Mathematical Analysis of Unsteady MHD Blood Flow through Parallel Plate Channel with Heat Source

Affiliation(s)

Basic Engineering Sciences Department, Faculty of Engineering, Elmenoufia University, Shibin El Kom, Egypt.

Basic Engineering Sciences Department, Faculty of Engineering, Elmenoufia University, Shibin El Kom, Egypt.

ABSTRACT

In the present study, a mathematical model of unsteady blood flow through parallel plate channel under the action of an applied constant transverse magnetic field is proposed. The model is subjected to heat source. Analytical expressions are obtained by choosing the axial velocity; temperature distribution and the normal velocity of the blood depend on y and t only to convert the system of partial differential equations into system of ordinary differential equations under the conditions defined in our model. The model has been analyzed to find the effects of various parameters such as, Hartmann number, heat source parameter and Prandtl number on the axial velocity, temperature distribution and the normal velocity. The numerical solutions of axial velocity, temperature distributions and normal velocity are shown graphically for better understanding of the problem. Hence, the present mathematical model gives a simple form of axial velocity, temperature distribution and normal velocity of the blood flow so that it will help not only people working in the field of Physiological fluid dynamics but also to the medical practitioners.

In the present study, a mathematical model of unsteady blood flow through parallel plate channel under the action of an applied constant transverse magnetic field is proposed. The model is subjected to heat source. Analytical expressions are obtained by choosing the axial velocity; temperature distribution and the normal velocity of the blood depend on y and t only to convert the system of partial differential equations into system of ordinary differential equations under the conditions defined in our model. The model has been analyzed to find the effects of various parameters such as, Hartmann number, heat source parameter and Prandtl number on the axial velocity, temperature distribution and the normal velocity. The numerical solutions of axial velocity, temperature distributions and normal velocity are shown graphically for better understanding of the problem. Hence, the present mathematical model gives a simple form of axial velocity, temperature distribution and normal velocity of the blood flow so that it will help not only people working in the field of Physiological fluid dynamics but also to the medical practitioners.

Cite this paper

nullI. Eldesoky, "Mathematical Analysis of Unsteady MHD Blood Flow through Parallel Plate Channel with Heat Source,"*World Journal of Mechanics*, Vol. 2 No. 3, 2012, pp. 131-137. doi: 10.4236/wjm.2012.23015.

nullI. Eldesoky, "Mathematical Analysis of Unsteady MHD Blood Flow through Parallel Plate Channel with Heat Source,"

References

[1] J. Singh and R. Rathee, “Analytical Solution of Two- Dimensional Model of Blood Flow with Variable Viscosity through an Indented Artery Due to LDL Effect in the Presence of Magnetic Field,” International Journal of Physical Sciences, Vol. 5, No. 12, 2010, pp. 1857-1868.

[2] O. Prakash, S. P. Singh, D. Kumar and Y. K. Dwivedi, “A Study of Effects of Heat Source on MHD Blood Flow through Bifurcated Arteries,” AIP Advances, Vol. 1, No. 4, 2011, pp. 1-7. doi:10.1063/1.3658616

[3] N. Verma and R. S. Parihar, “Effects of Magneto-Hydro- dynamic and Hematocrit on Blood Flow in an Artery with Multiple Mild Stenosis,” International Journal of Applied Mathematics and Computer Science, Vol. 1, No. 1, 2009, pp. 30-46.

[4] 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 Science, Vol. 11, 2007, pp. 43-56.

[5] E. E. Tzirtzilakis, “A Mathematical Model for Blood Flow in Magnetic Field,” Physics of Fluids, Vol. 17, No. 7, 2005, p. 077103. doi:10.1063/1.1978807

[6] G. Ramamurthy and B. Shanker, “Magnetohydrodynamic Effects on Blood Flow through Porous Channel,” Medi- cal and Biological Engineering and Computing, Vol. 32, No. 6, 1994, pp. 655-659. doi:10.1007/BF02524242

[7] K. Das and G. C. Saha, “Arterial MHD Pulsatile Flow of Blood under Periodic Body Acceleration,” Bulletin of Society of Mathematicians Banja Luka, Vol. 16, 2009, pp. 21-42.

[8] M. Jain, G. C. Sharma and A. Singh, “Mathematical Analysis of MHD Flow of Blood in Very Narrow Capillaries,” International Journal of Engineering Transactions B: Applications, Vol. 22, No. 3, 2009, pp. 307-315.

[9] V. P. Rathod and S. Tanveer, “Pulsatile Flow of Couple Stress Fluid through a Porous Medium with Periodic Body Acceleration and Magnetic Field,” Bulletin of the Malaysian Mathematical Sciences Society, Vol. 32, No. 2, 2009, pp. 245-259.

[10] J. Singh and R. Rathee, “Analytical Solution of Two- Dimensional Model of Blood Flow with Variable Viscosity through an Indented Artery Due to LDL Effect in the Presence of Magnetic Field,” International Journal of Physical Sciences, Vol. 5, No. 12, 2010, pp. 1857-1868.

[11] C. S. Dulal and B. Ananda, “Pulsatile Motion of Blood through an Axi-Symmetric Artery in Presence of Mag- netic Field,” Journal of Science and Technology of Assam University, Vol. 5, No. 2, 2010, pp. 12-20.

[12] M. Zamir and M. R. Roach, “Blood Flow Downstream of a Two-Dimensional Bifurcation,” Journal of Theoretical Biology, Vol. 42, No. 1, 1973, pp. 33-42. doi:10.1016/0022-5193(73)90146-X

[13] S. D. Adhikary and J. C. Misra, “Unsteady Two-Dimen- sional Hydromagnetic Flow and Heat Transfer of a Fluid,” International Journal of Applied Mathematics and Me- chanics, Vol. 7, No. 4, 2011, pp. 1-20.

[14] J. J. W. Lagendijk, “The Influence of Blood Flow in Large Vessels on the Temperature Distribution in Hyperthermia,” Physics in Medicine and Biology, Vol. 27, No. 1, 1982, p. 17. doi:10.1088/0031-9155/27/1/002

[15] C. Y. Wang, “Heat Transfer to Blood Flow in a Small Tube,” Journal of Biomechanical Engineering, Vol. 130, No. 2, 2008, p. 024501. doi:10.1115/1.2898722

[1] J. Singh and R. Rathee, “Analytical Solution of Two- Dimensional Model of Blood Flow with Variable Viscosity through an Indented Artery Due to LDL Effect in the Presence of Magnetic Field,” International Journal of Physical Sciences, Vol. 5, No. 12, 2010, pp. 1857-1868.

[2] O. Prakash, S. P. Singh, D. Kumar and Y. K. Dwivedi, “A Study of Effects of Heat Source on MHD Blood Flow through Bifurcated Arteries,” AIP Advances, Vol. 1, No. 4, 2011, pp. 1-7. doi:10.1063/1.3658616

[3] N. Verma and R. S. Parihar, “Effects of Magneto-Hydro- dynamic and Hematocrit on Blood Flow in an Artery with Multiple Mild Stenosis,” International Journal of Applied Mathematics and Computer Science, Vol. 1, No. 1, 2009, pp. 30-46.

[4] 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 Science, Vol. 11, 2007, pp. 43-56.

[5] E. E. Tzirtzilakis, “A Mathematical Model for Blood Flow in Magnetic Field,” Physics of Fluids, Vol. 17, No. 7, 2005, p. 077103. doi:10.1063/1.1978807

[6] G. Ramamurthy and B. Shanker, “Magnetohydrodynamic Effects on Blood Flow through Porous Channel,” Medi- cal and Biological Engineering and Computing, Vol. 32, No. 6, 1994, pp. 655-659. doi:10.1007/BF02524242

[7] K. Das and G. C. Saha, “Arterial MHD Pulsatile Flow of Blood under Periodic Body Acceleration,” Bulletin of Society of Mathematicians Banja Luka, Vol. 16, 2009, pp. 21-42.

[8] M. Jain, G. C. Sharma and A. Singh, “Mathematical Analysis of MHD Flow of Blood in Very Narrow Capillaries,” International Journal of Engineering Transactions B: Applications, Vol. 22, No. 3, 2009, pp. 307-315.

[9] V. P. Rathod and S. Tanveer, “Pulsatile Flow of Couple Stress Fluid through a Porous Medium with Periodic Body Acceleration and Magnetic Field,” Bulletin of the Malaysian Mathematical Sciences Society, Vol. 32, No. 2, 2009, pp. 245-259.

[10] J. Singh and R. Rathee, “Analytical Solution of Two- Dimensional Model of Blood Flow with Variable Viscosity through an Indented Artery Due to LDL Effect in the Presence of Magnetic Field,” International Journal of Physical Sciences, Vol. 5, No. 12, 2010, pp. 1857-1868.

[11] C. S. Dulal and B. Ananda, “Pulsatile Motion of Blood through an Axi-Symmetric Artery in Presence of Mag- netic Field,” Journal of Science and Technology of Assam University, Vol. 5, No. 2, 2010, pp. 12-20.

[12] M. Zamir and M. R. Roach, “Blood Flow Downstream of a Two-Dimensional Bifurcation,” Journal of Theoretical Biology, Vol. 42, No. 1, 1973, pp. 33-42. doi:10.1016/0022-5193(73)90146-X

[13] S. D. Adhikary and J. C. Misra, “Unsteady Two-Dimen- sional Hydromagnetic Flow and Heat Transfer of a Fluid,” International Journal of Applied Mathematics and Me- chanics, Vol. 7, No. 4, 2011, pp. 1-20.

[14] J. J. W. Lagendijk, “The Influence of Blood Flow in Large Vessels on the Temperature Distribution in Hyperthermia,” Physics in Medicine and Biology, Vol. 27, No. 1, 1982, p. 17. doi:10.1088/0031-9155/27/1/002

[15] C. Y. Wang, “Heat Transfer to Blood Flow in a Small Tube,” Journal of Biomechanical Engineering, Vol. 130, No. 2, 2008, p. 024501. doi:10.1115/1.2898722