Strain gradient beam model for dynamics of microscale pipes conveying fluid

Based on the strain gradient theory, we present a microstructure-dependent Bernoulli–Euler model to analyze the vibration and stability of microscale pipes conveying fluid. The equation of motion and boundary conditions are derived using Hamilton’s principle. The proposed strain gradient beam model contains three material length scale parameters to capture the size effect. This new model may be reduced to the modified couple stress beam model when two of these three material length scale parameters vanish and may be reduced to the classical beam model in the absence of all the material length scale parameters. From the numerical calculations for micropipes with both ends positively supported, it is found that the natural frequency and the critical flow velocity are size-dependent. The results show that the microscale pipe displays remarkable size effect when its outside diameter becomes comparable to the material length scale parameter, while the size effect is almost diminishing as the diameter is far greater than the material length scale parameter. Moreover, the size effect predicted by the current strain gradient beam model is stronger than that predicted by the modified couple stress beam model, since two other material length scale parameters have been accounted for in the former.