Analysis of conventional drag and lift models for multiphase CFD modeling of blood flow

  • Published : 2009.09.30

Abstract

This study analyzes especially drag and lift models recently developed for fluid-solid, fluid-fluid or liquid-liquid two-phase flows to understand their applicability on the computational fluid dynamics, CFD modeling of pulsatile blood flow. Virtual mass effect and the effect of red blood cells, RBCs aggregation on CFD modeling of blood flow are also shortly reviewed to recognize future tendencies in this field. Recent studies on two-phase flows are found as very useful to develop more powerful drag-lift models that reflect the effects of blood cell's shape, deformation, concentration, and aggregation.

Keywords

References

  1. Al-Taweel, A.M., S. Madhavan, K. Podila, M. Koksal, A. Troshko and Y.P. Gupta, 2006, CFD Simulation of Multiphase Flow: Closure Recommendations for Fluid-Fluid Systems, 12th European Conference on Mixing, Bologna, Italy, June 27-30
  2. Augier, F., O. Masbernat and P. Guiraud, 2003, Slip velocity and drag law in a liquid-liquid homogeneous dispersed flow, AIChE J. 49, 2300-2441 https://doi.org/10.1002/aic.690490907
  3. Auton, T.R., 1987, The lift force on a spherical body in a rotational flow, J. Fluid Mech. 183, 199-218 https://doi.org/10.1017/S002211208700260X
  4. Auton, T.R., 1984, The dynamics of bubbles, drops and particles in motion in liquids, PhD Dissertation, University of Cambridge
  5. Asmolov, E.S. and J.B. McLaughlin, 1999, The inertial lift on an oscillating sphere in a Iinear shear flow, Int. J. Multiphase Flow 25, 739-751 https://doi.org/10.1016/S0301-9322(98)00063-9
  6. Bagchi, P. and S. Balachandar, 2002a, Effects of free rotation on the motion of a solid sphere in linear shear flow at moderate Re, Phys. Fluids 14, 2719-2737 https://doi.org/10.1063/1.1487378
  7. Bagchi, P., and S. Balachandar, 2002b, Shear versus vortex-induced lift force on a rigid sphere at moderate Re, J. Fluid Mech. 473, 379-388
  8. Barnca, E. and J. Mizrahi, 1975, A generalized approach to the fluid dynamics of particulate systems. Part 2: sedimentation and fluidization of clouds of spherical liquid drops, Can. J. Chem. Eng. 53, 461-468 https://doi.org/10.1002/cjce.5450530501
  9. Baskurt, O.K. and H.J. Meiselman, 2007, Hemodynamic effects of red blood cell aggregation, Indian J. Exp. Biol. 45, 25-3
  10. Baumler, H., B. Neu, E. Donath and H. Kiesewetter, 1999, Basic phenomena of red blood cell rouleaux formation biorheologgy, Biorheology 36, 439-442
  11. Behzadi, A., R.I. Issa and H. Rusche, 2004, Modelling of dispersed bubble and droplet flow at high phase fractions, Chem. Eng. Sci. 59, 759-770 https://doi.org/10.1016/j.ces.2003.11.018
  12. Bothe, D., Schmidtke, M. and Warnecke, H.-J., 2006. VOF-simulation of the lift force for single bubbles in a simple shear flow, Chem. Eng. and Techn. 29, 1048-1053 https://doi.org/10.1002/ceat.200600168
  13. Buchanan, J.R., C. Kleinstreuer and J.K. Comer, 2000, Rheological effects on pulsatile hemodynamics in a stenosed tube, Comput. Fluid. 29, 695-724 https://doi.org/10.1016/S0045-7930(99)00019-5
  14. Buchanan, J.R., C. Kleinstreuer, S. Hyun and G.A. Truskey, 2003, Hemodynamics simulation and identification of susceptible sites of atherosclerotÎc lesion formation in a model abdominal aorta, J. Biomech. 36, 1185-1196 https://doi.org/10.1016/S0021-9290(03)00088-5
  15. Candelier, F., and J.R. Angilella, 2006, Analytical investigation of the combined affect of fluid inertia and unsteadiness on low-Re particle centrifugation, Phys. Rev. E 73, 047301 https://doi.org/10.1103/PhysRevE.73.047301
  16. Candelier, F. and M. Souhar, 2007, Time-dependent lift force acting on a particle moving arbitrarily in a pure shear flow at small reynolds numbers, Phys. Rev. E 76, 067301 https://doi.org/10.1103/PhysRevE.76.067301
  17. Carpinlioglu, M.O. and M.Y. Gundogdu, 2001, A critical review on pulsatile pipe flow studies directing towards future research topics, Flow. Meas. and Instrum. 12, 163-174 https://doi.org/10.1016/S0955-5986(01)00020-6
  18. Chakravarty, S. and S. Sen, 2005, Dynamic response of heat and mass transfer in blood flow through stenosed bifurcated arteries, Korea Aust. Rheol. J. 17, 47-62
  19. Cousins, R.R., 1970, A note on the shear Flow past a sphere, J. Fluid Mech. 40, 543-547
  20. Dandy, D.S. and Dwyer, H.A., 1990, A sphere in shear flow at finite Reynolds number effect of shear on particle lift, drag, and heat transfer, J. Fluid Mech. 216, 381-410 https://doi.org/10.1017/S0022112090000477
  21. Darwin, C. 1953, Note on hydrodynamics, Cambiridge Phil. Trans. 49, 342-354
  22. De Gruttola, S., K. Boomsma and D. Poulikakos, 2005, Computational simulation of a non-Newtonian model of the blood separation process, Artif Organs 29, 949-959 https://doi.org/10.1111/j.1525-1594.2005.00164.x
  23. De Vrics, A.W.G, A. Biesheuvel and L. van Wijngaarden, 2002, Notes on the path and wake of a gas bubble rising in pure water Intl. J. of Multiphase Flow 28, 1823-1835 https://doi.org/10.1016/S0301-9322(02)00036-8
  24. Drew, D. and R.T. Lahey, 1979, Applications of general constitutive principles to the derivation of multidimensional twophase flow equations, Intl J. Multiphase Flow 5, 243-264 https://doi.org/10.1016/0301-9322(79)90024-7
  25. Drew, D. and R.T. Lahey, 1987, The virtual mass and lift force on a sphere in rotating and straining inviscid flow, Intl J. Multiphase Flow 13, 113-121 https://doi.org/10.1016/0301-9322(87)90011-5
  26. Drew, D. and R.T. Lahey, 1990, Some supplemental analysis concerning the virtual mass and lift force on a sphere in a rotating and straining flow, Intl J. Multiphase Flow 16, 1127-1130 https://doi.org/10.1016/0301-9322(90)90110-5
  27. Drew, D. and R. T. Lahey, 1993, Analytical modeling of multiphase flow, Particulate Two-Phase Flow, Butterworth-Heinemann, Boston, 509-566
  28. Ervin, E.A. and G. Tryggvason, 1994, The rise of bubbles in an vertical shear flow, In: Proceedings of the ASME Winter Meeting, Fluids Engineering Div., Chicago
  29. Fischer, T.M., M. St$\ddot{o}$hr-Liesen and H. Schmid-Sch$\ddot{o}$nbein, 1978, The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow, Science 202, 894-896 https://doi.org/10.1126/science.715448
  30. Fluent User's Guide, ver. 6.3.26, Fluent Inc., 2006
  31. Fung, Y.C., 1993, Biomechanics: Mechanical Properties of Living Tissues, Second Edition, Springer-Verlag, New York
  32. Goldsmith, H.L., D.N. Bell, S. Spain and F.A. McIntosh, 1999, Effect of red blood cells and their aggregates on platelets and white cells in flowing blood, Biorheology 36, 461-468
  33. Gundogdu, M.Y. and M.O. Carpinlioglu, 1999a, Present state of art on pulsatile flow theory, part 1. laminar and transitional flow regimes, JSME Int. J. 42, 384-397 https://doi.org/10.1299/jsmeb.42.384
  34. Gundogdu, M.Y., and M. O. Carpinlioglu, 1999b, Present state of art on pulsatile flow theory, part 2. turbulent flow regime, JSME Int. J. 42, 398-410
  35. Guyton, A.C. and J.E. Hall, 2006, Textbook of Medical Physiology, Eleventh Edition, EIsevier Saunders
  36. Hall, I.M., 1956, The displacement effect of a sphere in a two-dimensional flow. J. Fluid Mech. 1, 142-161 https://doi.org/10.1017/S002211205600010X
  37. Hibiki, T., and M. Ishii, 1999, Experimental study on interfacial area transport in bubbly two-phase flows, Intl J. of Heat and Mass Transfer 42, 3019-3035 https://doi.org/10.1016/S0017-9310(99)00014-9
  38. Hibiki, T., M. Ishii and Z. Xiao, 2001, Axial interfacial area transport of vertical bubbly flows, Intl J. of Heat and Mass Transfer 44, 1869-1888 https://doi.org/10.1016/S0017-9310(00)00232-5
  39. Hibiki, T., R. Situ, Y. Mi and M. Ishii, 2003, Local flow measurements of vertical upward bubbly flow in an annulus, Intl J. of Heat and Mass Tranfer 46, 1479-1496 https://doi.org/10.1016/S0017-9310(02)00421-0
  40. Hibiki, T., and M. Ishii, 2007, Lift force in bubbly flow systems, Chem. Eng. Sci. 62, 6457-6474 https://doi.org/10.1016/j.ces.2007.07.034
  41. Hyun, S., C. Kleinstreuer and Jr. J.P. Archie, 2000a, Hemodynamics analyses of arterial expansions with implications to thrombosis and restenosis, Med. Eng. Phys. 22, 13-27 https://doi.org/10.1016/S1350-4533(00)00006-0
  42. Hyun, S., C. Kleinstreuer and J.P. Archie, 2000b, Computer simulation and geometric design of endarterectomized carotid artery bifurcations, Crit. Rev. Biomed. Eng. 28, 53-59
  43. Hyun, S., C. Kleinstreuer and Jr. J.P. Archie, 2001, Computational particle hemodynamics analysis and geometric reconstruction after carotid endarterectomy, Comput. Biol. Med. 31, 365-384 https://doi.org/10.1016/S0010-4825(01)00007-5
  44. Ishii, M. and T.C. Chawla, 1979, Local drag laws in dispersed two-phase flow, Argonne National Laboratory Report, ANL- 79-105 (NUREG/CR-1230)
  45. Ishii, M., and N. Zuber, 1979, Drag coefficient and relative velocity in bubbly, droplet or particulate flows, AIChE J. 25, 843-855 https://doi.org/10.1002/aic.690250513
  46. Ishii, M. and T. Hibiki, 2006, Thermo-fluid Dynamics of Two-phase Flow, Springer, New York
  47. Jung, J. and A. Hassanein, 2008, Three-phase CFD analytical modeling of blood flow, Med. Eng. Phys. 30, 91-103 https://doi.org/10.1016/j.medengphy.2006.12.004
  48. Jung, J., R.W. Lyczkowski, C.B. Panchal and A. Hassanein, 2006a, Multiphase hemodynamic simulation with pulsatile flow in a coronary artery, J. of Biomech. 39, 2064-2073 https://doi.org/10.1016/j.jbiomech.2005.06.023
  49. Jung, J., A. Hassanein and R.W. Lyczkowski, 2006b, hemodynamic computaion using multiphase flowdynamics in a right coronary artery, Ann. Biomed. Eng. 34, 393-407 https://doi.org/10.1007/s10439-005-9017-0
  50. Kariyasaki, A., 1987, Behavior of a single gas bubble in a liquid flow with a linear velocity profile. Preceedings of the 1987 ASME/JSME Thermal Engineering Conference, 261-267
  51. Klaseboer, E., J.P. Chavaillier, A. Mate, O. Masbernat, and C. Gourdon, 2001, Model on experiments of drop impinging on an lmmersεd wall, Phys. Fluids 13, 45-57 https://doi.org/10.1063/1.1331313
  52. Kleinstreuer, C., 2006, Biofluid Dynamics: Principles and Selected Applications, CRC Pres
  53. Komori, S. and R. Kurose, 1996, The effects of shear and spin on particle lift and drag in shear flow at high Reynolds numbers, Advances in Turbulence VI, (ed. S. Gavrilakis, L. Machiels and P. Monkewitz), Kluwer, 551-554
  54. Kulkarni, A.A., 2008, Lift force on bubbles in a bubble column reactor: Experimental analysis, Chem. Eng. Sci. 63, 1710-1723 https://doi.org/10.1016/j.ces.2007.10.029
  55. Kumar, A., and S. Hartland, 1985, Gravity settling in liquid-liquid dispersions, Can. J. Chem. Eng. 63, 368-376 https://doi.org/10.1002/cjce.5450630303
  56. Kurose, R., and S. Komori, 1999, Drag and lift forces on a rotating sphere in a linear shear flow, J. Fluid Mech. 384, 183-206 https://doi.org/10.1017/S0022112099004164
  57. Legendre, D. and J. Magnaudet, 1997, A note on the lift force on a spherical bubble or drop in a low-Reynolds-number shear flow, Physics of Fluids 9, 3572-3574 https://doi.org/10.1063/1.869466
  58. Legendre, D. and J. Magnaudet, 1998, The lift force on a spherical bubble in a viscous linear shear flow, J. Fluid Mech. 368, 81-126 https://doi.org/10.1017/S0022112098001621
  59. Levich, V.G., 1962, Physicochemical Hydrodynamics, Prentice-Hall, New York
  60. Lighthill, M.J., 1956, Drift, J. Fluid Mech. 1, 31-53 https://doi.org/10.1017/S0022112056000032
  61. Liu, T.J., 1993, Bubble size and entrance length effiects on void development in a vertical channel, Intl J. of Heat and Mass Transfer 19, 99-113
  62. Longest, P.W. and C. Kleinstreuer, 2003, Comparison of blood particle deposition models for non-parallel flow domains, J. Biomech. 36, 421-430 https://doi.org/10.1016/S0021-9290(02)00434-7
  63. Longest, P.W., C. Kleinstreuer and J.R. Buchanan, 2004, Efficient computation of micro-particle dynamics including wall effects, Comput. Fluid. 33, 577-601 https://doi.org/10.1016/j.compfluid.2003.06.002
  64. Longest, P.W., C. Kleinstreuer and A. Deanda, 2005, Numerical simulation of wall shear stress and particle-based hemodynamic parameters in pre-cuffed and streamlined end-to-side anastomoses, Ann. Biomed. Eng. 33, 1752-1766 https://doi.org/10.1007/s10439-005-7784-2
  65. McLaughlin, J.B., 1991, Inertial migration of a small sphere in linear shear flows, J. Fluid Mech. 224, 261-274 https://doi.org/10.1017/S0022112091001751
  66. McLaughlin, J.B., 1993, The lift on a small sphere in wallbounded linear shear flows, J. Fluid Mech. 246, 249-265 https://doi.org/10.1017/S0022112093000114
  67. Mei, R., C.J. Lawrence and R.J. Adrian, 1991, Unsteady drag on a sphere at finite Reynolds number with small-amplitude fluctuations in the free-stream velocity, J. Fluid Mech. 233, 613-631 https://doi.org/10.1017/S0022112091000629
  68. Meiselman, H.J., B. Neu, M.W. Rampling and O.K. Baskurt, 2007, RBC aggregation: Laboratory data and models, Indian J. Exp. Biol. 45, 9-17
  69. Miyazaki, K., D. Bedeaux and J. Bonet Avalos, 1995, Drag on a Sphere in a Slow Shear Flow, J. Fluid Mech. 296, 373-390 https://doi.org/10.1017/S0022112095002163
  70. Morsi, S. A. and A. J. Alexander, 1972, An investigation of particle trajectories in two-phase flow systems, J. Fluid Mech. 55, 193-208 https://doi.org/10.1017/S0022112072001806
  71. Mying, W., S. Hosokawa and A. Tomiyama, 2006, Terminal velocity of single drops in stagnant liquids, Journal of Fluid Science and Technology 1, 72-81 https://doi.org/10.1299/jfst.1.72
  72. Naciri, M.A., 1992, Contribution $\acute{a}$ l'$\acute{e}$tude des forces exerc$\acute{e}$es par un liquide sur une bulle de gaz: portance, masse ajout$\acute{e}$e et interactions hydrodynamiques, Ph.D. Thesis, L'ecole Centrale de Lyon, France
  73. Ookawara, S., D. Street and K. Ogawa, 2004, Practical application to micro-separator/classifier of the Euler-granular model, In: International Conference on Multiphase Flow 2004, Yokohama, Japan
  74. Ookawara, S., D. Street and K. Ogawa, 2005, Quantitative prediction of separation efficiency of a micro-separator/classifier by Euler-granular model, In: A.I.Ch.E. 2005 Spring National Meeting, Atlanta
  75. Ookawara, S., D. Street and K. Ogawa, 2006, Numerical study on development of particle concentration profiles in a curved microchannel, Chemical Engineering Science 61, 3714-3724 https://doi.org/10.1016/j.ces.2006.01.016
  76. Oakwara, S., M. Agrawal, D. Street and K. Ogawa, 2007, Quasidirect numerical simulation of lift force-induced particle separation in a curved microchannel by use of a macroscopic particle model, Chemical Engineering Science 62, 2454-2465 https://doi.org/10.1016/j.ces.2007.01.031
  77. Rampling, M.W., H.J. Meiselman, B. Neu and O.K. Baskurt, 2004, Influence of cell-specific factors on red blood cell aggregation, Biorheology 41, 91-112
  78. Rusche, H. and R.l. Issa, 2000, The effect of voidage on the drag force on particles in dispersed two-phase flow, Japanese European Two-Phase Flow Meeting, Tsukuba, Japan
  79. Saffman, P.G, 1965, The lift force on a small sphere in a slow shear flow, J. Fluid Mech. 22, 385-400 https://doi.org/10.1017/S0022112065000824
  80. Saffman, P.G., 1968, Corrigendum to 'The lift on a small sphere in a slow shear flow', J. Fluid Mech. 31, 624 https://doi.org/10.1017/S0022112068999990
  81. Segre, G and A. Silberberg, 1962a, Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams, J. Fluid Mech. 14, 115-135 https://doi.org/10.1017/S002211206200110X
  82. Segre, G. and A. Silberberg, 1962b, Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation, J. Fluid Mech. 14, 136-157 https://doi.org/10.1017/S0022112062001111
  83. Schiller, L. and Z. Naumann, 1935, A Drag Coefficient Correlation, Z. Ver. Deutsch. Ing. 77, 318-320
  84. Schmid-Sch$\ddot{o}$nbein, H. and R. Wells, 1969, Fluid drop-like transition of erythrocytεs under shear, Science 165, 288-291 https://doi.org/10.1126/science.165.3890.288
  85. Shew, W.L., S. Poncet and J.F. Pinton, 2006, Force measurements on rising bubbles, J. Fluid Mech. 569, 51-60 https://doi.org/10.1017/S0022112006002928
  86. Shin, S., Y. Yang and J.S. Suh, 2009, Measurement of erythrocyte aggregation in a microchip stirring system by light transmission, Clini.l Hemorheo. and Microcirc. 41, 197-207
  87. Soulis, J.V., G.D. Giannoglou, V. Papaioannou, G.E. Parcharidis and G.E. Louridas, 2008, Low-density lipoprotein concentration in the normal left coronary artery tree, Biomedical Engineering On line 7, 26 https://doi.org/10.1186/1475-925X-7-26
  88. Srivastava, V.P. and R. Srivastava, 2009, Particulate suspension blood flow through a narrow catheterized artery, Computers and Mathematics with Applications 58, 227-238 https://doi.org/10.1016/j.camwa.2009.01.041
  89. Steinman, D.A., 2002, Image-based computational fluid dynamics modeling in realistic arterial geometries, Ann. Biomed. Eng. 30, 483-497 https://doi.org/10.1114/1.1467679
  90. Steinman, D.A. and C.A. Taylor, 2005, Flow imaging and computing: Large artery hemodynamics, Ann. Biomed. Eng. 33, 1704-1709 https://doi.org/10.1007/s10439-005-8772-2
  91. Stijnen, J.M.A., J. de Hart, P.H.M. Bovendeerd and F.N. van de Vosse, 2004, Evaluation of a fictitious domain method for predicting dynamic response of mechanical heart valves, J. Fluid. Struct. 19, 835-850 https://doi.org/10.1016/j.jfluidstructs.2004.04.007
  92. Sankaranarayanan, K. and S. Sundaresan, 2002, Lift force in bubbly suspensions, Chemical Engineering Science 57, 3521-3542 https://doi.org/10.1016/S0009-2509(02)00269-5
  93. Takagi, S. and Y.Matsumoto, 1995, Three dimensional calculation of a rising bubble. In: Proceedings of the Second International Conference on Multiphase Flow, Kyoto
  94. Tees, D.F.J., O. Coenen and H.L. Goldsmith, 1993, Interaction forces between red-cells agglutinated by antibody. 4. Time and force dependence of break-up, Biophys. J. 65, 1318-1334 https://doi.org/10.1016/S0006-3495(93)81180-9
  95. Tha, S.P. and H.L. Goldsmith, 1986, Interaction forces between red-cells agglutinated by antibody. 1. Theoretical, Biophys. J. 50, 1109-1116 https://doi.org/10.1016/S0006-3495(86)83555-X
  96. Tha, S.P. and H.L. Goldsmith, 1988, Interaction forcesbetween red-cells agglutinated by antibody. 3. Micromanipulation, Biophys. J. 53, 677-687 https://doi.org/10.1016/S0006-3495(88)83149-7
  97. Tomiyama, A., H. Tamai, I. Zun and S. Hosokawa, 2002, Transverse migration of single bubbles in simple shear flows, Chemical Engineering Science 57, 1849-1858 https://doi.org/10.1016/S0009-2509(02)00085-4
  98. Tomiyama, A., 2004, Drag, lift and virtual mass forces acting on a single bubble, Proceedings of the Third lnternational Symposium on Two-phase Flow Modeling and Expenmentation, Pisa, Italy, 22-24
  99. Wakaba, L. and S. Balachandar, 2005, History force on a sphere in a weak linear shear flow, Int. J. Multiphase Flow 31, 996-1014 https://doi.org/10.1016/j.ijmultiphaseflow.2005.05.009
  100. Wallis, G.B., 1974, The terminal speed of single drops or bubbles in an infinite medium, Int. J. Multiphase Flow 1, 491-511 https://doi.org/10.1016/0301-9322(74)90003-2
  101. Wen, C. and Y. Yu, 1966, Mechanics of Fluidization, Chem. Eng. Prog. Symp. Ser. 62, 100-110
  102. Yilmaz, F. and M.Y. Gundogdu, 2008, A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions, Korea Aust. Rheol. J. 20, 197-21
  103. Zeng, L., F. Najjar, S. Blachandar and P. Fischer, 2009, Force on a finite-sized particle located close to a wall in a linear shear flow, Phys. Fluids 21, 033302 https://doi.org/10.1063/1.3082232
  104. Zhu, C., 2000, Kinetics and mechanics of cell adhesion, J. of Biomech. 33, 23-33 https://doi.org/10.1016/S0021-9290(99)00163-3
  105. Zhang, D. Z. and Prosperetti, A., 1994, Averaged equations for inviscid disperse two-phase flow. J.Fluid Mech. 267, 185-219 https://doi.org/10.1017/S0022112094001151
  106. Zun, I., 1988, Transition from wall void peaking to core void peaking in turbulent bubbly flow, Transient Phenomena in Multiphase Flow, Hemisphere, Washington, 225-245