# Fluid Modeling and Boolean Algebra for Arbitrarily Complex Topology in Two Dimensions

## TL;DR

This paper introduces a mathematical model for multiphase fluid flows in two dimensions that captures complex topologies and deformations efficiently using Boolean algebra and topological invariants, enabling robust analysis of fluid behavior.

## Contribution

It presents a novel framework combining regular open semianalytic sets with Boolean operations to handle complex fluid topologies and changes efficiently and accurately.

## Key findings

- Topological features like Betti numbers can be extracted in constant time.
- The model accurately captures topological changes via non-manifold boundary points.
- Boolean operations handle all degenerate cases and complex topologies efficiently.

## Abstract

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.