The stress singularity for Phan-Thien–Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (commonly termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like , where r is the radial distance from the contact point and λ0 is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like and dominates the polymer stresses, which are like for PTT and for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.
REFERENCES
1.
2.
R. G.
Larson
, “Instabilities in viscoelastic flows
,” Rheol. Acta
31
, 213
–263
(1992
).3.
M. M.
Denn
, “Extrusion instabilities and wall slip
,” Annu. Rev. Fluid Mech.
33
(1
), 265
–287
(2001
).4.
D. B.
Harmon
, “Drop sizes from low speed jets
,” J. Franklin Inst.
259
(6
), 519
–522
(1955
).5.
S.
Middleman
and J.
Gavis
, “Expansion and contraction of capillary jets of Newtonian liquids
,” Phys. Fluids
4
(3
), 355
–359
(1961
).6.
S. L.
Goren
and S.
Wronski
, “The shape of low-speed capillary jets of Newtonian liquids
,” J. Fluid Mech.
25
(1
), 185
–198
(1966
).7.
J.
Batchelor
, J. P.
Berry
, and F.
Horsfall
, “Die swell in elastic and viscous fluids
,” Polymer
14
(7
), 297
–299
(1973
).8.
R. L.
Gear
, M.
Keentok
, J. F.
Milthorpe
, and R. I.
Tanner
, “The shape of low Reynolds number jets
,” Phys. Fluids
26
(1
), 7
–9
(1983
).9.
W. W.
Graessley
, S. D.
Glasscock
, and R. L.
Crawley
, “Die swell in molten polymers
,” Trans. Soc. Rheol.
14
(4
), 519
–544
(1970
).10.
L. A.
Utracki
, Z.
Bakerdjian
, and M. R.
Kamal
, “A method for the measurement of the true die swell of polymer melts
,” J. Appl. Polym. Sci.
19
(2
), 481
–501
(1975
).11.
D. C.
Huang
and J. L.
White
, “Extrudate swell from slit and capillary dies: An experimental and theoretical study
,” Polym. Eng. Sci.
19
(9
), 609
–616
(1979
).12.
S.
Richardson
, “A ‘stick-slip’ problem related to the motion of a free jet at low Reynolds numbers
,” Math. Proc. Cambridge Philos. Soc.
67
(2
), 477
–489
(1970
).13.
R. I.
Tanner
and X.
Huang
, “Stress singularities in non-Newtonian stick-slip and edge flows
,” J. Non-Newtonian Fluid Mech.
50
(2
), 135
–160
(1993
).14.
L.
Sturges
, “Die swell: The separation of the free surface
,” J. Non-Newtonian Fluid Mech.
6
(2
), 155
–159
(1979
).15.
M.
Zidan
, “Zur rheologie des spinnprozesses
,” Rheol. Acta
8
, 89
–123
(1969
).16.
S. A.
Trogdon
and D. D.
Joseph
, “The stick-slip problem for a round jet
,” Rheol. Acta
19
, 404
–420
(1980
).17.
C. J.
Coleman
, “A note on the stick-slip and die-swell problems for a second-order fluid
,” J. Non-Newtonian Fluid Mech.
3
(3
), 288
–292
(1978
).18.
L. D.
Sturges
, “A theoretical study of extrudate swell
,” J. Non-Newtonian Fluid Mech.
9
(3
), 357
–378
(1981
).19.
S.
Richardson
, “The die swell phenomenon
,” Rheol. Acta
9
, 193
–199
(Apr 1970
).20.
J.
Gavis
and M.
Modan
, “Expansion and contraction of jets of Newtonian liquids in air: Effect of tube length
,” Phys. Fluids
10
(3
), 487
–497
(1967
).21.
D. D.
Joseph
, “Slow motion and viscometric motion; stability and bifurcation of the rest state of a simple fluid
,” Arch. Ration. Mech. Anal.
56
, 99
–157
(1974
).22.
A. S.
Lodge
, Elastic Liquids: An Introductory Vector Treatment of Finite-Strain Polymer Rheology
(Academic Press
, 1964
).23.
A. B.
Metzner
, W. T.
Houghton
, R. A.
Sailor
, and J. L.
White
, “A method for the measurement of normal stresses in simple shearing flow
,” Trans. Soc. Rheol.
5
(1
), 133
–147
(1961
).24.
J. C.
Slattery
and W. R.
Schowalter
, “Effect of surface tension in the measurement of the average normal stress at the exit of a capillary tube through an analysis of the capillary jet
,” J. Appl. Polym. Sci.
8
(5
), 1941
–1947
(1964
).25.
J. M.
Davies
, J. F.
Hutton
, and K.
Walters
, “Theory for normal stresses in slits and capillaries
,” J. Phys. D: Appl. Phys.
6
(18
), 2259
–2266
(1973
).26.
J. M.
Davies
, J. F.
Hutton
, and K.
Walters
, “A critical re-appraisal of the jet-thrust technique for normal stresses, with particular reference to axial velocity and stress rearrangement at the exit plane
,” J. Non-Newtonian Fluid Mech.
3
(2
), 141
–160
(1977
).27.
D. V.
Boger
and M. M.
Denn
, “Capillary and slit methods of normal stress measurements
,” J. Non-Newtonian Fluid Mech.
6
(3
), 163
–185
(1980
).28.
R. R.
Huilgol
, “Some qualitative and quantitative features of die swell
,” Polym. Eng. Sci.
18
(8
), 643
–648
(1978
).29.
R. I.
Tanner
, “Die-swell reconsidered: Some numerical solutions using a finite element program
,” Appl. Polym. Symp.
20
, 201
–208
(1973
).30.
R. E.
Nickell
, R. I.
Tanner
, and B.
Caswell
, “The solution of viscous incompressible jet and free-surface flows using finite-element methods
,” J. Fluid Mech.
65
(1
), 189
–206
(1974
).31.
K. R.
Reddy
and R. I.
Tanner
, “Finite element solution of viscous jet flows with surface tension
,” Comput. Fluids
6
(2
), 83
–91
(1978
).32.
M. J.
Crochet
and R.
Keunings
, “Die swell of a Maxwell fluid: numerical prediction
,” J. Non-Newtonian Fluid Mech.
7
(2-3
), 199
–212
(1980
).33.
P.
Chang
, T. W.
Patten
, and B. A.
Finlayson
, “Collocation and galerkin finite element methods for viscoelastic fluid flow—II. Die swell problems with a free surface
,” Comput. Fluids
7
(4
), 285
–293
(1979
).34.
B. J.
Omodei
, “Computer solutions of a plane Newtonian jet with surface tension
,” Comput. Fluids
7
(2
), 79
–96
(1979
).35.
B. J.
Omodei
, “On the die-swell of an axisymmetric Newtonian jet
,” Comput. Fluids
8
(3
), 275
–289
(1980
).36.
M. J.
Crochet
and R.
Keunings
, “Finite element analysis of die swell of a highly elastic fluid
,” J. Non-Newtonian Fluid Mech.
10
(3
), 339
–356
(1982
).37.
M. B.
Bush
, J. F.
Milthorpe
, and R. I.
Tanner
, “Finite element and boundary element methods for extrusion computations
,” J. Non-Newtonian Fluid Mech.
16
(1
), 37
–51
(1984
).38.
J.
Clermont
and M.
Normandin
, “Numerical simulation of extrudate swell for Oldroyd-B fluids using the stream-tube analysis and a streamline approximation
,” J. Non-Newtonian Fluid Mech.
50
(2
), 193
–215
(1993
).39.
S.
Tanoue
, T.
Kajiwara
, Y.
Iemoto
, and K.
Funatsu
, “High Weissenberg number simulation of an annular extrudate swell using the differential type constitutive equation
,” Polym. Eng. Sci.
38
(3
), 409
–419
(1998
).40.
G.
Karapetsas
and J.
Tsamopoulos
, “Steady extrusion of viscoelastic materials from an annular die
,” J. Non-Newtonian Fluid Mech.
154
(2
), 136
–152
(2008
).41.
M.
Normandin
, J.
Clermont
, J.
Guillet
, and C.
Raveyre
, “Three-dimensional extrudate swell experimental and numerical study of a polyethylene melt obeying a memory-integral equation
,” J. Non-Newtonian Fluid Mech.
87
(1
), 1
–25
(1999
).42.
G.
Russo
and T. N.
Phillips
, “Spectral element predictions of die-swell for Oldroyd-B fluids
,” Comput. Fluids
43
(1
), 107
–118
(2011
), part of special issue: Symposium on High Accuracy Flow Simulations. Special Issue Dedicated to Prof. Michel Deville.43.
G.
Russo
and T. N.
Phillips
, “Numerical prediction of extrudate swell of branched polymer melts
,” Rheol. Acta
49
(6
), 657
–676
(2010
).44.
V.
Ganvir
, A.
Lele
, R.
Thaokar
, and B. P.
Gautham
, “Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method
,” J. Non-Newtonian Fluid Mech.
156
(1
), 21
–28
(2009
).45.
Y.
Cao
, X.
Ren
, X.
Guo
, M.
Wang
, Q.
Wang
, X.
Xu
, and X.
Yang
, “A new method to simulate free surface flows for viscoelastic fluid
,” Adv. Mater. Sci. Eng.
2015
, 159831
.46.
J.
Fang
, R. G.
Owens
, L.
Tacher
, and A.
Parriaux
, “A numerical study of the SPH method for simulating transient viscoelastic free surface flows
,” J. Non-Newtonian Fluid Mech.
139
(1
), 68
–84
(2006
).47.
A.
Rafiee
, M. T.
Manzari
, and M.
Hosseini
, “An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows
,” Int. J. Non-Linear Mech.
42
(10
), 1210
–1223
(2007
).48.
X.
Xu
, J.
Ouyang
, T.
Jiang
, and Q.
Li
, “Numerical simulation of 3D-unsteady viscoelastic free surface flows by improved smoothed particle hydrodynamics method
,” J. Non-Newtonian Fluid Mech.
177-178
, 109
–120
(2012
).49.
M. F.
Tomé
, B.
Duffy
, and S.
McKee
, “A numerical technique for solving unsteady non-Newtonian free surface flows
,” J. Non-Newtonian Fluid Mech.
62
(1
), 9
–34
(1996
).50.
M. F.
Tomé
, A. C.
Filho
, J. A.
Cuminato
, N.
Mangiavacchi
, and S.
Mckee
, “GENSMAC3D: A numerical method for solving unsteady three-dimensional free surface flows
,” Int. J. Numer. Methods Fluids
37
(7
), 747
–796
(2001
).51.
M. F.
Tomé
, L.
Grossi
, A.
Castelo
, J. A.
Cuminato
, N.
Mangiavacchi
, V. G.
Ferreira
, F. S.
de Sousa
, and S.
McKee
, “A numerical method for solving three-dimensional generalized Newtonian free surface flows
,” J. Non-Newtonian Fluid Mech.
123
(2
), 85
–103
(2004
).52.
M. F.
Tomé
, A.
Castelo
, V. G.
Ferreira
, and S.
McKee
, “A finite difference technique for solving the Oldroyd-B model for 3D-unsteady free surface flows
,” J. Non-Newtonian Fluid Mech.
154
(2
), 179
–206
(2008
).53.
M. F.
Tomé
, G. S.
Paulo
, F. T.
Pinho
, and M. A.
Alves
, “Numerical solution of the PTT constitutive equation for unsteady three-dimensional free surface flows
,” J. Non-Newtonian Fluid Mech.
165
(5
), 247
–262
(2010
).54.
G.
Mompean
, L.
Thais
, M. F.
Tomé
, and A.
Castelo
, “Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models
,” Comput. Fluids
44
(1
), 68
–78
(2011
).55.
R. A.
Figueiredo
, C. M.
Oishi
, J. A.
Cuminato
, and M. A.
Alves
, “Three-dimensional transient complex free surface flows: Numerical simulation of XPP fluid
,” J. Non-Newtonian Fluid Mech.
195
, 88
–98
(2013
).56.
M. F.
Tomé
, J.
Bertoco
, C. M.
Oishi
, M. S. B.
Araujo
, D.
Cruz
, F. T.
Pinho
, and M.
Vynnycky
, “A finite difference technique for solving a time strain separable K-BKZ constitutive equation for two-dimensional moving free surface flows
,” J. Comput. Phys.
311
, 114
–141
(2016
).57.
J. L.
Favero
, A. R.
Secchi
, N. S. M.
Cardozo
, and H.
Jasak
, “Viscoelastic fluid analysis in internal and in free surface flows using the software OpenFOAM
,” Comput. Chem. Eng.
34
(12
), 1984
–1993
(2010
).58.
R.
Comminal
, F.
Pimenta
, J. H.
Hattel
, M. A.
Alves
, and J.
Spangenberg
, “Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods
,” J. Non-Newtonian Fluid Mech.
252
(C
), 1
–18
(2018
).59.
J. P.
Berry
and J.
Batchelor
, “Application of the momentum balance theory of die-swell to polymer melts
,” Res. Rep.
184
(1970
).60.
S.
Richardson
, “A ‘stick-slip’ problem related to the motion of a free jet at low Reynolds numbers
,” Math. Proc. Cambridge Philos. Soc.
67
(2
), 477
–489
(1970
).61.
P.
André
and J.
Clermont
, “Numerical simulation of the die swell problem of a Newtonian fluid by using the concept of stream function and a local analysis of the singularity at the corner
,” J. Non-Newtonian Fluid Mech.
23
, 335
–354
(1987
).62.
R. I.
Tanner
, “A theory of die-swell revisited
,” J. Non-Newtonian Fluid Mech.
129
(2
), 85
–87
(2005
).63.
N.
Phan-Thien
and R. I.
Tanner
, “A new constitutive equation derived from network theory
,” J. Non-Newtonian Fluid Mech.
2
(4
), 353
–365
(1977
).64.
N.
Phan-Thien
, “A nonlinear network viscoelastic model
,” J. Rheol.
22
(3
), 259
–283
(1978
).65.
H.
Giesekus
, “A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility
,” J. Non-Newtonian Fluid Mech.
11
(1
), 69
–109
(1982
).66.
D. H.
Michael
, “The separation of a viscous liquid at a straight edge
,” Mathematika
5
(1
), 82
–84
(1958
).67.
H. K.
Moffatt
, “Viscous and resistive eddies near a sharp corner
,” J. Fluid Mech.
18
(1
), 1
–18
(1964
).68.
G. C.
Georgiou
, L. G.
Olson
, W. W.
Schultz
, and S.
Sagan
, “A singular finite element for Stokes flow: The stick-slip problem
,” Int. J. Numer. Methods Fluids
9
(11
), 1353
–1367
(1989
).69.
G. C.
Georgiou
, W. W.
Schultz
, and L. G.
Olson
, Singular Finite Elements for Fluid Flow Problems with Stress Singularities
(Springer Netherlands
, Dordrecht
, 1990
), pp. 174
–176
.70.
M.
Elliotis
, G.
Georgiou
, and C.
Xenophontos
, “Solution of the planar Newtonian stick-slip problem with the singular function boundary integral method
,” Int. J. Numer. Methods Fluids
48
(9
), 1001
–1021
(2005
).71.
T. R.
Salamon
, D. E.
Bornside
, R. C.
Armstrong
, and R. A.
Brown
, “Local similarity solutions for the stress field of an Oldroyd-B fluid in the partial-slip/slip flow
,” Phys. Fluids
9
(8
), 2191
–2209
(1997
).72.
J. D.
Evans
, “Stick-slip and slip-stick singularities of the Phan-Thien–Tanner fluid
,” J. Non-Newtonian Fluid Mech.
199
, 12
–19
(2013
).73.
J. D.
Evans
, “Stick-slip singularity of the Giesekus fluid
,” J. Non-Newtonian Fluid Mech.
222
, 24
–33
(2015
), part of special issue: Rheometry (and General Rheology): Festschrift dedicated to Professor K. Walters FRS on the occasion of his 80th birthday.74.
J. D.
Evans
, I. L.
Palhares Junior
, and C. M.
Oishi
, “Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity
,” Phys. Fluids
29
(12
), 121604
(2017
).75.
T. R.
Salamon
, D. E.
Bornside
, R. C.
Armstrong
, and R. A.
Brown
, “The role of surface tension in the dominant balance in the die swell singularity
,” Phys. Fluids
7
(10
), 2328
–2344
(1995
).76.
H.
Lugt
and E.
Schwiderski
, “Flows around dihedral angles. I. Eigenmotion analysis
,” Proc. R. Soc. London, Ser. A
285
(1402
), 382
–399
(1965
).77.
E. J.
Hinch
, “The flow of an Oldroyd fluid around a sharp corner
,” J. Non-Newtonian Fluid Mech.
50
(2
), 161
–171
(1993
).78.
M.
Renardy
, “How to integrate the upper convected Maxwell (UCM) stresses near a singularity (and maybe elsewhere, too)
,” J. Non-Newtonian Fluid Mech.
52
(1
), 91
–95
(1994
).79.
J. M.
Rallison
and E. J.
Hinch
, “The flow of an Oldroyd fluid past a reentrant corner: The downstream boundary layer
,” J. Non-Newtonian Fluid Mech.
116
, 141
–162
(2004
).80.
M.
Renardy
, “The high Weissenberg number limit of the UCM model and the Euler equations
,” J. Non-Newtonian Fluid Mech.
69
(2
), 293
–301
(1997
).81.
82.
S. S.
Deshpande
, L.
Anumolu
, and M. F.
Trujillo
, “Evaluating the performance of the two-phase flow solver interFoam
,” Comput. Sci. Discovery
5
, 014016
(2012
).© 2019 Author(s).
2019
Author(s)
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