We study non-Newtonian fluid displacements in horizontal narrow eccentric annuli in the situation where the inner cylinder is moving. This represents a practically important extension of the model analysed by Carrasco-Teja et al. (J. Fluid Mech., vol. 605, 2008, pp. 293–327). When motion of the inner cylinder is included, the Hele-Shaw model closure becomes significantly more complex and extremely costly to compute, except for Newtonian fluids. In the first part of the paper we address the model derivation and closure relations. The second part of the paper considers the limit of large buoyancy number, in which the interface elongates along the annulus. We derive a lubrication-style model for this situation, showing that the leading-order interface is symmetric. Rotation of the inner cylinder only affects the length of the leading-order interface, and this occurs only for non-Newtonian fluids via shear-thinning effects. At first order, casing rotation manifests in an asymmetrical ‘shift’ of the interface in the direction of the rotation. We also derive conditions on the eccentricity, fluid rheology and inner cylinder velocity, under which we are able to find steady travelling wave displacement solutions.
