Singularities of the Navier-Stokes Equations in Differential Form at the Interface between Air and Water

The properties of the Navier-Stokes equations are extremely important for theoretical and numerical studies of fluid flow. This chapter investigates the properties of the differential and integral forms of the NavierStoke equations for immiscible air-water flow. The analysis reveals that unlike other fluids the immiscible air-water flow is so special that the Navier-Stokes equations in the differential form have singularities at the interface due to the continuous movement of interface and discontinuous density. In contrast to the differential form, the Navier-Stokes equations in integral form hold well every where including at the interface, indicating the integral form can be widely used in the computation of air-water flow.