. Non-Newton. Fluid Mech. 145(2–3), 92–101 (2007)
Inkson, N.J., Phillips, T.N., van Os, R.G.M.: Numerical simulation of flow past a cylinder using models of XPP type. J. Non-Newton. Fluid Mech. 159, 7–20 (2009)
Keshtiban, I.J., Bumroong, P., Tamaddon-Jahromi, H.R., Webster, M.F.: Generalised approach for transient computation of start-up pressure-driven viscoelastic flow. J. Non-Newton. Fluid Mech. 151, 2–20 (2008)
MacLeish, T.C.B.R., Larson, G.: Molecular constitutive equations for a class of branched polymers: The pom–pom polymer. J. Rheol. 42, 81–110 (1998)
Matallah, H., Townsend, P., Webster, M.F.: Recovery and stress-splitting schemes for viscoelastic flows. J. Non-Newton. Fluid Mech. 75, 139–166 (1998)
Öttinger, H.C.: Thermodynamic admissibility of the pompom model for branched polymers. Rheol. Acta 40, 317–321 (2001)
Tanner, R.I., Nasseri, S.: Simple constitutive models for linear and branched polymers. J. Non-Newton. Fluid Mech. 116, 1–17 (2003)
Tamaddon-Jahromi, H.R., Webster, M.F., Walters, K.: Predicting numerically the large increases in extra pressure drop when Boger fluids flow through axisymmetric contractions. J. Natural. Science 2, 1–11 (2010)
Van Meerveld, J.: Note on thermodynamic consistency of the integral pompom model. J. Non-Newton. Fluid Mech. 108, 291–299 (2002)
Verbeeten, W.M.H., Peters, G.W.M., Baaijens, F.T.P.: Differential constitutive equations for polymer melts: the extended Pom-Pom model. J. Rheol. 45, 823–843 (2001)
Verbeeten, W.M.H., Peters, G.W.M., Baaijens, F.T.P.: Viscoelastic analysis of complex polymer melt flows using the extended pom-pom model. J. Non-Newton. Fluid Mech. 108, 301–326 (2002)
Verbeeten, W.M.H., Peters, G.W.M., Baaijens, F.T.P.: Numerical simulations of the planar contraction flow for a polyethylene melt using the XPP model. J. Non-Newton. Fluid Mech. 117, 73–84 (2004)
Wapperom, P., Webster, M.F.: Second-order hybrid finite-element/volume method for viscoelastic flows. J. Non-Newton. Fluid Mech. 79, 405–431 (1998)
Wapperom, P., Webster, M.F.: Simulation for viscoelastic flow by a finite volume/element method. Comput. Methods Appl. Mech. Eng. 180, 281–304 (1999)
Wapperom, P., Keunings, R.: Numerical simulation of branched polymer melts in transient complex flowusing pom-pom models. J. Non-Newton. Fluid Mech. 97, 267–281 (2001)
Webster, M.F., Tamaddon-Jahromi, H.R., Aboubacar, M.: Transient viscoelastic flows in planar contractions. J. Non-Newton. Fluid Mech. 118, 83–101 (2004)
Webster, M.F., Tamaddon-Jahromi, H.R., Aboubacar, M.: Time-dependent algorithm for viscoelastic flow-finite element/volume schemes. Numer. Methods Partial Differ. Equ. 21, 272–296 (2005)
Zienkiewicz, O.C., Morgan, K., Peraire, J., Vandati, M., Läohner, R.: Finite elements for compressible gas flow and similar systems. In: 7th Int. Conf. Comput. Meth. Appl. Sci. Eng. Versailles, France (1985)
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Tamaddon Jahromi, H.R., Webster, M.F. Transient behaviour of branched polymer melts through planar abrupt and rounded contractions using pom–pom models. Mech Time-Depend Mater 15, 181–211 (2011). https://doi.org/10.1007/s11043-010-9130-9
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DOI: https://doi.org/10.1007/s11043-010-9130-9