© Published under licence by IOP Publishing Ltd. The classical boundary conditions at flat liquid-liquid interfaces are continuity of the velocity and of the tangential component of stress; for curved interfaces, one also demands that the jump in the normal stress at the interface is balanced by the product of the interfacial tension and curvature. While these conditions are widely accepted, and are often used at the macro scale, the recent interest in micro and nano-fluidics challenges their validity. At molten polymer-polymer interfaces, for instance, it has been consistently shown by direct and indirect measurements that, apparent, velocity jumps exist and can be modelled effectively via a Navier slip condition (NSC). Here, we discuss that if a viscosity, which accounts for the density and mole fraction distributions, is included in the Navier-Stokes equations, we can describe, naturally, and without recourse to ad-hoc models such as the NSC, the velocity profile in the interfacial region separating two fluids. This approach is supported by the observation that there is a relation between apparent slip and density distribution across the interface.