A leading order solution for thermocapillary, buoyancy, and shear-driven flow within a thin. fixed-length, annular fluid collar is determined. The solution is based on the assumption that C/epsilon3 = O(1), where C is the capillary number and epsilon is the collar aspect ratio. In contrast to earlier models of flow within bridges, boundary layer matching near the end regions is not required since end walls are absent. The leading order velocity field is self-similar in the radial direction with the size of the similarity profiles determined by an unspecified surface heat flux distribution, an unspecified surface shear distribution, and by the unknown capillary surface shape. Conditions leading to thermocapillary, buoyant, or shear driven single and multi-cellular flow are identified. The capillary surface shape is determined using a minimization technique, with the object function defined as the difference between known and iteratively calculated collar volumes. The surface and flow solutions are illustrated using an exponentially decaying surface heat flux and a constant external shear.