The linear stability analysis of Rivlin–Ericksen fluids of second order is investigated for boundary layer flows, where a semi-infinite wedge is placed symmetrically with respect to the flow direction. Second order fluids belong to a larger family of fluids called order fluids, which is one of the first classes proposed to model departures from Newtonian behavior. Second order fluids can model non-zero normal stress differences, which is an essential feature of viscoelastic fluids. The linear stability properties are studied for both signs of the elasticity number K, which characterizes the non-Newtonian response of the fluid. Stabilization is observed for the temporal and spatial evolution of two-dimensional disturbances when K > 0 in terms of increase of critical Reynolds numbers and reduction of growth rates, whereas the flow is less stable when K < 0. By extending the analysis to three-dimensional disturbances, we show that a positive elasticity number K destabilizes streamwise independent waves, while the opposite happens for K < 0. We show that, as for Newtonian fluids, the non-modal amplification of streamwise independent disturbances is the most dangerous mechanism for transient energy growth, which is enhanced when K > 0 and diminished when K < 0.
REFERENCES
1.
R. S.
Rivlin
and J. L.
Ericksen
, “Stress deformation relations for isotropic materials
,” J. Ration. Mech. Anal.
4
, 323
–425
(1955
).2.
R. G.
Owens
and T. N.
Phillips
, Computational Rheology
(Imperial College Press
, 2002
).3.
K. R.
Rajagopal
, A. S.
Gupta
, and A. S.
Wineman
, “On a boundary layer theory for non-Newtonian fluids
,” Int. J. Eng. Sci.
18
, 875
–883
(1980
).4.
K. R.
Rajagopal
and P. N.
Kaloni
, “Some remarks on boundary conditions for flows of fluids of the differential type
,” in Continuum Mechanics and its Applications
, edited by G. A. C.
Graham
and S. K.
Malik
(Hemisphere Press
, 1989
).5.
K.
Rajagopal
, “On boundary conditions for fluids of the differential type
,” in Navier–Stokes Equations and Related Nonlinear Problems
, edited by A.
Sequeira
(Plenum Press
, 1995
).6.
D. W.
Beard
and K.
Walters
, “Elastico-viscous boundary-layer flows I. Two-dimensional flow near a stagnation point
,” Math. Proc. Cambridge Philos. Soc.
60
, 667
–674
(1964
).7.
K. R.
Rajagopal
, A. S.
Gupta
, and T. Y.
Na
, “A note on the Falkner–Skan flows of a non-Newtonian fluid
,” Int. J. Non-Linear Mech.
18
, 313
–320
(1983
).8.
V. K.
Garg
and K. R.
Rajagopal
, “Stagnation point flow of a non-Newtonian fluid
,” Mech. Res. Commun.
17
, 415
–421
(1990
).9.
V. K.
Garg
and K. R.
Rajagopal
, “Flow of a non-Newtonian fluid past a wedge
,” Acta Mech.
88
, 113
–123
(1991
).10.
K.
Vajravelu
and T.
Roper
, “Flow and heat transfer in a second grade fluid over a stretching sheet
,” Int. J. Non-Linear Mech.
34
, 1031
–1036
(1999
).11.
K.
Vajravelu
and D.
Rollins
, “Hydromagnetic flow of a second grade fluid over a stretching sheet
,” Appl. Math. Comput.
148
, 783
–791
(2004
).12.
C. M.
White
and M. G.
Mungal
, “Mechanics and prediction of turbulent drag reduction with polymer additives
,” Annu. Rev. Fluid Mech.
40
, 235
–256
(2008
).13.
E.
De Angelis
, C. M.
Casciola
, V. S.
L’vov
, R.
Piva
, and I.
Procaccia
, “Drag reduction by polymers in turbulent channel flows: Energy redistribution between invariant empirical modes
,” Phys. Rev. E
67
, 056312
(2003
).14.
E.
De Angelis
, C. M.
Casciola
, V. S.
L’vov
, A.
Pomyalov
, I.
Procaccia
, and V.
Tiberkevich
, “Drag reduction by a linear viscosity profile
,” Phys. Rev. E
70
, 055301
(2004
).15.
L.
Xi
, “Turbulent drag reduction by polymer additives: Fundamentals and recent advances
,” Phys. Fluids
31
, 121302
(2019
).16.
L. N.
Trefethen
, A. E.
Trefethen
, S. C.
Reddy
, and T. A.
Driscoll
, “Hydrodynamic stability without eigenvalues
,” Science
261
, 578
–584
(1993
).17.
K. M.
Butler
and B. F.
Farrell
, “Three-dimensional optimal perturbations in viscous shear flow
,” Phys. Fluids A
4
, 1637
–1650
(1992
).18.
M. T.
Landahl
, “A note on an algebraic instability of inviscid parallel shear flows
,” J. Fluid Mech.
98
, 243
–251
(1980
).19.
L.
Brandt
, “The lift-up effect: The linear mechanism behind transition and turbulence in shear flows
,” Eur. J. Mech., B: Fluids
47
, 80
–96
(2014
).20.
J. E.
Dunn
and K. R.
Rajagopal
, “Fluids of differential type: Critical review and thermodynamic analysis
,” Int. J. Eng. Sci.
33
, 689
–729
(1995
).21.
J. E.
Dunn
and R. L.
Fosdick
, “Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade
,” Arch. Ration. Mech. Anal.
56
, 191
–252
(1974
).22.
R. L.
Fosdick
and K. R.
Rajagopal
, “Anomalous features in the model of “second order fluids
,”” Arch. Ration. Mech. Anal.
70
, 145
–152
(1979
).23.
K. C.
Porteous
and M. M.
Denn
, “Linear stability of plane Poiseuille flow of viscoelastic liquids
,” Trans. Soc. Rheol.
16
, 295
(1972
).24.
D. H.
Chun
and W. H.
Schwarz
, “Stability of a plane Poiseuille flow of a second order fluid
,” Phys. Fluids
11
, 5
–9
(1968
).25.
K.
Sadeghy
, S. M.
Taghavi
, N.
Khabazi
, M.
Mirzadeh
, and I.
Karimfazli
, “On the use of hydrodynamic instability test as an efficient tool for evaluating viscoelastic fluid models
,” Adv. Stud. Theor. Phys.
1
, 367
–379
(2007
).26.
R.
Sureshkumar
and A. N.
Beris
, “Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm
,” J. Non-Newtonian Fluid Mech.
56
, 151
–182
(1995
).27.
M.
Zhang
, I.
Lashgari
, T. A.
Zaki
, and L.
Brandt
, “Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids
,” J. Fluid Mech.
737
, 249
–279
(2013
).28.
G. K.
Batchelor
, An Introduction to Fluid Dynamics
(Cambridge Mathematical Library Cambridge University Press
, 2000
).29.
W.
Tollmien
, “Über die entstehung der turbulenz (the production of turbulence)
,” in Vorträge aus dem Gebiete der Aerodynamik und verwandter Gebiete
, edited by A.
Gilles
, L.
Hopf
, and T. V.
Kármán
(Springer
, Berlin, Heidelberg
, 1930
), pp. 18
–21
.30.
P. J.
Schmid
and D. S.
Henningson
, Stability and Transition in Shear Flows, Applied Mathematical Sciences
(Springer
, New York
, 2000
).31.
M.
Cracco
, “Linear stability and transient behaviour of viscoelastic fluids in boundary layers
,” Ph.D. thesis, Cardiff University
, 2019
.32.
T.
Ellingsen
and E.
Palm
, “Stability of linear flow
,” Phys. Fluids
18
, 487
–488
(1975
).33.
L. N.
Trefethen
, Spectral Methods in MATLAB
(Society for Industrial and Applied Mathematics
, 2000
).34.
W. O.
Criminale
, T. L.
Jackson
, and R. D.
Joslin
, in Theory and Computation of Hydrodynamic Stability
, Cambridge Monographs on Mechanics (Cambridge University Press
, 2003
).35.
N.
Hoda
, M. R.
Jovanović
, and S.
Kumar
, “Energy amplification in channel flows of viscoelastic fluids
,” J. Fluid Mech.
601
, 407
–424
(2008
).36.
B. K.
Lieu
, M. R.
Jovanović
, and S.
Kumar
, “Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids
,” J. Fluid Mech.
723
, 232
–263
(2013
).37.
J.
Page
and T. A.
Zaki
, “Streak evolution in viscoelastic Couette flow
,” J. Fluid Mech.
742
, 520
–551
(2014
).38.
N.
Burshtein
, K.
Zografos
, A. Q.
Shen
, R. J.
Poole
, and S. J.
Haward
, “Inertioelastic flow instability at a stagnation point
,” Phys. Rev. X
7
, 041039
(2017
).39.
L. H.
Gustavsson
, “Initial-value problem for boundary layer flows
,” Phys. Fluids
22
, 1602
–1605
(1979
).40.
P.
Corbett
and A.
Bottaro
, “Optimal perturbations for boundary layers subject to stream-wise pressure gradient
,” Phys. Fluids
12
, 120
–130
(2000
).41.
P. J.
Schmid
, “Linear stability theory and bypass transition in shear flows
,” Phys. Plasmas
7
, 1788
–1794
(2000
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
