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.
Highlights
The purpose of this work is to evaluate the effect of two coalesced spherical pores or insulating inhomogeneities on the overall conductive properties
The resistivity contribution tensor gives the extra temperature gradient produced by introduction of the inhomogeneity into a material subjected to otherwise uniform heat flux
We show that the contribution of the two overlapped spherical pores to the overall conductivity can be approximated with good accuracy by that given by a spheroid of a well definite aspect ratio that we explicitly calculate
Summary
The purpose of this work is to evaluate the effect of two coalesced spherical pores or insulating inhomogeneities on the overall conductive properties. The main goal of this paper is to obtain an analytical solution for components of the resistivity contribution tensor of a pore or insulating particle formed by two overlapping spheres. The resistivity contribution tensor gives the extra temperature gradient produced by introduction of the inhomogeneity into a material subjected to otherwise uniform heat flux This tensor has been introduced by Sevostianov and Kachanov (2002). Sevostianov et al (2014) proposed a simple method of estimating the effect of interaction of two spherical pores or inhomogeneities on the overall elastic and conductive properties They approximately evaluated components of compliance and resistivity contribution tensors and compare their results with numerical calculations. Pitkonen (2006, 2007) considered dielectric overlapping spheres and obtained solutions in the form of an integral of a parameter given by a Neumann series.
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