For certain values of the material parameters, certain viscoelastic fluid models allow for a nonmonotonic relationship between the shear stress and shear rate in simple flows. We consider channel flow of such a fluid, the Johnson-Segalman liquid, subjected to exothermic reactions. A numerical algorithm based on the finite difference method is implemented in time and space for the solution process of the highly nonlinear governing equations. The phenomenon of shear banding is observed and explained in terms of the jump discontinuities in shear rates. We demonstrate that for a reacting Johnson-Segalman fluid, the shear banding can be catastrophic as it leads to large temperature buildup within the fluid and hence makes it easily susceptible, say, to thermal runaway. We also demonstrate that the shear banding can be eliminated by making the walls porous and hence allowing for suction and injection. The suction/injection flow is shown to significantly decrease fluid temperatures for the nonmonotonic viscoelastic Johnson-Segalman model but leads to significant temperature increases for the monotonic viscoelastic Oldroyd-B model.

References

References
1.
Chinyoka
T.
, 2008, “
Computational Dynamics of a Thermally Decomposable Viscoelastic Lubricant Under Shear
,”
ASME J. Fluids Eng.
,
130
(
12
), pp.
1
7
.
2.
Chinyoka
T.
, 2010, “
Poiseuille Flow of Reactive Phan-Thien-Tanner Liquids in 1D Channel Flow
,”
ASME J. Heat Transf.
,
132
(
11
), pp.
1
7
.
3.
Fyrillasa
,
M. M.
,
Georgioua
,
G. C.
, and
Vlassopoulos
D.
, 1999, “
Time-Dependent Plane Poiseuille Flow of a Johnson-Segalman Fluid
,”
J. Non-Newtonian Fluid Mech.
,
82
, pp.
105
123
.
4.
Dressler
,
M.
,
Edwards
B. J.
,
Öttinger
H. C.
, 1999, “
Macroscopic Thermodynamics of Flowing Polymeric Liquids
,”
Rheol. Acta
,
38
, pp.
117
136
.
5.
Wapperom
,
P.
, and
Hulsen
,
M. A.
, 1998, “
Thermodynamics of Viscoelastic Fluids: The Temperature Equation
,”
J. Rheol.
,
42
, pp.
999
1019
.
6.
Hütter
,
M.
,
Luap
,
C.
, and
Öttinger
,
H. C.
, 2009, “
Energy Elastic Effects and the Concept of Temperature in Flowing Polymeric Liquids
,”
Rheol. Acta
,
48
, pp.
301
316
.
7.
Peters
,
G. W. M.
, and
Baaijens
F. P. T.
, 1997, “
Modelling of Non-Isothermal Viscoelastic Flows
,”
J. Non-Newton Fluid Mech.
,
68
, pp.
205
224
.
8.
Straughan
B.
, 1998,
Explosive Instabilities in Mechanics
,
Springer
,
New York
.
9.
Frank-Kamenetskii
,
D. A.
, 1969,
Diffusion and Heat Transfer in Chemical Kinetics
,
2nd edition
,
Plenum Press
,
New York.
10.
Bair
,
S.
,
Qureshi
,
F.
, and
Khonsari
M.
, 1994, “
Adiabatic Shear Localization in a Liquid Lubricant Under Pressure
,”
ASME J. Tribol.
,
116
, pp.
705
709
.
11.
Hutson
,
M. S.
,
Hauger
,
S. A.
, and
Edwards
,
G.
, 2002, “
Thermal Diffusion and Chemical Kinetics in Laminar Biomaterial Due to Heating by a Free-Electron Laser
,”
Phys. Rev. E
,
65
, p.
061906
.
12.
Adler
J.
, 1991, “
Thermal Explosion Theory with Arrhenius Kinetics: Homogeneous and Inhomo-geneous media
,”
Proc. R. Soc. London, A
,
433
(
1888
), pp.
329
335
.
13.
Boddington
,
T.
,
Gray
,
P.
, and
Wake
,
G. C.
, 1977, “
Criteria for Thermal Explosions with and Without Reactant Consumption
,”
Proc. R. Soc. London, A
,
357
(
1691
), pp.
403
422
.
14.
Olagunju
,
D. O.
,
Cook
,
L. P.
, and
McKinley
G. H.
, 2002, “
Effect of Viscous Heating on Linear Stability of Viscoelastic Cone-and-Plate Flow: Axisymmetric Case
,”
J. Non-Newtonian Fluid Mech.
,
102
(
2
), pp.
321
342
.
15.
Olagunju
,
D. O.
, 2005, “
Secondary Flow in Non-Isothermal Viscoelastic Parallel-Plate Flow
,”
J. Eng. Math.
,
51
, pp.
325
338
.
16.
Bird
,
R. B.
,
Curtiss
,
C. F.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
, 1987,
Dynamics of Polymeric Liquids Vol. 1 Fluid Mechanics
,
2nd edition
,
Wiley
,
New York
.
17.
Ferry
,
J. D.
, 1981,
Viscoelastic Properties of Polymers
,
3rd edition
,
Wiley
,
New York
.
18.
Rao
,
I. J.
, and
Rajagopal
,
K. R.
, 1999, “
Some simple flows of a Johnson-Segalman fluid
,”
Acta Mech.
,
132
, pp.
209
219
.
19.
Chinyoka
,
T.
,
Renardy
,
Y. Y.
,
Renardy
,
M.
, and
Khismatullin
,
D. B.
, 2005, “
Two-Dimensional Study of Drop Deformation Under Simple Shear for Oldroyd-B Liquids
,”
J. Non-Newtonian Fluid Mech.
,
31
, pp.
45
56
.
You do not currently have access to this content.