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The effect of shear stress on solitary waves in arteries

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Abstract

In the present work, we study the propagation of solitary waves in a prestressed thick walled elastic tube filled with an incompressible inviscid fluid. In order to include the geometric dispersion in the analysis the wall inertia and shear deformation effects are taken into account for the inner pressure-cross-sectional area relation. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is examined. It is shown that, contrary to thin tube theories, the present approach makes it possible to have solitary waves even for a Mooney-Rivlin (M-R) material. Due to dependence of the coefficients of the governing Korteweg-deVries equation on initial deformation, the solution profile changes with inner pressure and the axial stretch. The variation of wave profiles for a class of elastic materals are depicted in graphical forms. As might be seen from these illustrations, with increasing thickness ratio, the profile of solitary wave is steepened for a M-R material but it is broadened for biological tissues.

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Abbreviations

A i :

inner cross-sectional area

α:

a material constant for soft tissues

b :

a material constant for a M-R material

c kl :

Finger deformation tensor

γ:

thickness parameter for tube wall

λ i θ :

\(\left( {B_i = \lambda _\theta ^{i^2 } } \right)\) circumferential stretch on the inner surface of tube

λ z :

stretch in the axial direction

P :

hydrostatic pressure of elastic solid

P * i :

fluid pressure

ρ 0 :

mass density of solid

ρ f :

mass density of fluid

t kl :

Cauchy stress tensor

w * :

axial fluid velocity

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Authors and Affiliations

  1. Faculty of Sciences and Letters, Department of Engineering Sciences, Istanbul Technical University, 80626, Maslak-Istanbul, Turkey

    Hilmi Demiray

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  1. Hilmi Demiray
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Demiray, H. The effect of shear stress on solitary waves in arteries. Bltn Mathcal Biology 59, 993–1012 (1997). https://doi.org/10.1007/BF02460003

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  • DOI: https://doi.org/10.1007/BF02460003

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