Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 1501–1514 | Cite as

A Theoretical Analysis of Thixotropic Parameter’s Influence on Blood Flow Through Constriction

  • Nazish ShahidEmail author
Research Article - Mechanical Engineering


This study has been prepared to investigate the changes in the dynamics of blood flow through a stenosed tapered artery owing to a change in structural parameter \(\lambda \) of thixotropic model. Following the time evolution range of this parameter as [0,1] for transient shear flows, the effects of \(\lambda \) on axial velocity, shear stress, flow rate and resistance to flow have been probed. Analytical expressions of axial velocity and shear stress have been obtained along with numerical computation of pressure gradient by means of continuity equation. The evolution of system with respect to time, t has been investigated in order to study the changes in dynamics of flow at certain times. A comparison of axial velocity profiles for some values of \(\lambda \) has been made to obtain profiles for Power law fluid and Newtonian fluid model. The inclination of velocity profiles for \(0.5\le \lambda \le 1\) towards experimental velocity profiles has been suggested by means of comparison with available results in history. This analysis has also been prepared as a foundational step of construction of an artificial channel with constriction and of adaptation of most suitable modelling of blood flow such that the findings of the parameter \(\lambda \) and its influence on flow can be incorporated experimentally for induction of decreased wall stress.


Thixotropy Blood flow Time dependence Constricted channel Axial velocity Yield stress 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author thanks Dr.Howard Stone from Department of Mechanical and Aerospace Engineering, Princeton University for his insightful discussion on topic and Princeton University for providing a wonderful working atmosphere for completion of this project.


  1. 1.
    Young, D.F.: Effect of a time dependent stenosis of flow through a tube. J. Eng. Ind. 90, 248–254 (1968)CrossRefGoogle Scholar
  2. 2.
    Young, D.F.: Fluid mechanics of arterial stenoses. J. Biomech. Eng. 101(3), 157–175 (1979)CrossRefGoogle Scholar
  3. 3.
    Young, D.F.; Tsai, F.Y.: Flow characteristics in models of arterial stenoses: I. Steady flow. J. Biomech. 6(4), 395–410 (1973)CrossRefGoogle Scholar
  4. 4.
    Shahed, M-El: Pulsatile flow of blood through a stenosed porous medium under periodic body acceleration. Appl. Math. Comput. 138(2–3), 479–488 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Elshehawey, E.F.; Elbarbary, E.M.; Affifi, N.A.S.; Shahed, M-El: Pulsatile flow of blood through a porous medium under periodic body acceleration. Int. J. Theor. Phys. 39(1), 183–188 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sharma, M.K.; Bansal, K.; Bansal, S.: Pulsatile unsteady flow of blood through porous medium in a stenotic artery under the influence of transverse magnetic field. Korea Aust. Rheol. J. 24(3), 181–189 (2012)CrossRefGoogle Scholar
  7. 7.
    Tu, C.; Deville, M.; Dheur, L.; Vandershuren, L.: Finite element simulation of pulsatile flow through arterial stenosis. J. Biomech. 25, 1141–1152 (1992)CrossRefGoogle Scholar
  8. 8.
    Nerem, R.E.: Vascular fluid mechanics, the arterial wall and arteriosclerosis. J. Biomech. Eng. Trans. ASME 114, 274–282 (1992)CrossRefGoogle Scholar
  9. 9.
    Cavalcanti, S.: Hemodynamics of an artery with mild stenosis. J. Biomech. 28, 387–399 (1995)CrossRefGoogle Scholar
  10. 10.
    Siouffi, M.; Deplano, V.; Pelissra, R.: Experimental analysis of unsteady flows through a stenosis. J. Biomech. 31, 11–19 (1997)CrossRefGoogle Scholar
  11. 11.
    Zendehboodi, G.R.; Moayeri, M.S.: Comparison of physiological and simple pulsatile flows through stenosed arteries. J. Biomech. 32, 959–965 (1999)CrossRefGoogle Scholar
  12. 12.
    Chakravarty, S.; Mandal, P.K.: Two-dimensional blood flow through tapered arteries under stenotic conditions Int. J. Nonlinear Mech. 35, 779–793 (2000)CrossRefzbMATHGoogle Scholar
  13. 13.
    Long, Q.; Ku, X.Y.; Ramnarine, K.V.; Hoskins, P.: Numerical investigation of physiologically realistic pulsatile flow through arterial stenosis. J. Biomech. 34, 1229–1242 (2001)CrossRefGoogle Scholar
  14. 14.
    Smith, F.T.: The separation flow through a severely constricted symmetric tube. J. Fluid Mech. 90, 725–754 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Deshpande, M.D.; Giddens, P.D.; Mabon, F.R.: Steady laminar flow through modelled vascular stenoses. J. Biomech. 9, 165–174 (1976)CrossRefGoogle Scholar
  16. 16.
    Mandal, P.K.; Chakravarty, S.; Mandal, A.; Amin, N.: Effect of body acceleration on unsteady pulsatile flow of non-Newtonian fluid through a stenosed artery. Appl. Math. Comput. 189(1), 766–779 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Mansour, R.B.; Badr, H.; Shaik, A.Q.; Maalej, N.: Modeling of pulsatile blood flow in an axisymmetric tube with a moving indentation. Arab. J. Sci. Eng. 33(1), 5 (2008)Google Scholar
  18. 18.
    Das, K.: A mathematical model on the consistency coefficient of the Herschel–Bulkley flow of blood through narrow vessel arab. J. Sci. Eng. 36, 405 (2011). MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ponalagusamy, P.: Mathematical analysis on effect of non-Newtonian behavior of blood on optimal geometry of microvascular bifurcation system. J. Frankl. Inst. 349(9), 2861–2874 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ponalagusamy, P.: Pulsatile flow of Herschel–Bulkley fluid in tapered blood vessels. In: Proceedings of the International Conference Science Computing, and World Congress in Computer Science, Computer Engineering and Applied Computing, pp 67–73 (2013)Google Scholar
  21. 21.
    Liepsch, D.; Moravec, S.T.: Pulsatile flow of non-Newtonian fluid in distensible models of human arteries. Biorheology 21, 571–586 (1984)CrossRefGoogle Scholar
  22. 22.
    Chakravarty, S.: Effects of stenosis on the flow behaviour of blood in an artery Int. J. Eng. Sci. 25, 1003–1016 (1987)CrossRefzbMATHGoogle Scholar
  23. 23.
    Nakamura, M.; Swada, T.: Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis. J. Biomech. Eng. Trans. ASME 110, 137–143 (1988)CrossRefGoogle Scholar
  24. 24.
    Nakamura, M.; Swada, T.: Numerical study on the unsteady flow of non-Newtonian fluid. J. Biomech. Eng. Trans. ASME 112, 100–103 (1990)CrossRefGoogle Scholar
  25. 25.
    Pak, B.; Young, Y.I.; Choi, S.U.S.: Separation and re-attachment of non-Newtonian fluid flows in a sudden expansion pipe. J. Non-Newton. Fluid Mech. 37, 175–199 (1990)CrossRefGoogle Scholar
  26. 26.
    Misra, J.C.; Patra, M.K.; Misra, S.C.: A non-Newtonian fluid model for blood flow through arteries under stenotic conditions. J. Biomech. 26, 1129–1141 (1993)CrossRefGoogle Scholar
  27. 27.
    Tu, C.; Deville, M.: Pulsatile flow of non-Newtonian fluid through arterial stenosis. J. Biomech. 29, 899–908 (1996)CrossRefGoogle Scholar
  28. 28.
    Das, B.; Johnson, P.C.; Popel, A.S.: Effect of non axisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes. Biorheology 35, 69–87 (1998)CrossRefGoogle Scholar
  29. 29.
    Bureau, M.; Healy, J.C.; Bourgoin, D.; Joly, M.: Etude rheologique en regime transitoire de quelques echantillons de sangs humains artificiellement modifies. Rheol. Acta 18, 756–768 (1979)CrossRefGoogle Scholar
  30. 30.
    Bureau, M.; Healy, J.C.; Bourgoin, D.; Joly, M.: Rheological hysteresis of blood at low shear rate. Biorheology 17, 191–203 (1980)CrossRefGoogle Scholar
  31. 31.
    Dintenfass, L.: Thixotropy of blood and proneness to thrombus formation. Circ. Res. 11, 233–239 (1962)CrossRefGoogle Scholar
  32. 32.
    Cokelet, G.R.; Merrill, E.W.; Gilliland, E.R.; Shin, H.; Britten, A.; Wells, E.R.: The rheology of human blood measurement near and at zero shear rate. Trans. Soc. Rheol. 7, 303–317 (1963)CrossRefGoogle Scholar
  33. 33.
    Chien, S.; Usami, S.; Taylor, H.M.; Lundberg, J.L.; Gregerse, M.I.: Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. J. Appl. Physiol. 21, 81–87 (1966)CrossRefGoogle Scholar
  34. 34.
    Merrill, E.W.; Benis, A.M.; Gillian, E.R.; Sherwood, T.K.; Salzman, E.W.: Pressure-flow relations of human blood in hollow fibers at low rates. J. Appl. Physiol. 20, 954–967 (1965)CrossRefGoogle Scholar
  35. 35.
    Merrill, E.W.; Pelletier, G.A.: Viscosity of human blood: transition from Newtonian to non-Newtonian. J. Appl. Physiol. 23, 178–182 (1967)CrossRefGoogle Scholar
  36. 36.
    Merrill, E.W.: Rheology of blood. Physiol. Rev. 49, 863–888 (1969)CrossRefGoogle Scholar
  37. 37.
    Thurston, G.B.: Elastic effects in pulsatile blood-flow. Microvasc. Res. 9, 145–157 (1975)CrossRefGoogle Scholar
  38. 38.
    Thurston, G.B.: Effects of frequency of oscillatory flow on impedance of rigid, blood-filled tubes. Biorheology 13, 191–199 (1976)CrossRefGoogle Scholar
  39. 39.
    Owens, R.G.: A new micro structure-based constitutive model for human blood. J. Non Newton. Fluid Mech. 140, 57–70 (2006)CrossRefzbMATHGoogle Scholar
  40. 40.
    Fang, J.N.; Owens, R.G.: Numerical simulations of pulsatile blood flow using a new constitutive model. Biorheology 43, 637–660 (2006)Google Scholar
  41. 41.
    Moyers-Gonzalez, M.; Owens, R.G.; Fang, J.N.: A non-homogeneous constitutive model for human blood. Part 1. Model derivation and steady flow. J. Fluid Mech. 617, 327–354 (2008a)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Anand, M.J.; Rajagopal, K.: A shear-thinning viscoelastic fluid model for describing the flow of blood Int. J. Cardiovasc. Med. Sci. 4, 59–68 (2004)Google Scholar
  43. 43.
    Anand, M.J.; Kwack, J.; Masood, A.: A new generalized Oldroyd-B model for blood flow in complex geometries Int. J. Eng. Sci. 72, 78–88 (2013)CrossRefzbMATHGoogle Scholar
  44. 44.
    Pincombe, B.; Mazumdar, J.; Hamilton-Craig, I.: Effects of multiple stenoses and post-stenotic dilatation on non-Newtonian blood flow in small arteries. Med. Biol. Eng. Comput. 37(5), 595–599 (1999)CrossRefGoogle Scholar
  45. 45.
    Scott Blair, G.W.; Spanner, D.C.: An Introduction to Biorheology. Elsevier Scientific Publishing, Amsterdam (1974)Google Scholar
  46. 46.
    Priyadharshini, S.; Ponalagusamy, R.: Biorheological model on flow of Herschel–Bulkley fluid through a tapered arterial stenosis with dilatation. Appl. Bionic Biomech. (2015).
  47. 47.
    Whitemore, R.L.: Rheology of Circulation. Pergamon Press, Oxford (1968)Google Scholar
  48. 48.
    Manton, M.J.: Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451–459 (1971)CrossRefzbMATHGoogle Scholar
  49. 49.
    Mandal, P.K.: An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis Int. J. Non Linear Mech. 40(1), 151–164 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Mujumdar, A.; Anthony, N.B.; Metzner, A.B.: Transient phenomena in thixotropic systems. J. Non Newton. Fluid Mech. 102, 157–178 (2002)CrossRefzbMATHGoogle Scholar
  51. 51.
    Dullaert, K.; Mewis, J.: A structural kinetics model for thixotropy. J. Non Newton. Fluid Mech. 139, 21–30 (2006)CrossRefzbMATHGoogle Scholar
  52. 52.
    Mewis, J.; Wagner, N.J.: Colloidal Suspension Rheology. Cambridge University, Cambridge (2012)zbMATHGoogle Scholar
  53. 53.
    Apostolidis, A.J.; Anthony, N.B.: Modelling of the blood rheology in steady-state shear flows. J. Rheol. 58, 607–633 (2014)CrossRefGoogle Scholar
  54. 54.
    Mewis, J.: Thixotropy: a general review. J. Non Newton. Fluid Mech. 6, 1–20 (1979)CrossRefzbMATHGoogle Scholar
  55. 55.
    Bird, R.B.; Armstrong, R.C.; Hassager, O.: Dynamics of Polymeric Liquids 1 Fluid Mech. Wiley, New York (1977)Google Scholar
  56. 56.
    Apostolidis, A.J.; Armstrong, M.J.; Anthony, N.B.: Modelling of human blood rheology in transient shear flows. J. Rheol. 59, 275–298 (2015)CrossRefGoogle Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Forman Christian CollegeA Chartered UniversityLahorePakistan
  2. 2.Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations