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 torsional mode to any general spheroidal mode.
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