Abstract
This paper reports the flow behaviour of Newtonian and Boger fluids through various axisymmetric contraction configurations by means of numerical predictions. A principal aim has been to evaluate the geometrical design choice of the hyperbolic contraction flow. The FENE-CR model has been used to reflect the behaviour of Boger fluids, with constant shear viscosity, finite (yet large) extensional viscosity and less than quadratic first normal stress difference. Numerical calculations have been performed on six different contraction configurations to evaluate an optimized geometry for measuring extensional viscosity in uniaxial extensional flow. The influence of a sharp or rounded recess-corner on the nozzle has also been investigated. Few commercial measuring systems are currently available for measurement of the extensional rheology of medium-viscosity fluids, such as foods and other biological systems. In this context, a technique based on the hyperbolic contraction flow would be a suitable alternative. The pressure drop, the velocity field, the first normal stress difference and the strain rate across the geometry have each been evaluated for Newtonian and Boger fluids. This numerical study has shown that the hyperbolic configuration is superior to the other geometry choices in achieving a constant extension rate. In this hyperbolic configuration, no vortices are formed, the measuring range is broader and the strain rate is constant throughout the geometric domain, unlike in the alternative configurations tested. The difference between sharp and rounded recess-corner configurations proved to be negligible and a rise in excess pressure drop (epd) for increasing deformation rates has been observed.
Similar content being viewed by others
References
Aboubacar M, Webster MF (2001) A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows. J Non-Newt Fluid Mech 98(2–3):83–106
Aguayo JP, Tamaddon-Jahromi HR, Webster MF (2008) Excess pressure-drop estimation in contraction and expansion flows for constant shear-viscosity, extension strain-hardening fluids. J Non-Newt Fluid Mech 153(2–3):157–176. doi:10.1016/j.jnnfm.2008.05.004
Baird DG, Huang J (2006) Elongational viscosity measurements using a semi-hyperbolic die. Appl Rheol 16(6):312–320
Baird DG, Chan TW, McGrady CD, Mazahir SM (2010) Evaluation of the use of a semi-hyperbolic die for measuring elongational viscosity of polymer melts. Appl Rheol 20:34900–34912
Binding DM (1988) An approximate analysis for contraction and converging flows. J Non-Newt Fluid Mech 27:173–189
Binding DM (1991) Further considerations of axisymmetric contraction flows. J Non-Newt Fluid Mech 41:27–42
Binding DM, Couch MA, Walters K (1998) The pressure dependence of the shear and elongational properties of polymer melts. J Non-Newt Fluid Mech 79(2–3):137–155
Binding DM, Phillips PM, Phillips TN (2006) Contraction/expansion flows: the pressure drop and related issues. J Non-Newt Fluid Mech 137(1–3):31–38. doi:10.1016/j.jnnfm.2006.03.006
Boger DV (1987) Viscoelastic flows through contractions. Annu Rev Fluid Mech 19:157–182
Chilcott MD, Rallison JM (1988) Creeping flow of dilute polymer-solutions past cylinders and spheres. J Non-Newt Fluid Mech 29(1–3):381–432
Collier JR, Romanoschi O, Petrovan S (1998) Elongational rheology of polymer melts and solutions. J Appl Polym Sci 69(12):2357–2367
Debbaut B, Crochet MJ (1988) Extensional effects in complex flows. J Non-Newt Fluid Mech 30:169–184
Donea J (1984) A Taylor–Galerkin method for convective-transport problems. Int J Numer Methods Eng 20(1):101–119
Entov VM, Hinch EJ (1997) Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid. J Non-Newt Fluid Mech 72(1):31–53
Fuller GG, Cathey CA, Hubbard B, Zebrowski BE (1985) Extensional viscosity measurements for low-viscosity fluids. J Rheol 31:235–249
Hawken DM, Tamaddon-Jahromi HR, Townsend P, Webster MF (1990) A Taylor–Galerkin-based algorothim for viscous incompressible flow. Int J Numer Methods Fluids 10(3):327–351
James DF, Chandler GM, Armour SJ (1990) A converging channel rheometer for the measurement of extensional viscosity. J Non-Newt Fluid Mech 35(2–3):421–443
Kim HC, Pendse A, Collier JR (1994) Polymer melt lubricated elongational flow. J Rheol 38(4):831–845
Meissner J (1972) Development of a universal extensional rheometer for the uniaxial extension of polymer melts. Trans Soc Rheol 16:405–420
Meissner J, Hostettler J (1994) A new elongational rheometer for polymer melts and other highly viscoelastic liquids. Rheol Acta 33(1):1–21
Nguyen TH, Boger DV (1984) The influence of elasticity on die entry flows. J Rheol 28(5):654–654
Nigen S, Walters K (2001) On the two-dimensional splashing experiment for Newtonian and slightly elastic liquids. J Non-Newt Fluid Mech 97(2–3):233–250. doi:10.1016/s0377-0257(00)00221-4
Oom A, Pettersson A, Taylor JRN, Stading M (2008) Rheological properties of kafirin and zein prolamins. J Cereal Sci 47(1):109–116
Rothstein JP, McKinley GH (1999) Extensional flow of a polystyrene Boger fluid through a 4:1:4 axisymmetric contraction/expansion. J Non-Newt Fluid Mech 86(1–2):61–88
Rothstein JP, McKinley GH (2001) The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop. J Non-Newt Fluid Mech 98(1):33–63
Sridhar T, Tirtaatmadja V, Nguyen DA, Gupta RK (1991) Measurement of extensional viscosity of polymer-solutions. J Non-Newt Fluid Mech 40(3):271–280
Stading M, Bohlin L (2000) Measurements of extensional flow properties of semi-solid foods in contraction flow. Proceedings of the 2nd International Symposium on Food Rheology and Structure 2:117–120
Stading M, Bohlin L (2001) Contraction flow measurements of extensional properties. Trans Nordic Rheol Soc 8/9:147–150
Szabo P, Rallison JM, Hinch EJ (1997) Start-up of flow of a FENE-fluid through a 4:1:4 constriction in a tube. J Non-Newt Fluid Mech 72(1):73–86. doi:10.1016/s0377-0257(97)00023-2
Tamaddon-Jahromi HR, Webster MF, Walters K (2010) Predicting numerically the large increases in extra pressure drop when boger fluids flow through axisymmetric contractions. Nat Scie 2(1):1–11
Tamaddon Jahromi HR, Webster MF, Williams R (2011) Excess pressure drop and drag calculations for strain-hardening fluids with mild shear-thinning: contraction and falling sphere problems. J Non-Newt Fluid Mech 166:939–950
Walters K, Webster MF (2003) The distinctive CFD challenges of computational rheology. Int J Numer Method Fluid 43(5):577–596. doi:10.1002/fld.522
Wapperom P, Webster MF (1998) A second-order hybrid finite-element/volume method for viscoelastic flows. J Non-Newt Fluid Mech 79:405–431
Wapperom P, Webster MF (1999) Simulation for viscoelastic flow by a finite volume/element method. Comput Meth Appl Mech Eng 180(3–4):281–304
Webster MF, Matallah H, Sujatha KS (2005a) Sub-cell approximations for viscoelastic flows-filament stretching. J Non-Newt Fluid Mech 126:187–205
Webster MF, Tamaddon-Jahromi HR, Aboubacar M (2005b) Time-dependent algorithms for viscoelastic flow: finite element/volume schemes. Numer Meth Part Differ Equ 21(2):272–296. doi:10.1002/num.20037
White SA, Gotsis AD, Baird DG (1987) Review of the entry flow problem—experimental and numerical. J Non-Newt Fluid Mech 24(2):121–160
Wikström K, Bohlin L (1999a) Extensional flow studies of wheat flour dough. I. Experimental method for measurements in contraction flow geometry and application to flours varying in breadmaking performance. J Cereal Sci 29(3):217–226
Zatti D, Wiklund J, Vignali G, Stading M (2009) Determination of velocities profiles in hyperbolic contraction using ultrasound velocity profiling. Annu Trans Nordic Rheol Soc 17:277–280
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nyström, M., Tamaddon Jahromi, H.R., Stading, M. et al. Numerical simulations of Boger fluids through different contraction configurations for the development of a measuring system for extensional viscosity. Rheol Acta 51, 713–727 (2012). https://doi.org/10.1007/s00397-012-0631-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00397-012-0631-0