Using a new technique involving light reflection, the interfacial curvature of a meniscus formed by a well wetting liquid steadily displacing a gas in a glass capillary has been measured down to about 50 nm from the solid. Within the domain explored, the measured meniscus curvature increases strongly as the wall is approached, in agreement with classical models which make use of the continuum approximation, no slip, etc. The inner length scale, at which such models fail, is inferred from the measurements to be of the order of a molecular dimension, suggesting that non-continuum effects dominate. A comparison of measured dynamic contact angles of liquid-liquid pairs of large viscosity ratio with a model developed earlier by the authors, incorporating such an inner length scale, suggests that the true contact angle in the advancing fluid increases significantly with line speed in one of the cases.