Multiphase flow phenomena arise in a wide range of natural and engineering applications. When two or more macroscopically immiscible fluids, such as oil and water, come into contact, they are separated by a thin interfacial transition layer. Although the thickness of this layer is often much smaller than the characteristic length scales of the flow, it is commonly approximated as a zero-thickness interface, leading to the classical sharp-interface formulation. In this framework, governing equations are defined separately within each fluid domain, together with appropriate jump conditions imposed across the interface.

However, the sharp-interface description becomes inadequate when the interface undergoes topological changes, such as pinch-off or reconnection, where geometric quantities like curvature may become singular. Furthermore, numerical studies have demonstrated that the finite thickness of the interfacial layer significantly influences the dynamics when the radius of curvature or the spacing between interfaces approaches the interfacial thickness scale.

An alternative framework is provided by diffuse-interface, or phase-field, models. In these approaches, the interface possesses a finite thickness due to molecular or chemical diffusion between fluid components, resulting in continuous variation of field variables, such as the order parameter, across the interfacial region. Phase-field models naturally accommodate smooth topological transitions and are typically formulated through the coupling of a Cahn-Hilliard-type equation with the governing fluid equations, including capillary effects in a thermodynamically consistent manner.

From a computational perspective, phase-field models can be implemented on fixed grids without the need for explicit interface tracking. Nevertheless, their numerical solution remains challenging because the governing equations constitute a strongly coupled, nonlinear, and higher-order system with significant stiffness arising from steep gradients across the diffuse interface. In this context, our research focuses on the development and analysis of unconditionally stable, high-order, and decoupled numerical schemes for phase-field fluid models. We further investigate phase-field formulations for complex multiphase flow phenomena in a variety of physical settings.

Phase separation in binary-fluid systems through spinodal decomposition is a fundamental phenomenon encountered in materials science, soft matter physics, and multiphase fluid dynamics. When a homogeneous binary mixture is rapidly quenched into a thermodynamically unstable region, spontaneous amplification of concentration fluctuations leads to the formation of distinct phases without the need for nucleation. The resulting dynamics are governed by the intricate interplay between diffusion, interfacial energy, and hydrodynamic interactions, giving rise to complex coarsening patterns and evolving domain morphologies. In our research, we investigate the mathematical modeling and high-fidelity numerical simulation of spinodal decomposition using the phase-field approach, with particular emphasis on accurately capturing interfacial dynamics, topological transitions, and long-time coarsening behavior in binary-fluid systems.