Research Projects

Moving boundary problems arise in many physical and engineering systems such as melting, solidification, fluid–structure interaction, crack propagation, tumor growth, and multi-phase flows. These problems involve interfaces that change their shape and position over time, often in complex ways. Capturing such motion accurately is challenging, especially when the interface becomes sharp, breaks or merges, or interacts with irregular shapes. Traditional numerical methods often struggle in these cases due to the need for remeshing or manual tracking of interfaces. The goal of this research is to overcome these challenges by developing a novel numerical framework that combines a thermodynamically consistent, conservative, and energy-stable phase-field model with an adaptive wavelet-based grid refinement technique and a meshfree spatial discretization method using local Radial Basis Functions (RBFs). This method aims to accurately simulate moving boundaries without remeshing, while ensuring important physical properties like mass and energy are preserved, and is expected to work better and faster than traditional techniques. The research will begin with the development of the model and then validation using well-known benchmark problems such as the Stefan problem and Mullins–Sekerka instability. After validation, the framework will be applied to practical case studies: (a) solute dissolution problems such as sugar dissolving in water or hot tea, where the model will predict the time required for complete dissolution, the evolution of the dissolving shape and the resulting fluid flow patterns; (b) tumor growth modeling, where the interest will be to see how the tumor interface evolves over time and how biological factors like nutrient diffusion affect its shape and size; and (c) swimming in fluids, such as fish swimming in rivers or athletes in pools, where the model will help predict swimming speed, flow patterns and how body shape or movement affects performance. These applications will demonstrate how flexible and powerful the proposed method is for real-life problems. This study is expected to provide a simple and robust general-purpose simulation tool for accurately capturing complex moving boundaries without the need for remeshing which can be used across a wide range of physical, biological and engineering applications.