Study of the non-Newtonian behaviour of Reiner Rivlin relative to power law in arterial stenosis

Document Type : Research Paper


Department of Mathematics, School of Physical Sciences, Doon University, Dehradun, India


The present paper develops the solution of steady axi-symmetric Navier-Stokes conservation equations incorporating Reiner Rivlin stress and strain rate relation that represents generalized non-Newtonian fluid. Perturbation solution is obtained to determine the flow field for axially symmetric stenosed artery. The flow field obtained from the Perturbation solution is compared with the exact analytical solution. In perturbation solution, cross viscosity that represents non Newtonian characteristics is considered a perturbation parameter, and the result obtained is observed to be dependent on the perturbation parameter. At smaller values of cross viscosity, the perturbation result is significantly closer to the analytical solution. But, as the values of cross viscosity increase, the perturbation results show a wider deviation from analytical results. Further, in this paper, the results of Reiner Rivlin are compared with the results obtained from the Power Law stress and strain rate relation. Such comparison of results of Reiner Rivlin with Power law is utilized to study the flow characteristics of blood. The flow profile in the case of Reiner Rivlin is observed to be significantly closer to that of Power law. The study infers that Reiner Rivlin’s constitutive relation is fairly suitable in simulating blood flow in arterial stenosis.


  • [1]          Z. Abbas, M. S. Shabbir, and N. Ali, Analysis of rheological properties of Herschel-Bulkley fluid for pulsating flow of blood in ω-shaped stenosed artery, AIP Advances, 7 (2017), 1051123-1-12.
  • [2]          R. Ahmed, A. Farooki, R. Farooki, N. N. Hamadneh, M. F. Asad, I. Khan, M. Sajid, G. Bary, and M. F. S. Khan, An  Analytical Approach to Study the Blood Flow over a Nonlinear Tapering Stenosed Artery in Flow of Carreau    Fluid Model, Hindawi Complexity, Article ID 9921642, (2021).
  • [3]          N. S. Akbar, S. Nadeem, and Kh. S. Mekheimer, Rheological properties of Reiner-Rivlin fluid model for blood flow through tapered artery with stenosis, Journal of the Egyptian Mathematical Society, 24 (2016), 138–142.
  • [4]          N. S. Akbar and S. Nadeem, Carreau fluid model for blood flow through  a  tapered  artery  with  a  stenosis,  Ain  Shams Engineering Journal, 5 (2014), 1307–1316.
  • [5]          S. Akhtar, L. B. Mc Cash, N. Sohail, S. Saleem, and A. Issakhov, Mechanics of non-Newtonian blood flow in an artery having multiple stenosis and electroosmotic effects, Science Progress, 104(3) (2021), 1–15.
  • [6]          N. Antonova, On Some Mathematical Models in Hemorheology, Biotechnology & Biotechnological Equipment, 26(5) (2012), 3286-3291.
  • [7]          N. Dash and S. Singh, Analytical Study of Non-Newtonian Reiner–Rivlin Model for Blood flow through Tapered Stenotic Artery, Math. Biol. Bioinf., 15(2) (2020), 295-312.
  • [8]          Y. Egushi and T. Karino, Measurement of rheologic property of  blood  by  a  falling-ball  blood  viscometer,  Ann  Biomed Eng., 36(4) (2008), 545-553.
  • [9]          E. Fatahian, N. Kordani, and H. Fatahian, A review on rheology of non-Newtonian properties of blood, IIUM Engineering Journal, 19(1) (2018), 237-250.
  • [10]        A. K. Maiti, Casson flow of blood through an arterial tube with overlapping stenosis, IOSR Journal of Mathematics, 11(6) (2015), 26-31.
  • [11]        M. A. Massoudi, Generalization of Reiner’s mathematical model for wet sand, Mech. Res. Commun., 38 (2011), 378-381.
  • [12]        Kh. S. Mekheimer and M. A. El Kot, The micropolar fluid model for blood flow through a tapered artery with  a  stenosis, Acta Mech. Sin., 24 (2008), 637–644.
  • [13]        M. N. L. Narasimham, On steady laminar flow of certain non-Newtonian liquids through an elastic tube, Proceed- ings of Indian Academy of Sciences, Sec A, 43(2) (1956), 237-246.
  • [14]        G. Neeraja, P.  A. Dinesh, K. Vidya, and C. S. K. Raju, Peripheral layer viscosity on the stenotic blood  vessels  for Herschel-Bulkley fluid model, Informatics in Medicine Unlocked, 9 (2017), 161–165.
  • [15]        S. O’Callaghan, M. Walsh, and T. Mc Gloughlin, Numerical modelling of Newtonian and non-Newtonian repre- sentation of blood in a distal end-to-side vascular bypass graft anastomosis, Med. Eng. Phys., 28 (2006), 70-74.
  • [16]        S. L. Rathna and P. L. Bhatnagar, Weissenberg and Merrington effects in non-Newtonian fluids, Jl. of Indian Institute of Science, 45(2) (1962), 57-82.
  • [17]        R. Revellin, F. Rousset, D. Baud, and J. Bonjour, Extension of Murray’s law using a non-Newtonian model of blood flow, Theoretical Biology and Medical Modelling, 6(9) (2009), 1-9.
  • [18]        S. Sapna, Analysis of non-Newtonian fluid flow in a stenosed artery, International Journal of Physical Sciences, 4(11) (2009), 663-671.
  • [19]        S. Sreenadh, A. R. Pallavi, and B. H. Satyanarayana, Flow of a Casson fluid through an inclined tube of non-  uniform cross section with multiple stenoses, Adv Appl Sci Res., 2(5) (2011), 340–349.
  • [20]        B. Thomas and K. S. Suman, Blood Flow in Human Arterial System-A Review, Procedia Technology, 24 (2016), 339-346.
  • [21]        G. B. Thurston, Erythrocyte Rigidity as a Factor in Blood Rheology: Viscoelastic Dilatancy, Journal of Rheology, 23(6) (2000), 703.
  • [22]        G. B. Thurston and N. M. Henderson, Effects of flow geometry on blood viscoelasticity. Biorheology., 43(6) (2006), 729-746.
  • [23]        J. Venkatesan, D. S. Sankar, K. Hemalatha, and Y. Yatim, Mathematical Analysis  of  Casson  Fluid  Model  for  Blood Rheology in Stenosed Narrow Arteries, Journal of Applied Mathematics., Article ID 583809, (2013).
  • [24]        V. K. Verma, Study of Non-Newtonian Herschel-Bulkley Model through Stenosed Arteries, International Journal of Mathematics and its Application, 4(1-B) (2016), 105-111.
  • [25]        D. F. Young, Fluid mechanics of arterial stenosis, J Biomech Eng., 101 (1979), 157–175.
  • [26]        F. Yilmaz and M. Y. Gundogdu, A critical review on blood flow in large  arteries;  relevance  to  blood  rheology,  viscosity models, and physiologic conditions., Korea-Australia Rheology Journal, 20(4) (2008), 197-211.
Volume 10, Issue 4
October 2022
Pages 928-941
  • Receive Date: 30 July 2021
  • Revise Date: 18 April 2022
  • Accept Date: 19 April 2022
  • First Publish Date: 24 April 2022