This paper reports the results of an experimental investigation to determine the climbing constants of viscoelastic liquids in the rotating rod geometry with two layers of viscoelastic liquids and the results of an investigation which describes the effects of eccentricity on the free surface shape of a non-Newtonian liquid in motion between eccentric cylinders. The former set of results shows the rotating rod geometry with a layered medium of viscoelastic liquids may be used as a free surface viscometer yielding reliable and reproducible values for the sequentially determined climbing constants of those liquids. The latter set of results indicates that the concentric cylinder geometry is not a good free surface viscometer. Very small eccentricities generate large circumferential pressure gradients which overwhelm small normal stress effects. There is good agreement between experimental and second order predictions for small eccentricities and rotational speeds and fluids which display large normal stress effects. © 1984 Birkhäuser Verlag.