Theoretical analysis of a two-dimensional bilayer convection-diffusion-reaction-source problem
Guillermo Federico Umbricht, Diana Rubio, Domingo Alberto Tarzia
TL;DR
The paper addresses transient heat transfer in a two-dimensional bilayer with diffusion, advection, internal/external heat sources, and interfacial thermal resistance under general convective boundaries. It develops a Fourier-based analytical solution after removing advection via a moving-frame transformation, and complements it with a second-order explicit finite-difference scheme to simulate specific scenarios, including interface discontinuities. Validation is achieved by reducing to established 1D results under simplifying assumptions, and numerical experiments with realistic material pairs demonstrate center-focused heating and interface temperature jumps that grow with interfacial resistance. The work delivers analytical benchmarks and a practical numerical approach valuable for design and optimization of multilayer thermal systems such as thermal barriers and electronic packaging.
Abstract
This work investigates the two-dimensional thermal behavior of a bilayer medium subject to both internal and external heat sources. The model incorporates diffusion, advection, and temperature-dependent volumetric heat generation or absorption in each layer, as well as general convective conditions on the external boundaries. The influence of interfacial thermal resistance between the two materials is also considered. An analytical solution is developed using Fourier-based techniques, and a stable and convergent finite difference method is proposed to analyze particular scenarios. The theoretical results are validated against known solutions and numerical simulations, demonstrating consistency with the expected physical behavior. The findings contribute to a deeper understanding of heat transfer phenomena in layered systems and offer potential insights for optimizing thermal performance in engineering applications involving composite materials.