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Studies in two dimensions have been made of the evolution of an initially linear interface between two liquid areas by both continuum hydrodynamics and molecular dynamics. It was found that at both levels the interface evolves into a fractal curve. While the limiting fractal dimensionalities (D) of the continuum and molecular interfaces differ, this difference is no greater than that that can be induced in the hydrodynamic results by marked changes in the initial conditions and/or fluid parameters. There is, however, a marked qualitative difference in the initial values of D. This is discussed and shown to be a natural consequence of the molecular discreteness of the real liquids.