Resistivity contribution tensor for two non-conductive overlapping spheres having different radii

Abstract

We consider here the problem of a three-dimensional (3D) body subjected to an arbitrarily oriented and remotely applied stationary heat flux. The body includes a non-conductive inhomogeneity (or pore) having the shape of two intersecting spheres with different radii. Using toroidal coordinates, the steady-state temperature field and the heat flux have been expressed in terms of Mehler–Fock transforms. Then, by imposing Neumann BCs at the surface of the spheres, a system of two Fredholm integral equations is obtained and solved based on Gauss–Laguerre quadrature rule. It is shown that the components of the resistivity contribution tensor exhibit a non-monotonic trend with the distance between sphere centers. In particular, if the inhomogeneity has a symmetric dumbbell-shape, then the extrema of the resistivity contribution tensor components occur when the two overlapping spheres have the same size. Differently, when the inhomogeneity has a lenticular shape, then these extrema are attained for a non-symmetric configuration, namely, for different radii of the intersecting spheres.

Resistivity contribution tensor for nonconductive sphere doublets

“…Let x = (x, y, z) denote the position vector on the sphere surface, then (2.1) yields: where is the volume of the two tangent spheres. Note that for ρ = 1 then R zz = −1.3523/k and thus the result obtained by Lanzoni et al (2020) for two equal touching spheres is fully recovered. Note also that, keeping fixed the thermal conductivity k, one has R zz (ρ) = R zz (1/ρ) and that k R zz → 3/2 as ρ → 0, + ∞, as predicted for a single sphere (Kachanov & Sevostianov, 2018).…”

“…Kushch and Sevostianov (2014) showed that shape, orientation, and spatial arrangement of the inhomogeneities in particulate composites may produce macroscopic anisotropy of the overall conductivity, whereas volume content of inhomogeneities yields the change in the anisotropy extent. Radi and Sevostianov (2016) and Lanzoni et al (2018Lanzoni et al ( , 2020Lanzoni et al ( , 2022 worked out the components of the resistivity contribution tensor for insulating inhomogeneities having the shape of a torus as well as that two overlapping cylinders or spheres of arbitrary size. The analytical results were then compared with the predictions provided by equivalent spheroids, finding a reasonable agreement within a definite range of variation of the geometrical parameters.…”

“…To examine the case where strong interactions between particles exist, the problem of two equal spheres in contact was investigated in Lanzoniet al (2020) by using the tangent sphere coordinate system, as the limit case of two overlapping spheres. Moreover, the problem of two overlapping spheres of different sizes was recently solved in Lanzoni et al (2022). The analysis was also extended to assess the effective elastic properties of composites containing rigid inhomogeneities with the shapes of a torus or two overlapping cylinders (Krasnitckii et al, 2019;Lanzoni et al, 2019).…”

“…Therefore, a straightforward Euler shooting procedure has been implemented for finding the two unknown constants, thus obtaining the temperature distribution around the two spheres and, in turn, the transverse components of the resistivity contribution tensor. The approach developed here for the 3D non-axisymmetric problem takes inspiration from the method proposed by O'Neill (1969) and then employed by Latta and Hess (1973), Davis (1977), Cox and Cooker (2000) and Lanzoni et al (2020Lanzoni et al ( , 2022 for the case of two equal spheres, which has been generalized here to the problem of two spheres of different sizes.…”