Rheologica Acta

, Volume 48, Issue 5, pp 551–561 | Cite as

Effect of branching in cross-slot flow: the formation of “W cusps”

  • D. G. Hassell
  • D. Hoyle
  • D. Auhl
  • O. Harlen
  • M. R. Mackley
  • T. C. B. McLeish
Original Contribution


The sensitivity of the principal stress difference (PSD) profile to molecular architecture is demonstrated for flow in a cross-slot geometry. For materials with low levels of branching, the pattern along the outlet centre line exhibited “single cusps”, while an increase in molecular branching was found to lead to “W cusps”. The formation of these W cusps was found to be independent of extensional rate for the conditions probed, and they were formed initially at the stagnation point and travelled along the outlet centre line with time. Comparison with simulations performed using a multi-mode “pom-pom” model failed to predict W cusps, although the general level of PSD was accurately captured.


Flow modeling Extensional flow Birefringence 



We would like to thank S. Butler for useful input and discussions and Dow for materials. All authors would like to acknowledge funding under the EPSRC Microscale Polymer Processing (MUPP2) research project.


  1. Blackwell RJ, Harlen OG, McLeish TCB (2000) Molecular drag–strain coupling in branched polymer melts. J Rheol 44:121–136CrossRefADSGoogle Scholar
  2. Bogaerds ACB, Verbeeten WMH, Peters GWM, Baaijens FPT (1999) 3D Viscoelastic analysis of a polymer solution in a complex flow. J Non-Newton Fluid Mech 180:413–430MATHGoogle Scholar
  3. Cathey CA, Fuller GG (1990) The optical and mechanical response of flexible polymer solutions to extensional flow. J Non-Newton Fluid Mech 34:63–88CrossRefGoogle Scholar
  4. Checker N, Mackley MR, Mead DW (1983) On the flow of molten polymer into, within and out of ducts. Phil Trans R Soc Lond A 1504(308):451–477ADSGoogle Scholar
  5. Clemeur N, Rutgers RPG, Debbaut B (2004) Numerical evaluation of three dimensional effects in planar flow birefringence. J Non-Newton Fluid Mech 123:105–120MATHCrossRefGoogle Scholar
  6. Collis MW, Lele AK, Mackley MR, Graham RS, Groves DJ, Likhtman AE, Nicholson TM, Harlen OG, McLeish TCB, Hutchings L, Fernyhough CM, Young RN (2005) Constriction flows of monodisperse linear entangled polymers: multiscale modelling and flow visualization. J Rheol 49(2):501CrossRefADSGoogle Scholar
  7. Coventry KD (2006) Cross-slot rheology of polymers. PhD Thesis, Department of Chemical Engineering, University of CambridgeGoogle Scholar
  8. Coventry KD, Mackley MR (2008) Cross-slot extensional flow of polymer melts using a multi-pass rheometer. J Rheol 52(2):401CrossRefADSGoogle Scholar
  9. Cressely R, Hoqueart R, Scrivener O (1978) Lignes de biréfringence d’écoulement localisée dans un dispositif à deux rouleaux. Opt Acta 25:559–571Google Scholar
  10. Cressely R, Hoqueart R, Scrivener O (1979) Lignes de biréfringence d’écoulement localisée dans un dispositif à deux rouleaux tournant en sens contraire. Opt Acta 26:1173–1181Google Scholar
  11. Crowley DG, Frank FC, Mackley MR, Stephenson RG (1976) Localised flow birefringence of polyethylene oxide solutions in a four roll mill. J Polym Sci 14:1111–1119Google Scholar
  12. Das C, Inkson NJ, Read DJ, Kelmanson K (2006) Computational linear rheology of general branch-on-branch polymers. J Rheol 50(2):207–234CrossRefADSGoogle Scholar
  13. den Doelder CF, Koopmans R, Dees M, Mangnus M (2005) Pressure oscillations and periodic extrudate distortions of long-chain branched polyolefins. J Rheol 49(1):113–126CrossRefADSGoogle Scholar
  14. Frank FC, Mackley MR (1976) Localized flow birefringence of polyethylene oxide solutions in a two roll mill. J Polym Sci A2 14:1121–1131Google Scholar
  15. Gardner K, Pike ER, Miles MJ, Keller A, Tanaka K (1982) Polymer 23:1435–1442CrossRefGoogle Scholar
  16. Harlen OG, Rallison JM, Chilcott MD (1990) High-Deborah-number flows of dilute polymer solutions. J Non-Newton Fluid Mech 34:319–349MATHCrossRefGoogle Scholar
  17. Harlen OG, Hinch EJ, Rallison JM (1992) Birefringent pipes: the steady flow of a dilute polymer solution near a stagnation point. J Non-Newton Fluid Mech 44:229–265MATHCrossRefGoogle Scholar
  18. Harlen OG, Rallison JM, Szabo P (1995) A split Lagrangian–Eulerian method for simulating transient viscoelastic flows. J Non-Newton Fluid Mech 60:81CrossRefGoogle Scholar
  19. Harris OJ, Rallison JM (1993) Start-up of a strongly extensional flow of a dilute polymer solution. J Non-Newton Fluid Mech 50:89–124MATHCrossRefGoogle Scholar
  20. Hassell DG, Mackley MR (2008) Localised flow induced crystallisation of a polyethyelene melt. Rheol Acta 47(4):435–446CrossRefGoogle Scholar
  21. Hassell DG, Auhl D, McLeish TCB, Mackley MR (2008) The effect of viscoelasticity on stress fields within polyethylene melt flow for a cross-slot and contraction–expansion slit geometry. Rheol Acta 47:821–834CrossRefGoogle Scholar
  22. Hertel D, Vallette R, Münstedt H (2008) Three-dimensional entrance flow of a low-density polyethylene (LDPE) and a linear low-density polyethylene (LLDPE) into a slit die. J Non-Newton Fluid Mech 153(2–3):82–94CrossRefGoogle Scholar
  23. Inkson NJ, McLeish TCB, Harlen OG, Groves DG (1999) Predicting low density polyethylene melt rheology in elongational and shear flows with “pom-pom” constitutive equations. J Rheol 43:873–896CrossRefADSGoogle Scholar
  24. Kalpokaite-Dichkuvene R, Stravinskas G (2006) Behaviour of a fuel oil droplet on a hot surface. J Eng Phys Thermophys 79(1):10–17CrossRefGoogle Scholar
  25. Keller A, Muller AJ, Odell JA (1987) Entanglements in semi-dilute solutions as revealed by elongational flow studies. Prog Colloid Polym 75:179–200CrossRefGoogle Scholar
  26. Lee K, Mackley MR, Mcleish TCB, Nicholson TM, Harlen O (2001) Experimental observation and numerical simulation of transient stress fangs within flowing molten polyethylene. J Rheol 45(6):1261–1277CrossRefADSGoogle Scholar
  27. Luap C, Karlina M, Schweizer T, Venerus DC (2006) Limit of validity of the stress-optical rule from polystyrene melts: influence of polydispersity. J Non-Newton Fluid Mech 138(2–3):197–203CrossRefGoogle Scholar
  28. Lyazid A, Scrivener O, Teitgen R (1980) In: Astarita G, Marruci G, Nicolais L (eds) Rheology, vol 2. Plenum, New York, pp 260–265Google Scholar
  29. Mackley MR, Marshall RTJ, Smeulders JBAF (1995) The multipass rheometer. J Rheol 39(6):1293–1309CrossRefADSGoogle Scholar
  30. Macosko CW (1994) Rheology, principles, measurements and applications. Wiley-VCH, New YorkGoogle Scholar
  31. McLeish TCB (2002) Tube theory of entangled polymers. Adv Phys 51:1379–1527CrossRefADSGoogle Scholar
  32. McLeish TCB, Larson RC (1998) Molecular constitutive equations for a class of branched polymers: the pom-pom polymer. J Rheol 42(1):81–110CrossRefADSGoogle Scholar
  33. Martyn MT, Nakason G, Coates PD (2000a) Measurement of apparent extensional viscosities of polyolefin melts from process contraction flows. J Non-Newton Fluid Mech 92(2000):203–226MATHCrossRefGoogle Scholar
  34. Martyn MT, Groves DJ, Coates PD (2000b) In process measurement of apparent extensional viscosity of low density polyethylene melts using flow visualization. Plast Rubber Compos 29:14–22Google Scholar
  35. Meissner J, Hostettler J (1994) A new elongational rheometer for polymer melts and other highly viscoelastic liquids. Rheol Acta 33(1):1–21CrossRefGoogle Scholar
  36. Perera MGN, Lyazid A, Scrivener O (1982) Numerical simulation of elongational flow behaviour in the cross cell experiments. Rheol Acta 21(4–5):543–546CrossRefGoogle Scholar
  37. Schoonen J (1998) Determination of rheological constitutive equations using complex flows. PhD Thesis, Eindhoven Univeristy of Technology www.mate.tue.nl
  38. Schoonen JFM, Swartjes FHM, Peters GWM, Bjens FPT, Meijer HEH (1998) A 3D numerical/experimental study on a stagnation flow of a polyisobutylene solution. J Non-Newton Fluid Mech 79(2–3):529–561MATHCrossRefGoogle Scholar
  39. Scrivener O, Berner C, Cressely R, Hocquart R, Sellin R, Vlaches NS (1979) Dynamical behaviour of drag-reducing polymer solutions. J Non-Newton Fluid Mech 5:475–495CrossRefGoogle Scholar
  40. Schuberth S, Münstedt H (2008a) Simultaneous measurements of velocity and stress distributions in polyisobutylenes using laser-Doppler velocimetry and flow induced birefringence. Rheol Acta 47(2):111–119CrossRefGoogle Scholar
  41. Schuberth S, Münstedt H (2008b) Transient elongational viscosities of aqueous polyacrylamide solutions measured with an optical rheometer. Rheo Acta 47(2):139–147CrossRefGoogle Scholar
  42. Soulages J (2007) Flow birefringence and velocity measurements for polymer melts in a cross-slot flow channel. PhD Thesis no. 17180. ETH ZürichGoogle Scholar
  43. Soulages J, Schweizer T, Venerus DC, Hostettler J, Mettler F, Kroger M, Ottinger HC (2008) Lubricated optical rheometer for the study of two-dimensional complex flows of polymer melts. J Non-Newton Fluid Mech 150(1):42–47CrossRefGoogle Scholar
  44. Sridhar T, Tirtaatmadja V, Nguyan DA, Gupta RK (1991) Measurement of extensional viscosity of polymer solutions. J Non-Newton Fluid Mech 40(3):271–280CrossRefGoogle Scholar
  45. Taylor GI (1934) The formation of emulsions in definable fields of flow. Proc R Soc Lond A 146:501–523CrossRefADSGoogle Scholar
  46. Venerus DC, Zhu SH, Öttinger HC (1999) Stress and birefringence measurements during the uniaxial elongation of polystyrene melts. J Rheol 43(3):795–813CrossRefADSGoogle Scholar
  47. Verbeeten WMH (2001) Computational polymer melt rheology. PhD Thesis, Technische Universiteit EindhovenGoogle Scholar
  48. Wales JLS (1976) The application of flow birefringence to rheological studies of polymer melts. PhD Thesis, Delft University of Technology, DelftGoogle Scholar
  49. Wood-Adams P, Costeux S (2001) Thermorheological behaviour of polyethylene: effects of microstructure and long chain branching. Macromolecules 34:6281–6290CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • D. G. Hassell
    • 1
    • 4
  • D. Hoyle
    • 3
  • D. Auhl
    • 2
  • O. Harlen
    • 3
  • M. R. Mackley
    • 1
  • T. C. B. McLeish
    • 2
    • 5
  1. 1.Department of Chemical EngineeringUniversity of CambridgeCambridgeUK
  2. 2.IRC in Polymer Science and Technology, Department of Physics and AstronomyUniversity of LeedsLeedsUK
  3. 3.Department of MathematicsUniversity of LeedsLeedsUK
  4. 4.Department of Chemical EngineeringUniversity of Nottingham Malaysia CampusSemenyihMalaysia
  5. 5.Department of PhysicsUniversity of DurhamSouth RoadUK

Personalised recommendations