The flow of a viscoelastic fluid in the annular region between two infinite, eccentrically‐mounted, rotating cylinders is analyzed. The flow is assumed to be sufficiently slow, so that the second‐order Rivlin‐Ericksen constitutive equation is applicable and so that inertial effects may be neglected. The resultant forces on the cylinders, as well as the distribution of their normal and tangential components, are calculated. No restrictions are placed on the ratio of the radii or on the distance between the cylinder axes.