Averaged Navier-Stokes equations with turbulent coefficients and the continuity equation are applied to a theoretical investigation of the effects of inlet swirl velocity on the dynamic characteristics of annular plain seals employed in pumps. The turbulent coefficients are calculated based on the "law of the wall" and Prandtl's mixing length hypothesis. The numerical results show that for any eccentricity ratio at an equilibrium position of the rotor center and for any rotor spinning velocity, the inlet swirl velocity significantly influences the cross-coupled stiffness terms, whose magnitudes increase as the inlet swirl velocity increases in the direction of the rotor spin. This phenomenon yields an increase of the whirl-frequency ratio, resulting in a narrower stable range of the rotor spinning velocity.