A liquid film can flow between two solid surfaces in close proximity due to capillary effects. Such flow occurs in natural processes such as the wetting of soils, drainage through rocks, water rise in plants and trees, as well as in engineering applications such as liquid flow in nanofluidic systems and the development of liquid bridges within small-scale devices. In this work, a numerical model is formulated to describe the radial capillary-driven flow between two contacting, elastic, annular rough surfaces. A mixed lubrication equation with capillary-pressure boundary conditions is solved for the pressure within the liquid film and both macro- and micro-contact models are employed to account for solid–solid contact pressures and interfacial deformation. Measurements of interfacial spreading rate are performed for liquids of varying viscosity flowing between an optical flat and a metallic counter surface. Good agreement is found between modeling and experiment. A semi-analytical relation is developed for the capillary flow between the two contacting surfaces.