Stick-slip flow is a challenging viscoelastic benchmark problem due to the presence of a separation or transition point at the die exit where a sudden change in flow boundary conditions occurs. We present numerical simulations of transient planar stick-slip flow of the Phan-Thien–Tanner (PTT) and Giesekus fluids, investigating the polymer stress behavior around the stress singularity at the stick-slip point, confirming the asymptotic results presented by Evans et al. [“Stresses of the Oldroyd-B, PTT and Giesekus fluids in a Newtonian velocity field near the stick-slip singularity,” Phys. Fluids 29, 1–33 (2017)]. In order to improve the numerical knowledge about this viscoelastic benchmark problem, two distinct mathematical methodologies are used for comparison in the computational simulations: the Cartesian and natural stress formulations. The former is widely applied in computational rheology, while the latter is used for the first time in the context of this problem. The natural stress formulation gives improved convergence results both temporally and spatially near to the singularity while maintaining the same global flow characteristics as the Cartesian.
REFERENCES
1.
J. R. A.
Pearson
, Mechanics of Polymer Processing–Application to Polymer Processing (Videorecording)
(Center for Advanced Engineering Study
, M.I.T., Cambridge, MA
, 1974
).2.
R. I.
Tanner
, Engineering Rheology
, The Oxford Engineering Science Series (Clarendon Press
, 1985
).3.
S.
Richardson
, “A ‘stick-slip’ problem related to the motion of a free jet at low Reynolds numbers
,” Math. Proc. Cambridge Philos. Soc.
67
, 477
–489
(1970
).4.
S.
Richardson
, “The die swell phenomenon
,” Rheol. Acta
9
(2
), 193
–199
(1970
).5.
S. A.
Trogdon
and D. D.
Joseph
, “The stick-slip problem for a round jet
,” Rheol. Acta
19
(4
), 404
–420
(1980
).6.
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
).7.
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
).8.
C. J. S.
Petrie
and M. M.
Denn
, “Instabilities in polymer processing
,” AIChE J.
22
(2
), 209
–236
(1976
).9.
M. M.
Denn
, “Issues in viscoelastic fluid mechanics
,” Annu. Rev. Fluid Mech.
22
(1
), 13
–32
(1990
).10.
M. M.
Denn
, “Extrusion instabilities and wall slip
,” Annu. Rev. Fluid Mech.
33
(1
), 265
–287
(2001
).11.
R. G.
Larson
, “Instabilities in viscoelastic flows
,” Rheol. Acta
31
(3
), 213
–263
(1992
).12.
M. D.
Graham
, “The sharkskin instability of polymer melt flows
,” Chaos
9
(1
), 154
–163
(1999
).13.
Y.
Kwon
, “Numerical modelling of two-dimensional melt fracture instability in viscoelastic flow
,” J. Fluid Mech.
855
, 595
–615
(2018
).14.
O.
Hassager
, “Working group on numerical techniques
,” in Proceedings of the Vth Workshop on Numerical Methods in Non-Newtonian Flow
[J. Non-Newtonian Fluid Mech.
29
, 2
–5
(1988
)].15.
M.
Renardy
, Mathematical Analysis of Viscoelastic Flows
(Society for Industrial and Applied Mathematics, SIAM
, Philadelphia
, 2000
).16.
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
).17.
N.
Phan-Thien
, “A nonlinear network viscoelastic model
,” J. Rheol.
22
(3
), 259
–283
(1978
).18.
H.
Giesekus
, “A unified approach to a variety of constitutive models for polymer fluids based on the concept of configuration-dependent molecular mobility
,” Rheol. Acta
21
(4
), 366
–375
(1982
).19.
R. B.
Bird
, R. C.
Armstrong
, and O.
Hassager
, Dynamics of Polymeric Liquids
, Fluid Mechanics Vol. 1 (Wiley
, New York
, 1987
).20.
L. L.
Ferrás
, A. M.
Afonso
, J. M.
Nóbrega
, and F. T.
Pinho
, “A numerical and theoretical study on viscoelastic fluid slip flows
,” Phys. Fluids
29
(5
), 053102
(2017
).21.
L.
Campo-Deaño
, R. P. A.
Dullens
, D. G. A. L.
Aarts
, F. T.
Pinho
, and M. S. N.
Oliveira
, “Viscoelasticity of blood and viscoelastic blood analogues for use in polydymethylsiloxane in vitro models of the circulatory system
,” Biomicrofluidics
7
(3
), 034102
(2013
).22.
A.
Ramiar
, M. M.
Larimi
, and A. A.
Ranjbar
, “Investigation of blood flow rheology using second-grade viscoelastic model (Phan-Thien-Tanner) within carotid artery
,” Acta Bioeng. Biomech.
19
(3
), 27
–41
(2017
).23.
N. A.
Talib
and Ö.
Ertunç
, “Application of Giesekus model for capillary extrusion of rubber compound
,” J. Chem. Eng. Jpn.
52
(2
), 185
–193
(2019
).24.
D.
Tang
, F. H.
Marchesini
, D. R.
D’hooge
, and L.
Cardon
, “Isothermal flow of neat polypropylene through a slit die and its die swell: Bridging experiments and 3D numerical simulations
,” J. Non-Newtonian Fluid Mech.
266
, 33
–45
(2019
).25.
Z.
Khan
, R. A.
Shah
, S.
Islam
, B.
Jan
, M.
Imran
, and F.
Tahir
, “Steady flow and heat transfer analysis of Phan-Thein-Tanner fluid in double-layer optical fiber coating analysis with slip conditions
,” Sci. Rep.
6
, 34593
(2016
).26.
A.
Syrakos
, Y.
Dimakopoulos
, and J.
Tsamopoulos
, “Theoretical study of the flow in a fluid damper containing high viscosity silicone oil: Effects of shear-thinning and viscoelasticity
,” Phys. Fluids
30
(3
), 030708
(2018
).27.
L. L.
Ferrás
, A. M.
Afonso
, M. A.
Alves
, J. M.
Nóbrega
, and F. T.
Pinho
, “Electro-osmotic and pressure-driven flow of viscoelastic fluids in microchannels: Analytical and semi-analytical solutions
,” Phys. Fluids
28
(9
), 093102
(2016
).28.
H. S.
Gaikwad
, P.
Baghel
, R.
Sarma
, and P. K.
Mondal
, “Transport of neutral solutes in a viscoelastic solvent through a porous microchannel
,” Phys. Fluids
31
(2
), 022006
(2019
).29.
J.
Evans
, I. L.
Palhares Junior
, and C. M.
Oishi
, “Stresses of the Oldroyd-B, PTT and Giesekus fluids in a Newtonian velocity field near the stick-slip singularity
,” Phys. Fluids
29
, 121604
(2017
).30.
J.
Evans
, “Stick-slip and slip-stick singularities of the Phan-Thien–Tanner fluid
,” J. Non-Newtonian Fluid Mech.
199
, 12
–19
(2013
).31.
J.
Evans
, “Stick-slip singularity of the Giesekus fluid
,” J. Non-Newtonian Fluid Mech.
222
, 24
–33
(2015
).32.
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
).33.
J. D.
Evans
and C. M.
Oishi
, “Transient computations using the natural stress formulation for solving sharp corner flows
,” J. Non-Newtonian Fluid Mech.
249
, 48
–52
(2017
).34.
J. D.
Evans
, H. L.
Franca
, and C. M.
Oishi
, “Application of the natural stress formulation for solving unsteady viscoelastic contraction flows
,” J. Comput. Phys.
388
(1
), 462
–489
(2019
).35.
A.
Fortin
, A.
Zine
, and J.-F.
Agassant
, “Computing viscoelastic fluid flow problems at low cost
,” J. Non-Newtonian Fluid Mech.
45
(2
), 209
–229
(1992
).36.
F. P. T.
Baaijens
, “Application of low-order discontinuous Galerkin methods to the analysis of viscoelastic flows
,” J. Non-Newtonian Fluid Mech.
52
, 37
–57
(1994
).37.
S.-C.
Xue
, R. I.
Tanner
, and N.
Phan-Thien
, “Three-dimensional numerical simulations of viscoelastic flows–predictability and accuracy
,” Comput. Methods Appl. Mech. Eng.
180
, 305
–331
(1999
).38.
G.
Karapetsas
and J.
Tsamopoulos
, “On the stick-slip flow from slit and cylindrical dies of a Phan-Thien and Tanner fluid model. I. Steady state
,” Phys. Fluids
21
(12
), 123101
(2009
).39.
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
).40.
P.
Wapperom
and M.
Renardy
, “Numerical prediction of the boundary layers in the flow around a cylinder using a fixed velocity field
,” J. Non-Newtonian Fluid Mech.
125
(1
), 35
–48
(2005
).41.
M. I.
Gerritsma
and T. N.
Phillips
, “On the characteristics and compatibility equations for the UCM model fluid
,” Z. Angew. Math. Mech.
88
(7
), 523
–539
(2008
).42.
M. I.
Gerritsma
and T. N.
Phillips
, “On the use of characteristic variables in viscoelastic flow problems
,” IMA J. Appl. Math.
66
(2
), 127
–147
(2001
).43.
C. M.
Oishi
, F. P.
Martins
, M. F.
Tome
, J. A.
Cuminato
, and S.
Mckee
, “Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows
,” J. Non-Newtonian Fluid Mech.
166
(3
), 165
–179
(2011
).44.
A. J.
Chorin
and J. E.
Marsden
, A Mathematical Introduction to Fluid Mechanics
(Springer
, 1990
), Vol. 3.45.
D. L.
Brown
, R.
Cortez
, and M. L.
Minion
, “Accurate projection methods for the incompressible Navier–Stokes equations
,” J. Comput. Phys.
168
(2
), 464
–499
(2001
).46.
F. S.
Sousa
, C. M.
Oishi
, and G. C.
Buscaglia
, “Spurious transients of projection methods in microflow simulations
,” Comput. Methods Appl. Mech. Eng.
285
, 659
–693
(2015
).47.
G.
Tryggvason
, R.
Scardovelli
, and S.
Zaleski
, Direct Numerical Simulations of Gas–Liquid Multiphase Flows
(Cambridge University Press
, 2011
).48.
M. A.
Alves
, F. T.
Pinho
, and P. J.
Oliveira
, “Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions
,” J. Non-Newtonian Fluid Mech.
110
, 45
–75
(2003
).49.
Y.-C.
Ahn
and M. E.
Ryan
, “A finite difference analysis of the extrudate swell problem
,” Int. J. Numer. Methods Fluids
13
(10
), 1289
–1310
(1991
).50.
H. K.
Moffatt
, “Viscous and resistive eddies near a sharp corner
,” J. Fluid Mech.
18
(1
), 1
–18
(1964
).51.
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
).© 2019 Author(s).
2019
Author(s)
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