×

Parametric resonance in spherical immersed elastic shells. (English) Zbl 1339.74008

Summary: We perform a stability analysis for a fluid-structure interaction (FSI) problem in which a spherical elastic shell or membrane is immersed in a three-dimensional (3D) viscous, incompressible fluid. The shell is an idealized structure having zero thickness and has the same fluid lying both inside and outside. The problem is formulated mathematically using the immersed boundary framework, in which Dirac delta functions are employed to capture the two-way interaction between fluid and immersed structure. The elastic structure is driven parametrically via a time-periodic modulation of the elastic membrane stiffness. We perform a Floquet stability analysis in the case of both a viscous and inviscid fluid and demonstrate that the forced fluid-membrane system gives rise to parametric resonances in which the solution becomes unbounded even in the presence of viscosity. The analytical results are validated using numerical simulations with a 3D immersed boundary code for a range of wavenumbers and physical parameter values. Moreover, we propose a benchmark computation that is supported by our analytical results and which other FSI software developers can use to validate their simulations. Finally, potential applications to biological systems are discussed, with a particular focus on the human heart and investigating whether or not FSI-mediated instabilities could play a role in cardiac fluid dynamics.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
70K28 Parametric resonances for nonlinear problems in mechanics
74K15 Membranes
74K25 Shells
76D05 Navier-Stokes equations for incompressible viscous fluids
76Z05 Physiological flows

Software:

DistMesh
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Abramowitz and I. A. Stegun, {\it Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables}, Dover Publications, New York, 1965. · Zbl 0171.38503
[2] I. Avrahami and M. Gharib, {\it Computational studies of resonance wave pumping in compliant tubes}, J. Fluid Mech., 608 (2008), pp. 139-160. · Zbl 1145.76355
[3] R. Barrera, G. Estevez, and J. Giraldo, {\it Vector spherical harmonics and their application to magnetostatics}, Euro. J. Phys., 6 (1985), pp. 287-294.
[4] E. Braunwald, E. C. Brockenbrough, C. J. Frahm, and J. Ross, Jr., {\it Left atrial and left ventricular pressures in subjects without cardiovascular disease: Observations in eighteen patients studied by transseptal left heart catheterization}, Circulation, 24 (1961), pp. 267-269.
[5] A. R. Champneys, {\it The dynamics of parametric excitation}, in Encyclopedia of Complexity and Systems Science, R. A. Meyers, ed., Springer, New York, 2009, pp. 2323-2344.
[6] R. Cortez, C. S. Peskin, J. M. Stockie, and D. Varela, {\it Parametric resonance in immersed elastic boundaries}, SIAM J. Appl. Math., 65 (2004), pp. 494-520. · Zbl 1074.74024
[7] R. Cortez and D. Varela, {\it The dynamics of an elastic membrane using the impulse method}, J. Comput. Phys., 138 (1997), pp. 224-247. · Zbl 0910.73035
[8] G. Cottet and E. Maitre, {\it A level set method for fluid-structure interactions with immersed surfaces}, Math. Models Methods Appl. Sci., 16 (2006), pp. 415-438. · Zbl 1088.74050
[9] G. Cottet, E. Maitre, and T. Milcent, {\it An Eulerian method for fluid-structure coupling with biophysical applications}, in Proceedings of the European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2006), P. Wesseling, E. On͂ate, and J. Périaux, eds., Egmond aan Zee, The Netherlands, 2006. · Zbl 1163.76040
[10] A. Diaz, D. Barthès-Biesel, and N. Pelekasis, {\it Effect of membrane viscosity on the dynamic response of an axisymmetric capsule}, Phys. Fluids, 13 (2001), pp. 3835-3838. · Zbl 1184.76136
[11] B. U. Felderhof, {\it Jittery velocity relaxation of an elastic sphere immersed in a viscous incompressible fluid}, Phys. Rev. E, 89 (2014), 033001.
[12] E. Givelberg, {\it Modeling elastic shells immersed in fluid}, Comm. Pure Appl. Math., 57 (2004), pp. 283-330. · Zbl 1118.74014
[13] Z. Gong, H. Huang, and C. Lu, {\it Stability analysis of the immersed boundary method for a two-dimensional membrane with bending rigidity}, Commun. Comput. Phys., 3 (2008), pp. 704-723. · Zbl 1183.74136
[14] J. B. Grotberg, {\it Pulmonary flow and transport phenomena}, in Annu. Rev. Fluid Mech. 26, Annual Reviews, Palo Alto, CA, 1994, pp. 529-571. · Zbl 0802.76100
[15] A. Y. Gunawan, J. Molenaar, and A. A. F. van de Ven, {\it Does shear flow stabilize an immersed thread?}, Euro. J. Mech. B, 24 (2005), pp. 379-396. · Zbl 1060.76042
[16] W. W. Hansen, {\it A new type of expansion in radiation problems}, Phys. Rev., 47 (1935), pp. 139-143. · Zbl 0010.43202
[17] W. Helfrich, {\it Elastic properties of lipid bilayers: Theory and possible experiments}, Z. Naturforsch. C, 28 (1973), pp. 693-703.
[18] E. L. Hill, {\it The theory of vector spherical harmonics}, Amer. J. Phys., 22 (1954), pp. 211-214. · Zbl 0057.05303
[19] W. Huang and H. J. Sung, {\it An immersed boundary method for fluid-flexible structure interaction}, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 2650-2661. · Zbl 1228.74105
[20] R. W. James, {\it The spectral form of the magnetic induction equation}, Proc. R. Soc. Lond. A, 340 (1974), pp. 287-299.
[21] D. W. Jordan and P. Smith, {\it Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems}, Oxford University Press, Oxford, UK, 1999. · Zbl 0955.34001
[22] R. E. Kelly, {\it The flow of a viscous fluid past a wall of infinite extent with time-dependent suction}, Quart. J. Mech. Appl. Math., 18 (1965), pp. 287-298.
[23] C. Kleinstreuer, {\it Biofluid Dynamics: Principles and Selected Applications}, CRC Press, Boca Raton, FL, 2006.
[24] W. Ko, {\it Parametric Resonance in Immersed Elastic Structures, with Application to the Cochlea}, Ph.D. thesis, Department of Mathematics, Simon Fraser University, Burnaby, Canada, 2015.
[25] W. Ko and J. M. Stockie, {\it An immersed boundary model of the cochlea with parametric forcing}, SIAM J. Appl. Math., 75 (2015), pp. 1065-1089. · Zbl 1316.35031
[26] K. Kumar, {\it Linear theory of Faraday instability in viscous liquids}, Proc. R. Soc. Lond. A, 452 (1996), pp. 1113-1126. · Zbl 0885.76035
[27] K. Kumar and L. S. Tuckerman, {\it Parametric instability of the interface between two fluids}, J. Fluid Mech., 279 (1994), pp. 49-68. · Zbl 0823.76026
[28] M. Lai and Z. Li, {\it A remark on jump conditions for the three-dimensional Navier-Stokes equations involving an immersed moving membrane}, Appl. Math. Lett., 14 (2001), pp. 149-154. · Zbl 1013.76021
[29] H. Lamb, {\it Oscillations of a liquid globe, and of a bubble}, in Hydrodynamics, 6th ed., Cambridge University Press, Cambridge, UK, 1932, article 275, pp. 473-475.
[30] Sir J. Lighthill, ed., {\it Mathematical Biofluiddynamics}, CBMS-NSF Regional Conf. Ser. in Appl. Math. 17, SIAM, Philadelphia, 1975. · Zbl 0312.76076
[31] M. S. Link, {\it Evaluation and initial treatment of supraventricular tachycardia}, New Engl. J. Med., 367 (2012), pp. 1438-1448.
[32] L. Loumes, I. Avrahami, and M. Gharib, {\it Resonant pumping in a multilayer impedance pump}, Phys. Fluids, 20 (2008), 023103. · Zbl 1182.76475
[33] A. H. Nayfeh, {\it Introduction to Perturbation Techniques}, John Wiley & Sons, New York, 1993. · Zbl 0449.34001
[34] P.-O. Persson and G. Strang, {\it A simple mesh generator in MATLAB}, SIAM Rev., 46 (2004), pp. 329-345. · Zbl 1061.65134
[35] C. S. Peskin, {\it Numerical analysis of blood flow in the heart}, J. Comput. Phys., 25 (1977), pp. 220-252. · Zbl 0403.76100
[36] C. S. Peskin, {\it The immersed boundary method}, Acta Numer., 11 (2002), pp. 1-39. · Zbl 1123.74309
[37] R. E. Phillips and M. K. Feeney, {\it The Cardiac Rhythms: A Systematic Approach to Interpretation}, Saunders, Philadelphia, 1973.
[38] C. Pozrikidis, {\it Effect of surface viscosity on the deformation of liquid drops and the rheology of dilute emulsions in simple shearing flow}, J. Non-Newton. Fluid Mech., 51 (1994), pp. 161-178.
[39] S. Ramanujan and C. Pozrikidis, {\it Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: Large deformations and the effect of fluid viscosities}, J. Fluid Mech., 361 (1998), pp. 117-143. · Zbl 0921.76058
[40] C. Semler and M. P. Païdoussis, {\it Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe}, J. Fluids Struct., 10 (1996), pp. 787-825.
[41] J. M. Stockie and B. T. R. Wetton, {\it Stability analysis for the immersed fiber problem}, SIAM J. Appl. Math., 55 (1995), pp. 1577-1591. · Zbl 0839.35105
[42] D. Terzopoulos and K. Fleischer, {\it Deformable models}, Visual Computer, 4 (1988), pp. 306-331.
[43] S. Vogel, {\it Life in Moving Fluids: The Physical Biology of Flow}, 2nd ed., Princeton University Press, Princeton, NJ, 1994.
[44] W. Wang, D. Buehler, A. M. Martland, X. D. Feng, and Y. J. Wang, {\it Left atrial wall tension directly affects the restoration of sinus rhythm after Maze procedure}, Eur. J. Cardiothorac. Surg., 40 (2011), pp. 77-82.
[45] Y.-Y. L. Wang, M.-Y. Jan, C.-S. Shyu, C.-A. Chiang, and W.-K. Wang, {\it The natural frequencies of the arterial system and their relation to the heart rate}, IEEE Trans. Biomed. Eng., 51 (2004), pp. 193-195.
[46] J. K. Wiens, {\it An Efficient Parallel Immersed Boundary Algorithm, with Application to the Suspension of Flexible Fibers}, Ph.D. thesis, Department of Mathematics, Simon Fraser University, Burnaby, Canada, 2014.
[47] J. K. Wiens and J. M. Stockie, {\it An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver}, J. Comput. Phys., 281 (2015), pp. 917-941. · Zbl 1351.76130
[48] J. Wright, S. Yon, and C. Pozrikidis, {\it Numerical studies of two-dimensional Faraday oscillations of inviscid fluids}, J. Fluid Mech., 402 (2000), pp. 1-32. · Zbl 0972.76019
[49] A. Zehe and A. Ramírez, {\it Vibration of eukariotic cells in suspension induced by a low-frequency electric field: A mathematical model}, Trans. Biol. Biomed., 1 (2004), pp. 55-59.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.