, Volume 51, Issue 10, pp 2321–2330 | Cite as

A dynamic viscoelastic analogy for fluid-filled elastic tubes

  • Andrea Giusti
  • Francesco Mainardi


In this paper we evaluate the dynamic effects of the fluid viscosity for fluid filled elastic tubes in the framework of a linear uni-axial theory. Because of the linear approximation, the effects on the fluid inside the elastic tube are taken into account according to the Womersley theory for a pulsatile flow in a rigid tube. The evolution equations for the response variables are derived by means of the Laplace transform technique and they all turn out to be the very same integro-differential equation of the convolution type. This equation has the same structure as the one describing uni-axial waves in linear viscoelastic solids characterized by a relaxation modulus or by a creep compliance. In our case, the analogy is connected with a peculiar viscoelastic solid which exhibits creep properties similar to those of a fractional Maxwell model (of order 1 / 2) for short times, and of a standard Maxwell model for long times. The present analysis could find applications in biophysics concerning the propagation of pressure waves within large arteries.


Elastic tubes Viscoelasticity Transient waves  Bessel functions Dirichlet series 

Mathematics Subject Classification

35Q35 76A10 76D05 



The work of F.M. has been carried out in the framework of the activities of the National Group of Mathematical Physics (GNFM, INdAM) and of the Interdepartmental Center “L. Galvani” for integrated studies of Bioinformatics, Biophysics and Biocomplexity of the University of Bologna. The authors are grateful to the anonymous referees for their remarks and suggestions. In particular, we appreciate the comment of one referee who introduced us to the paper in [20].


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Physics and Astronomy, INFNUniversity of BolognaBolognaItaly

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