Fluid Dynamics

, Volume 10, Issue 4, pp 611–618 | Cite as

Fluid wave motion in pipes of viscoelastic material. Stationary nonlinear waves

  • I. M. Rutkevich


Steady blood flows in the small vessels can lose stability under conditions of realizing a dropping section on the curve of the pressure as a function of the radius. There are experiments verifying the existence of such a section [1]. The viscoelastic properties of the blood vessel walls, which have been determined in tests [2] and predicted by rheological models of the muscle tissue (see [3], for instance), play a major part in oscillation of the perturbations. A linear analysis of the flow stability in vessels with S- and N-shaped static characteristics is given in [4]. The formation of stationary waves of finite amplitude can be the result of the development of instability in nonlinear systems. Such motions are investigated below in an inertialess hydraulic approximation. Nonlinear model equations, relating the final nonstationary pressure perturbations and the deformations, are involved in the closure of the hydraulic description. A class of bounded solutions containing periodic waves and solitons is studied for S-type vessels. Existence conditions for the periodic solutions are determined in the case of the N-shaped characteristic; a description is given for the periodic waves in a harmonic approximation. It is detected that the lengths of bounded stationary waves in vessels of both types correspond to the linear instability domain.


Soliton Periodic Wave Stationary Wave Viscoelastic Material Rheological Model 
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Literature cited

  1. 1.
    P. C. Johnson, “Autoregulatory responses of cat mesenteric arterioles, measuredin vivo,∝ Circulat. Res.,22, No. 2 (1968).Google Scholar
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    Hydrodynamics of Blood Circulation [Russian translation], Mir, Moscow (1971).Google Scholar
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    P. I. Usik, “Continual mechanochemical model of muscle tissue,∝ Prikl. Mat. Mekh.,37, No. 3 (1973).Google Scholar
  4. 4.
    S. A. Regirer and I. M. Rutkevich, “Wave motions of a fluid in pipes of viscoelastic material. Small amplitude waves,∝ Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1 (1975).Google Scholar
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    F. G. Tricomi, Differential Equations, Hafner (1961).Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • I. M. Rutkevich
    • 1
  1. 1.Moscow

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