An anisotropic constitutive equation for the stress tensor of blood based on mixture theory

Mathematical Problems in Engineering

Volumn 2008, Issue , 2008, Pages

  Massoudi, Mehrdad ^a   Antaki, James F ^b  

^a NATIONAL ENERGY TECHNOLOGY LABORATORY   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ANISOTROPY; BIOCHEMISTRY; BLOOD; MIXTURES; PLASMAS;

CONSTITUTIVE MODELLING; CONSTITUTIVE RELATIONS; COUPLED EQUATIONS; DENSITY GRADIENTS; EXACT SOLUTIONS; MIXTURE THEORY; RED BLOOD CELLS; SIMPLE SHEAR FLOW; STRESS TENSORS; TWO-COMPONENT MIXTURES; VISCOUS FLUIDS;

NONLINEAR EQUATIONS;

EID: 52949132990     PISSN: 1024123X     EISSN: 15635147     Source Type: Journal    
DOI: 10.1155/2008/579172     Document Type: Article

References (130)

1. Kaloni P. N., Chan Man Fong C. F., De Kee D., Polymeric solutions as two-fluid-mixture continua Industrial Engineering Chemistry Research 1997 36 8 3205 3210
2. Massoudi M., On the importance of material frame-indifference and lift forces in multiphase flows Chemical Engineering Science 2002 57 17 3687 3701
3. Rajagopal K. R., Tao L., Mechanics of Mixtures 1995 35 River Edge, NJ, USA World Scientific xii+195 Series on Advances in Mathematics for Applied Sciences
4. Ishii M., Thermo-Fluid Dynamic Theory of Two-Phase Flow 1975 Paris, France Eyrolles
5. Anderson T. B., Jackson R., A fluid mechanical description of fluidized beds. Equations of motion Industrial Engineering Chemistry Fundamentals 1967 6 4 527 539
6. Drew D. A., Segel L. A., Averaged equations for two-phase flows Studies in Applied Mathematics 1971 50 3 205 231
7. Gidaspow D., Multiphase Flow and Fluidization 1994 Boston, Mass, USA Academic Press xx+467
8. Jung J., Hassanein A., Lyczkowski R. W., Hemodynamic computation using multiphase flow dynamics in a right coronary artery Annals of Biomedical Engineering 2006 34 3 393 407
9. Johnson G., Massoudi M., Rajagopal K. R., Flow of a fluid-solid mixture between flat plates Chemical Engineering Science 1991 46 7 1713 1723
10. Johnson G., Massoudi M., Rajagopal K. R., Flow of a fluid infused with solid particles through a pipe International Journal of Engineering Science 1991 29 6 649 661
11. Massoudi M., Rajagopal K. R., Ekmann J. M., Mathur M. P., Remarks on the modeling of fluidized systems AIChE Journal 1992 38 3 471 472
12. Massoudi M., Rajagopal K. R., Phuoc T. X., On the fully developed flow of a dense particulate mixture in a pipe Powder Technology 1999 104 3 258 268
13. Rajagopal K. R., On a hierarchy of approximate models for flows of incompressible fluids through porous solids Mathematical Models Methods in Applied Sciences 2007 17 2 215 252
14. Truesdell C., Sulle basi della termomeccanica. I, II Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1957 22 33 38, 158166
15. Truesdell C., Rational Thermodynamics 1984 2nd New York, NY, USA Springer xvii+578
16. Massoudi M., A note on the meaning of mixture viscosity using the classical continuum theories of mixtures International Journal of Engineering Science 2008 46 7 677 689
17. Atkin R. J., Craine R. E., Continuum theories of mixtures: applications Journal of the Institute of Mathematics and Its Applications 1976 17 2 153 207
18. Atkin R. J., Craine R. E., Continuum theories of mixtures: basic theory and historical development The Quarterly Journal of Mechanics Applied Mathematics 1976 29 2 209 244
19. Bowen R. M., Eringen A. C., Theory of mixtures Continuum Physics, Vol. 3 1976 New York, NY, USA Academic Press
20. Bedford A., Drumheller D. S., Theories of immiscible and structured mixtures International Journal of Engineering Science 1983 21 8 863 960
21. Samohyl I., Thermomechanics of Irreversible Processes in Fluid Mixtures 1987 Leipzig, Germany Tuebner
22. Ateshian G. A., Likhitpanichkul M., Hung C. T., A mixture theory analysis for passive transport in osmotic loading of cells Journal of Biomechanics 2006 39 3 464 475
23. Garikipati K., Arruda E. M., Grosh K., Narayanan H., Calve S., A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics Journal of the Mechanics and Physics of Solids 2004 52 7 1595 1625
24. Humphrey J. D., Rajagopal K. R., A constrained mixture model for growth and remodeling of soft tissues Mathematical Models Methods in Applied Sciences 2002 12 3 407 430
25. Klisch S. M., Lotz J. C., A special theory of biphasic mixtures and experimental results for human annulus fibrosus tested in confined compression Journal of Biomechanical Engineering 2000 122 2 180 188
26. Lemon G., King J. R., Byrne H. M., Jensen O. E., Shakesheff K. M., Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory Journal of Mathematical Biology 2006 52 5 571 594
27. Tao L., Humphrey J. D., Rajagopal K. R., A mixture theory for heat-induced alterations in hydration and mechanical properties in soft tissues International Journal of Engineering Science 2001 39 14 1535 1556
28. Ramtani S., Pre-stressed and reinforced hollow cylindrical mixture of non-linearly elastic solid and ideal fluid subjected to combined deformations: A study within the context of the theory of interacting continua International Journal of Non-Linear Mechanics 2006 41 5 736 750
29. Fung Y. C., Biomechanics: Mechanical Properties of Living Tissues 1993 2nd New York, NY, USA Springer
30. Thurston G. B., Viscoelasticity of human blood Biophysical Journal 1972 12 9 1205 1217
31. Thurston G. B., Frequency and shear rate dependence of viscoelasticity of human blood Biorheology 1973 10 3 375 381
32. Humphrey J. D., Delange S. L., An Introduction to Biomechanics 2004 New York, NY, USA SpringerScience/Business Media
33. Mills N., Incompressible mixtures of Newtonian fluids International Journal of Engineering Science 1966 4 2 97 112
34. Rajagopal K. R., Kaloni P. N., Some remarks on boundary conditions for flows of fluids of the differential type Continuum Mechanics and Its Applications (Burnaby, BC, 1988) 1989 New York, NY, USA Hemisphere 935 942
35. Atkin R. J., Cowin S. C., Fox N., On boundary conditions for polar materials Zeitschrift fr Angewandte Mathematik und Physik 1977 28 6 1017 1026
36. Leslie F. M., Theory of flow phenomenon in liquid crystals Advances in Liquid Crystals 1979 4 1 81
37. Ericksen J. L., Equilibrium theory of liquid crystals Advances in Liquid Crystals 1976 2 233 298
38. Rajagopal K. R., Wineman A. S., Gandhi M., On boundary conditions for a certain class of problems in mixture theory International Journal of Engineering Science 1986 24 8 1453 1463
39. Ramtani S., Saturation boundary condition within the context of the theory of interacting media with a certain distribution of fibers International Journal of Engineering Science 2003 41 19 2273 2280
40. Ravindran P., Anand N. K., Massoudi M., Steady free surface flow of a fluid-solid mixture down an inclined plane Particulate Science and Technology 2004 22 3 253 273
41. Massoudi M., Boundary conditions in mixture theory and in CFD applications of higher order models Computers Mathematics with Applications 2007 53 2 156 167
42. Gudhe R., Yalamanchili R. C., Massoudi M., Flow of granular materials down a vertical pipe International Journal of Non-Linear Mechanics 1994 29 1 1 12
43. Rajagopal K. R., On implicit constitutive theories Applications of Mathematics 2003 48 4 279 319
44. Massoudi M., Phuoc T. X., Effect of slip boundary condition on the flow of granular materials: a continuum approach International Journal of Non-Linear Mechanics 2000 35 4 745 761
45. Tao L., Rajagopal K. R., Rajagopal K. R., On boundary conditions in mixture theory Recent Advances in Elasticity and Viscoelasticity 1995 Singapore World scientific 130 149
46. Beevers C. E., Craine R. E., On the determination of response functions for a binary mixture of incompressible Newtonian fluids International Journal of Engineering Science 1982 20 6 737 745
47. Antman S. S., Nonlinear Problems of Elasticity 1995 107 New York, NY, USA Springer xviii+750 Applied Mathematical Sciences
48. Maugin G. A., The Thermomechanics of Nonlinear Irreversible Behaviors 1999 River Edge, NJ, USA World Scientific
49. Coussot P., Rheometry of Pastes, Suspensions, and Granular Materials 2005 Hoboken, NJ, USA John Wiley Sons
50. Liu I.-S., Continuum Mechanics 2002 Berlin, Germany Springer xii+297 Advanced Texts in Physics
51. Batra R. C., Elements of Continuum Mechanics 2006 Reston, Va, USA American Institute of Aeronautics and Astronautics
52. Rajagopal K. R., Srinivasa A. R., A thermodynamic frame work for rate type fluid models Journal of Non-Newtonian Fluid Mechanics 2000 88 3 207 227
53. Rajagopal K. R., Srinivasa A. R., Modeling anisotropic fluids within the framework of bodies with multiple natural configurations Journal of Non-Newtonian Fluid Mechanics 2001 99 2-3 109 124
54. Mller I., A thermodynamic theory of mixtures of fluids Archive for Rational Mechanics and Analysis 1968 28 1 1 39
55. Green A. E., Naghdi P. M., A theory of mixtures Archive for Rational Mechanics and Analysis 1967 24 4 243 263
56. Green A. E., Naghdi P. M., A note on mixtures International Journal of Engineering Science 1968 6 11 631 635
57. Adkins J. E., Nonlinear diffusion. I. Diffusion and flow of mixtures of fluids Philosophical Transactions of the Royal Society of London. Series A 1963 255 1064 607 633
58. Adkins J. E., Nonlinear diffusion II. Constitutive equations for mixtures of isotropic fluids Philosophical Transactions of the Royal Society of London. Series A 1963 255 1064 635 648
59. Massoudi M., Application of mixture theory to fluidized beds, Ph.D. dissertation 1986 May Pittsburgh, Pa, USA University of Pittsburgh
60. Massoudi M., Stability analysis of fluidized beds International Journal of Engineering Science 1988 26 7 765 769
61. Massoudi M., On the flow of granular materials with variable material properties International Journal of Non-Linear Mechanics 2001 36 1 25 37
62. Massoudi M., Constitutive relations for the interaction force in multicomponent particulate flows International Journal of Non-Linear Mechanics 2003 38 3 313 336
63. Rajagopal K. R., Troy W., Massoudi M., Existence of solutions to the equations governing the flow of granular materials European Journal of Mechanics. B 1992 11 3 265 276
64. Rajagopal K. R., Wineman A. S., Flow of electro-rheological materials Acta Mechanica 1992 91 1-2 57 75
65. Rajagopal K. R., Massoudi M., Wineman A. S., Flow of granular materials between rotating disks Mechanics Research Communications 1994 21 6 629 634
66. Massoudi M., Johnson G., On the flow of a fluid-particle mixture between two rotating cylinders, using the theory of interacting continua International Journal of Non-Linear Mechanics 2000 35 6 1045 1058
67. Massoudi M., Lakshmana Rao C., Vertical flow of a multiphase mixture in a channel Mathematical Problems in Engineering 2001 6 6 505 526
68. Cowin S. C., A theory for the flow of granular materials Powder Technology 1974 9 2-3 61 69
69. Rajagopal K. R., Massoudi M., A method for measuring material moduli of granular materials: flow in an orthogonal rheometer 1990 DOE/PETC/TR-90/3 Pennsylvania, Pa, USA USDOE Pittsburgh Energy Technology Center
70. Walton O. R., Braun R. L., Stress calculations for assemblies of inelastic speres in uniform shear Acta Mechanica 1986 63 14 73 86
71. Walton O. R., Braun R. L., Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks Journal of Rheology 1985 30 5 949 980
72. Reiner M., A mathematical theory of dilatancy American Journal of Mathematics 1945 67 3 350 362
73. Reiner M., Elasticity beyond the elastic limit American Journal of Mathematics 1948 70 2 433 446
74. Rivlin R. S., The hydrodynamics of non-Newtonian fluids. I Proceedings of the Royal Society of London. Series A 1948 193 1033 260 281
75. Rajagopal K. R., Gupta G., Yalamachili R. C., A rheometer for measuring the properties of granular materials Particulate Science and Technology 2000 18 1 39 55
76. Baek S., Rajagopal K. R., Srinivasa A. R., Measurements related to the flow of granular material in a torsional rheometer Particulate Science and Technology 2001 19 2 175 186
77. Massoudi M., Mehrabadi M. M., A continuum model for granular materials: considering dilatancy and the Mohr-Coulomb criterion Acta Mechanica 2001 152 14 121 138
78. Collins I. F., Elastic/plastic models for soils and sands International Journal of Mechanical Sciences 2005 47 4-5 493 508
79. Massoudi M., Antony S. J., Hoyle W., Ding Y., Constitutive modelling of flowing granular materials: a continuum approach Granular Materials: Fundamentals and Applications 2004 Cambridge, UK The Royal Society of Chemistry 63 107
80. Johnson G., Massoudi M., Rajagopal K. R., A review of interaction mechanisms in fluid-solid flows Topical report 1990 DOE/PETC/TR-90/9 Pittsburgh, Pa, USA Pittsburgh Energy Technology Center
81. Ericksen J. L., Anisotropic fluids Archive for Rational Mechanics and Analysis 1960 4 231 237
82. Leslie F. M., Some constitutive equations for anisotropic fluids The Quarterly Journal of Mechanics and Applied Mathematics 1966 19 3 357 370
83. Sorensen E. N., Burgreen G. W., Wagner W. R., Antaki J. F., Computational simulation of platelet deposition and activationI: model development and properties Annals of Biomedical Engineering 1999 27 4 436 448
84. Sorensen E. N., Burgreen G. W., Wagner W. R., Antaki J. F., Computational simulation of platelet deposition and activationII: results for Poiseuille flow over collagen Annals of Biomedical Engineering 1999 27 4 449 458
85. Jordan A., David T., Homer-Vanniasinkam S., Graham A., Walker P., The effects of margination and red cell augmented platelet diffusivity on platelet adhesion in complex flow Biorheology 2004 41 5 641 653
86. Anand M., Rajagopal K., Rajagopal K. R., A model for the formation and lysis of blood clots Pathophysiology of Haemostasis and Thrombosis 2005 34 2-3 109 120
87. Goodman P. D., Barlow E. T., Crapo P. M., Mohammad S. F., Solen K. A., Computational model of device-induced thrombosis and thromboembolism Annals of Biomedical Engineering 2005 33 6 780 797
88. Fahraeus R., The suspension stability of blood Physiological Reviews 1929 9 2 241 274
89. Fahraeus R., Lindqvist T., The viscosity of the blood in narrow capillary tubes The American Journal of Physiology 1931 96 3 562 568
90. Chien S., Usami S., Dellenback R. J., Gregersen M. I., Nanninga L. B., Guest M. M., Blood viscosity: influence of erythrocyte aggregation Science 1967 157 3790 829 831
91. Chien S., Usami S., Dellenback R. J., Gregersen M. I., Blood viscosity: influence of erythrocyte deformation Science 1967 157 3790 827 829
92. Chien S., Usami S., Dellenback R. J., Gregersen M. I., Shear-dependent deformation of erythrocytes in rheology of human blood The American Journal of Physiology 1970 219 1 136 142
93. Sutera S., Mehrjardi M., Mohandas N., Deformation of erythrocytes under shear Blood Cells 1975 1 2 369 374
94. Blackshear P. L. Jr., Dorman F. D., Steinbach J. H., Maybach E. J., Singh A., Collingham R. E., Shear, wall interaction and hemolysis Transactions of the American Society for Artificial Internal Organs 1966 12 113 120
95. Blackshear P. L. Jr., Forstrom R. J., Dorman F. D., Voss G. O., Effect of flow on cells near walls Federation Proceedings 1971 30 5 1600 1611
96. Goldsmith H. L., Red cell motions and wall interactions in tube flow Federation Proceedings 1971 30 5 1578 1590
97. Goldsmith H. L., Deformation of human red cells in tube flow Biorheology 1971 7 4 235 242
98. Goldsmith H. L., The flow of model particles and blood cells and its relation to thrombogenesis Progress in Hemostasis and Thrombosis 1972 1 97 127
99. Hochmuth R. M., Mohandas N., Spaeth E. E., Williamson J. R., Blackshear P. L. Jr., Johnson D. W., Surface adhesion, deformation and detachment at low shear of red cells and white cells Transactions of the American Society for Artificial Internal Organs 1972 18 0 325 334
100. Eckstein E. C., Bailey D. G., Shapiro A. H., Self-diffusion of particles in shear flow of a suspension Journal of Fluid Mechanics 1977 79 1 191 208
101. Eckstein E. C., Bilsker D. L., Waters C. M., Kippenhan J. S., Tilles A. W., Transport of platelets in flowing blood Annals of the New York Academy of Sciences 1987 516 442 452
102. Aarts P. A. M. M., Steendijk P., Sixma J. J., Heethaar R. M., Fluid shear as a possible mechanism for platelet diffusivity in flowing blood Journal of Biomechanics 1986 19 10 799 805
103. Truesdell C., Toupin R., The classical field theories Handbuch der Physik, Vol. III/1 1960 Berlin, Germany Springer 226 793
104. Hand G. L., A theory of dilute suspensions Archive for Rational Mechanics and Analysis 1961 7 1 81 86
105. Hand G. L., A theory of anisotropic fluids Journal of Fluid Mechanics 1962 13 1 33 46
106. Advani S. G., Tucker C. L. III, The use of tensors to describe and predict fiber orientation in short fiber composites Journal of Rheology 1987 31 8 751 784
107. Advani S. G., Flow and Rheology in Polymer Composites Manufacturing 1994 Amsterdam, The Netherlands Elsevier
108. Petrie C. J. S., The rheology of fibre suspensions Journal of Non-Newtonian Fluid Mechanics 1999 87 2-3 369 402
109. Truesdell C., Noll W., The Non-Linear Field Theories of Mechanics 1992 New York, NY, USA Springer x+591
110. Massoudi M., An anisotropic constitutive relation for the stress tensor of a rod-like (fibrous-type) granular material Mathematical Problems in Engineering 2005 2005 6 679 702
111. Rajagopal K. R., Ruika M., Mathematical modeling of electrorheological materials Continuum Mechanics and Thermodynamics 2001 13 1 59 78
112. Spencer A. J. M., Eringen A. C., Theory of invariants Continuum Physics 1971 New York, NY, USA Academic Press 239 353
113. Wang C. C., A new representation theorem for isotropic functions: an answer to Professor G. F. Smiths criticism of my papers on representations for isotropic functionsI: scalar-valued isotropic functions Archive for Rational Mechanics and Analysis 1970 36 166 197
114. Wang C. C., A new representation theorem for isotropic functions: an answer to Professor G. F. Smiths criticism of my papers on representations for isotropic functionsII: vector-valued isotropic functions, symmetric ten tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions Archive for Rational Mechanics and Analysis 1970 36 198 223
115. Wang C. C., Corrigendum to my recent papers on Representations for isotropic functions Archive for Rational Mechanics and Analysis 1971 43 5 392 395
116. Zheng Q. S., Theory of representations for tensor functionsa unified invariant approach to constitutive equations Applied Mechanics Review 1994 47 11 545 587
117. Pontrelli G., Blood flow through a circular pipe with an impulsive pressure gradient Mathematical Models Methods in Applied Sciences 2000 10 2 187 202
118. Yeleswarapu K. K., Evaluation of continuum models for characterizing the constitutive behavior of blood, Ph.D dissertation 1996 Pittsburgh, Pa, USA University of Pittsburgh
119. Kuharsky A. L., Fogelson A. L., Surface-mediated control of blood coagulation: the role of binding site densities and platelet deposition Biophysical Journal 2001 80 3 1050 1074
120. Anand M., Rajagopal K. R., A mathematical model to describe the change in the constitutive character of blood due to platelet activation Comptes Rendus Mecanique 2002 330 8 557 562
121. Anand M., Rajagopal K., Rajagopal K. R., A viscoelastic fluid model for describing the mechanics of a coarse ligated plasma clot Theoretical and Computational Fluid Dynamics 2006 20 4 239 250
122. Anand M., Rajagopal K., Rajagopal K. R., A model for the formation, growth, and lysis of clots in quiescent plasma. A comparison between the effects of antithrombin III deficiency and protein C deficiency Journal of Theoretical Biology 2008 253 4 725 738
123. Anand M., Rajagopal K. R., A shear-thinning viscoelastic fluid model for describing the flow of blood International Journal of Cardiovascular Medicine and Science 2004 4 2 59 68
124. Yeleswarapu K. K., Kameneva M. V., Rajagopal K. R., Antaki J. F., The flow of blood in tubes: theory and experiment Mechanics Research Communications 1998 25 3 257 262
125. Macosko C. W., Rheology: Principles, Measurements, and Applications 1994 New York, NY, USA Wiley-VCH
126. Larson R. G., The Structure and Rheology of Complex Fluids 1999 New York, NY, USA Oxford University Press
127. Massoudi M., Phuoc T. X., Pulsatile flow of blood using a modified second-grade fluid model Computers Mathematics with Applications 2008 56 1 199 211
128. Langtangen H. P., Munthe O., Solving systems of partial differential equations using object-oriented programming techniques with coupled heat and fluid flow as example ACM Transactions on Mathematical Software 2001 27 1 1 26
129. Ernst E. E. W., Matrai A., Intermittent claudication, exercise, and blood rheology Circulation 1987 76 5 1110 1114
130. Wannamethee S. G., Lowe G. D. O., Whincup P. H., Rumley A., Walker M., Lennon L., Physical activity and hemostatic and inflammatory variables in elderly men Circulation 2002 105 15 1785 1790