Heat conduction in a semi-infinite medium with time-periodic boundary temperature and a circular inhomogeneity

We solve the problem of heat conduction in a 2D homogeneous medium (of diffusivity α) below a boundary subjected to time-periodic temperature (of frequency ω), in the presence of a circular inhomogeneity (of radius R), whose center is at distance d > R (depth) from the boundary. This study is a continuation of a previous one which considers a 3D medium with a spherical inhomogeneity. The general solution depends on four dimensionless parameters: d/R, the heat conductivity ratio κ, the heat capacity ratio C and the displacement thickness δ/R=2α/(ωR2). An analytical solution is derived as an infinite series of eigenfunctions pertaining to the 2D Helmholtz equation. The solution converges quickly and is shown to be in agreement with a finite element numerical solution. The results are illustrated and analyzed for a given accuracy and for a few values of the governing parameters. A comparison is held with the previous 3D solution pointing out the differences between the two. To widen the range of possible applications, an extension of the solution to a domain of finite depth is also presented. The general solution can be simplified considerably for asymptotic values of the parameters. A first approximation, obtained for R/d≪1, pertains to an unbounded domain. A further approximate solution, for R/δ≪1, while κ and C are fixed, can be regarded as pertaining to a quasi-steady regime. However, its accuracy deteriorates for κ≪1, and a solution, coined as the insulated circle approximation, is derived for this case. Comparison with the exact solution shows that these approximations are accurate for a wide range of parameter values.