Based on a modified couple stress theory, the governing equation for the motion of the micro?scale cantilever pipe is derived by using Hamilton′s principle, where the geometric nonlinearity arising from the Lagrange strain tensors is taken into account. An eigenvalue analysis for linear equation is performed to examine the effect of internal material length scale parameter on the graph of critical flow velocity versus mass ratio and the complex frequencies of the four lowest modes of the micro?pipe. It is found that these curves of critical flow velocity versus mass ratio for the micro?scale pipes with different material length scale parameters may intersect each other. At each critical velocity, the first Lyapunov′s coefficient and the derivation of the degenerate eigenvalue with respect to flow velocity are calculated by employing the projection methods based on the center manifold theory and normal form method, which demonstrates that the bifurcation is supercritical. The dynamics at the hysteresis and intersection points of the curve of critical flow velocity versus mass ratio are also investigated, and the bifurcation diagrams inwards different directions are then detected.