Skip to main content
SpringerLink
Log in

Unsteady nonisothermal tangent hyperbolic fluid flow in a stenosed blood vessel with pulsatile pressure gradient and body acceleration

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

This article presents a numerical solution to the time-dependent blood flow in a ω-shaped stenosed vessel influenced by the body acceleration and pulsatile pressure gradient. The tangent hyperbolic fluid model is used to describe the blood rheology. The model under consideration is one of the non-Newtonian models whose constitutive equation is valid at both high and low shear rates. It falls into the class of generalized Newtonian fluids and is capable of describing the shear-thinning behaviour very well. It is proven that this model is significantly sensitive to modest changes in zero shear-rate viscosity and mildly sensitive to fluctuations in infinite shear-rate viscosity. As an accurate predictor of shear-thinning, this model is well suited for the rheology of blood. Scale analysis is used to simplify the basic equations of fluid flow and heat transport for the moderate constriction model. The effects of the vessel’s wall are immobilized via radial coordinate transformation. The resultant nonlinear system of differential equations with specified boundary conditions is computed by utilizing an explicit finite difference approach. The numerical analysis of the dominant quantities such as resistive impendence, flow rate, wall shear stress, axial velocity, and temperature field is performed with variations in the relevant non-dimensional parameters. These findings are represented graphically and described briefly in the discussion. The flow rate and blood velocity rise with escalating the amplitude of body acceleration; however, the reverse response is observed when the stenosis height, Weissenberg number, and power-law exponent are increased. In addition, the Brinkman number and Prandtl number are found to have significant effects on the temperature field.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Data availability statement

The data that supports the findings of this study are available within the article.

References

  1. D.F. Young, Effect of time-dependent stenosis on flow through a tube. J. Eng. 90, 248–254 (1968)

    Google Scholar 

  2. J.H. Forrester, D.F. Young, Flow through a converging-diverging tube and its implications in occlusive vascular disease—I: theoretical development. J. Biomech. 3(3), 297–305 (1970)

    Article  Google Scholar 

  3. J.H. Forrester, D.F. Young, Flow through a converging-diverging tube and its implications in occlusive vascular disease—II: theoretical and experimental results and their implications. J. Biomech. 3(3), 307–316 (1970)

    Article  Google Scholar 

  4. J.S. Lee, Y.C. Fung, Flow in a locally constricted tube at low Reynolds number. J. Appl. Mech. 37(1), 9–16 (1970)

    Article  ADS  MATH  Google Scholar 

  5. J.C.F. Chow, K. Soda, Laminar flow and blood oxygenation in channels with boundary irregularities. J. Appl. Mech. 40, 843–850 (1973)

    Article  ADS  Google Scholar 

  6. B.E. Morgan, D.F. Young, An integral method for the analysis of flow in arterial stenosis. Bull. Math. Biol. 36, 39–53 (1974)

    MATH  Google Scholar 

  7. T. Azuma, T. Fukushima, Flow patterns in stenotic blood vessel models. Biorheology 13, 337–355 (1976)

    Article  Google Scholar 

  8. D.A. MacDonald, On steady flow through modeled vascular stenosis. J. Biomech. 12, 13–30 (1979)

    Article  Google Scholar 

  9. W. Youngchareon, D.F. Young, Initiation of turbulence in models of arterial stenosis. J. Biomech. 12, 185–196 (1979)

    Article  Google Scholar 

  10. J. Doffin, F. Chagneau, Oscillating flow between a clot model and stenosis. J. Biomech. 14, 143–148 (1981)

    Article  Google Scholar 

  11. J.C. Mishra, B.K. Sahu, Flow through blood vessels under the action of a periodic acceleration field: a mathematical analysis. Comput. Math. Appl. 16, 993–1016 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Perktold, R. Peter, M. Resch, Pulsatile non-Newtonian blood flow simulation through a bifurcation with an aneurism. Biorheology 26, 1011–1030 (1989)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. R. Usha, K. Prema, Pulsatile flow of particle-fluid suspension model of blood under periodic body acceleration. ZAMP 50, 175–192 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  15. M. El-Shahed, Pulsatile flow of blood through a stenosed porous medium under periodic body acceleration. Appl. Math. Comput. 138, 479–488 (2003)

    MathSciNet  MATH  Google Scholar 

  16. P.K. Mandal, An unsteady of non-Newtonian blood flow through tapered arteries with stenosis. Int. J. Nonlinear Mech. 40, 151–164 (2005)

    Article  ADS  MATH  Google Scholar 

  17. F. Yilmaz, M.Y. Gundogdu, A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions. J. Korea-Aust. Rheol. 20, 197–211 (2008)

    Google Scholar 

  18. Kh.S. Mekheimer, M.A.E.I. Kot, Mathematical modeling of unsteady flow of Sisko fluid through anisotropically tapered elastic arteries with time-variant overlapping stenosis. Appl. Math. Model. 36, 5393–5407 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. D.S. Sankar, U. Lee, Mathematical modeling of the pulsatile flow of non-Newtonian fluid in stenosed arteries. Commun. Nonlinear Sci. Num. Simul. 14, 2971–2981 (2009)

    Article  MATH  Google Scholar 

  20. N. Ali, A. Zaman, M. Sajid, J.J. Nieto, A. Torres, Unsteady non-Newtonian blood flow through a tapered overlapping stenosed catheterized vessel. Math. Biosci. 269, 94–103 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Zaman, N. Ali, M. Sajid, Slip effects on unsteady non-Newtonian blood flow through an inclined catheterized overlapping stenotic artery. AIP Adv. (2016). https://doi.org/10.1063/1.4941358

    Article  Google Scholar 

  22. M. Roy, B.S. Sikarwar, M. Bhandwal, P. Ranjan, Modelling of blood flow in stenosed arteries. Procedia Comput. Sci. 115, 821–830 (2017)

    Article  Google Scholar 

  23. S. Charm, G. Kurland, Viscometry of human blood for shear rates of 0–100,000 sec-1. Nature 206(4984), 617–618 (1965)

    Article  ADS  Google Scholar 

  24. C.R. Huang, N. Siskovic, R.W. Robertson, W. Fabisiak, E.H. Smitherberg, A.L. Copley, Quantitative characterization of thixotropy of whole human blood. Biorheology 12(5), 279–282 (1975)

    Article  Google Scholar 

  25. G.B. Thurston, Viscoelasticity of human blood. Biophys. J. 12(9), 1205–1217 (1972)

    Article  ADS  Google Scholar 

  26. Gijsen, J.H. Frank, F.N. van de Vosse, J.D. Janssen, The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model. J. Biomech. 32(6), 601–608 (1999)

    Article  Google Scholar 

  27. F. Irgens, Rheology and Non-newtonian Fluids (Springer, New York, 2014), p. 190

    Book  MATH  Google Scholar 

  28. J.V. Soulis, G.D. Giannoglou, Y.S. Chatzizisis, K.V. Seralidou, G.E. Parcharidis, G.E. Louridas, Non-Newtonian models for molecular viscosity and wall shear stress in a 3D reconstructed human left coronary artery. Med. Eng. Phys. 30(1), 9–19 (2008)

    Article  Google Scholar 

  29. S. Karimi, M. Dabagh, P. Vasava, M. Dadvar, B. Dabir, P. Jalali, Effect of rheological models on the hemodynamics within human aorta: CFD study on CT image-based geometry. J. Nonnewton. Fluid Mech. 207, 42–52 (2014)

    Article  Google Scholar 

  30. M. Iasiello, K. Vafai, A. Andreozzi, N. Bianco, Analysis of non-Newtonian effects within an aorta-iliac bifurcation region. J. Biomech. 64, 153–163 (2017)

    Article  Google Scholar 

  31. A.B. Caballero, S. Lain, Numerical simulation of non-Newtonian blood flow dynamics in human thoracic aorta. Comput. Methods Biomech. Biomed. Eng. 18(11), 1200–1216 (2015)

    Article  Google Scholar 

  32. J. Moradicheghamahi, J. Sadeghiseraji, M. Jahangiri, Numerical solution of the pulsatile: non-Newtonian and turbulent blood ow in a patient specific elastic carotid artery. Int. J. Mech. Sci. 150, 393–403 (2019)

    Article  Google Scholar 

  33. S. O’Callaghan, M. Walsh, T. McGloughlin, Numerical modelling of Newtonian and non-Newtonian representation of blood in a distal end-to-side vascular bypass graft anastomosis. Med. Eng. Phys. 28(1), 70–74 (2006)

    Article  Google Scholar 

  34. M. Abbasian, M. Shams, Z. Valizadeh, A. Moshfegh, A. Javadzadegan, S. Cheng, Effects of different non-Newtonian models on unsteady blood flow hemodynamics in patient-specific arterial models with in-vivo validation. Comput. Methods Programs Biomed. 186, 105185 (2020)

    Article  Google Scholar 

  35. I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media (Pergamon, Amsterdam, NewYork, 2001)

    Google Scholar 

  36. S. Nadeem, S. Akram, Peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. ZNA 64a, 559–567 (2009)

    ADS  Google Scholar 

  37. S. Nadeem, S. Ijaz, Theoretical analysis of shear thinning hyperbolic tangent fluid model for blood flow in curved artery with stenosis. J. Appl. Fluid Mech. 9(5), 2217–2227 (2016)

    Google Scholar 

  38. S. Jyothi, M.S. Reddy, P. Gangavathi, Hyperbolic tangent fluid flow through a porous medium in an inclined channel with peristalsis. Int. J. Adv. Sci. Res. Manag. 1(4), 113–121 (2016)

    Google Scholar 

  39. M. Naseer, M.Y. Malik, S. Nadeem, A. Rehman, The boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder. Alex. Eng. J. 53, 747–750 (2014)

    Article  Google Scholar 

  40. S. Nadeem, H. Sadaf, N.S. Akbar, Effects of nanoparticles on the peristaltic motion of tangent hyperbolic fluid model in an annulus. Alex. Eng. J. 54(4), 843–851 (2015)

    Article  Google Scholar 

  41. M.A. Abbas, Y.Q. Bai, M.M. Bhatti, M.M. Rashidi, Three-dimensional peristaltic flow of hyperbolic tangent fluid in a non-uniform channel having flexible walls. Alex. Eng. J. 55(1), 653–662 (2016)

    Article  Google Scholar 

  42. T. Hayat, T. Abbas, M. Ayub, M. Farooq, A. Alsaedi, Flow of nanofluid due to convectively heated Riga plate with variable thickness. J. Mol. Liq. 222, 854–862 (2016)

    Article  Google Scholar 

  43. S. Charm, B. Paltiel, G.S. Kurland, Heat transfer coefficients in blood flow. Biorheology 5(2), 133–145 (1968)

    Article  Google Scholar 

  44. J.C. Chato, Heat transfer to blood vessels. J. Biomech. Eng. 102(2), 110–118 (1980)

    Article  Google Scholar 

  45. M.C. Kolios, M.D. Sherar, J.W. Hunt, Large blood vessel cooling in heated tissues: a numerical study. Phys. Med. Biol. 40, 477–494 (1995)

    Article  Google Scholar 

  46. A. Ogulu, M.A. Tamunoimi, Simulation of heat transfer on an oscillatory blood flow in an indented porous artery. Int. Commun. Heat Mass Transf. 32(7), 983–989 (2005)

    Article  Google Scholar 

  47. A.E. Garcia, D.N. Riahi, Two-phase blood flow and heat transfer in an inclined stenosed artery with or without a catheter. Int. J. Fluid Mech. Res. 41(1), 16–30 (2014)

    Article  Google Scholar 

  48. A. Zaman, N. Ali, O.A. Bég, Heat and mass transfer to blood through a tapered overlapping stenosed artery. Int. J. Heat Mass Transf. 95, 1084–1095 (2016)

    Article  Google Scholar 

  49. M.S.A. Jamali, Z. Ismail, Simulation of heat transfer on blood flow through a stenosed bifurcated artery. J. Adv. Res. Fluid Mech. Therm. Sci. 60(2), 310–323 (2019)

    Google Scholar 

  50. Y. Liu, W. Liu, Blood flow analysis in tapered stenosed arteries with the influence of heat and mass transfer. J. Appl. Math. Comput. 63(1), 523–541 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  51. M. Fahim, M. Sajid, N. Ali, Non-isothermal flow of Sisko fluid in a stenotic tube induced via pulsatile pressure gradient and periodic body acceleration. Phys. Scr. 96(8), 085211 (2021)

    Article  ADS  Google Scholar 

  52. M. Thiriet, Biology and Mechanics of Blood Flows: Part II: Mechanics and Medical Aspects (Springer, Berlin, 2007)

    MATH  Google Scholar 

  53. F. Yilmaz, M.Y. Gundogdu, A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions. Korea-Austr. Rheol. J. 20(4), 197–211 (2008)

    Google Scholar 

  54. P.K. Kundu, I.M. Cohen, D.R. Dowling, Fluid Mechanics (Academic Press, Cambridge, 2015)

    Google Scholar 

  55. C. Vlachopoulos, M. O’Rourke, W.W. Nichols, McDonald’s Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles (CRC Press, Cambridge, 2011)

    Book  Google Scholar 

  56. G.S. Malindzak, Fourier analysis of cardiovascular events. Math. Biosci. 7(3), 273–289 (1970)

    Article  Google Scholar 

  57. J. Alastruey, K. H. Parker, and S. J. Sherwin, Arterial pulse wave haemodynamics. in 11th international conference on pressure surges. Virtual PiE Led t/a BHR Group, (2012) 401–443.

Download references

Acknowledgements

The valuable feedback from the reviewers and editor is highly appreciated. The authors would like to express their gratitude to HEC Pakistan for the financial assistance granted under Project No: 5785/ Federal/NRPU/ R&D/HEC/2017.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    M. Fahim, M. Sajid, N. Ali & M. N. Sadiq

Authors
  1. M. Fahim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Sajid
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. N. Sadiq
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Fahim.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fahim, M., Sajid, M., Ali, N. et al. Unsteady nonisothermal tangent hyperbolic fluid flow in a stenosed blood vessel with pulsatile pressure gradient and body acceleration. Eur. Phys. J. Plus 137, 847 (2022). https://doi.org/10.1140/epjp/s13360-022-03035-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-022-03035-5

Search

Navigation