Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes

A. T. Il'ichev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 5, pp. 843-882

Résumé

This survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter $\eta$, is analytic in the right-hand complex half-plane $\Omega^+$, and its zeros in $\Omega^+$ coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed. Bibliography: 83 titles.

Détail

Keywords: axisymmetric elastic tube, elastic energy, ideal fluid, internal pressure, spectral parameter, spectral stability, Evans function.
Mots-clés : membrane, quasi-one-dimensional motion, bifurcation

Bibliographie

[1] T. Young, “Hydraulic investigations, subservient to an intended Croonian lecture on the motion of the blood”, Philos. Trans. R. Soc. Lond., 98 (1808), 164–186 | DOI

[2] J. Lighthill, Waves in fluids, Cambridge Univ. Press, Cambridge–New York, 1978, xv+504 pp. | MR | MR | Zbl | Zbl

[3] T. B. Moodie, R. J. Tait, J. B. Haddow, “Waves in compliant tubes”, Wave propagation in viscoelastic media (Taormina, 1980), Research Notes in Mathematics, 52, Pitman (Advanced Publishing Program), Boston, MA–London, 1982, 124–168 | MR | Zbl

[4] T. B. Moodie, F. Mainardi, R. J. Tait, “Pressure pulses in fluid filled distensible tubes”, Meccanica, 20 (1985), 33–37 | DOI

[5] D. L. Newman, S. E. Greenwald, T. B. Moodie, “Reflection from elastic discontinuities”, Med. Biol. Eng. Comput., 21 (1983), 697–701 | DOI

[6] R. S. Johnson, Nonlinear waves in fluid filled elastic tubes and related phenomenon, Ph. D. diss., Univ. of London, London, 1971

[7] Y. Hashizume, “Nonlinear pressure waves in a fluid-filled elastic tube”, J. Phys. Soc. Japan, 54 (1985), 3305–3312 | DOI

[8] S. J. Cowley, “On the wavetrains associated with elastic jumps on fluid-filled elastic tubes”, Quart. J. Mech. Appl. Math., 36:3 (1983), 289–312 | DOI | MR | Zbl

[9] S. Yomosa, “Solitary waves in large blood vessels”, J. Phys. Soc. Japan, 56:2 (1987), 506–520 | DOI | MR

[10] H. Demiray, “Solitary waves in initially stressed thin elastic tubes”, Internat. J. Non-Linear Mech., 32:6 (1997), 1165–1176 | DOI | MR | Zbl

[11] H. A. Erbay, S. Erbay, S. Dost, “Wave propagation in fluid filled nonlinear viscoelastic tubes”, Acta Mech., 95:1-4 (1992), 87–102 | DOI | MR | Zbl

[12] H. Demiray, “Solitary waves in prestressed elastic tubes”, Bull. Math. Biol., 58:5 (1996), 939–955 | DOI | Zbl

[13] H. Demiray, S. Dost, “Axial and transverse solitary waves in prestressed thin elastic tubes”, ARI, 50 (1997), 201–210 | DOI

[14] N. Antar, H. Demiray, “Weakly nonlinear waves in a prestressed thin elastic tube containing a viscous fluid”, Internat. J. Engrg. Sci., 37:14 (1999), 1859–1876 | DOI | MR | Zbl

[15] M. Epstein, C. R. Jonhston, “On the exact speed and amplitude of solitary waves in fluid-filled elastic tubes”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457:2009 (2001), 1195–1213 | DOI | MR | Zbl

[16] Y. B. Fu, S. P. Pearce, K. K. Liu, “Post-bifurcation analysis of a thin-walled hyperelastic tube under inflation”, Internat. J. Non-Linear Mech., 43:8 (2008), 697–706 | DOI | Zbl

[17] E. Chater, J. W. Hutchinson, “On the propagation of bulges and buckles”, ASME J. Appl. Mech., 51:2 (1984), 269–277 | DOI

[18] S. Kyriakides, Chang Yu-Chung, “The initiation and propagation of a localized instability in an inflated elastic tube”, Internat. J. Solids Structures, 27:9 (1991), 1085–1111 | DOI

[19] D. C. Pamplona, P. B. Gonçalves, S. R. X. Lopes, “Finite deformations of cylindrical membrane under internal pressure”, Int. J. Mech. Sci., 48:6 (2006), 683–696 | DOI

[20] S. Noubissié, R. A. Kraenkel, P. Woafo, “Disturbance and repair of solitary waves in blood vessels with aneurysm”, Commun. Nonlinear Sci. Numer. Simul., 14:1 (2009), 51–60 | DOI | MR | Zbl

[21] K. H. Parker, “An introduction to wave intensity analysis”, Med. Biol. Eng. Comput., 47 (2009), 175–188 | DOI

[22] G. A. Holzapfel, T. C. Gasser, R. W. Ogden, “A new constitutive framework for arterial wall mechanics and a comparative study of material models”, J. Elasticity, 60:1-3 (2000), 1–48 | DOI | MR | Zbl

[23] S. P. Pearce, Y. B. Fu, “Characterization and stability of localized bulging/necking in inflated membrane tubes”, IMA J. Appl. Math., 75:4 (2010), 581–602 | DOI | MR | Zbl

[24] Y. B. Fu, G. A. Rogerson, Y. T. Zhang, “Initiation of aneurysms as a mechanical bifurcation phenomenon”, Internat. J. Non-Linear Mech., 47:2 (2012), 179–184 | DOI

[25] Y. B. Fu, Y. X. Xie, “Effects of imperfections on localized bulging in inflated membrane tubes”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370:1965 (2012), 1896–1911 | DOI | MR | Zbl

[26] R. T. Shield, “On the stability of finitely deformed elastic membranes. Part II: Stability of inflated cylindrical and spherical membranes”, Z. Angew. Math. Phys., 23 (1972), 16–34 | DOI | Zbl

[27] D. M. Haughton, R. W. Ogden, “Bifurcation of inflated circular cylinders of elastic material under axial loading. I. Membrane theory for thin-walled tubes”, J. Mech. Phys. Solids, 27:3 (1979), 179–212 | DOI | MR | Zbl

[28] Yi-Chao Chen, “Stability and bifurcation of finite deformations of elastic cylindrical membranes. Part I. Stability analysis”, Internat. J. Solids Structures, 34:14 (1997), 1735–1749 | DOI | Zbl

[29] L. M. Zubov, D. N. Sheidakov, “Instability of a hollow elastic cylinder under tension, torsion, and inflation”, ASME J. Appl. Mech., 75 (2008), 011002, 6 pp. | DOI

[30] Juan Wang, Yibin Fu, “Effect of double-fibre reinforcement on localized bulging of an inflated cylindrical tube of arbitrary thickness”, J. Engrg. Math., 109 (2018), 21–30 | DOI | MR | Zbl

[31] Juan Wang, A. Althobaiti, Yibin Fu, “Localized bulging of rotating elastic cylinders and tubes”, J. Mech. Mater. Struct., 12:4 (2017), 545–561 | DOI | MR

[32] Y. B. Fu, J. L. Liu, G. S. Francisco, “Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness”, J. Mech. Phys. Solids, 90 (2016), 45–60 | DOI | MR

[33] Yang Ye, Yang Liu, Yibin Fu, “Weakly nonlinear analysis of localized bulging of an inflated hyperelastic tube of arbitrary wall thickness”, J. Mech. Phys. Solids, 135 (2020), 103804, 16 pp. | DOI | MR

[34] K. N. Karagiozis, M. P. Païdoussis, M. Amabili, “Effect of geometry on the stability of cylindrical clamped shells subjected to internal fluid flow”, Comput. Structures, 85:11-14 (2007), 645–659 | DOI

[35] Zhiming Guo, J. Gattas, Shibin Wang, Linan Li, F. Albermani, “Experimental and numerical investigation of bulging behaviour of hyperelastic textured tubes”, Int. J. Mech. Sci., 115/116 (2016), 665–675 | DOI

[36] Shibin Wang, Zhiming Guo, Lei Zhou, Linan Li, Yibin Fu, “An experimental study of localized bulging in inflated cylindrical tubes guided by newly emerged analytical results”, J. Mech. Phys. Solids, 124 (2019), 536–554 | DOI | MR

[37] Yang Ye, Yang Liu, A. Althobaiti, Yu-Xin Xie, “Localized bulging in an inflated bilayer tube of arbitrary thickness: Effects of the stiffness ratio and constitutive model”, Internat. J. Solids Structures, 176/177 (2019), 173–184 | DOI

[38] J. B. Grotberg, O. E. Jensen, “Biofluid mechanics in flexible tubes”, Annu. Rev. Fluid Mech., 36 (2004), 121–147 | DOI | MR | Zbl

[39] M. Heil, A. L. Hazel, “Fluid-structure interaction in internal physiological flows”, Annu. Rev. Fluid Mech., 43 (2011), 141–162 | DOI | MR | Zbl

[40] O. E. Jensen, T. J. Pedley, “The existence of steady flow in a collapsed tube”, J. Fluid Mech., 206 (1989), 339–374 | DOI | MR | Zbl

[41] O. E. Jensen, “Instabilities of flow in a collapsed tube”, J. Fluid Mech., 220 (1990), 623–659 | DOI | MR | Zbl

[42] T. J. Pedley, X. Y. Luo, “Modelling flow and oscillations in collapsible tubes”, Theor. Comput. Fluid Dyn., 10:1-4 (1998), 277–294 | DOI | Zbl

[43] R. J. Whittaker, M. Heil, O. E. Jensen, S. L. Waters, “Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466:2124 (2010), 3635–3657 | DOI | MR | Zbl

[44] V. S. Yushutin, “Stability of flow of a nonlinear viscous power-law hardening medium in a deformable channel”, Moscow Univ. Mech. Bull., 67:4 (2012), 99–102 | DOI

[45] A. B. Poroshina, V. V. Vedeneev, “Existence and uniqueness of steady state of elastic tubes conveying power law fluid”, Russ. J. Biomech., 22:2 (2018), 169–193

[46] V. V. Vedeneev, A. B. Poroshina, “Stability of an elastic tube conveying a non-Newtonian fluid and having a locally weakened section”, Proc. Steklov Inst. Math., 300 (2018), 34–55 | DOI | DOI | MR | Zbl

[47] V. V. Vedeneev, “Nonlinear steady states of hyperelatic membrane tubes conveying a viscous non-Newtonian fluid”, J. Fluids Struct., 98 (2020), 103113, 21 pp. | DOI

[48] A. T. Il'ichev, Y.-B. Fu, “Stability of aneurysm solutions in a fluid-filled elastic membrane tube”, Acta Mech. Sin., 28:4 (2012), 1209–1218 | DOI | MR | Zbl

[49] Y. B. Fu, A. T. Il'ichev, “Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow”, Math. Mech. Solids, 20:10 (2015), 1198–1214 | DOI | MR | Zbl

[50] A. T. Il'ichev, Y. B. Fu, “Stability of an inflated hyperelastic membrane tube with localized wall thinning”, Internat. J. Engrg. Sci., 80 (2014), 53–61 | DOI | MR | Zbl

[51] A. T. Il'ichev, V. A. Shargatov, Y. B. Fu, “Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube”, Acta Mech., 231 (2020), 4095–4110 | DOI

[52] Y. B. Fu, Y. X. Xie, “Stability of localized bulging in inflated membrane tubes under volume control”, Internat. J. Engrg. Sci., 48:11 (2010), 1242–1252 | DOI | MR | Zbl

[53] R. L. Pego, M. I. Weinstein, “Eigenvalues, and instabilities of solitary waves”, Philos. Trans. Roy. Soc. London Ser. A, 340:1656 (1992), 47–94 | DOI | MR | Zbl

[54] J. W. Evans, “Nerve axon equations. III. Stability of the nerve impulse”, Indiana Univ. Math. J., 22:6 (1972/73), 577–593 | DOI | MR | Zbl

[55] C. K. R. T. Jones, “Stability of the travelling wave solution of the FitzHugh–Nagumo system”, Trans. Amer. Math. Soc., 286:2 (1984), 431–469 | DOI | MR | Zbl

[56] J. Alexander, R. Gardner, C. Jones, “A topological invariant arising in the stability analysis of travelling waves”, J. Reine Angew. Math., 410 (1990), 167–212 | DOI | MR | Zbl

[57] T. Kapitula, “The Evans function and generalized Melnikov integrals”, SIAM J. Math. Anal., 30:2 (1999), 273–297 | DOI | MR | Zbl

[58] J. C. Alexander, R. Sachs, “Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation”, Nonlinear World, 2:4 (1995), 471–507 | MR | Zbl

[59] J. Swinton, J. Elgin, “Stability of traveling pulse solutions to a laser equation”, Phys. Lett. A, 145:8-9 (1990), 428–433 | DOI | MR

[60] R. L. Pego, P. Smereka, M. I. Weinstein, “Oscillatory instability of travelling waves for a KdV–Burgers equation”, Phys. D, 67:1-3 (1993), 45–65 | DOI | MR | Zbl

[61] J. C. Alexander, M. G. Grillakis, C. K. R. T. Jones, B. Sandstede, “Stability of pulses on optical fibers with phase-sensitive amplifiers”, Z. Angew. Math. Phys., 48:2 (1997), 175–192 | DOI | MR | Zbl

[62] R. A. Gardner, K. Zumbrun, “The gap lemma and geometric criteria for instability of viscous shock profiles”, Comm. Pure Appl. Math., 51:7 (1998), 797–855 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[63] T. Kapitula, “Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation”, Phys. D, 116:1-2 (1998), 95–120 | DOI | MR | Zbl

[64] T. Kapitula, B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations”, Phys. D, 124:1-3 (1998), 58–103 | DOI | MR | Zbl

[65] A. L. Afendikov, T. J. Bridges, “Instability of the Hocking–Stewartson pulse and its implications for three-dimensional Poiseuille flow”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457:2006 (2001), 257–272 | DOI | MR | Zbl

[66] T.-J. Bridges, G. Derks, G. Gottwald, “Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework”, Phys. D, 172:1-4 (2002), 190–216 | DOI | MR | Zbl

[67] A. T. Il'ichev, “Instability of solitary waves on Euler's elastica”, Z. Angew. Math. Phys., 57:4 (2006), 547–566 | DOI | MR | Zbl

[68] A. T. Il'ichev, “Stability of solitary waves in membrane tubes: a weakly nonlinear analysis”, Theoret. and Math. Phys., 193:2 (2017), 1593–1601 | DOI | DOI | MR | Zbl

[69] Y. B. Fu, A. T. Il'ichev, “Solitary waves in fluid-filled elastic tubes: existence, persistence, and the role of axial displacement”, IMA J. Appl. Math., 75:2 (2010), 257–268 | DOI | MR | Zbl

[70] R. W. Ogden, Non-linear elastic deformations, Dover Publications, New York, 1997, 932 pp.

[71] R. W. Ogden, “Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubber-like solids”, Proc. Roy. Soc. London Ser. A, 326:1567 (1972), 565–584 | DOI | Zbl

[72] A. N. Gent, “A new constitutive relation for rubber”, Rubber Chem. Technol., 69:1 (1996), 59–61 | DOI

[73] Dongho Chae, “Liouville-type theorems for the forced Euler equations and the Navier–Stokes equations”, Comm. Math. Phys., 326:1 (2014), 37–48 | DOI | MR | Zbl

[74] G. A. Seregin, T. N. Shilkin, “Liouville-type theorems for the Navier–Stokes equations”, Russian Math. Surveys, 73:4 (2018), 661–724 | DOI | DOI | MR | Zbl

[75] V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140 | DOI | DOI | MR | Zbl

[76] G. Iooss, K. Kirchgässner, “Water waves for small surface tension: an approach via normal form”, Proc. Roy. Soc. Edinburgh Sect. A, 122:3-4 (1992), 267–299 | DOI | MR | Zbl

[77] G. Iooss, M. Adelmeyer, Topics in bifurcation theory and applications, Adv. Ser. Nonlinear Dynam., 3, 2nd ed., World Scientific Publishing Co., Inc., River Edge, NJ, 1998, viii+186 pp. | MR | Zbl

[78] N. A. Kudryashov, Metody nelineinoi matematicheskoi fiziki, Uch. posobie, Izd. dom “Intellekt”, Dolgoprudnyi, 2010, 368 pp.

[79] S. A. Kaschenko, “Asymptotics of rapidly oscillating solutions of the generalized Korteweg–de Vries–Burgers equation”, Russian Math. Surveys, 74:4 (2019), 755–757 | DOI | DOI | MR | Zbl

[80] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, MA, 1965, viii+166 pp. | MR | Zbl

[81] T. B. Benjamin, “The stability of solitary waves”, Proc. Roy. Soc. London Ser. A, 328:1573 (1972), 153–183 | DOI | MR

[82] M. Grillakis, J. Shatah, W. Strauss, “Stability theory of solitary waves in the presence of symmetry. I”, J. Funct. Anal., 74:1 (1987), 160–197 | DOI | MR | Zbl

[83] A. T. Il'ichev, “Physical parameters of solitary wave packets in shallow basins under ice cover”, Theoret. and Math. Phys., 201:3 (2019), 1710–1722 | DOI | DOI | MR | Zbl