Acta Mechanica Solida Sinica

, Volume 30, Issue 4, pp 435–444

# Vibrational frequency analysis of finite elastic tube filled with compressible viscous fluid

• Ilyess Mnassri
• Adil El Baroudi
Article

## Abstract

The vibrational frequency analysis of finite elastic tube filled with compressible viscous fluid has received plenty of attention in recent years. To apply frequency analysis to defect detection for example, it is necessary to investigate the vibrational behavior under appropriate boundary conditions. In this paper, we present a detailed theoretical study of the three dimensional modal analysis of compressible fluid within an elastic cylinder. The dispersion equations of flexural, torsional and longitudinal modes are derived by elastodynamic theory and the unsteady Stokes equation. The symbolic software Mathematica is used in order to find the coupled vibration frequencies. The dispersion equation is deduced and analytically solved. The finite element results are compared with the present method for validation and an acceptable match between them are obtained.

## Keywords

Frequency analysis Compressible Stokes flow Coupled vibration Elastodynamic

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## References

1. 1.
H. Lamb, On the velocity of sound in a tube, as affected by the elasticity of the walls, Manchester Literary Philos. Soc. Mem. Proc. 42 (1898) 1–16.
2. 2.
J.R. Womersley, Oscillatory motion of a viscous liquid in a thin-walled elastic tube: the linear approximation for long waves, Philos. Mag. 46 (1955) 199–219.
3. 3.
R.H. Cox, Wave propagation through a Newtonian fluid contained within a thick-walled viscoelastic tube, Biophys. J. 8 (1968) 691–709.
4. 4.
K.D. Mahrer, F.J. Mauk, Seismic wave motion for a new model of hydraulic fracture with an induced low-velocity zone, J. Geophys. Res 92 (B9) (1987) 9293–9309.
5. 5.
J. Dvorkin, G. Mavko, A. Nur, The dynamics of viscous compressible fluid in a fracture, Geophysics 57 (5) (1992) 720–726.
6. 6.
A.S. Ashour, Wave motion in a viscous fluid-filled fracture, Int. J. Eng. Sci. 38 (2000) 505–515.
7. 7.
F. Behroozi, Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy, Eur. J. Phys. 25 (2004) 115–122.
8. 8.
C.Y. Wang, L.C. Zhang, Circumferential vibration of microtubules with long axial wavelength, J. Biomech. 41 (2008) 1892–1896.
9. 9.
O. San, A.E. Staples, Dynamics of pulsatile flows through elastic microtubes, Int. J. Appl. Mech. 4 (1) (2012) 1250006 (23 pages).
10. 10.
Y.Z. Wang, H.T. Cui, F.M. Li, K. Kishimoto, Effects of viscous fluid on wave propagation in carbon nanotubes, Phys. Lett. A 375 (2011) 2448–2451.
11. 11.
E.M. Abulwafa, E.K. El-Shewy, A.A. Mahmoud, Time fractional effect on pressure waves propagating through a fluid filled circular long elastic tube, Egypt. J. Basic Appl. Sci. 3 (2016) 35–43.
12. 12.
X. Zhou, Vibration and stability of ring-stiffened thin-walled cylindrical shells conveying fluid, Acta Mech. Solida Sin. 25 (2) (2012) 168–176.
13. 13.
Z. Zhang, Y. Liu, B. Li, Free vibration analysis of fluid-conveying carbon nanotube via wave method, Acta Mech. Solida Sin. 27 (6) (2014) 626–634.
14. 14.
Z. Zhang, Y. Liu, H. Zhao, W. Liu, Acoustic nanowave absorption through clustered carbon nanotubes conveying fluid, Acta Mech. Solida Sin. 29 (3) (2016) 257–270.
15. 15.
M. Hosseini, R. Bahaadini, Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory, Int. J. Eng. Sci. 101 (2016) 1–13.
16. 16.
V. Korneev, Slow waves in fractures filled with viscous fluid, Geophysics 73 (1) (2008), doi:10.1190/1.2802174.
17. 17.
L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, 1959.Google Scholar
18. 18.
M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1946.
19. 19.
J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, The Netherlands, 1975.
20. 20.
K.F. Graff, Wave Motion in Elastic Solids, Dover Publications, New York, 1975.
21. 21.
Comsol Multiphysics, Analysis User’s Manual Version 5.2, 2016.Google Scholar