In this paper, a novel analytical approach based on nonlocal strain gradient theory is proposed to investigate small-scale effects on the radial vibration of isotropic spherical nanoparticles interacting with a viscoelastic fluid. The viscoelastic fluid is assumed to be compressible and takes into account both compressional and shear relaxation processes. The frequency equation is derived by imposing the fluid-nanosphere interface continuity conditions. To demonstrate the validity and accuracy of the approach, a comparison is made with the literature results in some particular cases, which shows a good agreement. Numerical examples are finally conducted to reveal the significance of small-scale effects in the radial vibrations, which need to be included in the nonlocal strain gradient model of submerged spherical nanoparticles. It is found that the vibration behavior greatly depends on the nanosphere size, nonlocal parameter, strain gradient parameter and glycerol-water mixture. Particularly, the small-scale effects play a very pronounced role when the spherical gold nanoparticle radius is smaller than 3 nm. Thus, the obtained frequency equation is very useful to interpret the experimental measurements of vibrational characteristics of nanospheres submerged in a viscoelastic fluid.

在本文中，提出了一种基于非局部应变梯度理论的新型分析方法，用于研究各向同性球形纳米粒子与粘弹性流体相互作用的径向振动的小尺度效应。假设粘弹性流体是可压缩的，并考虑压缩和剪切松弛过程。通过施加流体-纳米球界面连续性条件导出频率方程。为了证明该方法的有效性和准确性，在一些特定案例中与文献结果进行了比较，结果吻合较好。最后通过数值例子揭示了径向振动中小尺度效应的重要性，这些效应需要包含在水下球形纳米粒子的非局部应变梯度模型中。研究发现，振动行为很大程度上取决于纳米球尺寸、非局部参数、应变梯度参数和甘油-水混合物。特别是当球形金纳米颗粒半径小于3 nm时，小尺度效应发挥着非常显着的作用。因此，所获得的频率方程对于解释浸没在粘弹性流体中的纳米球的振动特性的实验测量非常有用。