  We propose a mathematical model for fluids in multiphase flows in order to
establish a solid theoretical foundation for the study of their complex
topology, large geometric deformations, and topological changes such as
merging. Our modeling space consists of regular open semianalytic sets with
bounded boundaries, and is further equipped with constructive and algebraic
definitions of Boolean operations. Major distinguishing features of our model
include (a) topological information of fluids such as Betti numbers can be
easily extracted in constant time, (b) topological changes of fluids are
captured by non-manifold points on fluid boundaries, (c) Boolean operations on
fluids correctly handle all degenerate cases and apply to arbitrarily complex
topologies, yet they are simple and efficient in that they only involve
determining the relative position of a point to a Jordan curve and intersecting
a number of curve segments. Although the main targeting field is multiphase
flows, our theory and algorithms may also be useful for related fields such as
solid modeling, computational geometry, computer graphics, and geographic
information system.
