# Phase-Field Modeling of Two-Phase Flows: A Projection-Based Cahn-Hilliard-Navier-Stokes Framework

## TL;DR

This paper develops a projection-based phase-field model combining Cahn-Hilliard and Navier-Stokes equations for simulating complex two-phase flows with diffuse interfaces, addressing numerical challenges and validating against benchmark problems.

## Contribution

It introduces a decoupled pressure-projection numerical scheme for the coupled CH-NS system with variable properties, enabling stable and accurate multiphase flow simulations.

## Key findings

- Accurately predicts bubble rise velocity and interface shape.
- Successfully captures hydrodynamic instabilities like Plateau-Taylor.
- Framework is extendable to heat transfer and phase change phenomena.

## Abstract

The coupled Cahn-Hilliard and Navier-Stokes (CH-NS) equations provide a powerful framework for modeling multiphase flows with diffuse interfaces, enabling simulations of droplet breakup, bubble dynamics, and hydrodynamic instabilities. These capabilities are vital in boiling heat transfer, microfluidics, coating, additive manufacturing, and oil-water separation, where resolving fluid-fluid interactions is essential. Numerically, the CH-NS system is challenging: the Cahn-Hilliard equation involves higher-order derivatives and nonlinearities, and coupling with Navier-Stokes introduces strong two-way interactions. The velocity field advects the phase field, while the evolving interface alters density and viscosity, feeding back into the flow. Variable-density and variable-viscosity systems further increase complexity, requiring accurate treatment of property contrasts without losing stability or mass conservation. To address this, we employ a decoupled pressure-projection method with finite differences on staggered grids and explicit Euler time stepping. Our formulation extends the CH-NS system to homogeneous and variable-property fluids with consistent hydrodynamic-phase-field coupling. Validation against canonical benchmarks-including bubble rise and Plateau-Taylor instability shows excellent agreement in rise velocity, interface shape, and instability wavelength. This framework establishes a reproducible foundation for multiphysics extensions such as heat transfer, phase change, and electrohydrodynamics in boiling, droplet manipulation, and electronics cooling