Skip to main content
SpringerLink
Log in

FDM analysis for MHD flow of a non-Newtonian fluid for blood flow in stenosed arteries

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

A computational model is developed to analyze the effects of magnetic field in a pulsatile flow of blood through narrow arteries with mild stenosis, treating blood as Casson fluid model. Finite difference method is employed to solve the simplified nonlinear partial differential equation and an explicit finite difference scheme is obtained for velocity and subsequently the finite difference formula for the flow rate, skin friction and longitudinal impedance are also derived. The effects of various parameters associated with this flow problem such as stenosis height, yield stress, magnetic field and amplitude of the pressure gradient on the physiologically important flow quantities namely velocity distribution, flow rate, skin friction and longitudinal impedance to flow are analyzed by plotting the graphs for the variation of these flow quantities for different values of the aforesaid parameters. It is found that the velocity and flow rate decrease with the increase of the Hartmann number and the reverse behavior is noticed for the wall shear stress and longitudinal impedance of the flow. It is noted that flow rate increases and skin friction decreases with the increase of the pressure gradient. It is also observed that the skin friction and longitudinal impedance increase with the increase of the amplitude parameter of the artery radius. It is also found that the skin friction and longitudinal impedance increases with the increase of the stenosis depth. It is recorded that the estimates of the increase in the skin friction and longitudinal impedance to flow increase considerably with the increase of the Hartmann number.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. A. Hogan and M. Henriksen, An evaluation of a micropolar model for blood flow through an idealized stenosis, J. Biomech., 22(3) (1989) 21–218.

    Article  Google Scholar 

  2. M. D. Deshpande, D. P. Giddens and R. F. Mabon, Steady laminar flow through modeled vascular stenoses, J. Biomech., 9 (1976) 165–174.

    Article  Google Scholar 

  3. I. Marshall, S. Zhao, P. Papathanasopoulou, P. Hoskins and X. Y. Xu, MRI and CFD studies of pulsatile flow in healthy and stenosed carotid bifurcation models, J. Biomech., 27 (2004) 679–687.

    Article  Google Scholar 

  4. E. W. Merrill, A. M. Benis, E. R. Gilliland, T. K. Sherwood and E. W. Salzman, Pressure flow relations of human blood in hallow fibers at low shear rates, J. Appl. Physiol., 20 (1965) 954–967.

    Google Scholar 

  5. A. P. Dwivedi, T. S. Pal and L. Rakesh, Micropolar fluid model for blood flow through small tapered tube, Indian J. Technology, 20 (1982) 295–299.

    MATH  Google Scholar 

  6. C. Tu and M. Deville, Pulsatile flow of non-Newtonian fluids through arterial stenosis, J. Biomech., 29 (1996) 899–908.

    Article  Google Scholar 

  7. M. Motta, Y. Haik, A. Gandhari and C. J. Chen, High magnetic field effects on human deoxygenated hemoglobin light absorption, Bioelectrochem. Bioenerg., 47 (1998) 297–300.

    Article  Google Scholar 

  8. E. F. EL-Shehawey, E. M. E. Elbarbary, N. A. S. Afifi and M. Elshahed, MHD flow of an elastico-viscous fluid under periodic body acceleration, J. Math. & Math. Sci., 23 (2000) 795–799.

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Midya, G. C. Layek, A. S. Gupta and T. Roy Mahapatra, Magenetohydrodynamic viscous flow separation in a channel with constrictions, ASME. J. Fluid Eng., 125 (2003) 952–962.

    Article  Google Scholar 

  10. Md. A. Ikbal, S. Chakravarty, K. L. Kelvin Wong, J. Mazumdar and P. K. Mandal, Unsteady response of non-Newtonian blood flow through a stenosed artery in magnetic field, J. Computat. Appl. Math., 230 (2009) 243–259.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Ramachandra Rao and K. S. Deshikachar, Physiological type flow in a circular pipe in the presence of a transverse magnetic field, J. Indian Inst. Sci., 68 (1988) 247–260.

    MATH  Google Scholar 

  12. P. A. Voltairas, D. I. Fotiadis and L. K. Michalis, Hydrodynamics of magnetic drug targeting, J. Biomech., 35 (2002) 813–821.

    Article  Google Scholar 

  13. V. A. Vardayan, Effect of magnetic field on blood flow, Biotizaka, 18 (1973) 494–496.

    Google Scholar 

  14. R. Bhargava, S. Rawat, H. S. Takhar and O. A. Beg, Pulsatile magneto-biofluid flow and mass transfer in a non-Darcian porous medium channel, Meccanica, 42 (2007) 247–262.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Haldar and S. N. Ghosh, Effect of a magnetic field on blood flow through an indented tube in the presence of erythrocytes, Indian J. Pure. Appli. Math., 25 (1994) 345–352.

    MATH  Google Scholar 

  16. Y. Haik, V. Pai and C. J. Chen, Apparent viscosity of human blood in a high static magnetic field, J. Magn. Magn. Mater., 225 (2001) 180–186.

    Article  Google Scholar 

  17. E. Amos and A. Ogulu, Magentic effect on pulsatile flow in a constricted axis-symmetric tube, Indian J. Pure Appl. Math., 34(9) (2003) 1315–1326.

    MATH  Google Scholar 

  18. E. E. Tzirtzilakis, A mathematical model for blood flow in magnetic field, Phys. Fluids, 17 (2005) 077103–077115.

    Article  MathSciNet  Google Scholar 

  19. R. Bali and U. Awasthi, Effect of a magnetic field on the resistance to blood flow through stenotic artery, Appl. Math. Computat., 188 (2007) 1635–1641.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. L. Siauw, E. Y. K. Ng and J. Mazumdar, Unsteady stenosis flow prediction: a comparitve study of non-Newtonian models with operator splitting scheme, Med. Eng. Phys., 22 (2000) 265–277.

    Article  Google Scholar 

  21. P. K. Mandal, S. Chakravarty, A. Mandal and N. Amin, Effect of body acceleration on unsteady generalized pulsatile flow of non-Newtonian fluid through a stenosed artery, Appl. Math. Computat., 189 (2007) 766–799.

    Article  MathSciNet  MATH  Google Scholar 

  22. D. S. Sankar and Usik Lee, Two-phase non-linear model for the flow through stenosed blood vessels, J. Mech. Sci. Tech., 21 (2007) 678–689.

    Article  Google Scholar 

  23. J. C. Misra, B. Pal and A. S. Gupta, Hydromagnetic flow of a second-grade fluid in a channel — Some applications to physiological Systems, Math. Models Methods Appl. Sci., 8 (1998) 1323–1342.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Bhargava, Sugandha, H. S. Takhar and O. A. Beg, Computational simulation of biomagnetic micropolar blood flow in porous media, J. Biomech., 39 (2006) S648–S649.

    Article  Google Scholar 

  25. O. A. Beg, H. S. Takhar, R. Bhargava, S. Sharma and T. K. Hung, Mathematical modeling of biomagnetic flow in a micropolar fluid-saturated Darcian porous medium, Int. J. Fluid Mech. Res., 34 (2007) 403–424.

    Article  Google Scholar 

  26. H. A. Attia and M. E. S. Ahmed, Unsteady hydromagnetic generalized Couette flow of a non-Newtonian fluid with heat transfer between parallel porous plates, Journal of heat transfer, 130 (2008) 114504-1–5.

    Article  Google Scholar 

  27. N. Casson, In: Rheology of disperse systems (Edited by Mill, C. C). Pergamon Press, London, 1959.

    Google Scholar 

  28. G. W. Scott Blair, An equation for the flow of blood, Plasma and Serum through Glass Capillaries, Nature, 183 (1959) 613.

    Article  Google Scholar 

  29. A. L. Copley, In: Flow properties of blood and other biological systems (Edited by Copley, A. L., Stainsby, G) Pergamon Press, Oxford, 1960.

    Google Scholar 

  30. E. W. Merrill, A. M. Benis, E. R. Gilliland, T. K. Sherwood, E. W. Salzman, Pressure flow relations of human blood in hollow fibers at low shear rates, J. Appl. Physiol., 20 (1965) 954.

    Google Scholar 

  31. S. E. Charm and G. Kurland, Viscometry of human blood for shear rates of 0–100,000 sec−1, Nature, 206 (1965) 617.

    Article  Google Scholar 

  32. G. W. Scott Blair and D. C. Spanner, An introduction to Biorheology, Elsevier, Amsterdam, Oxford and New York, 1974.

    Google Scholar 

  33. J. J. Chiu, D. L. Wang, S. Chien, R. Skalak and S. Usami S. Effects of disturbed flow on endothelial cells, J. Biomech. Eng., 120 (1998) 2–8.

    Article  Google Scholar 

  34. G. G. Galbraith, R. Skalak and S. Chien S, Shear stress induces spatial reorganization of the endothelial cell cytoskeleton, Cell Motility and the Cytoskeleton, 40 (1988) 317–330.

    Article  Google Scholar 

  35. T. Karino and H. L. Goldsmith, Flow behavior of blood cells and rigid spheres in annular vortex, Philosophical Transactions Royal Society, London 1977; B279: 413–445.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Penang, Malaysia

    D. S. Sankar

  2. Department of Mechanical Engineering, Inha University, 253, Yonghyun-Dong, Nam-Gu, Incheon, 402-751, Korea

    Usik Lee

Authors
  1. D. S. Sankar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Usik Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Usik Lee.

Additional information

This paper was recommended for publication in revised form by Associate Editor Do Hyung Lee

D. S. Sankar received his B. Sc degree in Mathematics from the University of Madras, India, in 1989. He then received his M. Sc., M. Phil. And Ph.D degrees from Anna University, India, in 1991, 1992 and 2004 respectively. Dr. D. S. Sankar is currently an Associate Professor in the School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia. His serves as a referee for several reputed ISI journals and his research interest includes fluid dynamics, hemodynamics, differential equations and numerical analysis.

Usik Lee received his B.S. degree in Mechanical Engineering from Yonsei University, Korea in 1979. He then received his M.S. and Ph.D degrees in Mechanical Engineering from Stanford University, USA in 1982 and 1985, respectively. Dr. Lee is currently a Professor at the Department of Mechanical Engineering at Inha University in Incheon, Korea. He has served as a referee for many reputed international journals. Dr. Lee’s research interests include structural dynamics, biomechanics, and computational mechanics.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sankar, D.S., Lee, U. FDM analysis for MHD flow of a non-Newtonian fluid for blood flow in stenosed arteries. J Mech Sci Technol 25, 2573–2581 (2011). https://doi.org/10.1007/s12206-011-0728-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-011-0728-x

Keywords

Search

Navigation