Abstract
This paper presents a mathematical model and analytical study of the thermal dynamics in an electrical contact bridge under the influence of the Thomson effect and Joule heat generation. The model considers a bridge structure adjacent to a vapor zone, in which temperature evolution is governed by a nonlinear heat equation, featuring temperature-dependent thermal and thermoelectric coefficients, as well as an internal Joule heat source. The analysis introduces dimensionless variables and employs a self-similar transformation to reduce the problem to a boundary value problem for nonlinear ordinary differential and integral equations. The existence and uniqueness of the similarity solution are established via fixed point theory under appropriate conditions on the nonlinear coefficients. Analytical results are obtained for the case of constant coefficients, while the general nonlinear case is treated with an integral approach. Additionally, special cases such as linearly temperature-dependent Thomson and thermal coefficients are examined to illustrate parameter sensitivity. The results describe how variations in the Thomson effect, Joule heating, and material properties influence the temperature field, bridge opening, and boiling front propagation, providing a deeper understanding of coupled thermoelectric and phase-change processes in electrical contacts. The findings provide a rigorous mathematical basis for simulating temperature fields in electrical contacts with moving boundaries and for understanding the influence of thermoelectric effects in current-carrying devices.