New approach including Navier’s slip condition to predict vibration signature of immersed nanoparticles

The strain gradient theory is widely recognized for addressing scale effects. In this study, it is applied to predict the natural frequencies and quality factors of nanospheres immersed in a semi-infinite, incompressible, and homogeneous viscoelastic fluid. The study emphasizes the influence of the second gradient model’s sign on the vibrational behavior of nanospheres. The frequency equations are derived by enforcing continuity conditions at the fluid-nanosphere interface, including the Navier slip condition. The findings reveal that the Navier slip length has negligible impact when the nanosphere’s size exceeds \(200\,\textrm{nm}\). Additionally, a second gradient model with a negative sign produces physically unrealistic results, as the phase velocity surpasses that of the classical continuum and becomes unbounded. Consequently, the frequency equations presented in this paper provide valuable insights for interpreting Raman spectroscopy data of nanospheres in viscoelastic fluids.