Mathematical Model of Blood Flow in Small Blood Vessel in the Presence of Magnetic Field

ABSTRACT

A mathematical model for blood flow in the small blood vessel in the presence of magnetic field is presented in this paper. We have modeled the two phase model for the blood flow consists of a central core of suspended erythrocytes and cell-free layer surrounding the core. The system of differential equations has been solved analytically. We have obtained the result for velocity, flow rate and effective viscosity in presence of peripheral layer and magnetic field .All the result has been obtained and discussed through graphs.

A mathematical model for blood flow in the small blood vessel in the presence of magnetic field is presented in this paper. We have modeled the two phase model for the blood flow consists of a central core of suspended erythrocytes and cell-free layer surrounding the core. The system of differential equations has been solved analytically. We have obtained the result for velocity, flow rate and effective viscosity in presence of peripheral layer and magnetic field .All the result has been obtained and discussed through graphs.

Cite this paper

nullR. Bali and U. Awasthi, "Mathematical Model of Blood Flow in Small Blood Vessel in the Presence of Magnetic Field,"*Applied Mathematics*, Vol. 2 No. 2, 2011, pp. 264-269. doi: 10.4236/am.2011.22031.

nullR. Bali and U. Awasthi, "Mathematical Model of Blood Flow in Small Blood Vessel in the Presence of Magnetic Field,"

References

[1] G. R. Cokelet, Y. C. Fung, N. Perrone and M. Anliker, “Biomechanics-Its Foundations and Objectives,” Prentice-Hall, Englewood Cliffs, 1972.

[2] V. P. Srivastava and R.Srivastava, “Particulate Suspension Blood Flow Through a Narrow Catheterized Artery,” Computers & Mathematics with Applications, Vol. 58, No. 2, 2009, pp. 227-238. doi:10.1016/j.camwa.2009.01.041

[3] S. Chien, S. Usami and R. Skalak, “Blood Flow in Small Tubes,” E. M. Renkins and C. C. Michel, (Eds.), American Physiological Society Handbook of Physiology, Section 2, The Cardiovascular System, 4, Bethesda MD, American Physiological Society, 1984, pp. 217-249.

[4] R. Fahraeus and R. Lindqvist, “The Viscosity of the Blood in Narrow Capillary Tubes,” American Journal Physiology, Vol. 96, 1931, pp. 562-568.

[5] A. R. Pries, D. Neuhaus and P. Gaehtegnes, “Blood Viscosity in Tube Flow: Dependence on Diameter and Hematocrit,” American Journal of Physiology, Vol. 263, 1992, pp. 1170-1778.

[6] G. Bugliarello and J. Sevilla, “Velocity Distribution and Other Characteristics of Steady and Pulsatiles Blood Flow in Fine Glasss,” Biorheology, Vol. 7, No. 2, 1970, pp. 85-107.

[7] G. R. Cokelet, “The Rheology of Human Blood,” Y. C. Fung (Eds.), Biomechanics, Prentice-Hall, Englewood Cliffs, 1972.

[8] M. Sharan and A. S. Popel, “A Two-Phase Model for Flow of Blood in Narrow Tubes with Increased Effective Viscosity Near the Wall,” Biorheology, Vol. 38, No. 5-6, 2001, pp. 415-428.

[9] V. P. Srivastava, “A Theoretical Model for Blood Flow in Small Vessels,” International Journal of Application and Applied Mathematics, Vol. 2, No. 1, 2007, pp. 51-65.

[10] D. S. Sankar and U. Lee, “Two-Fluid Herchel-Bilkey Model for Blood Flow in Catheterized Arteries,” Journal of Mechanical Sciences and Technology, Vol. 22, No. 5, 2008, pp. 1008-1018. doi:10.1007/s12206-008-0123-4

[11] B. K. Mishra and N. Verma, “Magnetic Effect on Blood Flow in a Multi-Stenosised Artery,” Applied Mathematics and Computation, 2007.

[12] I. H. Chen, “Analysis of an Intensive Magnetic Field on Blood Flow: Part-2,” Journal of Bioelectronics, Vol. 4, No. 1, 1985, pp. 55-61.

[13] Y. Haik, V. Pai and C. J. Chen, “Apparent Viscosity of Human Blood in a High Statics Magnetic Field,” Journal of Magnetism and Magnetic Materials, Vol. 225, No. 1-2, 2001, pp. 180-186. doi:10.1016/S0304-8853(00)01249-X

[14] J. R. Womersley, “Method for the Calculation of Velocity Rate of Flow and Viscous Dragin Arteries When the Pressure Gradient is Known,” The Journal of Physiology, Vol. 127, No. 3, 1955, pp. 533-563.

[15] E. N. Lightfoot, “Transport Phenomenon in Living System,” Wiley, New York, 1974.

[16] V. K. Sud and G. S. Sekhon, “Arterial Flow under Periodic Body Acceleration,” Bulletin of Mathematical Biology, Vol. 47, No. 1, 1985, pp. 35-52.

[17] P. Chaturani and V. Palanisamy, “Pulsatile Flow of Bblood with Periodic Body Acceleration,” International Journal of Engineering Science, Vol. 29, No. 1, 1991, pp. 113-119. doi:10.1016/0020-7225(91)90081-D

[18] D. K. Wagh and S. D. Wagh, “Blood Flow Considered as Magnetic Flow,” Proceeding of Physiology of fluid Dynamics III, 1992, pp. 311-315.

[19] P. G. Saffman, “On the Stability of Laminar Flow of a Dusty Gas,” The Journal of Fluid Mechanics, Vol. 13, 1962, pp. 120-128. doi:10.1017/S0022112062000555

[20] A. H. Nayfeh, “Osicillatory Two-Phase Flow Through a Rigid Pipe,” AIAA Journal, Vol. 4, 1966, pp. 1868-1871. doi:10.2514/3.3804

[21] D. C. Sanyal, K. Das and S. Debnath, “Effect of Magnetic Field on Pulsatile Blood Flow Through an Inclined Circular Tube with Periodic Body Acceleration,” The Journal of Physiological Sciences, Vol. 11, 2007, pp. 43-56.

[22] T. Yamamoto, Y. Nagayama and M. Tamura, “A Blood–Oxygenation Dependent Increase in Blood Viscosity Due to a Static Magnetic Field,” Physics in Medicine and Biology, Vol. 49, No. 14, 2004, pp. 3267-3277. doi:10.1088/0031-9155/49/14/017

[1] G. R. Cokelet, Y. C. Fung, N. Perrone and M. Anliker, “Biomechanics-Its Foundations and Objectives,” Prentice-Hall, Englewood Cliffs, 1972.

[2] V. P. Srivastava and R.Srivastava, “Particulate Suspension Blood Flow Through a Narrow Catheterized Artery,” Computers & Mathematics with Applications, Vol. 58, No. 2, 2009, pp. 227-238. doi:10.1016/j.camwa.2009.01.041

[3] S. Chien, S. Usami and R. Skalak, “Blood Flow in Small Tubes,” E. M. Renkins and C. C. Michel, (Eds.), American Physiological Society Handbook of Physiology, Section 2, The Cardiovascular System, 4, Bethesda MD, American Physiological Society, 1984, pp. 217-249.

[4] R. Fahraeus and R. Lindqvist, “The Viscosity of the Blood in Narrow Capillary Tubes,” American Journal Physiology, Vol. 96, 1931, pp. 562-568.

[5] A. R. Pries, D. Neuhaus and P. Gaehtegnes, “Blood Viscosity in Tube Flow: Dependence on Diameter and Hematocrit,” American Journal of Physiology, Vol. 263, 1992, pp. 1170-1778.

[6] G. Bugliarello and J. Sevilla, “Velocity Distribution and Other Characteristics of Steady and Pulsatiles Blood Flow in Fine Glasss,” Biorheology, Vol. 7, No. 2, 1970, pp. 85-107.

[7] G. R. Cokelet, “The Rheology of Human Blood,” Y. C. Fung (Eds.), Biomechanics, Prentice-Hall, Englewood Cliffs, 1972.

[8] M. Sharan and A. S. Popel, “A Two-Phase Model for Flow of Blood in Narrow Tubes with Increased Effective Viscosity Near the Wall,” Biorheology, Vol. 38, No. 5-6, 2001, pp. 415-428.

[9] V. P. Srivastava, “A Theoretical Model for Blood Flow in Small Vessels,” International Journal of Application and Applied Mathematics, Vol. 2, No. 1, 2007, pp. 51-65.

[10] D. S. Sankar and U. Lee, “Two-Fluid Herchel-Bilkey Model for Blood Flow in Catheterized Arteries,” Journal of Mechanical Sciences and Technology, Vol. 22, No. 5, 2008, pp. 1008-1018. doi:10.1007/s12206-008-0123-4

[11] B. K. Mishra and N. Verma, “Magnetic Effect on Blood Flow in a Multi-Stenosised Artery,” Applied Mathematics and Computation, 2007.

[12] I. H. Chen, “Analysis of an Intensive Magnetic Field on Blood Flow: Part-2,” Journal of Bioelectronics, Vol. 4, No. 1, 1985, pp. 55-61.

[13] Y. Haik, V. Pai and C. J. Chen, “Apparent Viscosity of Human Blood in a High Statics Magnetic Field,” Journal of Magnetism and Magnetic Materials, Vol. 225, No. 1-2, 2001, pp. 180-186. doi:10.1016/S0304-8853(00)01249-X

[14] J. R. Womersley, “Method for the Calculation of Velocity Rate of Flow and Viscous Dragin Arteries When the Pressure Gradient is Known,” The Journal of Physiology, Vol. 127, No. 3, 1955, pp. 533-563.

[15] E. N. Lightfoot, “Transport Phenomenon in Living System,” Wiley, New York, 1974.

[16] V. K. Sud and G. S. Sekhon, “Arterial Flow under Periodic Body Acceleration,” Bulletin of Mathematical Biology, Vol. 47, No. 1, 1985, pp. 35-52.

[17] P. Chaturani and V. Palanisamy, “Pulsatile Flow of Bblood with Periodic Body Acceleration,” International Journal of Engineering Science, Vol. 29, No. 1, 1991, pp. 113-119. doi:10.1016/0020-7225(91)90081-D

[18] D. K. Wagh and S. D. Wagh, “Blood Flow Considered as Magnetic Flow,” Proceeding of Physiology of fluid Dynamics III, 1992, pp. 311-315.

[19] P. G. Saffman, “On the Stability of Laminar Flow of a Dusty Gas,” The Journal of Fluid Mechanics, Vol. 13, 1962, pp. 120-128. doi:10.1017/S0022112062000555

[20] A. H. Nayfeh, “Osicillatory Two-Phase Flow Through a Rigid Pipe,” AIAA Journal, Vol. 4, 1966, pp. 1868-1871. doi:10.2514/3.3804

[21] D. C. Sanyal, K. Das and S. Debnath, “Effect of Magnetic Field on Pulsatile Blood Flow Through an Inclined Circular Tube with Periodic Body Acceleration,” The Journal of Physiological Sciences, Vol. 11, 2007, pp. 43-56.

[22] T. Yamamoto, Y. Nagayama and M. Tamura, “A Blood–Oxygenation Dependent Increase in Blood Viscosity Due to a Static Magnetic Field,” Physics in Medicine and Biology, Vol. 49, No. 14, 2004, pp. 3267-3277. doi:10.1088/0031-9155/49/14/017