The present work develops a numerical method for the solution of rotating internal weakly viscoelastic flows in rectangular ducts for dimensionless parameters such as the Reynolds, Rossby and Weissenberg numbers, taken respectively in the intervals between 171 and 12 000, 0.047 and 1/12 and up to 1/10 000. It is shown that the usual counter-rotating double-vortex configuration of secondary flow breaks down with the increase of the Reynolds number (over the threshold of 171). For higher Reynolds numbers such as 7500 and 12 000 the secondary flow diffuses to the interior of the duct where it assumes a fully developed configuration and the transition to the turbulance structure is observed. The Sobolev norms increase almost proportionally to the increase of the Reynolds number, and play an essential role for more complex problems involving transition to turbulance modelling. Copyright © 2002 John Wiley & Sons, Ltd.