Size-dependent effects on critical flow velocity of fluid-conveying microtubes via nonlocal strain gradient theory

Size-dependent Timoshenko and Euler-Bernoulli models are derived for fluid-conveying microtubes in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions are deduced by employing the Hamilton principle. A flow-profile-modification factor, which is related to the flow velocity profile, is introduced to consider the size-dependent effects of flow. The analytical solutions of predicting the critical flow velocity of the microtubes with simply supported ends are derived. By choosing different values of the nonlocal parameter and the material length scale parameter, the critical flow velocity of the nonlocal strain gradient theory can be reduced to that of the nonlocal elasticity theory, the strain gradient theory, or the classical elasticity theory. It is shown that the critical flow velocity can be increased by increasing the flexural rigidity, decreasing the length of tube, decreasing the mass density of internal flow, or increasing the shear rigidity. The critical flow velocity can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter. The flow-profile-modification factor can decrease the critical flow velocity. The critical flow velocity predicted by classical elasticity theory is generally larger than that of nonlocal strain gradient theory when considering the size-dependent effect of flow.