Motivated by the ubiquitous flows of viscoelastic fluids possessing yield stress in industrial applications and natural settings, the linear stability of sliding and plane Couette flows of White and Metzner (WM) fluid, which exhibits elasticity and yield stress, is analyzed. After yielding, the WM fluid behaves as an Upper Convected Maxwell fluid. In the creeping-flow limit, in the absence of yield stress, two linearly stable discrete Gorodtsov and Leonov (GL) modes exist for an arbitrary high value of the Weissenberg number. As the yield stress effect (i.e., Bingham number) increases, the GL modes become unstable, leading to an unstable flow. Analysis reveals that these modes originate near the walls due to the ‘extra normal stress’ arising from the synergistic effects of fluid yield stress and elasticity. The extra normal stress is an inherent feature of a WM fluid. Thus, the predicted instability is expected to be present in wall-bounded flows of WM fluid in the creeping-flow limit. The dispersion curves exhibit weak Hadamard instability, which is removed by incorporating a stress diffusivity term into the constitutive equation. The outcomes of the present study are potentially relevant to the WM fluid flows in geophysical and biological settings and industrial processes.
REFERENCES
1.
Abdelgawad
M. S.
, Cannon
I.
, and Rosti
M. E.
, “Scaling and intermittency in turbulent flows of elastoviscoplastic fluids
,” Nat. Phys.
19
, 1059
(2023
).2.
Apostolidis
A. J.
and Beris
A. N.
, “Modeling of the blood rheology in steady-state shear flows
,” J. Rheol.
58
(3
), 607
–633
(2014
).3.
Balmforth
N. J.
, Frigaard
I. A.
, and Ovarlez
G.
, “Yielding to stress: Recent developments in viscoplastic fluid mechanics
,” Annu. Rev. Fluid Mech.
46
, 121
–146
(2014
).4.
Barry
B. W.
and Meyer
M. C.
, “Rheological properties of carbopol gels. i: Continuous shear and creep properties of carbopol gels
,” Int. J. Pharm.
2
, 1
–25
(1979
).5.
Beneitez
M.
, Page
J.
, and Kerswell
R. R.
, “Polymer diffusive instability leading to elastic turbulence in plane couette flow
,” Phys. Rev. Fluids
8
(10
), L101901
(2023
).6.
7.
Bird
R. B.
, Armstrong
R. C.
, and Hassager
O.
, Dynamics of Polymeric Liquids, Vol 1 Fluid Mechanics
(John Wiley
, New York
, 1977
).8.
Bodiguel
H.
, Beaumont
H.
, Machado
A.
, Martinie
L.
, Kellay
H.
, and Colin
A.
, “Flow enhancement due to elastic turbulence in channel flows of shear thinning fluids
,” Phys. Rev. Lett.
114
(5
), 028302
(2015
).9.
Buza
G.
, Beneitez
M.
, Page
J.
, and Kerswell
R. R.
, “Finite-amplitude elastic waves in viscoelastic channel flow from large to zero reynolds number
,” J. Fluid Mech.
951
, A3
(2022
).10.
Castillo
H.
and Wilson
H.
, “Towards a mechanism for instability in channel flow of highly shear-thinning viscoelastic fluids
,” J. Non-Newtonian Fluid Mech.
247
, 15
–21
(2017
).11.
Chandra
B.
, Mangal
R.
, Shankar
V.
, and Das
D.
, “An instability driven by the shear-thinning and elasticity in the fluid flow of polymer solutions through microtubes
,” Phys. Rev. Fluids
4
, 083301
(2019
).12.
Couchman
M. M. P.
, Buza
G.
, Beneitez
M.
, Page
J.
, and Kerswell
R. R.
, “Inertial enhancement of the polymer diffusive instability
,” J. Fluid Mech.
981
, A2
(2024
).13.
Datta
S. S.
et al, “Perspectives on viscoelastic flow instabilities and elastic turbulence
,” Phys. Rev. Fluids
7
(8
), 080701
(2022
).14.
Deguchi
K.
and Nagata
M.
, “Bifurcations and instabilities in sliding Couette flow
,” J. Fluid Mech.
678
, 156
–178
(2011
).15.
Dong
M.
and Zhang
M.
, “Asymptotic study of linear instability in a viscoelastic pipe flow
,” J. Fluid Mech.
935
, A28
(2022
).16.
Erken
O.
, Fazla
B.
, Muradoglu
M.
, Izbassarov
D.
, Romanò
F.
, and Grotberg
J. B.
, “Effects of elastoviscoplastic properties of mucus on airway closure in healthy and pathological conditions
,” Phys. Rev. Fluids
8
, 053102
(2023
).17.
Fielding
S. M.
, “Linear instability of planar shear banded flow
,” Phys. Rev. Lett.
95
, 134501
(2005
).18.
Frigaard
I. A.
, “Super-stable parallel flows of multiple visco-plastic fluids
,” J. Non-Newt. Fluid Mech.
100
, 49
–76
(2001
).19.
Frigaard
I. A.
, Howison
S. D.
, and Sobey
I. J.
, “On the stability of poiseuille flow of a Bingham fluid
,” J. Fluid Mech.
263
, 133
–150
(1994
).20.
Gittler
P.
, “Stability of axial Poiseuille-Couette flow between concentric cylinders
,” Acta Mech.
101
, 1
–13
(1993
).21.
Gorodtsov
V. A.
and Leonov
A. I.
, “On a linear instability of plane parallel Couette flow of viscoelastic fluid
,” J. Appl. Math. Mech.
31
, 310
–319
(1967
).22.
Johnson
M.
and Segalman
D.
, “A model for viscoelastic fluid behavior which allows non-affine deformation
,” J. Non-Newtonian Fluid Mech.
2
, 255
–270
(1977
).23.
Joseph
D. D.
and Saut
J. C.
, “Change of type and loss of evolution in the flow of viscoelastic fluids
,” J. Non-Newtonian Fluid Mech.
20
, 117
–141
(1986
).24.
Landry
M. P.
, Frigaard
I. A.
, and Martinez
D. M.
, “Stability and instability of taylor–couette flows of a bingham fluid
,” J. Fluid Mech.
560
, 321
–353
(2006
).25.
Larson
R. G.
, Constitutive Equations for Polymer Melts and Solutions
(Butterworths
, Boston, MA
, 1988
).26.
Larson
R. G.
, Shaqfeh
E. S. G.
, and Muller
S. J.
, “A purely elastic instability in taylor– couette flow
,” J. Fluid Mech.
218
, 573
–600
(1990
).27.
Liu
R.
and Liu
Q. S.
, “Non-modal stability in sliding Couette flow
,” J. Fluid Mech.
710
, 505
–544
(2012
).28.
Nouar
C.
, Kabouya
N.
, Dusek
J.
, and Mamou
M.
, “Modal and non-modal linear stability of the plane bingham-poiseuille flow
,” J. Fluid Mech.
577
, 211
(2007
).29.
Park
Y.
and Liu
P.
, “Oscillatory pipe flows of a yield-stress fluid
,” J. Fluid Mech.
658
, 211
–228
(2010
).30.
Patne
R.
, “Effect of inhaled air temperature on mucus layer in the proximal airways
,” J. Fluid Mech.
978
, A15
(2024
).31.
Patne
R.
, Mandloi
S.
, Shankar
V.
, and Subramanian
G.
, “Instabilities in strongly shear-thinning viscoelastic flows through channels and tubes
,” arXiv:2408.01004 (2024
).32.
Patne
R.
and Shankar
V.
, “Stability of flow through deformable channels and tubes: Implications of consistent formulation
,” J. Fluid Mech.
860
, 837
–885
(2019
).33.
Patne
R.
and Shankar
V.
, “Instability induced by wall deformability in sliding couette flow
,” Phys. Fluids
32
, 114102
(2020
).34.
Peng
J.
and Zhu
K.-Q.
, “Linear stability of bingham fluids in spiral couette flow
,” J. Fluid Mech.
512
, 21
–45
(2004
).35.
Picaut
L.
, Ronsin
O.
, Caroli
C.
, and Baumberger
T.
, “Experimental evidence of a helical, supercritical instability in pipe flow of shear thinning fluids
,” Phys. Rev. Fluids
2
, 083303
(2017
).36.
Poole
R. J.
, “Elastic instabilities in parallel shear flows of a viscoelastic shear-thinning liquid
,” Phys. Rev. Fluids
1
, 041301
(2016
).37.
Preziosi
L.
and Rosso
F.
, “Stability of a viscous liquid between sliding pipes
,” Phys. Fluids A: Fluid Dyn.
2
, 1158
(1990
).38.
Sahu
K. C.
, Valluri
P.
, Spelt
P. D. M.
, and Matar
O. K.
, “Linear instability of pressure-driven channel flow of a newtonian and a herschel-bulkley fluid
,” Phys. Fluids
19
, 122101
(2007
).39.
Schmid
P. J.
and Henningson
D. S.
, Stability and Transition in Shear Flows
(Springer
, New York
, 2001
).40.
Shemilt
J. D.
, Horsley
A.
, Jensen
O. E.
, Thompson
A. B.
, and Whitfield
C. A.
, “Surface-tension-driven evolution of a viscoplastic liquid coating the interior of a cylindrical tube
,” J. Fluid Mech.
944
, A22
(2022
).41.
Shemilt
J. D.
, Horsley
A.
, Jensen
O. E.
, Thompson
A. B.
, and Whitfield
C. A.
, “Surfactant amplifies yield-stress effects in the capillary instability of a film coating a tube
,” J. Fluid Mech.
971
, A24
(2023
).42.
43.
Wen
C.
, Poole
R. J.
, Willis
A. P.
, and Dennis
D. J. C.
, “Experimental evidence of symmetry-breaking supercritical transition in pipe flow of shear-thinning fluids
,” Phys. Rev. Fluids
2
, 031901
(2017
).44.
White
J. L.
and Metzner
A. B.
, “Development of constitutive equations for polymeric melts and solutions
,” J. Appl. Polym. Sci.
7
, 1867
–1889
(1963
).45.
Wilson
H.
and Loridan
V.
, “Linear instability of a highly shear-thinning fluid in channel flow
,” J. Non-Newtonian Fluid Mech.
223
, 200
–208
(2015
).46.
Wilson
H.
and Rallison
J.
, “Instability of channel flow of a shear-thinning White-Metzner fluid
,” J. Non-Newtonian Fluid Mech.
87
, 75
–96
(1999
).© 2025 Author(s). Published under an exclusive license by AIP Publishing.
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