A rational derivation of a tube law from shell theory

Quarterly Journal of Mechanics and Applied Mathematics

Volumn 63, Issue 4, 2010, Pages 465-496

  Whittaker, Robert J ^a   Heil, Matthias ^b   Jensen, Oliver E ^c   Waters, Sarah L ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

^c UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

AMPLITUDE DEFORMATIONS; ASYMPTOTIC LIMITS; AXIAL FORCES; AZIMUTHAL MODES; CROSS SECTIONAL AREA; ELASTIC TUBES; ELLIPTICAL CROSS SECTION; IN-PLANE SHEARING; NUMERICAL SOLUTION; PRE-STRESS; RESTORING FORCES; SHELL THEORY; THIN-WALLED; TRANSMURAL PRESSURE;

DEFORMATION;

TUBES (COMPONENTS);

EID: 78149455371     PISSN: 00335614     EISSN: 14643855     Source Type: Journal    
DOI: 10.1093/qjmam/hbq020     Document Type: Article

References (22)

1. M. Heil and O. E. Jensen, Flows in deformable tubes and channels: theoretical models and biological applications, Flow Past Highly Compliant Boundaries and in Collapsible Tubes (ed. P. W. Carpenter & T. J. Pedley; Kluwer, Dordrecht 2003), 15-49.
2. J. B. Grotberg and O. E. Jensen, Biofluid mechanics in flexible tubes, Annu. Rev. Fluid Mech. 36 (2004) 121-147.
3. S. J. Sherwin, V. Franke, J. Peiro and K. Parker, One-dimensional modelling of vascular network in space-time variables, J. Engng Math. 47 (2003) 217-250.
4. J.-M. Fullana and S. Zaleski, A branched one-dimensional model of vessel networks, J. Fluid Mech. 621 (2009) 183-204.
5. D. Elad, R. D. Kamm and A. H. Shapiro, Choking phenomena in a lung like model, J. Biomech. Engng 109 (1987) 1-9.
6. K. Berkouk, P.W. Carpenter and A. D. Lucey, Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 1: basic theory, ibid. 125 (2003) 852-856.
7. R. J. Whittaker, S. L.Waters, O. E. Jensen, J. Boyle and M. Heil, The energetics of flow through a rapidly oscillating tube. Part 1. General theory, J. Fluid Mech. 648 (2010) 83-121.
8. R. J. Whittaker, M. Heil, J. Boyle, O. E. Jensen and S. L.Waters, The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube, ibid. 648 (2010) 123-153.
9. A. H. Shapiro, Steady flow in collapsible tubes. J. Biomech. Engng 99 (1977) 126-147.
10. I. Kececioglu, M. E. McClurken, R. D. Kamm and A. H. Shapiro, Steady, supercritical flow in collapsible tubes. Part 1. Experimental observations, ibid. 109 (1981) 367-389.
11. I. Tadjbakhsh and F. Odeh, Equilibrium states of elastic rings, J. Math. Anal. Appl. 18 (1967) 59-74.
12. J. E. Flaherty, J. Keller and S. I. Rubinow, Post-buckling behaviour of elastic tubes and rings with opposite sides in contact, SIAM J. Appl. Math. 23 (1972) 446-455.
13. M. E. McClurken, I. Kececioglu, R. D. Kamm and A. H. Shapiro, Steady, supercritical flow in collapsible tubes. Part 2. Theoretical studies, J. Fluid Mech. 109 (1981) 391-415.
14. J. W. Reyn, Multiple solutions and flow limitation for steady flow through a collapsible tube held open at the ends, ibid. 174 (1987) 467-493.
15. M. Heil and T. J. Pedley, Large post-buckling deformations of cylindrical shells conveying viscous fluid, J. Fluids Struct. 10 (1996) 565-599.
16. M. Heil, Stokes flow in collapsible tubes: computation and experiment, J. Fluid Mech. 353 (1997) 285-312.
17. W. Flu¨gge, Tensor Analysis and Continuum Mechanics (Springer, Berlin 1972).
18. G. Wempner, Mechanics of Solids with Applications to Thin Bodies (McGraw-Hill, New York 1973).
19. A. E. H. Love, The small free vibrations and deformations of a thin elastic shell, Philos. Trans. R. Soc. A 179 (1888) 491-546.
20. N. Sndergaard, Moving frames applied to shell elasticity, J. Phys. A: Math. & Theor. 40 (2007) 5067-5081.
21. M. Heil and A. Hazel, Oomph-lib: an object-oriented multi-physics finite-element library. Open-source software; available from http://www.oomph-lib.org/ (2007).
22. R. J. Whittaker, M. Heil, O. E. Jensen and S. L. Waters, Predicting the onset of highfrequency self-excited oscillations in elastic-walled tubes. Proc. R. Soc. A. doi: 10.1098/rspa.2009.0641.