Investigation of the Effect of Fluid Field on the Vibrations of a Curved Microbeam Based on Nonclassical Continuum Mechanics under Elastic Supports

Microbeams play a crucial role in mechanical processes and in the design of micro- and nanoscale structures. In many engineering applications, these structures are supported by elastic foundations, which can significantly affect their vibrational and stability characteristics. When subjected to external loading and internal fluid flow, the interaction between the microbeam and its elastic support must be accurately modeled to predict realistic dynamic behavior. The mechanical stability analysis of such structures, especially in the presence of internal fluid flow, is of great significance. In this study, the stability of a curved microbeam resting on an elastic foundation and containing an internal fluid flow is investigated. To account for the nanoscale effects, the nonlocal couple stress theory is employed for the solid domain, while the modified velocity theory is adopted for the fluid part. The internal fluid flow is analyzed using the Navier–Stokes equations, and the governing equations of the microbeam are derived from Hamilton’s principle. The fluid–structure interaction is modeled as a two-way coupled phenomenon. The governing differential equations are solved numerically using the Galerkin method combined with Gauss quadrature integration. The results reveal that size effects play a significant role in the stability analysis, and neglecting nonclassical continuum mechanics may result in considerable errors. Small curvatures are found to have a noticeable influence on the vibrational behavior of the microbeam system. Increasing the fluid flow velocity enhances system instability and reduces the natural frequency. Furthermore, the type of fluid and the stiffness of the elastic foundation have a remarkable impact on the system’s dynamic response. Overall, this study demonstrates that accurate mechanical modeling of microbeams, incorporating both size-dependent effects and elastic foundation interactions, can provide valuable insights for the design and optimization of micro- and nanoscale structures.