Abstract: On the basis of some studies of elastic pipe conveying fluid, the dynamic behavior and stability of Maxwell viscoelastic pipes conveying fluid with both ends simply supported, which are gyroscopic conservative system, were investigated by using the finite difference method and the corresponding recurrence formula. The effect of relaxation time of viscoelastic materials on the variation curve between dimensionless flow velocity and the real part and imaginary part of dimensionless complex frequencies in the first-three-order modes were analyzed concretely. It is found that critical flow velocities of divergence instability of Maxwell viscoelastic pipes conveying fluid with both ends simply supported decrease with the decrease of the relaxation time, while after the onset of divergence instability (buckling) critical flow velocities of coupled-mode flutter increase with the decrease of the relaxation time. Particularly, in the case of greater mass ratio, with the decrease of relaxation time, the onset of coupled-mode flutter delays, and even does not take place. When the relaxation time is greater than 10³, stability behavior of viscoelastic pipes conveying fluid is almost similar to the elastic pipes conveying fluid.