Hostname: page-component-5c6d5d7d68-vt8vv Total loading time: 0.001 Render date: 2024-08-16T02:11:30.624Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional vibrations of multilayered hollow spheres submerged in a complex fluid

Published online by Cambridge University Press:  01 October 2019

B. Wu ,
Y. Gan ,
E. Carrera  and
W. Q. Chen Open the ORCID record for W. Q. Chen [Opens in a new window]
B. Wu
Affiliation:
Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, PR China State Key Laboratory of Computer-Aided Design and Computer Graphics, Zhejiang University, Hangzhou 310058, PR China Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino 10129, Italy Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, PR China
Y. Gan
Affiliation:
Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, PR China
E. Carrera*
Affiliation:
Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino 10129, Italy
W. Q. Chen*
Affiliation:
Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, PR China State Key Laboratory of Computer-Aided Design and Computer Graphics, Zhejiang University, Hangzhou 310058, PR China Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, PR China Soft Matter Research Center, Zhejiang University, Hangzhou 310027, PR China
*
†Email addresses for correspondence: erasmo.carrera@polito.it, chenwq@zju.edu.cn
†Email addresses for correspondence: erasmo.carrera@polito.it, chenwq@zju.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Fluid–structure interaction is fundamental to the characteristics of the induced flows due to the motion of structures in fluids and also is crucial to the performance of submerged structures. This paper presents a three-dimensional analytical study of the intrinsic free vibration of an elastic multilayered hollow sphere interacting with an exterior non-Newtonian fluid medium. The fluid is assumed to be characterized by a compressible linear viscoelastic model accounting for both the shear and compressional relaxation processes. For small-amplitude vibrations, the equations governing the viscoelastic fluid can be linearized, which are then solved by introducing appropriate potential functions. The solid is assumed to exhibit a particular material anisotropy, i.e. spherical isotropy, which includes material isotropy as a special case. The equations governing the anisotropic solid are solved in spherical coordinates using the state-space formalism, which finally establishes two separate transfer relations correlating the state vectors at the innermost surface with those at the outermost surface of the multilayered hollow sphere. By imposing the continuity conditions at the fluid–solid interface, two separate analytical characteristic equations are derived, which characterize two independent classes of vibration. Numerical examples are finally conducted to validate the theoretical derivation as well as to investigate the effects of various factors, including fluid viscosity and compressibility, fluid viscoelasticity, solid anisotropy and surface effect, as well as solid intrinsic damping, on the vibration characteristics of the submerged hollow sphere. Particularly, our theoretically predicted vibration frequencies and quality factors of gold nanospheres with intrinsic damping immersed in water agree exceptionally well with the available experimentally measured results. The reported analytical solution is truly and fully three-dimensional, covering from the purely radial breathing mode to the torsional mode to any general spheroidal mode as well as being applicable to various simpler situations, and hence can be a broad-spectrum benchmark in the study of fluid–structure interaction.

JFM classification

Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , pp. 682 - 715
Check for updates
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Babincová, M., Sourivong, P. & Babinec, P. 2000 Resonant absorption of ultrasound energy as a method of HIV destruction. Med. Hypotheses 55, 450–451.10.1054/mehy.2000.1088Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Behara, S., Borazjani, I. & Sotiropoulos, F. 2011 Vortex-induced vibrations of an elastically mounted sphere with three degrees of freedom at Re = 300: hysteresis and vortex shedding modes. J. Fluid Mech. 686, 426–450.10.1017/jfm.2011.339Google Scholar
Boyko, E., Bercovici, M. & Gat, A. D. 2017 Viscous-elastic dynamics of power-law fluids within an elastic cylinder. Phys. Rev. Fluids 2, 073301.10.1103/PhysRevFluids.2.073301Google Scholar
Cammarata, R. C. 1997 Surface and interface stress effects on interfacial and nanostructured materials. Mater. Sci. Engng A 237 (2), 180–184.10.1016/S0921-5093(97)00128-7Google Scholar
Chakraborty, D., Hartland, G. V., Pelton, M. & Sader, J. E. 2018 When can the elastic properties of simple liquids be probed using high-frequency nanoparticle vibrations? J. Phys. Chem. C 122 (25), 13347–13353.10.1021/acs.jpcc.7b09951Google Scholar
Chakraborty, D., van Leeuwen, E., Pelton, M. & Sader, J. E. 2013 Vibration of nanoparticles in viscous fluids. J. Phys. Chem. C 117 (16), 8536–8544.10.1021/jp401141bGoogle Scholar
Chakraborty, D. & Sader, J. E. 2015 Constitutive models for linear compressible viscoelastic flows of simple liquids at nanometer length scales. Phys. Fluids 27 (5), 052002.10.1063/1.4919620Google Scholar
Chang, W. S., Wen, F., Chakraborty, D., Su, M. N., Zhang, Y., Shuang, B., Nordlander, P., Sader, J. E., Halas, N. J. & Link, S. 2015 Tuning the acoustic frequency of a gold nanodisk through its adhesion layer. Nat. Commun. 6, 7022.10.1038/ncomms8022Google Scholar
Chen, W. Q. & Ding, H. J. 2001 Free vibration of multi-layered spherically isotropic hollow spheres. Intl J. Mech. Sci. 43 (3), 667–680.10.1016/S0020-7403(00)00044-8Google Scholar
Chen, W. Q. & Ding, H. J. 2002 On free vibration of a functionally graded piezoelectric rectangular plate. Acta Mech. 153 (3/4), 207–216.10.1007/BF01177452Google Scholar
Chen, W. Q., Wang, X. & Ding, H. J. 1999 Free vibration of a fluid-filled hollow sphere of a functionally graded material with spherical isotropy. J. Acoust. Soc. Am. 106 (5), 2588–2594.10.1121/1.428090Google Scholar
Chen, W. Q., Wu, B., Zhang, C. L. & Zhang, Ch. 2014 On wave propagation in anisotropic elastic cylinders at nanoscale: surface elasticity and its effect. Acta Mech. 225 (10), 2743–2760.10.1007/s00707-014-1211-4Google Scholar
Cohen, H., Shah, A. H. & Ramakrishnan, C. V. 1972 Free vibrations of a spherically isotropic hollow sphere. Acta Acust. 26 (6), 329–333.Google Scholar
Crut, A., Maioli, P., Del Fatti, N. & Vallée, F. 2009 Anisotropy effects on the time-resolved spectroscopy of the acoustic vibrations of nanoobjects. Phys. Chem. Chem. Phys. 11 (28), 5882–5888.10.1039/b902107hGoogle Scholar
Crut, A., Maioli, P., Del Fatti, N. & Vallée, F. 2015 Acoustic vibrations of metal nano-objects: time-domain investigations. Phys. Rep. 549, 1–43.10.1016/j.physrep.2014.09.004Google Scholar
Dey, A. A., Modarres-Sadeghi, Y. & Rothstein, J. P. 2017 Experimental observation of viscoelastic fluid–structure interactions. J. Fluid Mech. 813, R5.10.1017/jfm.2017.15Google Scholar
Ding, H. J. & Chen, W. Q. 1996 Natural frequencies of an elastic spherically isotropic hollow sphere submerged in a compressible fluid medium. J. Sound Vib. 192 (1), 173–198.10.1006/jsvi.1996.0182Google Scholar
Ding, H. J. & Chen, W. Q. 1998 Exact shell theory analysis of free vibrations of submerged thin spherical shells. Intl J. Solids Struct. 35 (33), 4381–4389.Google Scholar
Ding, H. J., Chen, W. Q. & Zhang, L. C. 2006 Elasticity of Transversely Isotropic Materials. Springer.Google Scholar
Dowell, E. H. & Hall, K. C. 2001 Modeling of fluid–structure interaction. Annu. Rev. Fluid Mech. 33, 445–490.10.1146/annurev.fluid.33.1.445Google Scholar
Faran, J. J. 1951 Sound scattering by solid cylinders and spheres. J. Acoust. Soc. Am. 23 (4), 405–418.10.1121/1.1906780Google Scholar
Galstyan, V., Pak, O. S. & Stone, H. A. 2015 A note on the breathing mode of an elastic sphere in Newtonian and complex fluids. Phys. Fluids 27, 032001.10.1063/1.4914045Google Scholar
Gao, T., Howard, H. H. & Ponte Castañeda, P. 2011 Rheology of a suspension of elastic particles in a viscous shear flow. J. Fluid Mech. 687, 209–237.10.1017/jfm.2011.347Google Scholar
Guz, A. N. 1980 Hydroelasticity problems for viscous compressible liquids in a spherical coordinate system. Sov. Appl. Mech. 16, 937–943.10.1007/BF00884872Google Scholar
Hou, G., Wang, J. & Layton, A. 2012 Numerical methods for fluid–structure interaction – a review. Commun. Comput. Phys. 12 (2), 337–377.10.4208/cicp.291210.290411sGoogle Scholar
Howe, M. S 1998 Acoustics of Fluid–Structure Interactions. Cambridge University Press.10.1017/CBO9780511662898Google Scholar
Hsueh, C. C., Gordon, R. & Rottler, J. 2018 Dewetting during terahertz vibrations of nanoparticles. Nano Lett. 18 (2), 773–777.10.1021/acs.nanolett.7b03984Google Scholar
Janela, J., Moura, A. & Sequeira, A. 2010 A 3D non-Newtonian fluid–structure interaction model for blood flow in arteries. J. Comput. Appl. Maths 234 (9), 2783–2791.10.1016/j.cam.2010.01.032Google Scholar
Jiang, J. Q., Zhu, J. & Chen, W. Q. 2019 Dispersion curves of magneto-electro-elastic imperfect cylinders filled with fluid. Math. Mech. Solids 24 (1), 195–211.10.1177/1081286517735147Google Scholar
Junger, J. C. 1952a Sound scattering by thin elastic shells. J. Acoust. Soc. Am. 24 (4), 366–373.10.1121/1.1906905Google Scholar
Junger, J. C. 1952b Vibrations of elastic shells in a fluid medium and the associated radiation of sound. Trans. ASME J. Appl. Mech. 19, 439–445.10.1115/1.4010540Google Scholar
Junger, J. C. & Feit, D. 1986 Sound, Structures, and Their Interaction. MIT Press.Google Scholar
Khaderi, S. N. & Onck, P. R. 2012 Fluid–structure interaction of three-dimensional magnetic artificial cilia. J. Fluid Mech. 708, 303–328.10.1017/jfm.2012.306Google Scholar
Lai, W. M., Krempl, E. & Rubin, D. 2009 Introduction to Continuum Mechanics. Butterworth-Heinemann.Google Scholar
Lamb, H. 1881 On the vibrations of an elastic sphere. Proc. Lond. Math. Soc. s1‐13 (1), 189–212.10.1112/plms/s1-13.1.189Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.10.1088/0034-4885/72/9/096601Google Scholar
Li, G. J. & Ardekani, A. M. 2015 Undulatory swimming in non-Newtonian fluids. J. Fluid Mech. 784, R4.10.1017/jfm.2015.595Google Scholar
Liao, S. J. 2003 On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189–212.10.1017/S0022112003004865Google Scholar
Liu, T. M., Chen, H. P., Wang, L. T., Wang, J. R., Luo, T. N., Chen, Y. J., Liu, S. I. & Sun, C. K. 2009 Microwave resonant absorption of viruses through dipolar coupling with confined acoustic vibrations. Appl. Phys. Lett. 94, 043902.Google Scholar
Muller, D. E. 1956 A method for solving algebraic equations using an automatic computer. Math. Tables Other Aids to Comput. 10, 208–215.10.2307/2001916Google Scholar
Nasouri, B., Khot, A. & Elfring, G. J. 2017 Elastic two-shpere swimmer in Stokes flow. Phys. Rev. Fluids 2, 043101.10.1103/PhysRevFluids.2.043101Google Scholar
Païdoussis, M. P. 2003 Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 2. Academic Press.Google Scholar
Pelton, M., Chakraborty, D., Malachosky, E., Guyot-Sionnest, P. & Sader, J. E. 2013 Viscoelastic flows in simple liquids generated by vibrating nanostructures. Phys. Rev. Lett. 111, 244502.10.1103/PhysRevLett.111.244502Google Scholar
Pelton, M., Wang, Y., Gosztola, D. & Sader, J. E. 2011 Mechanical damping of longitudinal acoustic oscillations of metal nanoparticles in solution. J. Phys. Chem. C 115 (48), 23732–23740.10.1021/jp207971tGoogle Scholar
Quintanilla, F. H., Fan, Z., Lowe, M. J. S. & Craster, R. V. 2015 Guided waves’ dispersion curves in anisotropic viscoelastic single- and multi-layered media. Proc. R. Soc. Lond. A 471, 20150268.Google Scholar
Rajamuni, M. M., Thompson, M. C. & Hourigan, K. 2018 Transverse flow-induced vibrations of a sphere. J. Fluid Mech. 837, 931–966.10.1017/jfm.2017.881Google Scholar
Ruijgrok, P. V., Zijlstra, P., Tchebotareva, A. L. & Orrit, M. 2012 Damping of acoustic vibrations of single gold nanoparticles optically trapped in water. Nano Lett. 12 (2), 1063–1069.10.1021/nl204311qGoogle Scholar
Sareen, A., Zhao, J., Lo Jacono, D., Sheridan, J., Hourigan, K. & Thompson, M. C. 2018 Vortex-induced vibration of a rotating sphere. J. Fluid Mech. 837, 258–292.10.1017/jfm.2017.847Google Scholar
Saviot, L., Netting, C. H. & Murray, D. B. 2007 Damping by bulk and shear viscosity of confined acoustic phonons for nanostructures in aqueous solution. J. Phys. Chem. B 111 (25), 7457–7461.10.1021/jp071765xGoogle Scholar
Timoshenko, S. P. & Goodier, J. N. 1973 Theory of Elasticity, 3rd edn. McGraw-Hill.Google Scholar
Wu, B., Chen, W. Q. & Zhang, Ch. 2018 On free vibration of piezoelectric nanospheres with surface effect. Mech. Adv. Mater. Struct. 25 (13), 1101–1114.10.1080/15376494.2017.1365986Google Scholar
Wu, B., Su, Y. P., Chen, W. Q. & Zhang, Ch. 2017 On guided circumferential waves in soft electroactive tubes under radially inhomogeneous biasing fields. J. Mech. Phys. Solids 99, 116–145.10.1016/j.jmps.2016.11.004Google Scholar
Wu, B., Zhang, C. L., Chen, W. Q. & Zhang, Ch. 2015 Surface effects on anti-plane shear waves propagating in magneto-electro-elastic nanoplates. Smart Mater. Struct. 24 (9), 095017.10.1088/0964-1726/24/9/095017Google Scholar
Yong, W. A. 2014 Newtonian limit of Maxwell fluid flows. Arch. Rat. Mech. Anal. 214 (3), 913–922.10.1007/s00205-014-0769-2Google Scholar
Yu, K., Major, T. A., Chakraborty, D., Devadas, M. S., Sader, J. E. & Hartland, G. V. 2015 Compressible viscoelastic liquid effects generated by the breathing modes of isolated metal nanowires. Nano Lett. 15 (6), 3964–3970.10.1021/acs.nanolett.5b00853Google Scholar
Zhu, J., Chen, W. Q., Ye, G. R. & Fu, J. Z. 2013 Waves in fluid-filled functionally graded piezoelectric hollow cylinders: a restudy based on the reverberation-ray matrix formulation. Wave Motion 50 (3), 415–427.10.1016/j.wavemoti.2012.10.006Google Scholar
Supplementary material: File

Wu et al. supplementary material

Wu et al. supplementary material

Download Wu et al. supplementary material(File)
File 525.8 KB