Analytical Modeling of Thermal Spreading/Constriction From an Isothermal Source Into a Multilayer Orthotropic Semi-Infinite Flux Tube

Thermal spreading and constriction have been widely studied due to relevance in heat transfer across interfaces with imperfect contact and problems such as microelectronics thermal management. Much of the past work in this field addresses an iso-flux source, with relatively lesser work on the isothermal source problem, which is of much relevance to heat transfer across rough interfaces. This work presents an analytical solution for thermal spreading/constriction resistance that governs heat flow from an isothermal source into a multilayer orthotropic semi-infinite flux tube. The mixed boundary condition due to the isothermal source is accounted for by writing a convective boundary condition with an appropriately chosen spatially-varying Biot number. A series solution for the temperature field is derived, along with a set of linear algebraic equations for the series coefficients. An expression for the non-dimensional thermal spreading resistance is derived for Cartesian and cylindrical problems. It is shown that, depending on the values of various non-dimensional parameters, heat transfer in either the thin film or the flux tube may dominate and govern the overall thermal spreading resistance. Results for a single-layered isotropic flux tube are derived as a special case of the general result, for which, good agreement with past work is demonstrated. An easy-to-use polynomial fit for this special case is presented. This work contributes a novel technique for solving mixed boundary problems involving an isothermal source. Results may contribute towards solving practical problems involving thermal management and interfacial heat transfer.