Skip to main content

Effects of Rheology of Non-Newtonian Fluid and Chemical Reaction on a Dispersion of a Solute and Implications to Blood Flow


The process of dispersion of soluble matter in blood flow has been investigated in the present study. The constitutive equation of blood obeys the law of K-L fluid model. The first order homogeneous chemical reaction is taken in the analysis which has been studied by Taylor’s dispersion method in a circular tube. The influences of the reaction rate constant, the yield stress and K-L parameters on the equivalent dispersion coefficient are discussed. A decrease in the value of dispersion coefficient has been observed in Newtonian as well as non-Newtonian fluids with increase in the rate of chemical reaction. The dispersion coefficient is further decreased with the enhancement of yield stress. It is pertinent to point out that one of K-L parameters tends to decrease the equivalent dispersion coefficient while another K-L parameter enhances the equivalent dispersion coefficient. From the present investigation, many rheological models for blood such as Newtonian, Bingham plastic and Casson can be obtained by giving appropriate values to yield stress and parameters of K-L fluid. The present analytical study provides useful information to the bio-chemical processing and physiological process in the cardiovascular system.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Data availibility

Enquiries about data availability should be directed to the authors.


  1. Griffth, A.: On the movement of a coloured index along a capillary tube, and its application to the measurement of the circulation of water in a closed circuit. Proc. Phys. Soc. Lond. 23, 190 (1911)

    Article  Google Scholar 

  2. Taylor, G.I.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 219, 186–203 (1953)

    Google Scholar 

  3. Taylor, G.I.: The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 223(1155), 446–468 (1954).

    Article  Google Scholar 

  4. Aris, R.: On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 235, 67–77 (1956)

    Google Scholar 

  5. Welstenholme, G.E.W., Knight, J.W.: Circulatory and Respiratory Mass Transfer. Churchill London (1969)

  6. Fan, L.T., Hwang, W.S.: Dispersion of Ostwald-de Waele fluid in laminar flow through a cylindrical tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 283, 576–582 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, L.T., Wang, C.B.: Dispersion of matter in non-Newtonian laminar flow through a circular tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 292, 203–208 (1966).

    Article  Google Scholar 

  8. Ghoshal, S.: Dispersion of solutes in non-Newtonian flows through a circular tube. Chem. Eng. Sci. 26, 185–188 (1971).

    Article  Google Scholar 

  9. Shah, S.H., Cox, K.E.: Dispersion of solutes in non-Newtonian laminar flow through a circular tube-Eyring model fluid. Chem. Eng. Sci. 29, 1282–1286 (1974).

    Article  Google Scholar 

  10. Prenosil, J.E., Jarvis, P.E.: Note on Taylor diffusion for a power Law fluid. Chem. Eng. Sci. 29, 1290 (1974)

    Article  Google Scholar 

  11. Sharp, M.K.: Shear-augmented dispersion in non-Newtonian fluids. Ann. Biomed. Eng. 21(4), 407–415 (1993).

    Article  Google Scholar 

  12. Sankar, D.S., Jaafar, N.A.B., Yatim, Y.M.: Nonlinear analysis for shear augmented dispersion of solutes in blood flow through narrow arteries. J. Appl. Math. (2012).

    Article  MathSciNet  MATH  Google Scholar 

  13. Katz, S.: Chemical reactions catalysed on a tube wall. Chem. Eng. Sci. 10, 202–211 (1959).

    Article  Google Scholar 

  14. Walker, R.E.: Chemical reaction and diffusion in a catalytic tubular reactor. Phys. Fluids 4, 1211–1216 (1961).

    Article  Google Scholar 

  15. Soloman, R.L., Hudson, J.L.: Homogeneous and heterogeneous reactions in a tubular reactor. Am. Inst. Chem. Eng. J. 13, 545–550 (1967).

    Article  Google Scholar 

  16. Gill, W.N., Shankarasubramaniam, R.: Exact analysis of unsteady convective diffusion. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 316, 341–350 (1970).

    Article  Google Scholar 

  17. Gill, W.N., Shankarasubramaniam, R.: Dispersion of a non-uniform slug in time-dependent flow. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 322, 101–117 (1971).

    Article  MATH  Google Scholar 

  18. Gupta, P.S., Gupta, A.S.: Effect of homogeneous and heterogeneous reactions on the dispersion of a solute in the laminar flow between two plates. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 330, 59–63 (1972).

    Article  MATH  Google Scholar 

  19. Scherer, P.W., Shendalman, L.H., Greene, N.M.: Simultaneous diffusion and convection in single breath lung washout. Bull. Math. 34, 393–412 (1972).

    Article  MATH  Google Scholar 

  20. Shukla, J.B., Parihar, R.S., Rao, B.R.P.: Dispersion in non-Newtonian fluids: effects of chemical reaction. Rheol. Acta 18, 740–748 (1979).

    Article  MATH  Google Scholar 

  21. Singh, S.P., Chadda, G.C., Sinha, A.K.: A study of sectionally related dispersion and chemical reaction effects. Defence Sci. J. 39(3), 305–318 (1989)

    Article  Google Scholar 

  22. Jaafar, N.A.B., Yatim, Y.M., Sankar, D.S.: Effect of chemical reaction in solute dispersion in Herschel–Bulkley fluid flow with applications to blood flow. Adv. Appl. Fluid Mech. 20(2), 279–310 (2017).

    Article  MATH  Google Scholar 

  23. Chein, S.: Hemorheology in clinical medicine. Recent Adv. Cardiovasc. Dis. 2, 21–26 (1981)

    Google Scholar 

  24. Luo, X.Y., Kuang, Z.B.: A study on the constitutive equation of blood. J. Biomech. 25(8), 929–934 (1992).

    Article  Google Scholar 

  25. Cokelet, G.R., Merrill, E.W., Gilliland, E.R., Shin, H., Britten, A., Wells, R.E.: Rheology of human blood: measurement near and at zero shear rate. Trans. Sot. Rheol. 7, 303–317 (1963)

    Article  Google Scholar 

  26. Cokelet, G.R.: In: Fung, Y.C., Perrone, N., Anliker, M. (eds.) Bildmeckmics: Its Foundation and Objectives, pp. 63–103. Prentice Hall, Englewood Cliffs (1972)

  27. Bate, H.: Blood viscosity at different shear rates in canillarv tubes. Biorheoloav 14, 267–275 (1977)

    Article  Google Scholar 

  28. Easthope, P.L., Brooks, D.E.: A comparison of rheological constitutive functions for whole human blood. Biorheology 17(3), 235–247 (1980)

    Google Scholar 

  29. Nagarani, P., Sebastian, B.T.: Dispersion of a solute in pulsatile non-Newtonian fluid flow through a tube. Acta Mech. 224, 571–585 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  30. Rana, J., Murthy, P.V.S.N.: Unsteady solute dispersion in Herschel–Bulkley fluid in a tube with wall absorption. Phys. Fluids 28(11), 111903 (2016).

    Article  Google Scholar 

  31. Bugliarello, G., Sevilla, J.: Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 7(2), 85–107 (1970).

    Article  Google Scholar 

  32. Zhang, J.B., Kuang, Z.B.: Study on blood constitutive parameters in different blood constitutive equations. J. Biomech. 33(3), 355–360 (2000)

    Article  Google Scholar 

  33. Ashrafizaadeh, M., Bakhshaei, H.: A comparison of non-Newtonian models for lattice Boltzmann blood flow simulations. Comput. Math. Appl. 58(5), 1045–1054 (2009)

    Article  MathSciNet  Google Scholar 

  34. Sriyab, S.: Mathematical analysis of Non-Newtonian Blood flow in stenosis narrow arteries. Comput. Math. Methods Med. (2014).

    Article  MathSciNet  MATH  Google Scholar 

  35. Bali, Rekha, Gupta, Nivedita: Study of transport of nanoparticles with K-L model through a stenosed microvessels. Appl. Appl. Math. Int. J. (AAM) 13(2), 1157–1170 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Ponalagusamy, R., Manchi, R.: Mathematical study on two fluid model for flow of K-L fluid in a stenosed artery with porous wall. SN Appl. Sci. 3, 1–21 (2021)

    Article  Google Scholar 

Download references


The authors have not disclosed any funding.

Author information

Authors and Affiliations


Corresponding author

Correspondence to S. Priyadharshini.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author. My manuscript has no associated data.

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

Verify currency and authenticity via CrossMark

Cite this article

Ponalagusamy, R., Murugan, D. & Priyadharshini, S. Effects of Rheology of Non-Newtonian Fluid and Chemical Reaction on a Dispersion of a Solute and Implications to Blood Flow. Int. J. Appl. Comput. Math 8, 109 (2022).

Download citation

  • Accepted:

  • Published:

  • DOI:


  • Dispersion
  • K-L fluid
  • Molecular diffusion
  • Non-Newtonian fluid
  • Chemical reaction

Mathematics Subject Classification

  • 92C10