The lattice Boltzmann method is modified to allow the simulation of non-Newtonian shear-dependent viscosity models. Casson and Carreau-Yasuda non-Newtonian blood viscosity models are implemented and are used to compare two-dimensional Newtonian and non-Newtonian flows in the context of simple steady flow and oscillatory flow in straight and curved pipe geometries. It is found that compared to analogous Newtonian flows, both the Casson and Carreau-Yasuda flows exhibit significant differences in the steady flow situation. In the straight pipe oscillatory flows, both models exhibit differences in velocity and shear, with the largest differences occurring at low Reynolds and Womersley numbers. Larger differences occur for the Casson model. In the curved pipe Carreau-Yasuda model, moderate differences are observed in the velocities in the central regions of the geometries, and the largest shear rate differences are observed near the geometry walls. These differences may be important for the study of atherosclerotic progression.
REFERENCES
1.
C. J. L.
Murray
and A. D.
Lopez
, The Global Burden of Disease: A Comprehensive Assessment of Mortality and Disability from Diseases, Injuries, and Risk Factors in 1990 and Projected to 2020
(Harvard School of Public Health
, Boston, MA
, 1996
).2.
C. G.
Caro
, “Vascular fluid dynamics and vascular biology and disease
,” Math. Methods Appl. Sci.
24
, 1311
(2001
).3.
A. M.
Malek
, S. L.
Alper
, and S.
Izumo
, “Hemodynamic shear stress and its role in atherosclerosis
,” J. Am. Med. Assoc.
282
, 2035
(1999
).4.
T.
Asakura
and T.
Karino
, “Flow patterns and spatial distribution of atherosclerotic lesions in human coronary arteries
,” Circ. Res.
66
, 1045
(1990
).5.
A.
Gnasso
, C.
Irace
, C.
Carallo
, M. S.
De Franceschi
, C.
Motti
, P. L.
Mattioli
, and A.
Pujia
, “In vivo association between low wall shear stress and plaque in subjects with asymmetrical carotid atherosclerosis
,” Stroke
28
, 993
(1997
).6.
D. N.
Ku
, D. P.
Giddens
, C. K.
Zarins
, and S.
Glagov
, “Pulsatile flow and atherosclerosis in the human carotid bifurcation: Positive correlation between plaque location and low oscillating shear stress
,” Arteriosclerosis (Dallas)
5
, 293
(1985
).7.
D. C.
Chappell
, S. E.
Varner
, R. M.
Nerem
, R. M.
Medford
, and R. W.
Alexander
, “Oscillatory shear stress stimulates adhesion molecule expression in cultured human endothelium
,” Circ. Res.
82
, 532
(1998
).8.
D. A.
Wolf-Gladrow
, Lattice-Gas Cellular Automata and Lattice Boltzmann Models, An Introduction
(Springer
, Berlin
, 2000
).9.
S.
Chen
and G. D.
Doolen
, “Lattice Boltzmann method for fluid flows
,” Annu. Rev. Fluid Mech.
30
, 329
(1998
).10.
S.
Succi
, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond
(Oxford University Press
, Oxford, 2001
).11.
Y. H.
Qian
, D.
d’Humières
, and P.
Lallemand
, “Lattice BGK models for Navier–Stokes equation
,” Europhys. Lett.
17
, 479
(1992
).12.
J. A
Cosgrove
, J. M.
Buick
, S. J.
Tonge
, C. G.
Munro
, C. A.
Greated
, and D. M.
Campbell
, “Application of the lattice Boltzmann method to transition in oscillatory channel flow
,” J. Phys. A
36
, 2609
(2003
).13.
S.
Chen
, H.
Chen
, D.
Martinez
, and W.
Matthaeus
, “Lattice Boltzmann model for simulation of magnetohydrodynamics
,” Phys. Rev. Lett.
67
, 3776
(1991
).14.
X.
Shan
and H.
Chen
, “Lattice Boltzmann model for simulating flows with multiple phases and components
,” Phys. Rev. E
47
, 1815
(1993
).15.
H.
Fang
, Z.
Wang
, Z.
Lin
, and M.
Liu
, “Lattice Boltzmann simulation of viscous fluid systems with elastic boundaries
,” Phys. Rev. E
57
, R25
(1998
).16.
H.
Fang
, Z.
Wang
, Z.
Lin
, and M.
Liu
, “Lattice Boltzmann method for simulating the viscous flow in large distensible blood vessels
,” Phys. Rev. E
65
, 051925
(2002
).17.
A. J. C.
Ladd
and R.
Verberg
, “Lattice Boltzmann simulations of particle-fluid suspensions
,” J. Stat. Phys.
104
, 1191
(2001
).18.
Z.
Guo
, C.
Zheng
, and B.
Shi
, “An extrapolation method for boundary conditions in lattice Boltzmann method
,” Phys. Fluids
14
, 2007
(2002
).19.
M.
Krafczyk
, M.
Cerrolaza
, M.
Schulz
, and E.
Rank
, “Analysis of 3D transient blood flow passing through and artificial aortic valve by lattice-Boltzmann methods
,” J. Biomech.
31
, 453
(1998
).20.
M.
Krafczyk
, J.
Tölke
, E.
Rank
, and M.
Schulz
, “Two-dimensional simulation of the fluid-structure interaction using lattice-Boltzmann methods
,” J. Comp. Struct.
79
, 2031
(2001
).21.
R. G.
Belleman
and P. M. A.
Sloot
, “Simulated vascular reconstruction in a virtual operating theatre
,” Proceedings of CARS 2001
(Elsevier Science B. V., Amsterdam
, 2001
), p. 928
.22.
J.
Boyd
, J. M.
Buick
, and S.
Green
, “A second order accurate lattice Boltzmann non-Newtonian flow model
,” J. Phys. A
39
, 14241
(2006
).23.
A. M.
Artoli
, A. G.
Hoekstra
, and P. M. A.
Sloot
, “Accuracy of 2D pulsatile flow in the lattice Boltzmann BGK method
,” Proceedings of the International Conference in Computational Science (ICCS)
(Springer-Verlag
, London
, 2002
), p. 361.24.
A. M.
Artoli
, A. G.
Hoekstra
, and P. M. A.
Sloot
, “3D pulsatile flow with the lattice Boltzmann BGK method
,” Int. J. Mod. Phys. C
13
, 1119
(2002
).25.
A. M.
Artoli
, A. G.
Hoekstra
, and P. M. A.
Sloot
, “Simulation of a systolic cycle in a realistic artery with the lattice Boltzmann BGK method
,” Int. J. Mod. Phys. B
17
, 95
(2003
).26.
J.
Boyd
, J. M.
Buick
, J. A.
Cosgrove
, and P.
Stansell
, “Application of the lattice Boltzmann method to arterial flow simulation: Investigation of boundary conditions for complex arterial geometries
,” Australas. Phys. Eng. Sci. Med.
27
, 147
(2004
).27.
J.
Boyd
, J. M.
Buick
, J. A.
Cosgrove
, and P.
Stansell
, “Application of the lattice Boltzmann model to simulated stenosis growth in a two-dimensional carotid artery
,” Phys. Med. Biol.
50
, 4783
(2005
).28.
R.
Ouared
and B.
Chopard
, “Lattice Boltzmann simulations of blood flow: Non-Newtonian rheology and clotting processes
,” J. Stat. Phys.
121
, 209
(2005
).29.
A. M.
Artoli
, A. G.
Hoekstra
, and P. M. A.
Sloot
, “Mesoscopic simulations of systolic flow in the human abdominal aorta
,” J. Biomech.
39
, 873
(2006
).30.
H. B.
Li
, H.
Fang
, Z.
Lin
, S. X.
Xu
, and S.
Chen
, “Lattice Boltzmann simulation on particle suspensions in a two-dimensional symmetric stenotic artery
,” Phys. Rev. E
69
, 031919
(2004
).31.
A.
Quarteroni
, M.
Tuveri
, and A.
Veneziani
, “Computational vascular fluid dynamics: Problems, models and methods
,” Computing and Visualisation in Science
2
, 163
(2000
).32.
33.
K.
Perktold
, R.
Peter
, and M.
Resch
, “Pulsatile non-Newtonian blood flow simulation through a bifurcation with an aneurysm
,” Biorheology
26
, 1011
(1989
).34.
K.
Perktold
, M.
Resch
, and H.
Florian
, “Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model
,” J. Biomech. Eng.
113
, 464
(1991
).35.
X.
Li
, G.
Wen
, and D.
Li
, “Computer simulation of non-Newtonian flow and mass transport through coronary arterial stenosis
,” Appl. Math. Mech.
22
, 409
(2001
).36.
P.
Neofytou
, “Comparison of blood rheological models for physiological flow simulation
,” Biorheology
41
, 693
(2004
).37.
C. S.
Kim
, C.
Kiris
, and D.
Kwak
, “Numerical simulation of local blood flow in the carotid and cerebral arteries under altered gravity
,” J. Biomech. Eng.
128
, 194
(2006
).38.
A. M.
Artoli
and A.
Sequeira
, “Mesoscopic simulations of unsteady shear-thinning flows
,” Lect. Notes Comput. Sci.
3992
, 78
(2006
).39.
A. M.
Artoli
, J.
Janela
, and A.
Sequeira
, “The role of the Womersley number in shear-thinning fluids
,” WSEAS Trans. Fluid Mech.
1
, 133
(2006
).40.
A. M.
Artoli
, J.
Janela
, and A.
Sequeira
, “A comparative numerical study of a non-Newtonian blood flow model
,” Proceedings of the 2006 IAMSE/WSEAS International Conference on Continuum Mechanics
(WSEAS Press
, Athens, Greece
, 2006
), pp. 91
–96
.41.
F.
Abraham
, M.
Behr
, and M.
Heinkenschloss
, “Shape optimisation in steady blood flow: A numerical study of non-Newtonian effects
,” Comput. Methods Biomech. Biomed. Eng.
8
, 127
(2005
).42.
P. L.
Bhatnagar
, E. P.
Gross
, and M.
Krook
, “A model for collision processes in gases. I: Small amplitude processes in charged and neutral one-component system
,” Phys. Rev.
94
, 511
(1954
).43.
S.
Chen
, Z.
Wang
, X.
Shan
, and G. D.
Doolen
, “Lattice Boltzmann computational fluid dynamics in three dimensions
,” J. Stat. Phys.
68
, 379
(1992
).44.
A.
Artoli
, Mesoscopic Computational Haemodynamics
(Ponsen and Looijen
, Wageningen, Netherlands
, 2003
).45.
A.
Quarteroni
and A.
Veneziani
, “Computational science for the 21st century
,” Modeling and Simulation of Blood Flow Problems
(Wiley
, Chichester, UK
, 1997
).46.
F.
Gijsen
, Modeling of Wall Shear Stress in Large Arteries
(Eindhoven University of Technology
, Eindhoven, 1998
).47.
J. M.
Buick
and C. A.
Greated
, “Gravity in a lattice Boltzmann model
,” Phys. Rev. E
61
, 5307
(2000
).48.
T.
Inamuro
, M.
Yoshino
, and F.
Ogino
, “A non-slip boundary condition for lattice Boltzmann simulations
,” Phys. Fluids
7
, 2928
(1995
).49.
P.
Neofytou
and D.
Drikakis
, “Non-Newtonian flow instability in a channel with a sudden expansion
,” J. Non-Newtonian Fluid Mech.
111
, 127
(2003
).50.
M. C.
Potter
and D. C.
Wiggert
, Mechanics of Fluids
(Prentice Hall
, Englewood Cliffs, NJ,(1991
).51.
E. W.
Merrill
, “Rheology of blood
,” Physiol. Rev.
49
, 863
(1969
).52.
Q. D.
Nguyen
and D. V.
Boger
, “Measuring the flow properties of yield stress fluids
,” Annu. Rev. Fluid Mech.
24
, 47
(1992
).53.
54.
J. R.
Womersley
, “Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known
,” J. Physiol. (London)
127
, 553
(1955
).55.
R. L.
Whitmore
, “The flow behaviour of blood in the circulation
,” Nature
215
, 123
(1967
).56.
L.
Stoner
, M.
Sabatier
, K.
Edge
, and K.
McCully
, “Relationship between blood velocity and conduit artery diameter and the effects of smoking on vascular responsiveness
,” J. Appl. Physiol.
96
, 2139
(2004
).57.
B. O.
Haugen
, S.
Berg
, K. M.
Brecke
, H.
Torp
, S. A.
Slørdahl
, T.
Skjærpe
, and S. O.
Samstad
, “Blood flow velocity profiles in the aortic annulus: A 3-dimensional freehand color flow Doppler imaging study
,” J. Am. Soc. Echocardiogr
15
, 328
(2002
).© 2007 American Institute of Physics.
2007
American Institute of Physics
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