Mathematical Analysis of a Fully Coupled Nonlinear Dynamical System with Phase Transition Effects

Source abstract

This study presents a fully coupled nonlinear dynamical model describing the interaction between a rigid spherical body and an actively heated plate undergoing phase change. The formulation integrates mechanical motion, thermal transport, and melt-layer evolution within a unified framework, capturing the feedback mechanisms arising from thin-film viscous resistance and latent heat effects. The governing equations consist of a system of nonlinear ordinary differential equations for the position, velocity, plate temperature, melt thickness, and sphere temperature. A lubrication-based model is employed to characterize viscous resistance, while heat transfer in both the plate and sphere is treated using lumped thermal approximations. Melting dynamics are incorporated through a latent heat balance with an activation mechanism governed by a temperature-dependent switching function. The system is subsequently non-dimensionalized, revealing key parameters controlling viscous dissipation, thermal coupling, and phase change intensity. Steady-state solutions are derived, and a linear stability analysis is performed using the Jacobian matrix of the coupled system. The analysis reveals a block triangular structure, allowing decomposition into mechanical and thermal subsystems. The resulting eigenvalue spectrum exhibits multiple zero eigenvalues alongside a negative mode, indicating marginal stability of the equilibrium state. This degeneracy highlights the absence of sufficient dissipative mechanisms in the baseline model. The results provide new insight into the interplay between thermal activation, viscous resistance, and nonlinear coupling in melting-driven systems. The framework establishes a foundation for extended models incorporating additional dissipative effects and offers a mathematically consistent approach to analyzing thermo-mechanical phase-change dynamics.