- A homogeneous thin polymer film with a stepped height profile levels due to the presence of Laplace pressure gradients. Here we report on studies of polymeric samples with precisely controlled, spatially inhomogeneous molecular weight distributions. The viscosity of a polymer melt strongly depends on the chain length distribution; thus, we learn about thin-film hydrodynamics with viscosity gradients. These gradients are achieved by stacking two films with different molecular weights atop one another. After a sufficient time these samples can be well described as having one dimensional viscosity gradients in the plane of the film, with a uniform viscosity normal to the film. We develop a hydrodynamic model that accurately predicts the shape of the experimentally observed self-similar profiles. The model allows for the extraction of a capillary velocity, the ratio of the surface tension and the viscosity, in the system. The results are in excellent agreement with capillary velocity measurements of uniform mono- and bi-disperse stepped films and are consistent with bulk polymer rheology.