At interfaces between different fluids, properties such as density, viscosity, and molecular cohesion are discontinuous. To animate small-scale details of incompressible viscous multi-phase fluids realistically, we focus on the discontinuities in the state variables that express these properties. Surface tension of both free and bubble surfaces is modeled using the jump condition in the pressure field; and discontinuities in the velocity gradient field, driven by viscosity differences, were also considered. To obtain derivatives of the pressure and velocity fields with sub-grid accuracy, they are extrapolated across interfaces using continuous variables based on physical properties. The numerical methods that we present are easy to implement and do not impact the performance of existing solvers. Small-scale fluid motions, such as capillary instability, breakup of liquid sheets, and bubbly water can all be successfully animated.