Abstract
Fountain flow is the phenomenon of deceleration and outward motion of fluid particles as they approach a slower moving interface. Numerical simulations have been undertaken for the flow of viscoelastic fluids, obeying an integral constitutive equation of the K-BKZ type, capable of describing the behavior of polymer melts. Two polyethylene melts are considered, a branched LDPE and a linear HDPE. Their rheology is well captured by the integral model. The flow simulations are performed for planar and axisymmetric geometries and show the shape and extent of the free surface, as well as the stresses and pressures in the system. The semicircle is a good rough approximation for the free surface of fountain flow, but detailed computations show the effect of elasticity on the free surface, which is non-monotonic for the LDPE as the elasticity level (or apparent shear rate) increases. The less elastic HDPE shows a monotonic decrease in the extent of the flow front as elasticity increases. In all cases, the excess pressure losses (front pressure correction) increase with increasing flow rate. The effect of a nonzero second normal stress difference is to extend the flow front and increase the pressure losses.
References
Bagley, E. B., Birks, A. M., “Flow of Polyethylene into a Capillary”, J. Appl. Phys., 31, 556–561(1960) DOI: 10.1063/1.1735627Search in Google Scholar
Beaulne, M., Mitsoulis, E., “Effect of Viscoelasticity in the Film-Blowing Process”, J. Appl. Polym. Sci., 105, 2098–2112(2007) DOI: 10.1002/app.26325Search in Google Scholar
Behrens, R. A., et al., “Transient Free-Surface Flows: Motion of a Fluid Advancing in a Tube”, AIChE J., 33, 1178–1186(1987) DOI: 10.1002/aic.690330712Search in Google Scholar
Beris, A. N., “Fluid Elements Deformation behind an Advancing Flow Front”, J. Rheol., 31, 121–124(1987) DOI: 10.1122/1.549918Search in Google Scholar
Bogaerds, A. C. B., et al., “Stability Analysis of Injection Molding Flows”, J. Rheol., 48, 765–785(2004) DOI: 10.1122/1.1753276Search in Google Scholar
Bulters, M., Schepens, A., “The Origin of the Surface Defect ‘Slip-Stick’ on Injection Moulded Products”, 16th Annual Meeting of the Polymer Processing Society, Shanghai, 144–145 (2000)Search in Google Scholar
Coyle, D. J., et al., “The Kinematics of Fountain Flow in Mold-Filling”, AIChE J., 33, 1168–1177(1987) DOI: 10.1002/aic.690330711Search in Google Scholar
Dimakopoulos, Y., Tsamopoulos, J., “On the Transient Coating of a Straight Tube with a Viscoelastic Material”, J. Non-Newtonian Fluid Mech., 159, 95–114(2009) DOI: 10.1016/j.jnnfm.2009.02.001Search in Google Scholar
Georgiou, G. C., Boudouvis, A. G., “Converged Solutions for the Newtonian Extrudate-Swell Problem”, Int. J. Numer. Meth. Fluids, 29, 363–371(1999) DOI: 10.1002/(SICI)1097-0363(19990215)29:3<363::AID-FLD792>3.0.CO;2-DSearch in Google Scholar
Goublomme, A., Crochet, M. J., “Numerical Prediction of Extrudate Swell of a High-Density Polyethylene: Further Results”, J. Non-Newtonian Fluid Mech., 47, 281–287(1993) DOI: 10.1016/0377-0257(93)80055-GSearch in Google Scholar
Kamal, M. R., Tan, V., “Orientation in Injection Molded Polystyrene”, Polym. Eng. Sci., 19, 558–563(1979) DOI: 10.1002/pen.760190806Search in Google Scholar
Keunings, R., “On the High Weissenberg Number Problem”, J. Non-Newtonian Fluid Mech., 20, 209–226(1986) DOI: 10.1016/0377-0257(86)80022-2Search in Google Scholar
Luo, X.-L., Mitsoulis, E., “A Numerical Study of the Effect of Elongational Viscosity on Vortex Growth in Contraction Flows of Polyethylene Melts”, J. Rheol., 34, 309–342(1990a) DOI: 10.1122/1.550131Search in Google Scholar
Luo, X.-L., Mitsoulis, E., “An Efficient Algorithm for Strain History Tracking in Finite Element Computations of Non-Newtonian Fluids with Integral Constitutive Equations”, Int. J. Num. Meth. Fluids, 11, 1015–1031(1990b) DOI: 10.1002/fld.1650110708Search in Google Scholar
Luo, X.-L., Tanner, R. I., “A Computer Study of Film Blowing”, Polym. Eng. Sci., 25, 620–629(1985) DOI: 10.1002/pen.760251008Search in Google Scholar
Luo, X.-L., Tanner, R. I., “Finite Element Simulation of Long and Short Circular Die Extrusion Experiments Using Integral Models”, Int. J. Num. Meth. Eng., 25, 9–22(1988) DOI: 10.1002/nme.1620250104Search in Google Scholar
Mavridis, H., et al., “Finite Element Simulation of Fountain Flow in Injection Molding”, Polym. Eng. Sci., 26, 449–454(1986) DOI: 10.1002/pen.760260702Search in Google Scholar
Mavridis, H., et al., “Transient Free-Surface Flows in Injection Mold Filling”, AIChE J., 34, 403–410(1988a) DOI: 10.1002/aic.690340307Search in Google Scholar
Mavridis, H., et al., “The Effect of Fountain Flow on Molecular Orientation in Injection Molding”, J. Rheol., 32, 639–663(1988b) DOI: 10.1122/1.549984Search in Google Scholar
Meissner, J., “Basic Parameters, Melt Rheology, Processing and End-Use Properties of Three Similar Low Density Polyethylene Samples”, Pure Appl. Chem., 42, 551–612(1975) DOI: 10.1351/pac197542040551Search in Google Scholar
Mitsoulis, E., “Numerical Simulation of Planar Entry Flow for a Polyisobutylene Solution Using an Integral Constitutive Equation”, J. Rheol., 37, 1029–1040(1993) DOI: 10.1122/1.550407Search in Google Scholar
Mitsoulis, E., “Flow of Polyethylene Melts through Contractions: Comparison of Simulations with Experiments”, XIIIth International Congress on Rheology, Binding, D.M.et al. (Eds.), British Society of Rheology, Cambridge, UK, 2, 184–186(2000)Search in Google Scholar
Mitsoulis, E., “Numerical Simulation of Entry Flow of the IUPAC-LDPE Melt”, J. Non-Newtonian Fluid Mech., 97, 13–30(2001) DOI: 10.1016/S0377-0257(00)00183-XSearch in Google Scholar
Mitsoulis, E., “Numerical Simulation of Fountain Flow of Pseudoplastic and Viscoplastic Fluids”, submitted to J. Non-Newtonian Fluid Mech.(2009)Search in Google Scholar
Mitsoulis, E., et al., “Simulation of Extrudate Swell from Long Slit and Capillary Dies”, Polym. Proc. Eng., 2, 153–177(1984)Search in Google Scholar
Mitsoulis, E., et al., “Entry Flow of LDPE Melts in a Planar Contraction”, J. Non-Newtonian Fluid Mech., 111, 41–61(2003) DOI: 10.1016/S0377-0257(03)00012-0Search in Google Scholar
Papanastasiou, A. C., et al., “An Integral Constitutive Equation for Mixed Flows: Viscoelastic Characterization”, J. Rheol., 27, 387–410(1983) DOI: 10.1122/1.549712Search in Google Scholar
Rose, W., “Fluid-Fluid Interfaces in Steady Motion”, Nature, 191, 242–243(1961) DOI: 10.1038/191242a0Search in Google Scholar
Sato, T., Richardson, S. M., “Numerical Simulation of the Fountain Flow Problem for Viscoelastic Fluids”, Polym. Eng. Sci., 35, 805–812(1995) DOI: 10.1002/pen.760351003Search in Google Scholar
Schmidt, R. L., “A Special Mold and Tracer Technique for Studying Shear and Extensional Flows in a Mold Cavity during Injection Molding”, Polym. Eng. Sci., 14, 797–800(1974) DOI: 10.1002/pen.760141111Search in Google Scholar
Tadmor, Z., Gogos, C. G.: Principles of Polymer Processing, SPE Monograph Series, Wiley, New York(1979)Search in Google Scholar
Taliadorou, E., et al., “Numerical Simulation of the Extrusion of Strongly Compressible Newtonian Liquids”, Rheol. Acta, 47, 49–62(2008) DOI: 10.1007/s00397-007-0207-6Search in Google Scholar
Tanner, R. I.: Engineering Rheology, 2nd Edition, Oxford University Press, Oxford(2000)Search in Google Scholar
Verbeeten, W. M. H., et al., “Differential Constitutive Equations for Polymer Melts: The Extended Pom-Pom Model”, J. Rheol., 45, 823–844(2001) DOI: 10.1122/1.1380426Search in Google Scholar
Wood-Adams, P. M., “The Effect of Long Chain Branches on the Shear Flow Behavior of Polyethylene”, J. Rheol., 45, 203–210(2001) DOI: 10.1122/1.1332785Search in Google Scholar
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