A Casson Fluid Model for Multiple Stenosed Artery in the Presence of Magnetic Field

ABSTRACT

The flow of blood through a multistenosed artery under the influence of external applied magnetic field is studied. The artery is modeled as a circular tube. The effect of non-Newtonian nature of blood in small blood vessels has been taken into account by modeling blood as a Casson fluid. The effect of magnetic field, height of stenosis, parameter determin- ing the shape of the stenosis on velocity field, volumetric flow rate in stenotic region and wall shear stress at surface of stenosis are obtained and shown graphically. Some important observations regarding the flow of blood in multi stenosed artery are obtained leading to medical interest.

The flow of blood through a multistenosed artery under the influence of external applied magnetic field is studied. The artery is modeled as a circular tube. The effect of non-Newtonian nature of blood in small blood vessels has been taken into account by modeling blood as a Casson fluid. The effect of magnetic field, height of stenosis, parameter determin- ing the shape of the stenosis on velocity field, volumetric flow rate in stenotic region and wall shear stress at surface of stenosis are obtained and shown graphically. Some important observations regarding the flow of blood in multi stenosed artery are obtained leading to medical interest.

Cite this paper

R. Bali and U. Awasthi, "A Casson Fluid Model for Multiple Stenosed Artery in the Presence of Magnetic Field,"*Applied Mathematics*, Vol. 3 No. 5, 2012, pp. 436-441. doi: 10.4236/am.2012.35066.

R. Bali and U. Awasthi, "A Casson Fluid Model for Multiple Stenosed Artery in the Presence of Magnetic Field,"

References

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[2] J. B. Shukla, R. S. Parihar and S. P. Gupta, “Effects of Peripheral Layer Viscosity on Blood Flow through the Artery,” Bulletin of Mathematical Biology, Vol. 42, No. 6, 1980, pp. 797-805.

[3] R. M. Nerem, “Fluid Dynamics Aspects of Arterial Diseases,” Proceedings of a Specialists Meeting, Columbus, 19-20 September 1974.

[4] I. I. H. Chen, “Analysis of an Intensive Magnetic Field on Blood Flow: Part 2,” Electromagnetic Biology and Medicine, Vol. 4, No. 1, 1985, pp. 55-61. doi:10.3109/15368378509040360

[5] A. T. Ogulua and T. M. Abbey, “Simulation of Heat Transfer on an Oscillatory Blood Flow in an Indented Porous Artery,” International Communications in Heat and Mass Transfer, Vol. 32, No. 7, 2005, pp. 983-989. doi:10.1016/j.icheatmasstransfer.2004.08.028

[6] M. J. Manton, “Low Reynolds Number Flow in Slowly Varying Axisymmetric Tubes,” Journal of Fluid Mechanics, Vol. 49, No. 3, 1971, pp. 451-459. doi:10.1017/S0022112071002192

[7] A. Ramachandra Rao and R. Devanathan, “Pulsatile Flow in Tubes of Varying Cross-Section,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 24, No. 22, 1973, pp. 203-213. doi:10.1007/BF01590913

[8] P. Hall, “Unsteady Viscous Flow in a Pipe of Slowing Varying Cross-Section,” Journal of Fluid Mechanics, Vol. 64, No. 2, 1974, pp. 209-226. doi:10.1017/S0022112074002369

[9] F. T. Smith, “Flow through Constricted or Dilated Pipes and Channels: Part-2,” The Quarterly Journal of Mechanics & Applied Mathematics, Vol. 29, No. 3, 1976, pp. 365-376. doi:10.1093/qjmam/29.3.365

[10] P. W. Duck, “Separation of Jets or Thermal Boundary Layers from a Wall,” Proceedings of the Royal Society A, Vol. 363, No. 2, 1976, p. 33.

[11] N. Padmanabhan, “Mathematical Model of Arterial Stenosis,” Medical Biological Engineering and Computing, Vol. 18, No. 3, 1980, pp. 281-286. doi:10.1007/BF02443380

[12] R. Mehrotra, G. Jayaraman and N. Padmanabhan, “Pulsatile Blood Flow in a Stenosed Artery a Theoretical Model,” Medical Biological Engineering and Computing, Vol. 23, No. 1, 1985, pp. 55-62. doi:10.1007/BF02444028

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[14] J. B. Shukla, R. S. Parihar and B. R. P. Rao, “Effects of Stenosis on Non-Newtonian Flow of the Blood in an Artery,” Bulletin of Mathematical Biology, Vol. 42, No. 3, 1980, pp. 283-294.

[15] S. Chakravorthy, “Effects of Stenosis on the Flow Behavior of Blood in an Artery,” International Journal of Engineering Science, Vol. 25, No. 8, 1987, pp. 1003-1016. doi:10.1016/0020-7225(87)90093-0

[16] N. Padmanabhan and R. Devanathan, “Mathematical Model of an Arterial Stenosis, Allowing for Tethering,” Medical Biological Engineering and Computing, Vol. 19, No. 4, 1981, pp. 385-390. doi:10.1007/BF02441299

[17] R. Devanathan and S. Parvathamma, “Flow of Micropolar Fluid through a Tube with Stenosis,” Medical Biological Engineering and Computing, Vol. 21, No. 4, 1983, pp. 438-445. doi:10.1007/BF02442631

[18] P. Chaturani and R. P. Swamy, “A Study of Non-Newtonian Aspects of Blood Flow through Stenosed Arteries and Its Applications in Arterial Diseases,” Biorheology, Vol. 22, No. 6, 1985, pp. 512-531.

[19] L. M. Srivastava, “Flow of Couple Stress Fluid through Stenotic Blood Vessels,” Journal of Biomechanics, Vol. 18, No. 7, 1985, pp. 479-485. doi:10.1016/0021-9290(85)90662-1

[20] P. Chaturani and R. P. Swamy, “Pulsatile Flow of Casson’s Fluid through Stenosed Arteries with Applications to Blood Flow,” Biorheology, Vol. 23, No. 5, 1986, pp. 499-511.

[21] K. Haldar, “Oscillatory Flow of Blood in Stenosed Artery,” Bulletin of Mathematical Biology, Vol. 49, No. 3, 1987, pp. 279-287.

[22] D. C. Sanyal and N. K. Maji, “Unsteady Blood Flow through an Indented Tube with Atherosclerosis,” Indian Journal of Pure and Applied Mathematics, Vol. 30, No. 10, 1999, pp. 951-959.

[23] K. Haldar and K. N. Dey, “Effect of Erythrocytes on the Flow Characteristic of Blood in an Indented Tube,” Archives of Mechanics, Vol. 42, No. 1, 1990, p. 109.

[24] K. Haldar and S. N. Ghosh, “Effects of a Magnetic Field on Blood Flow through an Indented Tube in the Presence of Erythrocytes,” Journal of Pure and Applied Mathematics, Vol. 25, No. 3, 1994, pp. 345-352.

[25] P. K. Mandal, S. Chakravarthy, A. Mandal and N. Amin, “Effect of Body Acceleration on Unsteady Pulsatile Flow of Non-Newtonian Fluid through a Stenosed Artery,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 766-779. doi:10.1016/j.amc.2006.11.139

[26] P. K. Suri and R. Pushpa, “Effect of Static Magnetic Field on Blood Flow in a Branch,” Journal of Pure and Applied Mathematics, Vol. 12, No. 7, 1981, pp. 907-918.

[27] E. Amos, “Magnetic Effect of Pulsatile Flow in a Constricted Axis Symmetric Tube,” Journal of Pure and Applied Mathematics, Vol. 34, No. 9, 2003, pp. 1315-1326.

[28] M. A. Elnaby, M. T. N. Eldabe, M. Y. Abou Zied and D. C. Sanyal, “Mathematical Analysis on MHD Pulsatile Flow of a Non-Newtonian Fluid through a Tube with Varying Cross-Section,” Journal of Institute of Mathematics and Computer Science, Vol. 20, No. 1, 2007, pp. 29-42.

[29] K. Das and G. C. Saha, “Arterial MHD Pulsatile Flow of Blood under Periodic Body Accelaration,” Bulletin of. Mathematic Society, Vol. 16, 2009, pp. 21-42.

[30] Y. C. Fung, “Mechanical Properties of Living Tissue,” Biomechanics, Vol. 4, No. 2, 1986, pp. 68-81.

[31] R. Ponalagusamy, “Blood Flow through an Artery with Mild Stenosis: A Two-Layered Model Different Shape of Stenoses and Velocity at Wall,” Journal of Applied Sci ences, Vol. 7, No. 7, 2007, pp. 1071-1077.

[32] 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,” Journal of Physical Sciences, Vol. 11, No. 1, 2007, pp. 43-56.

[33] D. Biswas and U. S. Chakraborty, “Pulsatile Blood Flow through a Catheterized Artery with an Axially Nonsymmetrical Stenosis,” Applied Mathematical Sciences, Vol. 4, No. 58, 2010, pp. 2865-2880.

[34] E. E. Tzirtzilakis, “A Mathematical Model for Blood Flow in Magnetic Field,” Physics of Fluids, Vol. 17, 2005, pp. 077107-077115. doi:10.1063/1.1978807

[35] Y. Haik, V. Pai and C. J. Chen, “Apparent Viscosity of Human Blood in a High Static Magnetic Field,” Magnetism and Magnetic Material, Vol. 225, No. 14, 2001, pp. 180-186. doi:10.1016/S0304-8853(00)01249-X

[36] V. P. Srivastava and M. Saxena, “Two-Layered Model of Casson Fluid Flow through Stenotic Blood Vessels; Apllications to the Cardiovascular System,” Journal of Biomechanics, Vol. 27, 1994, pp. 921-928. doi:10.1016/0021-9290(94)90264-X

[37] D. F. Fry, “Acute Vascular Emdothelial Changes Associated with Increased Blood Velocity Gradients,” Circulation Research, Vol. 22, 1968, pp. 165-197.

[1] D. F. Young, “Fluid Mechanics of Arterial Stenosis,” Journal of Biomechanical Engineering, Vol. 101, No. 3, 1979, pp. 157-175. doi:10.1115/1.3426241

[2] J. B. Shukla, R. S. Parihar and S. P. Gupta, “Effects of Peripheral Layer Viscosity on Blood Flow through the Artery,” Bulletin of Mathematical Biology, Vol. 42, No. 6, 1980, pp. 797-805.

[3] R. M. Nerem, “Fluid Dynamics Aspects of Arterial Diseases,” Proceedings of a Specialists Meeting, Columbus, 19-20 September 1974.

[4] I. I. H. Chen, “Analysis of an Intensive Magnetic Field on Blood Flow: Part 2,” Electromagnetic Biology and Medicine, Vol. 4, No. 1, 1985, pp. 55-61. doi:10.3109/15368378509040360

[5] A. T. Ogulua and T. M. Abbey, “Simulation of Heat Transfer on an Oscillatory Blood Flow in an Indented Porous Artery,” International Communications in Heat and Mass Transfer, Vol. 32, No. 7, 2005, pp. 983-989. doi:10.1016/j.icheatmasstransfer.2004.08.028

[6] M. J. Manton, “Low Reynolds Number Flow in Slowly Varying Axisymmetric Tubes,” Journal of Fluid Mechanics, Vol. 49, No. 3, 1971, pp. 451-459. doi:10.1017/S0022112071002192

[7] A. Ramachandra Rao and R. Devanathan, “Pulsatile Flow in Tubes of Varying Cross-Section,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 24, No. 22, 1973, pp. 203-213. doi:10.1007/BF01590913

[8] P. Hall, “Unsteady Viscous Flow in a Pipe of Slowing Varying Cross-Section,” Journal of Fluid Mechanics, Vol. 64, No. 2, 1974, pp. 209-226. doi:10.1017/S0022112074002369

[9] F. T. Smith, “Flow through Constricted or Dilated Pipes and Channels: Part-2,” The Quarterly Journal of Mechanics & Applied Mathematics, Vol. 29, No. 3, 1976, pp. 365-376. doi:10.1093/qjmam/29.3.365

[10] P. W. Duck, “Separation of Jets or Thermal Boundary Layers from a Wall,” Proceedings of the Royal Society A, Vol. 363, No. 2, 1976, p. 33.

[11] N. Padmanabhan, “Mathematical Model of Arterial Stenosis,” Medical Biological Engineering and Computing, Vol. 18, No. 3, 1980, pp. 281-286. doi:10.1007/BF02443380

[12] R. Mehrotra, G. Jayaraman and N. Padmanabhan, “Pulsatile Blood Flow in a Stenosed Artery a Theoretical Model,” Medical Biological Engineering and Computing, Vol. 23, No. 1, 1985, pp. 55-62. doi:10.1007/BF02444028

[13] J. C. Mishra and S. Chakravorty, “Flow in Arteries in the Presence of Stenosis,” Journal of Biomechanics, Vol. 19, No. 11, 1986, pp. 1907-1918.

[14] J. B. Shukla, R. S. Parihar and B. R. P. Rao, “Effects of Stenosis on Non-Newtonian Flow of the Blood in an Artery,” Bulletin of Mathematical Biology, Vol. 42, No. 3, 1980, pp. 283-294.

[15] S. Chakravorthy, “Effects of Stenosis on the Flow Behavior of Blood in an Artery,” International Journal of Engineering Science, Vol. 25, No. 8, 1987, pp. 1003-1016. doi:10.1016/0020-7225(87)90093-0

[16] N. Padmanabhan and R. Devanathan, “Mathematical Model of an Arterial Stenosis, Allowing for Tethering,” Medical Biological Engineering and Computing, Vol. 19, No. 4, 1981, pp. 385-390. doi:10.1007/BF02441299

[17] R. Devanathan and S. Parvathamma, “Flow of Micropolar Fluid through a Tube with Stenosis,” Medical Biological Engineering and Computing, Vol. 21, No. 4, 1983, pp. 438-445. doi:10.1007/BF02442631

[18] P. Chaturani and R. P. Swamy, “A Study of Non-Newtonian Aspects of Blood Flow through Stenosed Arteries and Its Applications in Arterial Diseases,” Biorheology, Vol. 22, No. 6, 1985, pp. 512-531.

[19] L. M. Srivastava, “Flow of Couple Stress Fluid through Stenotic Blood Vessels,” Journal of Biomechanics, Vol. 18, No. 7, 1985, pp. 479-485. doi:10.1016/0021-9290(85)90662-1

[20] P. Chaturani and R. P. Swamy, “Pulsatile Flow of Casson’s Fluid through Stenosed Arteries with Applications to Blood Flow,” Biorheology, Vol. 23, No. 5, 1986, pp. 499-511.

[21] K. Haldar, “Oscillatory Flow of Blood in Stenosed Artery,” Bulletin of Mathematical Biology, Vol. 49, No. 3, 1987, pp. 279-287.

[22] D. C. Sanyal and N. K. Maji, “Unsteady Blood Flow through an Indented Tube with Atherosclerosis,” Indian Journal of Pure and Applied Mathematics, Vol. 30, No. 10, 1999, pp. 951-959.

[23] K. Haldar and K. N. Dey, “Effect of Erythrocytes on the Flow Characteristic of Blood in an Indented Tube,” Archives of Mechanics, Vol. 42, No. 1, 1990, p. 109.

[24] K. Haldar and S. N. Ghosh, “Effects of a Magnetic Field on Blood Flow through an Indented Tube in the Presence of Erythrocytes,” Journal of Pure and Applied Mathematics, Vol. 25, No. 3, 1994, pp. 345-352.

[25] P. K. Mandal, S. Chakravarthy, A. Mandal and N. Amin, “Effect of Body Acceleration on Unsteady Pulsatile Flow of Non-Newtonian Fluid through a Stenosed Artery,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 766-779. doi:10.1016/j.amc.2006.11.139

[26] P. K. Suri and R. Pushpa, “Effect of Static Magnetic Field on Blood Flow in a Branch,” Journal of Pure and Applied Mathematics, Vol. 12, No. 7, 1981, pp. 907-918.

[27] E. Amos, “Magnetic Effect of Pulsatile Flow in a Constricted Axis Symmetric Tube,” Journal of Pure and Applied Mathematics, Vol. 34, No. 9, 2003, pp. 1315-1326.

[28] M. A. Elnaby, M. T. N. Eldabe, M. Y. Abou Zied and D. C. Sanyal, “Mathematical Analysis on MHD Pulsatile Flow of a Non-Newtonian Fluid through a Tube with Varying Cross-Section,” Journal of Institute of Mathematics and Computer Science, Vol. 20, No. 1, 2007, pp. 29-42.

[29] K. Das and G. C. Saha, “Arterial MHD Pulsatile Flow of Blood under Periodic Body Accelaration,” Bulletin of. Mathematic Society, Vol. 16, 2009, pp. 21-42.

[30] Y. C. Fung, “Mechanical Properties of Living Tissue,” Biomechanics, Vol. 4, No. 2, 1986, pp. 68-81.

[31] R. Ponalagusamy, “Blood Flow through an Artery with Mild Stenosis: A Two-Layered Model Different Shape of Stenoses and Velocity at Wall,” Journal of Applied Sci ences, Vol. 7, No. 7, 2007, pp. 1071-1077.

[32] 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,” Journal of Physical Sciences, Vol. 11, No. 1, 2007, pp. 43-56.

[33] D. Biswas and U. S. Chakraborty, “Pulsatile Blood Flow through a Catheterized Artery with an Axially Nonsymmetrical Stenosis,” Applied Mathematical Sciences, Vol. 4, No. 58, 2010, pp. 2865-2880.

[34] E. E. Tzirtzilakis, “A Mathematical Model for Blood Flow in Magnetic Field,” Physics of Fluids, Vol. 17, 2005, pp. 077107-077115. doi:10.1063/1.1978807

[35] Y. Haik, V. Pai and C. J. Chen, “Apparent Viscosity of Human Blood in a High Static Magnetic Field,” Magnetism and Magnetic Material, Vol. 225, No. 14, 2001, pp. 180-186. doi:10.1016/S0304-8853(00)01249-X

[36] V. P. Srivastava and M. Saxena, “Two-Layered Model of Casson Fluid Flow through Stenotic Blood Vessels; Apllications to the Cardiovascular System,” Journal of Biomechanics, Vol. 27, 1994, pp. 921-928. doi:10.1016/0021-9290(94)90264-X

[37] D. F. Fry, “Acute Vascular Emdothelial Changes Associated with Increased Blood Velocity Gradients,” Circulation Research, Vol. 22, 1968, pp. 165-197.