Effect of spherical pores coalescence on the overall conductivity of a material.

Abstract

The problem about steady-state temperature distribution in a homogeneous isotropic medium containing a pore or an insulating inhomogeneity formed by two coalesced spheres of the same radius, under arbitrarily oriented uniform heat flux, is solved analytically. The limiting case of two touching spheres is analyzed separately. The solution is obtained in the form of converged integrals that can be calculated using Gauss-Laguerre quadrature rule. The temperature on the inhomogeneity's surface is used to determine components of the resistivity contribution tensor for the insulating inhomogeneity of the mentioned shape. An interesting observation is that the extreme values of these components are achieved when the spheres are already slightly coalesced.

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

“…The components of the resistivity contribution tensor are finally assessed as surface integrals involving the temperature distributions on the spheres calculated for the two basic cases. The study generalizes the results reported in Lanzoni et al (2020) for the case of media embedding two overlapping spherical pores of equal size.…”

“…and the result (A.5) derived in the Appendix of Lanzoni et al (2020) one gets the following Fredholm integral equation of the second kind for the unknown functions u1 and u2 that must be imposed at  = 1, 2:…”

“…On the other hand, the (quite usual) approximation of inhomogeneities of various shape as "equivalent" ellipsoids of identical aspect ratio and/or equal volume can lead to rough predictions (Kachanov and Sevostianov, 2018). For these motivations, accurate investigations were already performed to define the effective conductive properties of composites containing inhomogeneities of complex shapes (Radi and Sevostianov, 2016;Lanzoni et al, 2018Lanzoni et al, , 2020. In particular, the components of the resistivity contribution tensor of such inhomogeneities were calculated analytically and then compared with the predictions obtained for equivalent spheroids.…”

Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry-Breaking of a Polarizable Dimer in Non-Uniform Fields

“…Let us first evaluate the corresponding far-field dipole d 0 (0) defined in Equation (11) by substituting the exact solution given in Equation (9), which renders (see [ 53 ], 3.523.3 & 3.557.3): where denotes the Euler–Riemann zeta function . Note that Equation (12) coincides with the longitudinal resistivity parameter, corresponding to two touching insulating identical spheres [ 38 ], obtained in the context of a dimer’s heat conduction and effective conductivity. Finally, we provide below an approximate solution for the frequency-dependent dipole term by substituting the solution of Equation (10) into Equation (11) which renders (see [ 53 ], 3.552.3): …”

“…The special arrangement of two (dimer) or more (chain) touching spherical particles often occurs in many branches of mathematical physics and nanotechnology, such as electrostatic [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 ] and optics [ 22 , 23 , 24 , 25 ]. The tangent-sphere coordinate system can be effectively used for analytically tackling some related problems involving particle-wall interactions in various electrokinetic [ 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 ], heat transfer [ 36 , 37 , 38 ], inviscid [ 39 , 40 , 41 , 42 , 43 ], and viscous [ 44 , 45 , 46 , 47 , 48 , 49 , 50 ] flow scenarios. Note that the corresponding tangent-sphere formulation can also be used as the leading-order (‘outer’) near-contact solution of a sphere lying next to an isothermal wall or a planar electrode, both for DC and AC (high-frequency) electrokinetic problems [ 32 , 34 , 36 ].…”