Nonlinear waves and shock calculations for hyperelastic fluid-filled tubes

Authors:
T. Bryant Moodie and Gordon E. Swaters

Journal:
Quart. Appl. Math. **47** (1989), 705-722

MSC:
Primary 76B99; Secondary 73D99, 73G05, 73K70, 73P05, 76B15, 76L05, 76Z05

DOI:
https://doi.org/10.1090/qam/1031686

MathSciNet review:
MR1031686

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Abstract | References | Similar Articles | Additional Information

Abstract: The propagation of both finite amplitude and weakly nonlinear waves in fluid-filled hyperelastic tethered tubes subjected to axial strain is investigated in some detail. Procedures based upon the methods of characteristics and relatively undistorted waves are deployed to compute the time and location of first shock formation in tubes having both constant and variable properties ahead of the wave. The influence of wall thickness changes in shock formation is explored and it is further found that if the transmural pressure ahead of the wave is zero then no shock can form on the lead characteristic for any model based on a rational theory of finite elasticity. This latter result is in disagreement with several earlier studies.

**[1]**G. Rudinger,*Shock waves in mathematical models of the aorta*, J. Appl. Mech.**37**, 34-37 (1970)**[2]**J. W. Lambert,*Fluid Flow in a Nonrigid Tube*, Doctoral Dissertation Series**19**, 418 pp., University Microfilms Inc., Ann Arbor, Mich., 1956**[3]**T. B. Moodie and J. B. Haddow,*Waves in thin-walled elastic tubes containing an incompressible inviscid fluid*, Internat. J. Non-Linear Mech.**12**, 223-231 (1977)**[4]**J. R. Womersley,*An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries*, Wright Air Development Center Technical Report TR56-614 (1957)**[5]**R. Skalak,*Wave propagation in blood flow*, Biomechanics Symposium (Ed. Y. C. Fund), New York, A.S.M.E.**[6]**P. J. Chen and M. F. McCarthy,*Unsteady flow of incompressible fluids in deformable tubes*, Acta Mech.**33**(1979), no. 3, 189–197 (English, with German summary). MR**546738**, https://doi.org/10.1007/BF01175915**[7]**M. Anliker, R. L. Rockwell, and E. Ogden,*Non-linear analysis of flow pulses and shock waves in arteries*, Parts I and II, Z. Angew. Math. Phys.**22**, 217-246 and 563-581 (1971)**[8]**S. J. Cowley,*Elastic jumps on fluid-filled elastic tubes*, J. Fluid Mech.**116**(1982), 459–473. MR**653238**, https://doi.org/10.1017/S002211208200055X**[9]**S. J. Cowley,*On the wavetrains associated with elastic jumps on fluid-filled elastic tubes*, Quart. J. Mech. Appl. Math.**36**(1983), no. 3, 289–312. MR**714304**, https://doi.org/10.1093/qjmam/36.3.289-a**[10]**R. J. Tait and T. Bryant Moodie,*Waves in nonlinear fluid filled tubes*, Wave Motion**6**(1984), no. 2, 197–203. MR**734480**, https://doi.org/10.1016/0165-2125(84)90015-5**[11]**E. Varley and E. Cumberbatch,*Non-linear, high frequency sound waves*, J. Inst. Maths. Applics.**2**, 133-143 (1966)**[12]**B. R. Seymour and M. P. Mortell,*Nonlinear geometrical acoustics*, Mechanics Today, Vol. 2 (edited by S. Nemat-Nasser), Pergamon, New York, 1975, pp. 251-312**[13]**R. W. Ogden,*Non-Linear Elastic Deformations*, Ellis Horwood, Chichester, 1984**[14]**A. J. Chorin and J. E. Marsden,*A mathematical introduction to fluid mechanics*, Springer-Verlag, New York-Heidelberg, 1979. MR**551053****[15]**G. B. Whitham,*The propagation of weak spherical shocks in stars*, Comm. Pure. Appl. Math.**6**(1953), 397–414. MR**0060940**, https://doi.org/10.1002/cpa.3160060305**[16]**G. B. Whitham,*On the propagation of weak shock waves*, J. Fluid Mech.**1**(1956), 290–318. MR**0082322**, https://doi.org/10.1017/S0022112056000172**[17]**Akira Isihara, Natsuki Hashitsume, and Masao Tatibana,*Statistical theory of rubber-like elasticity. IV. Two-dimensional stretching*, J. Chem. Phys.**19**(1951), 1508–1512. MR**0048271**, https://doi.org/10.1063/1.1748111**[18]**Gordon E. Swaters,*Resonant three-wave interactions in nonlinear hyper-elastic fluid-filled tubes*, Z. Angew. Math. Phys.**39**(1988), no. 5, 668–681. MR**963639**, https://doi.org/10.1007/BF00948729

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Additional Information

DOI:
https://doi.org/10.1090/qam/1031686

Article copyright:
© Copyright 1989
American Mathematical Society