A thin-film approximation is used to study the effects of surface tension on a thin liquid layer lining the interior of a cylindrical tube, where the tube has radius a and a centreline with weak, uniform curvature δ/a. Centreline curvature induces a pressure gradient in the fluid layer, analogous to that due to a weak gravitational field, that drives fluid from the inner to the outer wall of the tube, i.e. away from the centre of centreline curvature. The resulting draining flow is computed numerically under the assumption of axial uniformity, and the large-time asymptotic draining regimes and flow structures are identified. In the absence of destabilizing intermolecular interactions, the inner wall remains wet, covered with a vanishingly thin fluid layer, while a near-equilibrium lobe forms on the outer wall. The stability of this quasi-static lobe to axial variations is then investigated by using numerical and perturbation methods to solve the linearized Young–Laplace equation, prescribing zero contact angle at the lobe's free boundary. Conditions on δ, the fluid volume a3V and the tube length aL are identified separating axially uniform lobes (which are stable for low V/(δL) or small L), wavy lobes (some with a solitary structure) and localized fluid droplets (which exist for sufficiently large V/δ and L). Hysteresis is demonstrated between multiple equilibria, the topology of which can change dramatically as parameters are varied. The application of these results to lung airways is discussed.