Geodesic
Stability of an aneurysm in a membrane tube filled with an ideal fluid
Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 2, pp. 236-248 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The stability of standing localized structures formed in an axisymmetric membrane tube filled with fluid is studied. It is assumed that the tube wall is heterogeneous and is subjected to localized thinning. Because the problem has no translational invariance in the case of an inhomogeneous wall the stability of a standing localized structure located in the center of the inhomogeneity of the tube wall is understood as the usual, and not the orbital stability up to a shift. The spectral stability of localized elevation waves is considered in the context of aneurysm formation on human vessels. The fluid flowing inside the tube is assumed to be ideal and incompressible and its longitudinal velocity profile is assumed constant along the vertical section of the tube. Spectral stability is established by the proof of the absence of eigenvalues with a positive real part corresponding to exponentially time-increasing perturbations that are solutions of the linearized equations of the problem. The stability analysis is carried out by constructing the Evans function, which depends only on the spectral parameter and is analytic in the right complex half-plane $\Omega^+$. The zeros of the Evans function in $\Omega^+$ coincide with the unstable eigenvalues of the problem. The absence of zeros in $\Omega^+$ is proved by applying the argument principle from complex analysis.
Mots-clés : soliton
Keywords: instability, Evans function. 
@article{TMF_2022_211_2_a4,
     author = {A. T. Il'ichev and V. A. Shargatov},
     title = {Stability of an aneurysm in a~membrane tube filled with an~ideal fluid},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {236--248},
     year = {2022},
     volume = {211},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a4/}
}
TY  - JOUR
AU  - A. T. Il'ichev
AU  - V. A. Shargatov
TI  - Stability of an aneurysm in a membrane tube filled with an ideal fluid
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2022
SP  - 236
EP  - 248
VL  - 211
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a4/
LA  - ru
ID  - TMF_2022_211_2_a4
ER  - 
%0 Journal Article
%A A. T. Il'ichev
%A V. A. Shargatov
%T Stability of an aneurysm in a membrane tube filled with an ideal fluid
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2022
%P 236-248
%V 211
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a4/
%G ru
%F TMF_2022_211_2_a4
A. T. Il'ichev; V. A. Shargatov. Stability of an aneurysm in a membrane tube filled with an ideal fluid. Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 2, pp. 236-248. http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a4/

[1] M. Epstein, C. Jonhston, “On the exact speed and amplitude of solitary waves in fluid-filled elastic tubes”, Proc. Roy. Soc. London Ser. A, 457:2009 (2001), 1195–1213 | DOI | MR

[2] 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

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

[4] 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

[5] 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

[6] 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

[7] 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., 231:10 (2020), 4095–4110 | DOI | MR

[8] 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

[9] A. T. Ilichev, “Dinamika i spektralnaya ustoichivost solitonopodobnykh struktur v membrannykh trubkakh s zhidkostyu”, UMN, 75:5(455) (2020), 59–100 | DOI | DOI | MR

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

[11] R. W. Ogden, Non-Linear Elastic Deformations, Dover, New York, 1997 | MR

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

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

[14] C. O. Horgan, G. Saccomandi, “A description of arterial wall mechanics using limiting chain extensibility constitutive models”, Biomech. Model. Mechanobiol., 1 (2003), 251–266 | DOI

[15] Y. B. Fu, Y. X. Xie, “Effects of inperfections on localized buldging in inflated membrane tubes”, Phil. Trans. Roy. Soc. London Ser. A, 370:1965 (2012), 1896–1911 | DOI | MR

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

[17] 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

[18] V. V. Vedeneev, A. B. Poroshina, “Ustoichivost uprugoi trubki, soderzhaschei tekuschuyu nenyutonovskuyu zhidkost i imeyuschei lokalno oslablennyi uchastok”, Trudy MIAN, 300 (2018), 42–64 | DOI | DOI | MR

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