Skip to main content
SpringerLink
Log in

Analysis of the Flow Dynamics of Blood Through Viscoelastic Constricted Artery

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we focused on the flow through an axisymmetric constricted artery of the pulmonary region to study the condition of stenosis. Theory of dust particles suspended in gas is applied on blood flow through the artery, where the “particles” represent “cells” suspended in plasma. The flow is governed by two dimensional Navier–Stokes’ equations by including Darcy–Forchheimer drag force caused by non-Darcian effect. The material of the artery is approximated as a linear elastic and simplest rheological equation that includes viscosity and elasticity (considered lung as a Voigt body) is used. Effect of various parameters, such as Reynolds number (Re), Forchheimer number (\(F_s\)), Darcy number (Da), aspect ratio (\(\beta \)), shape factor (\(S_f\)), porosity (\(\epsilon \)), aerodynamic diameter (\(d_p\)), bulk compression (\(\phi \)) of elasticity, shear (\(\eta \)) and bulk (\(\zeta \)) coefficients of parenchymal viscosity are obtained on the radial and axial velocities of blood and particles graphically. We found that the fluid (blood) and particle (cells suspended in plasma) velocities along both the axes (axial and radial) increase by increasing Reynolds number, the pulsating amplitude, aspect ratio, and porosity of walls. While by increasing Forchheimer number, velocities of blood and particles in both the axes decreases gradually. The present analysis is also indicate that the viscoelasticity of walls are affected by the amplitude of pulsatile flow of blood and for a large value of amplitude, the viscoelastic effect decreases.

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

Similar content being viewed by others

References

  1. Al-Mayah, A., Moseley, J., Brock, K.K.: Contact surface and material nonlinearity modeling of human lungs. Phys. Med. Biol. 53(1), 305–317 (2008)

    Article  Google Scholar 

  2. Alimohamadi, H., Imani, M.: Finite element simulation of two-dimensional pulsatile blood flow through a stenosed artery in the presence of external magnetic field. Int. J. Comput. Methods Eng. Sci. Mech. 15(4), 390–400 (2014)

    Article  MathSciNet  Google Scholar 

  3. Chandran, K.B., Swanson, W.M., Ghista, D.N., Vayo, H.W.: Oscillatory flow in thin-walled curved elastic tubes. Ann. Biomed. Eng. 2, 392–412 (1974)

    Article  Google Scholar 

  4. Darcy, H.: Les fontaines publiques de la volle de dijon. Dalmont, V, Lyon Public Library (1856)

  5. Dyachenko, A.I., Lyubimov, G.A.: Propagation of sound in pulmonary parenchyma. Plenum Publ. Corp. N Y Washington 23, 641–652 (1988)

    MathSciNet  MATH  Google Scholar 

  6. Dyachenko, A.I., Manyuhina, O.V.: Modeling of weak blast wave propagation in the lung. J. Biomech. 39(11), 2113–2122 (2006)

    Article  Google Scholar 

  7. Eom, J., Xu, X.G., De, S., Shi, C.: Predictive modeling of lung motion over the entire respiratory cycle using measured pressure-volume data, 4dct images, and finite-element analysis. Med. Phys. 37(8), 4389–4400 (2010)

    Article  Google Scholar 

  8. Fuchs, N.A.: The mechanics of aerosols. American Association for the Advancement of Science (1994)

  9. Fulford, G.R., Blake, J.R.: Muco-ciliary transport in the lung. J. Theor. Biol. 121(4), 381–402 (1986)

    Article  Google Scholar 

  10. Fung, Y.C.: A theory of elasticity of the lung. J. Appl. Mech. 41(1), 8–14 (1974)

    Article  Google Scholar 

  11. Hinds, W.C.: Aerosol Technology: Properties, Behavior, and Measurement of Airborne Particles, 2nd edn. Wiley, London (1999)

    Google Scholar 

  12. Ilegbusi, O.J., Seyfi, B., Salvin, R.: Patient-specific model of lung deformation using spatially dependent constitutive parameters. Math. Comput. Model. Dyn. Syst. 20(6), 546–556 (2014)

    Article  MathSciNet  Google Scholar 

  13. Jahed, M., Lai-Fook, S.J., Bhagat, P.K., Kraman, S.S.: Propagation of stress waves in inflated sheep lungs. J. Appl. Physiol.: Respir. Environ. Exercise Physiol. 66(6), 2675–2680 (1989)

    Article  Google Scholar 

  14. Khaled, A.R.A., Vafai, K.: The role of porous media inmodeling flowand heat transfer in biological tissues. Int. J. Heat Mass Transf. 46(26), 4989–5003 (2003)

    Article  Google Scholar 

  15. Khanafer, K., Cook, K., Marafie, A.: The role of porous media in modeling fluid flow within hollow fiber membranes of the total artificial lung. J. Porous Media 15(2), 113–122 (2012)

    Article  Google Scholar 

  16. Kumar, B.V.R., Naidu, K.B.: A pulsatile suspension flow simulation in a stenosed vessel. Math. Comput. Model. 23(5), 75–86 (1996)

    Article  MathSciNet  Google Scholar 

  17. Lee, T.S., Liu, X., Li, G.C., Low, H.T.: Numerical study on sinusoidal fluctuated pulsatile laminar flow through various constrictions. Commun. Comput. Phys. 2(1), 99–122 (2007)

    MathSciNet  Google Scholar 

  18. Liu, B., Tang, D.: Computer simulations of atherosclerotic plaque growth in coronary arteries. Mol. Cell. Biomech. 7(4), 193–202 (2010)

    Google Scholar 

  19. Mehmood, O.U., Mustapha, N., Shafie, S.: Unsteady two-dimensional blood flow in porous artery, with multi-irregular stenoses. Transp. Porous Media 92(2), 259–275 (2012)

    Article  MathSciNet  Google Scholar 

  20. Murphy, M.J., Pokhrel, D.: Optimization of an adaptive neural network to predict breathing. Med. Phys. 36(1), 40–47 (2009)

    Article  Google Scholar 

  21. Ozkaya, N., Nordin, M., Goldsheyder, D., Leger, D.: Fundamentals of Biomechanics: Equilibrium, Motion, and Deformation, 3rd edn. Springer, New York (2012)

    Book  Google Scholar 

  22. Ponalagusamy, R.: A two-layered suspension (particle-fluid) model for non-newtonian fluid flow in a catheterized arterial stenosis with slip condition at the wall of stenosed artery. Korea-Australia Rheol. J. 29(2), 87–100 (2017)

    Article  MathSciNet  Google Scholar 

  23. Preziosi, L., Farina, A.: On darcy’s law for growing porous media. Int. J. Non-Linear Mech. 37(3), 485–491 (2002)

    Article  Google Scholar 

  24. Reddy, J.V.R., Srikanth, D., Mandal, P.K.: Computational hemodynamic analysis of flow through flexible permeable stenotic tapered artery. Int. J. Appl. Comput. Math. 3(1), 1261–1287 (2017)

    Article  MathSciNet  Google Scholar 

  25. Saffman, P.G.: On the stability of laminar flow of a dusty gas. J. Fluid Mech. 13(1), 120–128 (1962)

    Article  MathSciNet  Google Scholar 

  26. Saini, A., Katiyar, V.K., Parida, M.: Two dimensional model of pulsatile flow of a dusty fluid through a tube with axisymmetric constriction. World J. Model. Simul. 12(1), 70–78 (2016)

    Google Scholar 

  27. Sankar, M., Do, Y.: Numerical simulation of free convection heat transfer in a vertical annular cavity with discrete heating. Int. Commun. Heat Mass Transf. 37(6), 600–606 (2010)

    Article  Google Scholar 

  28. Sankar, M., Kim, Beomseok, Lopez, J.M., Do, Younghae: Thermosolutal convection from a discrete heat and solute source in a vertical porous annulus. Int. J. Heat Mass Transf. 55, 4116–4128 (2012)

    Article  Google Scholar 

  29. Sankar, M., Kiran, S., Ramesh, G.K., Makinde, O.D.: Natural convection in a non-uniformly heated vertical annular cavity. Defect Diffus. Forum 377, 189–199 (2017)

    Article  Google Scholar 

  30. Sankar, M., Park, J., Do, Y.: Natural convection in a vertical annuli with discrete heat sources. Numer. Heat Transf. Part A: Appl. 59(8), 594–615 (2011)

    Article  Google Scholar 

  31. Sankar, M., Park, Youngyong, Lopez, J.M., Do, Younghae: Numerical study of natural convection in a vertical porous annulus with discrete heating. Int. J. Heat Mass Transf. 54, 1493–1505 (2011)

    Article  Google Scholar 

  32. Smith, G.D.: Numerical Solution of Partial Differential Equations, 3rd edn. Oxford University Press, USA (1986)

    Google Scholar 

  33. Sorek, S., Sideman, S.: A porous medium approach for modelling heart mechanics. Math. Biosci. 81(1), 1–14 (1986)

    Article  MathSciNet  Google Scholar 

  34. Sturm, R., Hofmann, W.: A theoretical approach to the deposition and clearance of fibers with variable size in the human respiratory tract. J. Hazard. Mater. 170(1), 210–218 (2009)

    Article  Google Scholar 

  35. Vankan, W.J., Huyghe, J.M., Janssen, J.D., Huson, A., Hacking, W.J.G., Schrenner, W.: Finite element analysis of blood flow through biological tissue. Int. J. Eng. Sci. 35(4), 375–385 (1997)

    Article  Google Scholar 

  36. Weibel, E.R.: Morphometry of the Human Lung, chapter Introduction, pp. 1–4 (1963)

  37. Wong, K.K.L., Tu, J., Mazumdar, J., Abbott, D.: Modelling of blood flow resistance for an atherosclerotic artery with multiple stenoses and poststenotic dilatations. ANZIAM J. 51, 66–82 (2010)

    Article  MathSciNet  Google Scholar 

  38. Zendehbudi, G.R., Moayeri, M.S.: Comparison of physiological and simple pulsatile flows through stenosed arteries. J. Biomech. 32(5), 959–965 (1999)

    Article  Google Scholar 

Download references

Acknowledgements

The author, Jyoti Kori, is thankful to Ministry of Human Resource Development (Grant Code:- MHR-02-23-200-44) India for providing fund and support while writing this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    Jyoti Kori &  Pratibha

Authors
  1. Jyoti Kori
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pratibha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jyoti Kori.

Ethics declarations

Conflict of interest

No conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kori, J., Pratibha Analysis of the Flow Dynamics of Blood Through Viscoelastic Constricted Artery. Int. J. Appl. Comput. Math 6, 41 (2020). https://doi.org/10.1007/s40819-020-0796-7

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-020-0796-7

Keywords

Search

Navigation