Skip to main content
SpringerLink
Log in

One-Dimensional Model of Viscoelastic Blood Flow Through a Thin Elastic Vessel

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Based on the three-dimensional Oldroyd viscoelastic fluid model, we develop a simple linear one-dimensional model of blood flow through a thin blood vessel with an elastic multilayer cylindrical wall. Unlike known models, the obtained system of integrodifferential equations with respect to the variables z and t (the longitudinal coordinate ant time) includes the Volterra operator in t, which takes into account the relaxation effect of stresses in a pulsating flow of blood regarded as a many-component viscoelastic fluid. We construct a simplified differential model corresponding to “short-term memory.” We study the effect of high-amplitude longitudinal oscillations of wall under a dissection (lamination). Bibliography: 24 titles. Illustrations: 1 figure.

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

Similar content being viewed by others

References

  1. V. A. Kozlov and S. A. Nazarov, “Surface enthalpy and elastic properties of blood vessels” [in Russian], Dokl. RAN 441, No. 1, 38–43 (2011); English transl.: Dokl. Phys. 56, No. 11, 560–566 (2011).

  2. V. A. Kozlov and S. A. Nazarov, “Asymptotic models of anisotropic elastic walls of blood vessels,” Math. Methods Appl. Sci. [submitted]

  3. V. A. Kozlov and S. A. Nazarov, “Asymptotic model of interaction of blood flow with vein walls and the surrounding muscular tissue” [in Russian], Dokl. RAN 446, No. 6, 631–636 (2012); English transl.: Dokl. Phys. 57, No. 10, 411–416 (2012).

  4. V. A. Kozlov and S. A. Nazarov, “Asymptotic models of the blood flow in arteries and veins” [in Russian], Zap. Nauchn. Semin. POMI 409, 80–106 (2012); English transl.: J. Math. Sci., New York 194, No. 1, 44–57 (2013).

  5. J. Oldroyd, “On the formulation of rheological equations of state,” Proc. Roy. Soc. London, Ser. A 200, 523–541 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Astarita and G. Maricci, Principles of Non-Newtonian Fluid Hydromechanics, McGraw-Hill, New York (1974).

    Google Scholar 

  7. V. G. Litvinov, Motion of a Nonlinear-Viscous Fluid [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  8. M. A. Rao, Rheology of Fluid and Semisolid Foods: Principles and Applications, Springer, New York (2010).

    Google Scholar 

  9. K. R. Radzhagopal, “On some unsolved problems in nonlinear fluid dynamics” [in Russian], Usp. Mat. Nauk 58, No. 2, 111–121 (2003); English transl.: Russ. Math. Surv. 58, No. 2, 319–330 (2003).

  10. V. Volterra, “Sulle equazioni integro-differenziali della theoria dell’elasticity,” Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Natur., Rend. 18, 295–300 (1909).

    MATH  Google Scholar 

  11. G. A. Holzafel, “Collagen in arterial walls: Biomechanical aspects,” In: Collagen: Structure and Mechanics, pp. 285–324, Springer, New York etc. (2008).

  12. Y. C. Fung, Biomechanics. Mechanical Properties of Living Tissues, Springer, New York etc. (1993).

  13. Y. C. Fung, Biomechanics. Circulation, Springer, Berlin (2011).

    Google Scholar 

  14. S. A. Nazarov, G. H. Sweers, and A. S. Slutskij, “The flexural rigidity of a thin plate reinforced with periodic systems of separated rods” [in Russian], Prikl. Mat. Mekh. 74, No. 3, 441–454 (2010); English transl.: J. Appl. Math. Mech. 74, No. 3, 313–322 (2010).

  15. S. A. Nazarov, G. H. Sweers, and A. S. Slutskij, “Homogenization of a thin plate reinforced with periodic families of rigid rods” Mat. Sb. 202, No. 8, 41–80 (2011); English transl.: Math. Sb. 202, No. 8, 1127–1168 (2011).

  16. S. Čanič and A. Mikelič, “Effective equations modeling the flow of viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries,” SIAM J. Appl. Dyn. Syst. 2, No. 3, 431–463 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. C. Fung, Biomechanics. Motion. Flow, Stress, and Growth, Springer, New York etc. (1990).

  18. W. Flügge, Viscoelasticity, Springer, Berlin (1975).

    Book  MATH  Google Scholar 

  19. S. A. Nazarov and K. I. Piletskas, “The Reynolds flow of a fluid in a thin three-dimensional channel” [in Russian], Litovsk. Mat. Sb. 30, No. 4, 772–783 (1990).

    MATH  MathSciNet  Google Scholar 

  20. E. S. Baranovskii, “On steady motion of viscoelastic fluid of Oldroyd type” [in Russian], Mat. Sb. 205, No. 6, 3–18 (2014); English transl.: Sb. Math. 205, No. 6, 763–776 (2014).

  21. J. Boyd, J. M. Buick, and S. Green, “Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method,” Phys. Fluid. 19, 093103 (2007).

    Article  Google Scholar 

  22. Yu. N. Rabotnov, Creep of Construction Elements [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  23. Yu. N. Rabotnov and S. T. Mileiko, Short-Time Creep [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  24. N. Kh. Arutyunyan and V. B. Kolmanovskii, Creep Theory of Inhomogeneous Bodies [in Russian], Nauka, Moscow (1983).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Linköping University, SE-581 83, Linköping, Sweden

    V. A. Kozlov’

  2. St. Petersburg State University, St. Petersburg State Polytechnical University, Institute for Problems in Mechanical Engineering RAS, 61, Bolshoi pr. V.O., St. Petersburg, 199178, Russia

    S. A. Nazarov

Authors
  1. V. A. Kozlov’
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Nina Nikolaevna Uraltseva who taught us a lot

Translated from Problemy Matematicheskogo Analiza 78, January 2015, pp. 123-140.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozlov’, V.A., Nazarov, S.A. One-Dimensional Model of Viscoelastic Blood Flow Through a Thin Elastic Vessel. J Math Sci 207, 249–269 (2015). https://doi.org/10.1007/s10958-015-2370-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-015-2370-0

Keywords

Search

Navigation