Thermal conductivity of solids with coalescing spherical pores

The overall thermal properties of media containing insulating cylindrical inhomogeneities has been addressed recently in [1] making reference to a 2D layout. There, the cross section of the fibers is formed by two intersecting circles to simulate a variety of nonconductive fibers (e.g. electrospinned polystyrene fibers). On the other hand, intersecting circles is a relevant layout to assess the physical properties of a variety of porous materials (e.g. Gasar metals) during the processes of pore coalescence and growth. In this work we extend the analysis addressed in [1] to a 3D framework by considering the effect of insulating inhomogeneities having the shape of intersecting spheres. The analysis aims at assessing the second-order resistivity contribution tensor, which provides the corrective temperature gradient induced by the volume V * of the inhomogeneity over the reference volume V of the background material subjected the a remotely applied heat flux q. Owing to the geometric setting, reference is made to toroidal coordinates and Mehler–Fock transforms are used to represents the perturbation temperature field due to the inhomogeneity [2]. As remarked in [3] for coalescing spheres having the same diameter, the components of tensor R display a non-monotonic trend varying the distance between their centers. In particular, unlike what is observed for spheroids, the stationary values of the components of tensor R occur when spherical pores are slightly intersecting.