A mathematical model of blood flow and convective diffusion processes in constricted bifurcated arteries

S. Chakravarty; S. Sen

Korea-Australia Rheology Journal, Vol.18, No.2, 51-65, June, 2006

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A mathematical model of blood flow and convective diffusion processes in constricted bifurcated arteries

S. Chakravarty^†, and S. Sen

Department of Mathematics, Visva-Bharati University, Santiniketan 731 235, India

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Of concern in the present theoretical investigation is the study of blood flow and convection-dominated diffusion processes in a model bifurcated artery under stenotic conditions. The geometry of the bifurcated arterial segment having constrictions in both the parent and its daughter arterial lumen frequently appearing in the diseased arteries causing malfunction of the cardiovascular system, is constructed mathematically with the introduction of suitable curvatures at the lateral junction and the flow divider. The streaming blood contained in the bifurcated artery is treated to be Newtonian. The flow dynamical analysis applies the twodimensional unsteady incompressible nonlinear Navier-Stokes equations for Newtonian fluid while the mass transport phenomenon is governed by the convection diffusion equation. The motion of the arterial wall and its effect on local fluid mechanics is, however, not ruled out from the present model. The main objective of this study is to demonstrate the effects of constricted flow characteristics and the wall motion on the wall shear stress, the concentration profile and on the mass transfer. The ultimate numerical solutions of the coupled flow and diffusion processes following a radial coordinate transformation are based on an appropriate finite difference technique which attain appreciable stability in both the flow phenomena and the convection-dominated diffusion processes.

[References]

Back LH, Radbill JR, Crawford DW, J. Biomech., 10, 763 (1977)
Bassingthwaitghte JB, Deussen A, Beyer D, Chan IS, Raymond GR, King RB, Bukowski TR, Ploger JD, Kroll K, Revenaugh J, Link JM, Krohn KA, Adv. Biological Heat and Mass Transfer ASME, 231, 113 (1992)
Burton AC, Physiology and Biophysics of the Circulation: Introductory Text, Year Book Medical Publisher, Chicago (1996)
Chakravarty S, Mandal PK, Int. J. Eng. Sci., 35, 409 (1997)
Deshpande M, Giddens D, Mabon R, J. Biomech., 9, 165 (1976)
Fischer GM, Swain ML, Cherian K, Blood Vessels, 17, 215 (1980)
Fridman MH, Ehrlich LW, Cric. Res., 37, 446 (1975)
Fry DL, Cric. Res., 22, 165 (1968)
Imaeda K, Goodman FO, J. Biomech., 13, 1007 (1980)
Jo M, Dull RO, Hollis TM, Tarbell JM, Am. J. Physiol., 260, H1992 (1991)
Kashihara M, Matsumoto K, Cervos-Navarro J, Observations on carotid bifurcation, atherosclerosis in human cadavers, In: Role of Blood Flow in Atherogenesis (Edited by Yoshide, Y., Yamaguel, T., Caro, C.G. and Nevem, R.M.) Proc. Int. Symp. 19, Springer-Verleg, Tokyo (1988)
Liepsch DW, Effect of blood flow parameters on flow patterns in arterial bifurcations: studies in models, in Blood Flow in Large Arteries: Applications to Atherogenesis and Clinical Medicine, D.W. Liepsch, Ed., S. Karger, Basel, 63 (1990)
Liepsch DW, Moravec ST, Biorheology, 21, 571 (1984)
Ling SC, Atabek HB, J. Fluid Mech., 55, 492 (1972)
Lou Z, Yang WJ, Crit. Rev. Biomed. Engng., 19, 455 (1992)
Ma PP, Li XM, Ku DN, Int. J. Heat Mass Transf., 37(17), 2723 (1994)
McDonald DA, Blood Flow in Arteries, 2nd ed. Edward Arnold, London. (1974)
McIntire LV, Tay RTS, Concentration of materials released from mural platelet aggregates: flow effects, in Biomedical Engineering, Eds., W.J. Yang and J.L. Chun, Hemisphere, New York, 229-245 (1989)
Milnor WR, Haemodynamics, Williams and Williams, Baltimore (1982)
Pedley TJ, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge (1980)
Reidy MA, Bowyer DE, Atherosclerosis, 26, 181 (1997)
Rappitsch G, Perktold K, J. Biomech., 39, 207 (1996)
Singh MP, Sharan RK, Saxena RK, Malhotra I, A theoretical model for the exchange of gases in the systemic capillaries and the surrounding tissues under compression an application in underwater physiology, in Finite Element Flow Analysis, Ed. T. Kawai, University of Tokyo Press, North Holland, Amsterdam, 867-875 (1982)
Tarbell JM, BMES Bulletin, 17, 35 (1993)
Womersley JR, An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, Wright Air Development Centre Technical Report TR 56-6114 (1957)

[1] Back LH, Radbill JR, Crawford DW, J. Biomech., 10, 763 (1977)

[2] Bassingthwaitghte JB, Deussen A, Beyer D, Chan IS, Raymond GR, King RB, Bukowski TR, Ploger JD, Kroll K, Revenaugh J, Link JM, Krohn KA, Adv. Biological Heat and Mass Transfer ASME, 231, 113 (1992)

[3] Burton AC, Physiology and Biophysics of the Circulation: Introductory Text, Year Book Medical Publisher, Chicago (1996)

[4] Chakravarty S, Mandal PK, Int. J. Eng. Sci., 35, 409 (1997)

[5] Deshpande M, Giddens D, Mabon R, J. Biomech., 9, 165 (1976)

[6] Fischer GM, Swain ML, Cherian K, Blood Vessels, 17, 215 (1980)

[7] Fridman MH, Ehrlich LW, Cric. Res., 37, 446 (1975)

[8] Fry DL, Cric. Res., 22, 165 (1968)

[9] Imaeda K, Goodman FO, J. Biomech., 13, 1007 (1980)

[10] Jo M, Dull RO, Hollis TM, Tarbell JM, Am. J. Physiol., 260, H1992 (1991)

[11] Kashihara M, Matsumoto K, Cervos-Navarro J, Observations on carotid bifurcation, atherosclerosis in human cadavers, In: Role of Blood Flow in Atherogenesis (Edited by Yoshide, Y., Yamaguel, T., Caro, C.G. and Nevem, R.M.) Proc. Int. Symp. 19, Springer-Verleg, Tokyo (1988)

[12] Liepsch DW, Effect of blood flow parameters on flow patterns in arterial bifurcations: studies in models, in Blood Flow in Large Arteries: Applications to Atherogenesis and Clinical Medicine, D.W. Liepsch, Ed., S. Karger, Basel, 63 (1990)

[13] Liepsch DW, Moravec ST, Biorheology, 21, 571 (1984)

[14] Ling SC, Atabek HB, J. Fluid Mech., 55, 492 (1972)

[15] Lou Z, Yang WJ, Crit. Rev. Biomed. Engng., 19, 455 (1992)

[16] Ma PP, Li XM, Ku DN, Int. J. Heat Mass Transf., 37(17), 2723 (1994)

[17] McDonald DA, Blood Flow in Arteries, 2nd ed. Edward Arnold, London. (1974)

[18] McIntire LV, Tay RTS, Concentration of materials released from mural platelet aggregates: flow effects, in Biomedical Engineering, Eds., W.J. Yang and J.L. Chun, Hemisphere, New York, 229-245 (1989)

[19] Milnor WR, Haemodynamics, Williams and Williams, Baltimore (1982)

[20] Pedley TJ, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge (1980)

[21] Reidy MA, Bowyer DE, Atherosclerosis, 26, 181 (1997)

[22] Rappitsch G, Perktold K, J. Biomech., 39, 207 (1996)

[23] Singh MP, Sharan RK, Saxena RK, Malhotra I, A theoretical model for the exchange of gases in the systemic capillaries and the surrounding tissues under compression an application in underwater physiology, in Finite Element Flow Analysis, Ed. T. Kawai, University of Tokyo Press, North Holland, Amsterdam, 867-875 (1982)

[24] Tarbell JM, BMES Bulletin, 17, 35 (1993)

[25] Womersley JR, An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, Wright Air Development Centre Technical Report TR 56-6114 (1957)