This précis is aimed as a practical field guide to situations in which shear banding might be expected in complex fluids subject to an applied shear flow. Separately for several of the most common flow protocols, it summarizes the characteristic signatures in the measured bulk rheological signals that suggest the presence of banding in the underlying flow field. It does so both for a steady applied shear flow and for the time-dependent protocols of shear startup, step stress, finite strain ramp, and large amplitude oscillatory shear. An important message is that banding might arise rather widely in flows with a strong enough time dependence, even in fluids that do not support banding in a steadily applied shear flow. This suggests caution in comparing experimental data with theoretical calculations that assume a homogeneous shear flow. In a brief postlude, we also summarize criteria in similar spirit for the onset of necking in extensional filament stretching.
References
1.
Goveas
, J.
, and P.
Olmsted
, “A minimal model for vorticity and gradient banding in complex fluids
,” Eur. Phys. J. E
6
, 79
–89
(2001
).2.
Olmsted
, P. D.
, “Perspectives on shear banding in complex fluids
,” Rheol. Acta
47
, 283
–300
(2008
).3.
Britton
, M. M.
, and P. T.
Callaghan
, “Two-phase shear band structures at uniform stress
,” Phys. Rev. Lett.
78
, 4930
–4933
(1997
).4.
Salmon
, J.-B.
, S.
Manneville
, and A.
Colin
, “Shear banding in a lyotropic lamellar phase. I. time-averaged velocity profiles
,” Phys. Rev. E
68
, 051503
(2003
).5.
Manneville
, S.
, A.
Colin
, G.
Waton
, and F.
Schosseler
, “Wall slip, shear banding, and instability in the flow of a triblock copolymer micellar solution
,” Phys. Rev. E
75
, 061502
(2007
).6.
Rogers
, S.
, D.
Vlassopoulos
, and P.
Callaghan
, “Aging, yielding, and shear banding in soft colloidal glasses
,” Phys. Rev. Lett.
100
, 128304
(2008
).7.
Divoux
, T.
, D.
Tamarii
, C.
Barentin
, and S.
Manneville
, “Transient shear banding in a simple yield stress fluid
,” Phys. Rev. Lett.
104
, 208301
(2010
).8.
Martin
, J. D.
, and Y. T.
Hu
, “Transient and steady-state shear banding in aging soft glassy materials
,” Soft Matter
8
, 6940
–6949
(2012
).9.
Coussot
, P.
, J. S.
Raynaud
, F.
Bertrand
, P.
Moucheront
, J. P.
Guilbaud
, H. T.
Huynh
, S.
Jarny
, and D.
Lesueur
, “Coexistence of liquid and solid phases in flowing soft-glassy materials
,” Phys. Rev. Lett.
88
, 218301
(2002
).10.
Li
, Y.
, M.
Hu
, G. B.
McKenna
, C. J.
Dimitriou
, G. H.
McKinley
, R. M.
Mick
, D. C.
Venerus
, and L. A.
Archer
, “Flow field visualization of entangled polybutadiene solutions under nonlinear viscoelastic flow conditions
,” J. Rheol.
57
, 1411
–1428
(2013
).11.
Wang
, S.-Q.
, G.
Liu
, S.
Cheng
, P. E.
Boukany
, Y.
Wang
, and X.
Li
, “Letter to the editor: Sufficiently entangled polymers do show shear strain localization at high enough weissenberg numbers
,” J. Rheol.
58
, 1059
–1069
(2014
).12.
Wang
, S.-Q.
, S.
Ravindranath
, and P. E.
Boukany
, “Homogeneous shear, wall slip, and shear banding of entangled polymeric liquids in simple-shear rheometry: A roadmap of non-linear rheology
,” Macromolecules
44
, 183
–190
(2011
).13.
Ravindranath
, S.
, S.-Q.
Wang
, M.
Ofechnowicz
, and R. P.
Quirk
, “Banding in simple steady shear of entangled polymer solutions
,” Macromolecules
41
, 2663
–2670
(2008
).14.
Divoux
, T.
, M. A.
Fardin
, S.
Manneville
, and S.
Lerouge
, “Shear banding of complex fluids
,” Annu. Rev. Fluid Mech.
48
, 81
–103
(2016
).15.
Manneville
, S.
, “Recent experimental probes of shear banding
,” Rheol. Acta
47
, 301
–318
(2008
).16.
Fielding
, S. M.
, “Shear banding in soft glassy materials
,” Rep. Prog. Phys.
77
, 102601
(2014
).17.
Moorcroft
, R. L.
, and S. M.
Fielding
, “Criteria for shear banding in time-dependent flows of complex fluids
,” Phys. Rev. Lett.
110
, 086001
(2013
).18.
Moorcroft
, R. L.
, and S. M.
Fielding
, “Shear banding in time-dependent flows of polymers and wormlike micelles
,” J. Rheol.
58
, 103
–147
(2014
).19.
Adams
, J. M.
, and P. D.
Olmsted
, “Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions
,” Phys. Rev. Lett.
102
, 067801
(2009
).20.
Moorcroft
, R. L.
, M. E.
Cates
, and S. M.
Fielding
, “Age-dependent transient shear banding in soft glasses
,” Phys. Rev. Lett.
106
, 055502
(2011
).21.
Manning
, M. L.
, J. S.
Langer
, and J. M.
Carlson
, “Strain localization in a shear transformation zone model for amorphous solids
,” Phys. Rev. E
76
, 056106
(2007
).22.
Manning
, M. L.
, E. G.
Daub
, J. S.
Langer
, and J. M.
Carlson
, “Rate-dependent shear bands in a shear-transformation-zone model of amorphous solids
,” Phys. Rev. E
79
, 016110
(2009
).23.
Jagla
, E. A.
, “Shear band dynamics from a mesoscopic modeling of plasticity
,” J. Stat. Mech.: Theory Exp.
2010
, P12025
.24.
Adams
, J. M.
, S. M.
Fielding
, and P. D.
Olmsted
, “Transient shear banding in entangled polymers: A study using the rolie-poly model
,” J. Rheol.
55
, 1007
–1032
(2011
).25.
Boukany
, P. E.
, and S.-Q.
Wang
, “Use of particle-tracking velocimetry and ow birefringence to study nonlinear ow behavior of entangled wormlike micellar solution: From wall slip, bulk disentanglement to chain scission
,” Macromolecules
41
, 1455
–1464
(2008
).26.
Hu
, Y. T.
, C.
Palla
, and A.
Lips
, “Comparison between shear banding and shear thinning in entangled micellar solutions
,” J. Rheol.
52
, 379
–400
(2008
).27.
Boukany
, P. E.
, and S.-Q.
Wang
, “Shear banding or not in entangled DNA solutions depending on the level of entanglement
,” J. Rheol.
53
, 73
–83
(2009
).28.
Hu
, Y.
, L.
Wilen
, A.
Philips
, and A.
Lips
, “Is the constitutive relation for entangled polymers monotonic?
,” J. Rheol.
51
, 275
–295
(2007
).29.
Boukany
, P. E.
, and S.-Q.
Wang
, “Exploring the transition from wall slip to bulk shearing banding in well-entangled dna solutions
,” Soft Matter
5
, 780
–789
(2009
).30.
Divoux
, T.
, C.
Barentin
, and S.
Manneville
, “Stress overshoot in a simple yield stress fluid: An extensive study combining rheology and velocimetry
,” Soft Matter
7
, 9335
–9349
(2011
).31.
Cao
, J.
, and A. E.
Likhtman
, “Shear banding in molecular dynamics of polymer melts
,” Phys. Rev. Lett.
108
, 028302
(2012
).32.
Shi
, Y.
, M. B.
Katz
, H.
Li
, and M. L.
Falk
, “Evaluation of the disorder temperature and free-volume formalisms via simulations of shear banding in amorphous solids
,” Phys. Rev. Lett.
98
, 185505
(2007
).33.
Fielding
, S. M.
, R. L.
Moorcroft
, R. G.
Larson
, and M. E.
Cates
, “Modeling the relaxation of polymer glasses under shear and elongational loads
,” J. Chem. Phys.
138
, 12A504
(2013
).34.
Kurokawa
, A.
, V.
Vidal
, K.
Kurita
, T.
Divoux
, and S.
Manneville
, “Avalanche-like fluidization of a non-brownian particle gel
,” Soft Matter
11
, 9026
–9037
(2015
).35.
Hu
, Y. T.
, and A.
Lips
, “Kinetics and mechanism of shear banding in an entangled micellar solution
,” J. Rheol.
49
, 1001
–1027
(2005
).36.
Hu
, Y. T.
, “Steady-state shear banding in entangled polymers?
,” J. Rheol.
54
, 1307
–1323
(2010
).37.
Divoux
, T.
, C.
Barentin
, and S.
Manneville
, “From stress-induced fluidization processes to herschel-bulkley behaviour in simple yield stress fluids
,” Soft Matter
7
, 8409
–8418
(2011
).38.
Gibaud
, T.
, D.
Frelat
, and S.
Manneville
, “Heterogeneous yielding dynamics in a colloidal gel
,” Soft Matter
6
, 3482
–3488
(2010
).39.
Hyun
, K.
, M.
Wilhelm
, C. O.
Klein
, K. S.
Cho
, J. G.
Nam
, K. H.
Ahn
, S. J.
Lee
, R. H.
Ewoldt
, and G. H.
McKinley
, “A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (laos)
,” Prog. Polym. Sci.
36
, 1697
–1753
(2011
).40.
Carter
, K. A.
, J. M.
Girkin
, and S. M.
Fielding
, “Shear banding in large amplitude oscillatory shear (LAOStrain and LAOStress)of polymers and wormlike micelles
,” J. Rheol.
60
, 883
–904
(2016
); e-print arxiv.org/abs/1510.00191.41.
Yerushalmi
, J.
, S.
Katz
, and R.
Shinnar
, “The stability of steady shear flows of some viscoelastic fluids
,” Chem. Eng. Sci.
25
, 1891
–1902
(1970
).42.
Spenley
, N. A.
, M. E.
Cates
, and T. C. B.
McLeish
, “Nonlinear rheology of wormlike micelles
,” Phys. Rev. Lett.
71
, 939
–942
(1993
).43.
Olmsted
, P.
, O.
Radulescu
, and C.
Lu
, “Johnson-segalman model with a diffusion term in cylindrical couette flow
,” J. Rheol.
44
, 257
–275
(2000
).44.
Grand
, C.
, J.
Arrault
, and M.
Cates
, “Slow transients and metastability in wormlike micelle rheology
,” J. Phys. II
7
, 1071
–1086
(1997
).45.
Schmitt
, V.
, C. M.
Marques
, and F.
Lequeux
, “Shear-induced phase-separation of complex fluids – the role of flow-concentration coupling
,” Phys. Rev. E
52
, 4009
–4015
(1995
).46.
Brochard
, F.
, and P. G.
Degennes
, “Dynamical scaling for polymers in theta-solvents
,” Macromolecules
10
, 1157
–1161
(1977
).47.
Helfand
, E.
, and G. H.
Fredrickson
, “Large fluctuations in polymer-solutions under shear
,” Phys. Rev. Lett.
62
, 2468
–2471
(1989
).48.
Doi
, M.
, and A.
Onuki
, “Dynamic coupling between stress and composition in polymer-solutions and blends
,” J. Phys. II (France)
2
, 1631
–1656
(1992
).49.
Milner
, S. T.
, “Dynamical theory of concentration fluctuations in polymer-solutions under shear
,” Phys. Rev. E
48
, 3674
–3691
(1993
).50.
Wu
, X. L.
, D. J.
Pine
, and P. K.
Dixon
, “Enhanced concentration fluctuations in polymer-solutions under shear-flow
,” Phys. Rev. Lett.
66
, 2408
–2411
(1991
).51.
Beris
, A. N.
, and V. G.
Mavrantzas
, “On the compatibility between various macroscopic formalisms for the concentration and flow of dilute polymer solutions
,” J. Rheol.
38
, 1235
–1250
(1994
).52.
Sun
, T.
, A. C.
Balazs
, and D.
Jasnow
, “Dynamics of phase separation in polymer solutions under shear flow
,” Phys. Rev. E
55
, R6344
–R6347
(1997
).53.
Fielding
, S.
, and P.
Olmsted
, “Kinetics of the shear banding instability in startup flows
,” Phys. Rev. E
68
, 036313
(2003
).54.
Fielding
, S.
, and P.
Olmsted
, “Early stage kinetics in a unified model of shear-induced demixing and mechanical shear banding instabilities
,” Phys. Rev. Lett.
90
, 224501
(2003
).55.
Fielding
, S.
, and P.
Olmsted
, “Flow phase diagrams for concentration-coupled shear banding
,” Eur. Phys. J. E
11
, 65
–83
(2003
).56.
Cromer
, M.
, M. C.
Villet
, G. H.
Fredrickson
, and L. G.
Leal
, “Shear banding in polymer solutions
,” Phys. Fluids
25
, 051703
(2013
).57.
Cromer
, M.
, G. H.
Fredrickson
, and L. G.
Leal
, “A study of shear banding in polymer solutions
,” Phys. Fluids
26
, 063101
(2014
).58.
Besseling
, R.
, L.
Isa
, P.
Ballesta
, G.
Petekidis
, M. E.
Cates
, and W. C. K.
Poon
, “Shear banding and flow-concentration coupling in colloidal glasses
,” Phys. Rev. Lett.
105
, 268301
(2010
).59.
Jin
, H.
, K.
Kang
, K. H.
Ahn
, and J. K. G.
Dhont
, “Flow instability due to coupling of shear-gradients with concentration: non-uniform flow of (hard-sphere) glasses
,” Soft Matter
10
, 9470
–9485
(2014
).60.
Ragouilliaux
, A.
, B.
Herzhaft
, F.
Bertrand
, and P.
Coussot
, “Flow instability and shear localization in a drilling mud
,” Rheol. Acta
46
, 261
–271
(2006
).61.
Bandyopadhyay
, R.
, G.
Basappa
, and A. K.
Sood
, “Observation of chaotic dynamics in dilute sheared aqueous solutions of ctat
,” Phys. Rev. Lett.
84
, 2022
–2025
(2000
).62.
Ganapathy
, R.
, and A. K.
Sood
, “Intermittency route to rheochaos in wormlike micelles with flow-concentration coupling
,” Phys. Rev. Lett.
96
, 108301
(2006
).63.
Fielding
, S.
, and P.
Olmsted
, “Spatiotemporal oscillations and rheochaos in a simple model of shear banding
,” Phys. Rev. Lett.
92
, 084502
(2004
).64.
Aradian
, A.
, and M.
Cates
, “Instability and spatiotemporal rheochaos in a shear-thickening fluid model
,” Europhys. Lett.
70
, 397
–403
(2005
).65.
Aradian
, A.
, and M. E.
Cates
, “Minimal model for chaotic shear banding in shear thickening fluids
,” Phys. Rev. E
73
, 041508
(2006
).66.
Derec
, C.
, G.
Ducouret
, A.
Ajdari
, and F.
Lequeux
, “Aging and nonlinear rheology in suspensions of polyethylene oxide-protected silica particles
,” Phys. Rev. E
67
, 061403
(2003
).67.
Rogers
, S. A.
, P. T.
Callaghan
, G.
Petekidis
, and D.
Vlassopoulos
, “Time-dependent rheology of colloidal star glasses
,” J. Rheol.
54
, 133
–158
(2010
).68.
Koumakis
, N.
, and G.
Petekidis
, “Two step yielding in attractive colloids: transition from gels to attractive glasses
,” Soft Matter
7
, 2456
–2470
(2011
).69.
Dimitriou
, C. J.
, and G. H.
McKinley
, “A comprehensive constitutive law for waxy crude oil: a thixotropic yield stress fluid
,” Soft Matter
10
, 6619
–6644
(2014
).70.
Likhtman
, A. E.
, and R. S.
Graham
, “Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-poly equation
,” J. Non-Newtonian Fluid Mech.
114
, 1
–12
(2003
).71.
Gibaud
, T.
, C.
Barentin
, and S.
Manneville
, “Influence of boundary conditions on yielding in a soft glassy material
,” Phys. Rev. Lett.
101
, 258302
(2008
).72.
Mohagheghi
, M.
, and B.
Khomami
, “Elucidating the flow-microstructure coupling in entangled polymer melts: Part II. Molecular mechanisms of shear banding
,” J. Rheol.
60
, 861
–872
(2016
).73.
Colombo
, J.
, and E.
Del Gado
, “Stress localization, stiffening, and yielding in a model colloidal gel
,” J. Rheol.
58
, 1089
–1116
(2014
).74.
Varnik
, F.
, L.
Bocquet
, and J. L.
Barrat
, “A study of the static yield stress in a binary Lennard-Jones glass
,” J. Chem. Phys.
120
, 2788
–2801
(2004
).75.
Shrivastav
, G. P.
, P.
Chaudhuri
, and J.
Horbach
, “Heterogeneous dynamics during yielding of glasses: Effect of aging
,” J. Rheol.
60
, 835
–847
(2016
).76.
Kabla
, A.
, J.
Scheibert
, and G.
Debregeas
, “Quasi-static rheology of foams. part 2. continuous shear flow
,” J. Fluid Mech.
587
, 45
–72
(2007
).77.
Barry
, J. D.
, D.
Weaire
, and S.
Hutzler
, Rheol. Acta
49
, 687
(2010
); 5th Annual European Rheology Conference (AERC 2009), Cardiff Univ, Cardiff, Wales
, Apr. 15–17, 2009
.78.
Lehtinen
, A.
, A.
Puisto
, X.
Illa
, M.
Mohtaschemi
, and M. J.
Alava
, “Transient shear banding in viscoelastic maxwell fluids
,” Soft Matter
9
, 8041
–8049
(2013
).79.
Hinkle
, A. R.
, and M. R.
Falk
, “A small-gap effective-temperature model of transient shear band formation during flow
,” J. Rheol.
60
, 873
–882
(2016
).80.
Divoux
, T.
, V.
Grenard
, and S.
Manneville
, “Rheological hysteresis in soft glassy materials
,” Phys. Rev. Lett.
110
, 018304
(2013
).81.
Magnin
, A.
, and J.
Piau
, “Cone-and-plate rheometry of yield stress fluids – study of an aqueous gel
,” J. Non-Newtonian Fluid Mech.
36
, 85
–108
(1990
).82.
Grenard
, V.
, T.
Divoux
, N.
Taberlet
, and S.
Manneville
, “Timescales in creep and yielding of attractive gels
,” Soft Matter
10
, 1555
–1571
(2014
).83.
Sentjabrskaja
, T.
, P.
Chaudhuri
, M.
Hermes
, W. C. K.
Poon
, J.
Horbach
, S. U.
Egelhaaf
, and M.
Laurati
, “Creep and flow of glasses: strain response linked to the spatial distribution of dynamical heterogeneities
,” Sci. Rep.
5
, 11884
(2015
).84.
Chaudhuri
, P.
, and J.
Horbach
, “Onset of ow in a confined colloidal glass under an imposed shear stress
,” Phys. Rev. E
88
, 040301
(2013
).85.
Agimelen
, O. S.
, and P. D.
Olmsted
, “Apparent fracture in polymeric fluids under step shear
,” Phys. Rev. Lett.
110
, 204503
(2013
).86.
Marrucci
, G.
, “Dynamics of entanglements: A nonlinear model consistent with the cox-merz rule
,” J. Non-Newtonian Fluid Mech.
62
, 279
–289
(1996
).87.
Ianniruberto
, G.
, and G.
Marrucci
, “Convective constraint release (ccr) revisited
,” J. Rheol.
58
, 89
–102
(2014
).88.
Marrucci
, G.
, and N.
Grizzuti
, “The free energy function of the Doi-Edwards theory: analysis of the instabilities in stress relaxation
,” J. Rheol.
27
, 433
–450
(1983
).89.
Li
, X.
, and S.-Q.
Wang
, “Elastic yielding after step shear and during laos in the absence of meniscus failure
,” Rheol. Acta
49
, 985
–991
(2010
).90.
Boukany
, P. E.
, and S.-Q.
Wang
, “Exploring origins of interfacial yielding and wall slip in entangled linear melts during shear or after shear cessation
,” Macromolecules
42
, 2222
–2228
(2009
).91.
Wang
, S.-Q.
, S.
Ravindranath
, P.
Boukany
, M.
Olechnowicz
, R. P.
Quirk
, A.
Halasa
, and J.
Mays
, “Nonquiescent relaxation in entangled polymer liquids after step shear
,” Phys. Rev. Lett.
97
, 187801
(2006
).92.
Ravindranath
, S.
, and S.-Q.
Wang
, “What are the origins of stress relaxation behaviors in step shear of entangled polymer solutions?
,” Macromolecules
40
, 8031
–8039
(2007
).93.
Fang
, Y.
, G.
Wang
, N.
Tian
, X.
Wang
, X.
Zhu
, P.
Lin
, G.
Ma
, and L.
Li
, “Shear inhomogeneity in poly(ethylene oxide) melts
,” J. Rheol.
55
, 939
–949
(2011
).94.
Archer
, L. A.
, Y.-L.
Chen
, and R. G.
Larson
, “Delayed slip after step strains in highly entangled polystyrene solutions
,” J. Rheol.
39
, 519
–525
(1995
).95.
Ravindranath
, S.
, S.-Q.
Wang
, M.
Olechnowicz
, V. S.
Chavan
, and R. P.
Quirk
, “How polymeric solvents control shear inhomogeneity in large deformations of entangled polymer mixtures
,” Rheol. Acta
50
, 97
–105
(2011
).96.
Boukany
, P. E.
, S.-Q.
Wang
, and X.
Wang
, “Step shear of entangled linear polymer melts: New experimental evidence for elastic yielding
,” Macromolecules
42
, 6261
–6269
(2009
).97.
Zhou
, L.
, L. P.
Cook
, and G. H.
McKinley
, “Probing shear-banding transitions of the vcm model for entangled wormlike micellar solutions using large amplitude oscillatory shear (laos) deformations
,” J. Non-Newtonian Fluid Mech.
165
, 1462
–1472
(2010
).98.
Zhou
, L.
, R. H.
Ewoldt
, L. P.
Cook
, and G. H.
McKinley
, “Probing shear-banding transitions of entangled liquids using large amplitude oscillatory shearing (laos) deformations. In A Co, LG Leal, RH Colby, and AJ Giacomin, editors, XVTH International Congress on Rheology – The Society of Rheology 80th Annual Meeting, Pts 1 and 2
,” AIP Conf. Proc.
1027
, 189
–191
(2008
).99.
Tapadia
, P.
, S.
Ravindranath
, and S. Q.
Wang
, “Banding in entangled polymer fluids under oscillatory shearing
,” Phys. Rev. Lett.
96
, 196001
(2006
).100.
Cohen
, I.
, B.
Davidovitch
, A. B.
Schofield
, M. P.
Brenner
, and D. A.
Weitz
, “Slip, yield, and bands in colloidal crystals under oscillatory shear
,” Phys. Rev. Lett.
97
, 215502
(2006
).101.
Perge
, C.
, N.
Taberlet
, T.
Gibaud
, and S.
Manneville
, “Time dependence in large amplitude oscillatory shear: A rheo-ultrasonic study of fatigue dynamics in a colloidal gel
,” J. Rheol.
58
, 1331
–1357
(2014
).102.
Gibaud
, T.
, C.
Perge
, S. B.
Lindstrom
, N.
Taberlet
, and S.
Manneville
, “Multiple yielding processes in a colloidal gel under large amplitude oscillatory stress
,” Soft Matter
12
, 1701
–1712
(2016
).103.
Rouyer
, F.
, S.
Cohen-Addad
, R.
Hoehler
, P.
Sollich
, and S. M.
Fielding
, “The large amplitude oscillatory strain response of aqueous foam: Strain localization and full stress fourier spectrum
,” Eur. Phys. J. E
27
, 309
–321
(2008
).104.
Guo
, Y.
, W.
Yu
, Y.
Xu
, and C.
Zhou
, “Correlations between local flow mechanism and macroscopic rheology in concentrated suspensions under oscillatory shear
,” Soft Matter
7
, 2433
–2443
(2011
).105.
Dimitriou
, C. J.
, L.
Casanellas
, T. J.
Ober
, and G. H.
McKinley
, “Rheo-piv of a shear-banding wormlike micellar solution under large amplitude oscillatory shear
,” Rheol. Acta
51
, 395
–411
(2012
).106.
Gurnon
, A. K.
, and N. J.
Wagner
, “Large amplitude oscillatory shear (laos) measurements to obtain constitutive equation model parameters: Giesekus model of banding and non-banding wormlike micelles
,” J. Rheol.
56
, 333
–351
(2012
).107.
Calabrese
, M. A.
, S. A.
Rogers
, L.
Porcar
, and N. J.
Wagner
, “Understanding steady and dynamic shear banding in a wormlike micellar solution
,” J. Rheol.
60
, 1001
–1017
(2016
).108.
Fielding
, S. M.
, “Criterion for extensional necking instability in polymeric fluids
,” Phys. Rev. Lett.
107
, 258301
(2011
).109.
Hoyle
, D. M.
, and S. M.
Fielding
, “Age-dependent modes of extensional necking instability in soft glassy materials
,” Phys. Rev. Lett.
114
, 158301
(2015
).110.
Larson
, R.
, Constitutive Equations for Polymer Melts and Solutions
(Butterworth
, Stoneham, MA
, 1988
).111.
McLeish
, T. C. B.
, and R. G.
Larson
, “Molecular constitutive equations for a class of branched polymers: the pom-pom polymer
,” J. Rheol.
42
, 81
–110
(1998
).112.
113.
Hassager
, O.
, M. I.
Kolte
, and M.
Renardy
, “Failure and nonfailure of fluid filaments in extension
,” J. Non-Newtonian Fluid Mech.
76
, 137
–151
(1998
).© 2016 The Society of Rheology.
2016
The Society of Rheology
You do not currently have access to this content.
