Abstract: The dynamic stability of Kelvin viscoelastic pipes conveying fluid with both ends simply supported is investigated. The effect of dimensionless delay time of the viscoelastic material on the curves between dimensionless flow velocity, and real and imaginary components of dimensionless complex frequencies of the pipe conveying fluid in the first two modes are also analyzed. It is shown that when the dimensionless delay time is not greater than 10^-5, Kelvin viscoelastic pipe conveying fluid with both ends simply supported can be considered as elastic pipe conveying fluid. When the dimensionless delay time is greater than 10^-4, the coupled-mode flutter in the Kelvin viscoelastic pipe conveying fluid with both ends simply supported will not take place, but single-mode flutter will do. This is the remarkable difference between the Kelvin viscoelastic pipes conveying fluid and elastic or Maxwell viscoelastic pipes conveying fluid with both ends simply supported.