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Analytical approach for predicting vibration characteristics of an embedded elastic sphere in complex fluid

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Abstract

Vibration characteristics of elastic nanostructures embedded in fluid medium have been used for biological and mechanical sensing and also to investigate the materials mechanical properties. The fluid medium surrounding the nanostructure is typically modeled as a Newtonian fluid. A novel approach based on the exact theory has been developed in this paper, to accurately predict the various vibration scenarios of an elastic sphere, in a compressible viscous fluid. Then, the analysis is extended to a viscoelastic medium using the Maxwell fluid model. To demonstrate the accuracy of the present approach, a comparison is made with the published theoretical results in the literature in some particular cases, which shows a very good agreement. The effects of fluid compressibility and viscoelasticity are discussed in details, and we demonstrate that the fluid compressibility plays a significant role in the vibration modes of an elastic sphere. Results also show that the different vibration modes of a sphere trigger a viscoelastic response in water–glycerol mixtures similar to that of literature. In addition, the obtained results can serve as benchmark solution in design of liquid sensors.

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Authors and Affiliations

  1. LAMPA, Arts et Metiers Institute of Technology, 2 Boulevard du Ronceray, 49035, Angers, France

    Xi Yang, Adil El Baroudi & Jean Yves Le Pommellec

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  1. Xi Yang
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  2. Adil El Baroudi
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Appendix A

Appendix A

Table 7 Values of material parameters. \(\chi \) is the concentration of glycerol in water
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Table 8 Fluid parameters for different glycerol mass fractions \(\chi \)
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\(\bullet \) Matrix elements given in Eqs. (18) and (19) for the torsional and spheroidal vibrations (see Tables 7, 8):

$$\begin{aligned} T_{11}&=I_{n+\frac{1}{2}}\left( k_{s}a\right) \\ T_{12}&=K_{n+\frac{1}{2}}\left( k_{s}a\right) \\ T_{13}&=J_{n+\frac{1}{2}}\left( K_{s}a\right) \\ T_{21}&=(n+2)I_{n+\frac{1}{2}}\left( k_{s}a\right) -k_{s}aI_{n-\frac{1}{2}}\left( k_{s}a\right) \\ T_{22}&=(n+2)K_{n+\frac{1}{2}}\left( k_{s}a\right) +k_{s}aK_{n-\frac{1}{2}}\left( k_{s}a\right) \\ T_{23}&=(n+2)J_{n+\frac{1}{2}}\left( K_{s}a\right) -K_{s}aJ_{n-\frac{1}{2}}\left( K_{s}a\right) \\ T_{31}&=I_{n+\frac{1}{2}}\left( k_{s}b\right) \\ T_{32}&=K_{n+\frac{1}{2}}\left( k_{s}b\right) \\ S_{11}&=nJ_{n+\frac{1}{2}}\left( k_{c}a\right) -k_{c}aJ_{n+\frac{3}{2}}\left( k_{c}a\right) \\ S_{12}&=nY_{n+\frac{1}{2}}\left( k_{c}a\right) -k_{c}aY_{n+\frac{3}{2}}\left( k_{c}a\right) \\ S_{13}&=n(n+1)I_{n+\frac{1}{2}}\left( k_{s}a\right) \\ S_{14}&=n(n+1)K_{n+\frac{1}{2}}\left( k_{s}a\right) \\ S_{15}&=nJ_{n+\frac{1}{2}}\left( K_{c}a\right) -K_{c}aJ_{n+\frac{3}{2}}\left( K_{c}a\right) \\ S_{16}&=n(n+1)J_{n+\frac{1}{2}}\left( K_{s}a\right) \\ S_{21}&=nJ_{n+\frac{1}{2}}\left( k_{c}a\right) \\ S_{22}&=nY_{n+\frac{1}{2}}\left( k_{c}a\right) \\ S_{23}&=k_{s}aI_{n-\frac{1}{2}}\left( k_{s}a\right) -nI_{n+\frac{1}{2}}\left( k_{s}a\right) \\ S_{24}&=-k_{s}aK_{n-\frac{1}{2}}\left( k_{s}a\right) -nK_{n+\frac{1}{2}}\left( k_{s}a\right) \\ S_{25}&=nJ_{n+\frac{1}{2}}\left( K_{c}a\right) \\ S_{26}&=K_{s}aJ_{n-\frac{1}{2}}\left( K_{s}a\right) -nJ_{n+\frac{1}{2}}\left( K_{s}a\right) \\ S_{31}&=2k_{c}aJ_{n-\frac{1}{2}}\left( k_{c}a\right) -\left( n^2+3n+2+\frac{k_{c}^{2}a^2}{2}\right) J_{n+\frac{1}{2}}\left( k_{c}a\right) \\ S_{32}&=2k_{c}aY_{n-\frac{1}{2}}\left( k_{c}a\right) -\left( n^2+3n+2+\frac{k_{c}^{2}a^2}{2}\right) Y_{n+\frac{1}{2}}\left( k_{c}a\right) \\ S_{33}&=n(n+1)\left[ \left( n+2\right) I_{n+\frac{1}{2}}\left( k_{s}a\right) -k_{s}aI_{n-\frac{1}{2}}\left( k_{s}a\right) \right] \\ S_{34}&=n(n+1)\left[ \left( n+2\right) K_{n+\frac{1}{2}}\left( k_{s}a\right) +k_{s}aK_{n-\frac{1}{2}}\left( k_{s}a\right) \right] \\ S_{35}&=2K_{c}aJ_{n-\frac{1}{2}}\left( K_{c}a\right) -\left( n^2+3n+2-\frac{K_{s}^{2}a^2}{2}\right) J_{n+\frac{1}{2}}\left( K_{c}a\right) \\ S_{36}&=n(n+1)\left[ \left( n+2\right) J_{n+\frac{1}{2}}\left( K_{s}a\right) -K_{s}aJ_{n-\frac{1}{2}}\left( K_{s}a\right) \right] \\ S_{41}&=\left( n+2\right) J_{n+\frac{1}{2}}\left( k_{c}a\right) -k_{c}aJ_{n-\frac{1}{2}}\left( k_{c}a\right) \\ S_{42}&=\left( n+2\right) Y_{n+\frac{1}{2}}\left( k_{c}a\right) -k_{c}aY_{n-\frac{1}{2}}\left( k_{c}a\right) \\ S_{43}&=k_{s}aI_{n-\frac{1}{2}}\left( k_{s}a\right) -\left( n^2+2n+\frac{k_{s}^2a^2}{2}\right) I_{n+\frac{1}{2}}\left( k_{s}a\right) \\ S_{44}&=-k_{s}aK_{n-\frac{1}{2}}\left( k_{s}a\right) -\left( n^2+2n+\frac{k_{s}^2a^2}{2}\right) K_{n+\frac{1}{2}}\left( k_{s}a\right) \\ S_{45}&=\left( n+2\right) J_{n+\frac{1}{2}}\left( K_{c}a\right) -K_{c}aJ_{n-\frac{1}{2}}\left( K_{c}a\right) \\ S_{46}&=K_{s}aJ_{n-\frac{1}{2}}\left( K_{s}a\right) -\left( n^2+2n-\frac{K_{s}^2a^2}{2}\right) J_{n+\frac{1}{2}}\left( K_{s}a\right) \\ S_{51}&=nJ_{n+\frac{1}{2}}\left( k_{c}b\right) -k_{c}bJ_{n+\frac{3}{2}}\left( k_{c}b\right) \\ S_{52}&=nY_{n+\frac{1}{2}}\left( k_{c}b\right) -k_{c}bY_{n+\frac{3}{2}}\left( k_{c}b\right) \\ S_{53}&=n(n+1)I_{n+\frac{1}{2}}\left( k_{s}b\right) \\ S_{54}&=n(n+1)K_{n+\frac{1}{2}}\left( k_{s}b\right) \\ S_{61}&=nJ_{n+\frac{1}{2}}\left( k_{c}b\right) \\ S_{62}&=nY_{n+\frac{1}{2}}\left( k_{c}b\right) \\ S_{63}&=k_{s}bI_{n-\frac{1}{2}}\left( k_{s}b\right) -nI_{n+\frac{1}{2}}\left( k_{s}b\right) \\ S_{64}&=-k_{s}bK_{n-\frac{1}{2}}\left( k_{s}b\right) -nK_{n+\frac{1}{2}}\left( k_{s}b\right) \end{aligned}$$

\(\bullet \) Material parameters used in this work were derived from [26, 40] :

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Yang, X., El Baroudi, A. & Le Pommellec, J.Y. Analytical approach for predicting vibration characteristics of an embedded elastic sphere in complex fluid. Arch Appl Mech 90, 1399–1414 (2020). https://doi.org/10.1007/s00419-020-01674-7

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