Mathematical Modeling and Analysis of Simultaneous Melting, Solidification Phenomena With Nonlinear Stefan Problem Including Thomson Effect and Heat Source

Abstract This paper presents a comprehensive mathematical analysis of a three-phase Stefan problem that captures the complex thermal and phase-change phenomena occurring in electrical contact materials during arcing events. The model addresses the physically critical interplay between simultaneous vaporization, melting, and solidification processes, explicitly incorporating the Thomson effect–a reversible thermoelectric phenomenon where heat is absorbed or released when electric current flows through temperature gradients–and Joule heating from electrical resistance. Unlike classical Stefan formulations, the model accounts for temperature-dependent thermal conductivity, specific heat, and density across vapor, liquid, and solid phases, along with variable cross-section geometry that represents realistic micro-asperity contact structures. The physical significance of this work lies in its ability to predict localized thermal behavior and material degradation mechanisms under extreme conditions. By integrating the Thomson effect into the generalized heat equation, the model captures non-classical heat transport phenomena at the microscale, explaining asymmetric heating patterns and thermal explosions observed experimentally in electrical contacts. The bidirectional coupling between electrical current and temperature gradients creates heat sources and sinks that fundamentally alter the phase transition dynamics, leading to erosion patterns and contact degradation that cannot be explained by classical heat conduction alone. The variable geometry formulation accounts for current concentration at contact asperities, where intense localized heating drives the formation and evolution of molten pools and vapor regions. From a mathematical perspective, the governing system of nonlinear partial differential equations with two moving boundaries is transformed using dimensionless variables and self-similar transformations into a boundary value problem for nonlinear ordinary differential equations. Integral equations corresponding to the liquid and solid phases are derived, converting the free-boundary problem into a fixed-domain formulation. The existence and uniqueness of solutions are rigorously established using the Banach fixed-point theorem, with sufficient conditions for contraction mappings proven and bounds for the integral operators obtained. These a priori estimates provide quantitative control over solution behavior and guarantee the well-posedness of the physical model. The results provide a rigorous theoretical foundation for understanding and predicting the behavior of electrical contacts under high current densities, with direct implications for the design and optimization of switching devices, circuit breakers, and electrical connectors. The model enables quantitative prediction of material loss rates, erosion patterns, and contact lifetime, supporting the development of next-generation electrical contact materials with enhanced durability and performance. By capturing the coupled thermal-electrical-phase-change processes, including the often-neglected Thomson effect, this framework offers insights into microscale phenomena that govern macroscopic device reliability and efficiency. The mathematical methodology developed here–combining self-similar transformations, fixed-domain reformulation, and operator-theoretic analysis–is generalizable to other multi-phase free-boundary problems in physics and engineering, including welding, additive manufacturing, and materials processing applications involving rapid phase transitions and coupled thermal-electrical effects.