publication . Article . Other literature type . Preprint . 2019

Geometric Theory of Flexible and Expandable Tubes Conveying Fluid: Equations, Solutions and Shock Waves

François Gay-Balmaz; Vakhtang Putkaradze;
Open Access
  • Published: 01 Apr 2019 Journal: Journal of Nonlinear Science, volume 29, pages 377-414 (issn: 0938-8974, eissn: 1432-1467, Copyright policy)
  • Publisher: Springer Science and Business Media LLC
  • Country: France
International audience; We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive the conservation laws and Rankine-Hugoniot conditions in arbitrary spatial configuration of the tubes, and compute several examples of particular solutions. The theory is derived from a variational treatment of Cosserat rod theory extended to incorporate expandable walls and moving flow inside the...
arXiv: Physics::Fluid Dynamics
free text keywords: General Engineering, Modelling and Simulation, Applied Mathematics, [MATH]Mathematics [math], [NLIN]Nonlinear Sciences [physics], Physics - Fluid Dynamics, Mathematics - Dynamical Systems, Nonlinear Sciences - Chaotic Dynamics, Compressibility, Geometric group theory, Spatial configuration, Fluid equation, Shock wave, Conservation law, Flow (psychology), Physics, Elasticity (economics), Mechanics
69 references, page 1 of 5

[1] T. J. Pedley and X. Y. Luo. The e ects of wall inertia on ow in a two-dimensional collapsible channel. Journal of Fluid Mechanics, 363:253{280, 1998. [OpenAIRE]

[2] A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascular uid dynamics: problems, models and methods. Comput Visual Sci, 2:163{197, 2000.

[3] L. Formaggia, D. Lamponi, and A. Quarteroni. One-dimensional models for blood ow in arteries. Journal of Engineering Mathematics, 47:251{276, 2003. [OpenAIRE]

[4] P. S. Stewart, S. L. Waters, and O. E. Jensen. Local and global instabilities of ow in a exiblewalled channel. Eur. J. Mech. B: Fluids, 28:541{557, 2009.

[5] D. Tang, Y. Yang, C. Yang, and D. N. Ku. A nonlinear axisymmetric model with uid-wall interactions for steady viscous ow in stenotic elastic tubes. Transactions of the ASME, 121:494{ 501, 2009.

[6] D. Elad, R. D. Kamm, and A. H. Shapiro. Steady compressible ow in collapsible tubes: application to forced expiration. J. Applied Physiology, 203:401{418, 1989. [OpenAIRE]

[8] G. Donovan. Systems-level airway models of bronchoconstriction. Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 8:459, 2016.

[9] T. J. Pedley. Longitudinal tension variation in collapsible channels: a new mechanism for the breakdown of steady ow. Journal of Biomechanical Engineering, 114:60{67, 1992. [OpenAIRE]

[10] K. Kounanis and D. S. Mathioulakis. Experimental ow study within a self oscillating collapsible tube. J. Fluids and Struct., 13:61{73, 1999. [OpenAIRE]

[11] X. Y. Luo and T. J. Pedley. Modelling ow and oscillations in collapsible tubes. Theoretical and Computational Fluid Dynamics, 10:277{294, 1998.

[12] J. B. Grotberg and O. E. Jensen. Bio uid mechanics in exible tubes. Ann. Rev. of Fluid Mech, 36:121{147, 2004.

[13] M. Heil and A. L. Hazel. Fluid-structure interaction in internal physiological ows. Ann. Rev. Fluid Mech., 43:141{62, 2011.

[14] H. Ashley and G. Haviland. Bending vibrations of a pipe line containing owing uid. J. Appl. Mech., 17:229{232, 1950.

[15] B. T. Benjamin. Dynamics of a system of articulated pipes conveying uid I. Theory. Proc. Roy. Soc. A, 261:457{486, 1961.

[16] B. T. Benjamin. Dynamics of a system of articulated pipes conveying uid II. Experiments. Proc. Roy. Soc. A, 261:487{499, 1961.

69 references, page 1 of 5
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