The equations governing multiphase flows, where a sharp interface separates immiscible fluids or phases, are presented in this chapter. We first derive the equations for flows without interfaces, in a relatively standard manner. Then we discuss the mathematical representation of a moving interface and the appropriate jump conditions needed to couple the equations across the interfaces. Finally, we introduce the so-called “one-fluid” approach, where the interface is introduced as a singular distribution in equations written for the whole flow field. The “one-fluid” form of the equations plays a fundamental rôle for the numerical methods discussed in the rest of the book.

General principles

The derivation of the governing equations is based on three general principles: the continuum hypothesis, the hypothesis of sharp interfaces, and the neglect of intermolecular forces. The assumption that fluids can be treated as a continuum is usually an excellent approximation. Real fluids are, of course, made of atoms or molecules. To understand the continuum hypothesis, consider the density or amount of mass per unit volume. If this amount were measured in a box of sufficiently small dimensions ℓ, it would be a wildly fluctuating quantity (see Batchelor (1970), for a detailed discussion). However, as the box side ℓ increases, the density becomes ever smoother, until it is well approximated by a smooth function ρ. For liquids in ambient conditions this happens for ℓ above a few tens of nanometers (1 nm = 10−9 m).