A REVIEW ON RHEOLOGY OF NON-NEWTONIAN PROPERTIES OF BLOOD
DOI:
https://doi.org/10.31436/iiumej.v19i1.826Abstract
Human blood is composed of red cells, white blood cells, and platelets in a fluid called plasma that contains organic and mineral salts and protein. The rheological characteristics of blood are determined by the properties of these combinations and their interaction with each other. The plasma is essentially a Newtonian fluid, but the blood as a whole behaves as a non-Newtonian fluid showing all signs of non-Newtonian rheology including deformation rate dependency, viscoelasticity, yield stress, and thixotropy. The purpose of this study is mainly a review based on past work on blood rheology, determinants of blood viscosity, yield stress, thixotropy, blood viscosity measurement, and heat transfer in blood flow to better understand the non-Newtonian effect in the blood circulation system.
ABSTRAK: Darah manusia terdiri daripada sel-sel merah, sel darah putih dan platlet dalam cecair yang dipanggil plasma, mengandungi garam organik dan mineral dan protein. Ciri-ciri sifat reologi darah ditentukan oleh sifat-sifat kombinasi ini dan interaksi antara satu sama lain. Plasma adalah pada dasarnya Newtonian cecair tetapi secara keseluruhannya darah bersifat sebagai bukan suatu cecair Newtonian yang menunjukkan tanda-tanda bukan Newtonian reologi, mengandungi pergantungan kadar ubah bentuk, kelikatkenyalan, tegasan alah dan thixotropi. Tujuan kajian ini adalah berkenaan ulasan kajian-kajian lepas berkaitan rheology darah, penentu viskositi darah, tegasan alah dan thixotropi, ukuran kelikatan darah dan pemindahan haba dalam aliran darah bagi memahami kesan bukan Newtonian dalam sistem peredaran darah.
Downloads
Download data is not yet available.
Metrics
Metrics Loading ...
References
[1] Da Silva JL, Rao, MA. (2007) Rheological behavior of food gels. In Rheology of Fluid and Semisolid Foods (pp. 339-401). Springer US.
[2] Kordani, N., & Vanini, A. S. (2014) Optimizing the ethanol content of shear thickening fluid/fabric composites under impact loading. Journal of Mechanical Science and Technology, 28(2):663-667.
[3] Edward, R. (1969) Merrill, Rheology of Blood. Physiological reviews, 49(4):30-43.
[4] Cokelet GR, Merrill EW, Gilliland ER, Shin H, Britten A, Wells Jr RE. (1963) The rheology of human blood—measurement near and at zero shear rate. Transactions of the Society of Rheology, 7(1):303-317.
[5] Merrill EW, Gilliland ER, Lee TS, Salzman EW. (1966) Blood rheology: effect of fibrinogen deduced by addition. Circulation Research, 18(4):437-446.
[6] Merrill EW, Cheng CS, Pelletier GA. (1969) Yield stress of normal human blood as a function of endogenous fibrinogen. Journal of Applied Physiology, 26(1):1-3.
[7] Dintenfass L. (1962) Thixotropy of blood and proneness to thrombus formation. Circulation Research, 11(2):233-239.
[8] Merrill EW, Pelletier GA. (1967) Viscosity of human blood: transition from Newtonian to non-Newtonian. Journal of Applied Physiology, 23(2):178-182.
[9] Chien S, Usami S, Taylor HM, Lundberg JL, Gregersen MI. (1966) Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. Journal of Applied Physiology, 21(1):81-87.
[10] Picart C, Piau JM, Galliard H, Carpentier P. (1998) Human blood shear yield stress and its hematocrit dependence. Journal of Rheology, 42(1):1-12.
[11] Dintenfass L. (1962) Thixotropy of blood and proneness to thrombus formation. Circulation Research, 11(2):233-239.
[12] Sochi T. (2010) Flow of nonâ€Newtonian fluids in porous media. Journal of Polymer Science Part B: Polymer Physics, 48(23):2437-2767.
[13] Davenport, P., & Roath, S. (1981) Blood thixotropy. Journal of clinical pathology, 34(1):106-117.
[14] Mandal P K. (2005) An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis. International Journal of Non-Linear Mechanics, 40(1):151-164.
[15] Fung YC. (1993) Biomechanics–Mechanical Properties of Living Tissues, 2nd edn.
[16] Guyton AC, Hall JE. (1996) Textbook of Medical Physiology, 9th edition, W.B. Sanders Company, Philadelphia.
[17] Dinnar U. (1981) Cardiovascular fluid dynamics. CRC Press, Boca Raton, FL.
[18] Chien S, Dormandy JA, Ernst E, Matrai, A. (2012) Clinical hemorheology: Applications in cardiovascular and hematological disease, diabetes, surgery and gynecology (vol. 74). Springer Science & Business Media.
[19] Cheng DC, Evans F. (1965) Phenomenological characterization of the rheological behaviour of inelastic reversible thixotropic and antithixotropic fluids. British Journal of Applied Physics, 16(11):1599-1614.
[20] Mewis J, Spaull AJB. (1976). Rheology of concentrated dispersions. Advances in Colloid and interface Science, 6(3):173-200.
[21] Nakamura M, Sawada T. (1988) Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis. Journal of Biomechanical Engineering, 110(2):137-143.
[22] Stoltz JF, Singh M, Riha P. (1999) Hemorheology in practice (vol. 30). IOS press.
[23] Barbee JH. (1973) The effect of temperature on the relative viscosity of human blood. Biorheology, 10(1):1-5.
[24] Cokelet GR. (1972) The rheology of human blood. Biomechanics: its foundations and objectives, 72:63-103.
[25] Rathod VP, Tanveer S. (2009) Pulsatile flow of couple stress fluid through a porous medium with periodic body acceleration and magnetic field. Bulletin of the Malaysian Mathematical Sciences Society, 32(2):245-259.
[26] Boyd W. (1964) A Text-book of Pathology: Structure and Function in Diseases. Lea & Febiger: Philadelphia. pp. 130-133.
[27] Chandran KB, Rittgers SE, Yoganathan AP. (2012) Biofluid mechanics: the human circulation. CRC press.
[28] Blair GS. (1959) An equation for the flow of blood, plasma and serum through glass capillaries. Nature, 183(4661):613-614.
[29] Copley AL. (1960) Apparent viscosity and wall adherence of blood systems. Flow properties of blood and other biological systems, Pergamon Press, Oxford, UK, p. 97.
[30] Casson N. (1959) A flow equation for pigment-oil suspensions of the printing ink type. In:Mill CC, editor. Rheology of disperse systems. New York: Pergamon Press. pp. 82-104.
[31] Venkatesan J, Sankar DS, Hemalatha K, Yatim, Y. (2013) Mathematical analysis of Casson fluid model for blood rheology in stenosed narrow arteries. Journal of Applied Mathematics, 2013, pp. 1-11.
[32] Misra JC, Shit GC. (2006) Blood flow through arteries in a pathological state: A theoretical study. International Journal of Engineering Science, 44(10):662-671.
[33] Bodnár T, Sequeira A, Prosi M. (2011) On the shear-thinning and viscoelastic effects of blood flow under various flow rates. Applied Mathematics and Computation, 217(11):5055-5067.
[34] Fisher C, Rossmann J S. (2009) Effect of non-Newtonian behavior on hemodynamics of cerebral aneurysms. Journal of biomechanical engineering, 131(9):091004.
[35] Replogle RL, Meiselman HJ, Merrill EW. (1967) Clinical implications of blood rheology studies. Circulation, 36(1):148-160.
[36] Edward, R. (1969) Merrill, Rheology of Blood. Physiological reviews, 49(4):30-43.
[37] Morris CL, Rucknagel DL, Shukla R, Gruppo RA, Smith CM, Blackshear P. (1989) Evaluation of the yield stress of normal blood as a function of fibrinogen concentration and hematocrit. Microvascular research, 37(3):323-338.
[38] Lee BK, Xue S, Nam J, Lim H, Shin S. (2011) Determination of the blood viscosity and yield stress with a pressure-scanning capillary hemorheometer using constitutive models. Korea-Australia Rheology Journal, 23(1):1-6.
[39] Fedosov DA, Pan W, Caswell B, Gompper G, Karniadakis GE. (2011) Predicting human blood viscosity in silico. Proceedings of the National Academy of Sciences, 108(29):11772-11777.
[40] Barnes HA. (1997) Thixotropy—a review. Journal of Non-Newtonian fluid mechanics, 70(1):1-33.
[41] How TV. (ed.). (1996). Advances in Hemodynamics and Hemorheology (vol. 1). Elsevier.
[42] Huang CR, Siskovic N, Robertson RW, Fabisiak W, Smitherberg EH, Copley AL. (1975) Quantitative characterization of thixotropy of whole human blood. Biorheology, 12(5):279-282.
[43] Gaspar-Rosas A, Thurston GB. (1988) Erythrocyte aggregate rheology by transmitted and reflected light. Biorheology, 25(3):471-487.
[44] Pop GAM, Duncker DJ, Gardien M, Vranckx P, Versluis S, Hasan D, Slager CJ. (2002) The clinical significance of whole blood viscosity in (cardio) vascular medicine. Netherlands Heart Journal, 10(12):512-524.
[45] Cho YI, Cho DJ, Rosenson RS. (2014) Endothelial shear stress and blood viscosity in peripheral arterial disease. Current atherosclerosis reports, 16(4):404-419.
[46] Grotta J, Ackerman R Correia J, Fallick G, Chang J. (1982) Whole blood viscosity parameters and cerebral blood flow. Stroke, 13(3):296-301.
[47] Brown M, Marshall J. (1985) Regulation of cerebral blood flow in response to changes in blood viscosity. The Lancet, 325(8429):604-609.
[48] Tanner RI. (1985) Engineering Rheology, Clarendon. Oxford, 15, pp. 355-370.
[49] Ferguson J, Kemblowski Z. (1991) Applied Fluid Rheology, Elsevier, London, pp. 47–115.
[50] Macosko CW. (1994) Rheology: principles, measurements, and applications. VHC Publishers Inc., New York.
[51] Rosencranz R, Bogen SA. (2006) Clinical laboratory measurement of serum, plasma, and blood viscosity. Pathology Patterns Reviews, 125(suppl-1): S78-S86.
[52] Shin S, Keum DY, Ku YH. (2002) Blood viscosity measurements using a pressure-scanning capillary viscometer. Journal of Mechanical Science and Technology, 16(12):1719-1724.
[53] Shin S, Hou JX, Suh JS, Singh M. (2007) Validation and application of a microfluidic ektacytometer (RheoScan-D) in measuring erythrocyte deformability. Clinical hemorheology and microcirculation, 37(4):319-328.
[54] Lee BK, Xue S, Nam J, Lim H, Shin S. (2011). Determination of the blood viscosity and yield stress with a pressure-scanning capillary hemorheometer using constitutive models. Korea-Australia Rheology Journal, 23(1):1-6.
[55] Jun Kang Y, Yeom E, Lee SJ. (2013) A microfluidic device for simultaneous measurement of viscosity and flow rate of blood in a complex fluidic network. Biomicrofluidics, 7(5):054111.
[56] Kang D, Wang W, Lee J, Tai YC, Hsiai TK. (2015, June) Measurement of viscosity of adult zebrafish blood using a capillary pressure-driven viscometer. In Solid-State Sensors, Actuators and Microsystems (Transducers), 2015 Transducers-2015 18th International Conference on (pp. 1661-1664). IEEE.
[57] Nemeth N, Alexy T, Furka A, Baskurt OK, Meiselman HJ, Furka I, Miko I. (2009) Inter-species differences in hematocrit to blood viscosity ratio. Biorheology, 46(2):155-165.
[58] Furukawa K, Abumiya T, Sakai K, Hirano M, Osanai T, Shichinohe H, Houkin K. (2016) Measurement of human blood viscosity by an electromagnetic spinning sphere viscometer. Journal of medical engineering & technology, 40(6):285-292.
[59] Pennes HH. (1948) Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of applied physiology, 1(2):93-122.
[60] Incropera FP. (2007) Introduction to Heat Transfer, 5th ed. John Wiley and Sons Inc., New York, NY, 581-619.
[61] Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A, Larsen J. (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Annals of biomedical engineering, 28(11):1281-1299.
[62] He Y, Liu H, Himeno R. (2004) A one-dimensional thermo-fluid model of blood circulation in the human upper limb. International Journal of Heat and Mass Transfer, 47(12):2735-2745.
[63] Atabek HB, Chang CC. (1961) Oscillatory flow near the entry of a circular tube. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 12(3):185-201.
[64] Ling SC, Atabek HB. (1972) A nonlinear analysis of pulsatile flow in arteries. Journal of Fluid Mechanics, 55(3):493-511.
[65] Craciunescu OI, Clegg ST. (2001) Pulsatile blood flow effects on temperature distribution and heat transfer in rigid vessels. Transaction-American Society of Mechanical Engineers Journal of Biomechanical Engineering, 123(5):500-505.
[66] Back LH, Cho YI, Crawford DW, Cuffel RF. (1984) Effect of mild atherosclerosis on flow resistance in a coronary artery casting of man. Journal of biomechanical engineering, 106(1):48-53.
[67] Milnor WR. (1992) Hemodynamics, Baltimore: Williams and Williams, p. 20.
[68] Chakravarty S, Mandal PK. (2009) Effect of heat and mass transfer on non-Newtonian flow–Links to atherosclerosis. International Journal of Heat and Mass Transfer, 52(25):5719-5730.
[69] Ahmad, F., Hussain, S., & Ansari, A. (2015) Unsteady MHD blood flow with micropolar fluid characteristics and heat source through parallel plate channel. Journal of Applied Environment & Biological Sciences, 5(4):80-86.
[2] Kordani, N., & Vanini, A. S. (2014) Optimizing the ethanol content of shear thickening fluid/fabric composites under impact loading. Journal of Mechanical Science and Technology, 28(2):663-667.
[3] Edward, R. (1969) Merrill, Rheology of Blood. Physiological reviews, 49(4):30-43.
[4] Cokelet GR, Merrill EW, Gilliland ER, Shin H, Britten A, Wells Jr RE. (1963) The rheology of human blood—measurement near and at zero shear rate. Transactions of the Society of Rheology, 7(1):303-317.
[5] Merrill EW, Gilliland ER, Lee TS, Salzman EW. (1966) Blood rheology: effect of fibrinogen deduced by addition. Circulation Research, 18(4):437-446.
[6] Merrill EW, Cheng CS, Pelletier GA. (1969) Yield stress of normal human blood as a function of endogenous fibrinogen. Journal of Applied Physiology, 26(1):1-3.
[7] Dintenfass L. (1962) Thixotropy of blood and proneness to thrombus formation. Circulation Research, 11(2):233-239.
[8] Merrill EW, Pelletier GA. (1967) Viscosity of human blood: transition from Newtonian to non-Newtonian. Journal of Applied Physiology, 23(2):178-182.
[9] Chien S, Usami S, Taylor HM, Lundberg JL, Gregersen MI. (1966) Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. Journal of Applied Physiology, 21(1):81-87.
[10] Picart C, Piau JM, Galliard H, Carpentier P. (1998) Human blood shear yield stress and its hematocrit dependence. Journal of Rheology, 42(1):1-12.
[11] Dintenfass L. (1962) Thixotropy of blood and proneness to thrombus formation. Circulation Research, 11(2):233-239.
[12] Sochi T. (2010) Flow of nonâ€Newtonian fluids in porous media. Journal of Polymer Science Part B: Polymer Physics, 48(23):2437-2767.
[13] Davenport, P., & Roath, S. (1981) Blood thixotropy. Journal of clinical pathology, 34(1):106-117.
[14] Mandal P K. (2005) An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis. International Journal of Non-Linear Mechanics, 40(1):151-164.
[15] Fung YC. (1993) Biomechanics–Mechanical Properties of Living Tissues, 2nd edn.
[16] Guyton AC, Hall JE. (1996) Textbook of Medical Physiology, 9th edition, W.B. Sanders Company, Philadelphia.
[17] Dinnar U. (1981) Cardiovascular fluid dynamics. CRC Press, Boca Raton, FL.
[18] Chien S, Dormandy JA, Ernst E, Matrai, A. (2012) Clinical hemorheology: Applications in cardiovascular and hematological disease, diabetes, surgery and gynecology (vol. 74). Springer Science & Business Media.
[19] Cheng DC, Evans F. (1965) Phenomenological characterization of the rheological behaviour of inelastic reversible thixotropic and antithixotropic fluids. British Journal of Applied Physics, 16(11):1599-1614.
[20] Mewis J, Spaull AJB. (1976). Rheology of concentrated dispersions. Advances in Colloid and interface Science, 6(3):173-200.
[21] Nakamura M, Sawada T. (1988) Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis. Journal of Biomechanical Engineering, 110(2):137-143.
[22] Stoltz JF, Singh M, Riha P. (1999) Hemorheology in practice (vol. 30). IOS press.
[23] Barbee JH. (1973) The effect of temperature on the relative viscosity of human blood. Biorheology, 10(1):1-5.
[24] Cokelet GR. (1972) The rheology of human blood. Biomechanics: its foundations and objectives, 72:63-103.
[25] Rathod VP, Tanveer S. (2009) Pulsatile flow of couple stress fluid through a porous medium with periodic body acceleration and magnetic field. Bulletin of the Malaysian Mathematical Sciences Society, 32(2):245-259.
[26] Boyd W. (1964) A Text-book of Pathology: Structure and Function in Diseases. Lea & Febiger: Philadelphia. pp. 130-133.
[27] Chandran KB, Rittgers SE, Yoganathan AP. (2012) Biofluid mechanics: the human circulation. CRC press.
[28] Blair GS. (1959) An equation for the flow of blood, plasma and serum through glass capillaries. Nature, 183(4661):613-614.
[29] Copley AL. (1960) Apparent viscosity and wall adherence of blood systems. Flow properties of blood and other biological systems, Pergamon Press, Oxford, UK, p. 97.
[30] Casson N. (1959) A flow equation for pigment-oil suspensions of the printing ink type. In:Mill CC, editor. Rheology of disperse systems. New York: Pergamon Press. pp. 82-104.
[31] Venkatesan J, Sankar DS, Hemalatha K, Yatim, Y. (2013) Mathematical analysis of Casson fluid model for blood rheology in stenosed narrow arteries. Journal of Applied Mathematics, 2013, pp. 1-11.
[32] Misra JC, Shit GC. (2006) Blood flow through arteries in a pathological state: A theoretical study. International Journal of Engineering Science, 44(10):662-671.
[33] Bodnár T, Sequeira A, Prosi M. (2011) On the shear-thinning and viscoelastic effects of blood flow under various flow rates. Applied Mathematics and Computation, 217(11):5055-5067.
[34] Fisher C, Rossmann J S. (2009) Effect of non-Newtonian behavior on hemodynamics of cerebral aneurysms. Journal of biomechanical engineering, 131(9):091004.
[35] Replogle RL, Meiselman HJ, Merrill EW. (1967) Clinical implications of blood rheology studies. Circulation, 36(1):148-160.
[36] Edward, R. (1969) Merrill, Rheology of Blood. Physiological reviews, 49(4):30-43.
[37] Morris CL, Rucknagel DL, Shukla R, Gruppo RA, Smith CM, Blackshear P. (1989) Evaluation of the yield stress of normal blood as a function of fibrinogen concentration and hematocrit. Microvascular research, 37(3):323-338.
[38] Lee BK, Xue S, Nam J, Lim H, Shin S. (2011) Determination of the blood viscosity and yield stress with a pressure-scanning capillary hemorheometer using constitutive models. Korea-Australia Rheology Journal, 23(1):1-6.
[39] Fedosov DA, Pan W, Caswell B, Gompper G, Karniadakis GE. (2011) Predicting human blood viscosity in silico. Proceedings of the National Academy of Sciences, 108(29):11772-11777.
[40] Barnes HA. (1997) Thixotropy—a review. Journal of Non-Newtonian fluid mechanics, 70(1):1-33.
[41] How TV. (ed.). (1996). Advances in Hemodynamics and Hemorheology (vol. 1). Elsevier.
[42] Huang CR, Siskovic N, Robertson RW, Fabisiak W, Smitherberg EH, Copley AL. (1975) Quantitative characterization of thixotropy of whole human blood. Biorheology, 12(5):279-282.
[43] Gaspar-Rosas A, Thurston GB. (1988) Erythrocyte aggregate rheology by transmitted and reflected light. Biorheology, 25(3):471-487.
[44] Pop GAM, Duncker DJ, Gardien M, Vranckx P, Versluis S, Hasan D, Slager CJ. (2002) The clinical significance of whole blood viscosity in (cardio) vascular medicine. Netherlands Heart Journal, 10(12):512-524.
[45] Cho YI, Cho DJ, Rosenson RS. (2014) Endothelial shear stress and blood viscosity in peripheral arterial disease. Current atherosclerosis reports, 16(4):404-419.
[46] Grotta J, Ackerman R Correia J, Fallick G, Chang J. (1982) Whole blood viscosity parameters and cerebral blood flow. Stroke, 13(3):296-301.
[47] Brown M, Marshall J. (1985) Regulation of cerebral blood flow in response to changes in blood viscosity. The Lancet, 325(8429):604-609.
[48] Tanner RI. (1985) Engineering Rheology, Clarendon. Oxford, 15, pp. 355-370.
[49] Ferguson J, Kemblowski Z. (1991) Applied Fluid Rheology, Elsevier, London, pp. 47–115.
[50] Macosko CW. (1994) Rheology: principles, measurements, and applications. VHC Publishers Inc., New York.
[51] Rosencranz R, Bogen SA. (2006) Clinical laboratory measurement of serum, plasma, and blood viscosity. Pathology Patterns Reviews, 125(suppl-1): S78-S86.
[52] Shin S, Keum DY, Ku YH. (2002) Blood viscosity measurements using a pressure-scanning capillary viscometer. Journal of Mechanical Science and Technology, 16(12):1719-1724.
[53] Shin S, Hou JX, Suh JS, Singh M. (2007) Validation and application of a microfluidic ektacytometer (RheoScan-D) in measuring erythrocyte deformability. Clinical hemorheology and microcirculation, 37(4):319-328.
[54] Lee BK, Xue S, Nam J, Lim H, Shin S. (2011). Determination of the blood viscosity and yield stress with a pressure-scanning capillary hemorheometer using constitutive models. Korea-Australia Rheology Journal, 23(1):1-6.
[55] Jun Kang Y, Yeom E, Lee SJ. (2013) A microfluidic device for simultaneous measurement of viscosity and flow rate of blood in a complex fluidic network. Biomicrofluidics, 7(5):054111.
[56] Kang D, Wang W, Lee J, Tai YC, Hsiai TK. (2015, June) Measurement of viscosity of adult zebrafish blood using a capillary pressure-driven viscometer. In Solid-State Sensors, Actuators and Microsystems (Transducers), 2015 Transducers-2015 18th International Conference on (pp. 1661-1664). IEEE.
[57] Nemeth N, Alexy T, Furka A, Baskurt OK, Meiselman HJ, Furka I, Miko I. (2009) Inter-species differences in hematocrit to blood viscosity ratio. Biorheology, 46(2):155-165.
[58] Furukawa K, Abumiya T, Sakai K, Hirano M, Osanai T, Shichinohe H, Houkin K. (2016) Measurement of human blood viscosity by an electromagnetic spinning sphere viscometer. Journal of medical engineering & technology, 40(6):285-292.
[59] Pennes HH. (1948) Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of applied physiology, 1(2):93-122.
[60] Incropera FP. (2007) Introduction to Heat Transfer, 5th ed. John Wiley and Sons Inc., New York, NY, 581-619.
[61] Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A, Larsen J. (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Annals of biomedical engineering, 28(11):1281-1299.
[62] He Y, Liu H, Himeno R. (2004) A one-dimensional thermo-fluid model of blood circulation in the human upper limb. International Journal of Heat and Mass Transfer, 47(12):2735-2745.
[63] Atabek HB, Chang CC. (1961) Oscillatory flow near the entry of a circular tube. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 12(3):185-201.
[64] Ling SC, Atabek HB. (1972) A nonlinear analysis of pulsatile flow in arteries. Journal of Fluid Mechanics, 55(3):493-511.
[65] Craciunescu OI, Clegg ST. (2001) Pulsatile blood flow effects on temperature distribution and heat transfer in rigid vessels. Transaction-American Society of Mechanical Engineers Journal of Biomechanical Engineering, 123(5):500-505.
[66] Back LH, Cho YI, Crawford DW, Cuffel RF. (1984) Effect of mild atherosclerosis on flow resistance in a coronary artery casting of man. Journal of biomechanical engineering, 106(1):48-53.
[67] Milnor WR. (1992) Hemodynamics, Baltimore: Williams and Williams, p. 20.
[68] Chakravarty S, Mandal PK. (2009) Effect of heat and mass transfer on non-Newtonian flow–Links to atherosclerosis. International Journal of Heat and Mass Transfer, 52(25):5719-5730.
[69] Ahmad, F., Hussain, S., & Ansari, A. (2015) Unsteady MHD blood flow with micropolar fluid characteristics and heat source through parallel plate channel. Journal of Applied Environment & Biological Sciences, 5(4):80-86.
