 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| M. V. Vilde and Yu. P. Gulyaev, "Low-Frequency Axisymmetric Waves in Blood Vessels of Constant Cross-Section: an Asymptotic Approach," Mech. Solids. 44 (4), 608-620 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
4 |
Pages |
608-620 |
| DOI |
10.3103/S0025654409040116 |
| Title |
Low-Frequency Axisymmetric Waves in Blood Vessels of Constant Cross-Section: an Asymptotic Approach |
| Author(s) |
M. V. Vilde (Chernyshevskii Saratov State University, Astrakhanskaya 83, Saratov, 410012 Russia, mv_wilde@mail.ru)
Yu. P. Gulyaev (Chernyshevskii Saratov State University, Astrakhanskaya 83, Saratov, 410012 Russia, gulvis@yandex.ru) |
| Abstract |
The asymptotic methods of shell theory are used to study the
propagation of axisymmetric waves in blood vessels of constant cross-section. The initial equations are simplified using the assumption that the shell radius is small compared with the wave length. We show that the terms corresponding to the shell inertia cannot be omitted if it is required to describe not only the pressure wave but also the longitudinal wave. We study the influence of external fixation on the pressure wave. In this case, we compare the following two models: in the first model, the ambient medium is modelled by elastic and damping elements uniformly distributed over the shell exterior surface and by additional masses; in the second model, the ambient medium is represented by an infinite elastic space with a cylindrical cavity where the vessel is placed. On the interface between the elastic space and the vessel, we pose the full contact conditions. We show that, from the qualitative standpoint, both models lead to the same result: the pressure wave in the first approximation is a wave in the shell whose walls cannot move in the longitudinal direction. We asymptotically integrate the original equations and hence obtain a one-dimensional equation for the bulk blood flow. |
| Keywords |
dispersion equation, wave number, long-wave asymptotics, thin-wall parameter, interface, pressure wave, longitudinal wave |
| References |
| 1. | T. J. Pedley,
The Fluid Mechanics of Large Blood Vessels
(Cambridge University Press, Cambridge, 1980; Mir, Moscow, 1983). |
| 2. | S. A. Regirer (Editor),
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in Collection of translations
(Mir, Moscow, 1971)
[in Russian]. |
| 3. | J. R. Womersley,
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| 4. | D. J. Patel and D. L. Fry,
"Longitudinal Tethering of Articles in Dogs,"
Circ. Res.
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| 5. | H. B. Atabek,
"Wave Propagation through a Viscous Liquid Contained in a Tethered, Initially Stressed, Orthotropic Elastic Tube,"
Biophys. J.
8, 626-649 (1968). |
| 6. | D. J. Patel and R. H. Vaishnav,
"The Rheology of Large Blood Vessels,"
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(London-New York, 1972),
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| 7. | S. A. Ambartsumyan and L. A. Movsisyan,
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[Polymer Mech. (Engl. Transl.)
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| 8. | J. A. Maxwell and M. Anliker,
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8 (8), 920-950 (1968). |
| 9. | J. C. Misra and K. Roy Choudhury (sen),
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Rheol. Acta
23 (5), 548-555 (1984). |
| 10. | A. S. Vol'mir and M. S. Gershtein,
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Mekh. Polimer.,
No. 2, 373-379 (1970)
[Polymer Mech. (Engl. Transl.)
6 (2), 322-326 (1970)]. |
| 11. | Yu. A. Ustinov,
"A Model of the Helical Pulsed Flow of Blood in Arteries,"
Dokl. Ross. Akad. Nauk
398 (3), 344-348 (2004)
[Dokl. Phys. (Engl. Transl.)
49 (9), 543-547 (2004)]. |
| 12. | J. D. Kalpunov, L. Yu. Kossovich, and E. V. Nolde,
Dynamics of Thin Walled Elastic Bodies
(Acad. Press, San Diego, 1998). |
| 13. | M. Abramovich and I. M. Stegun (Editors)
Handbook of Special Functions with Formulas, Graphs,
and Mathematical Tables,
(Nauka, Moscow, 1979)
[in Russian]. |
| 14. | Yu. P. Gulyaev and L. Yu. Kossovich,
Mathematical Models of Biomechanics in Medicine
(Izd-vo Saratov Univ., Saratov, 2001)
[in Russian]. |
|
| Received |
13 December 2006 |
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