Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 80 / 2

Within each constant-conductivity region the temperature is harmonic. For $u=v(r)\cos \theta$, the spherical Laplacian givesTrying $v=r^s$ yields $s(s-1)+2s-2=(s-1)(s+2)=0$. Thus $v=Cr+D{r}^{-2}$. Boundedness at the origin removes the interior $r^{-2}$ term, while the prescribed far gradient fixes the exterior growing term:Continuity of temperature and normal flux at $r=r_1$ giveSolving these two equations yields $B=-r_1^3(a_1-a_0)/(a_1+2a_0)$ and $C_{\mathrm{in}}=3a_0/(a_1+2a_0)$. ConsequentlyPositive conductivities make the denominator nonzero. The matched solution is unique in the usual finite-energy decaying-perturbation class: the difference of two solutions has zero far data and interface jumps, and an energy integration gives zero gradient.

For the polarization field of a conductivity inclusion, use the sign convention consistent with the perturbation equation:ThusFor a general incident constant gradient $E$, rotational symmetry and linearity replace $e_3$ by $E$. The polarization vanishes outside the sphere because its local conductivity equals the reference value. The temperature perturbation outside is dipolar; this is the spherical conductivity inclusion response used in the remaining parts.

Let $E=⟨\nabla u⟩$ and retain only the independent response of each well-separated small sphere. In the leading dilute spherical-inclusion conductivity approximation,Averaging gives $⟨P⟩\simeq p\beta E$, where $p={\sum }_j|B_j|/|\Omega |$. Since $a\nabla u=a_0\nabla u+P$,The spherical response holds for every $E$, soThis retains the order-$p$ term and neglects interactions. A precise dilute regime can be obtained by shrinking the spheres while holding separated centres fixed. Volume fraction alone is not enough if the inclusions instead remain in close clusters whose collective polarization persists at leading order.

Set $E=e_3$ and $\beta =3a_0(a_1-a_0)/(a_1+2a_0)$. The incident field relevant to each sphere is its exciting field of a conductivity inclusion: the macroscopic field plus the regular field due to other inclusions and periodic images, excluding its own singular dipole. In the first interaction approximation writeThe interior response from part (i) then gives the improved polarizationAveraging, using $⟨{\chi }_{B_j}⟩=p_j$, givesTherefore the dipole interaction correction to effective conductivity isThe factors $p_j$ cancel against the denominator in the given field correction. Linearity permits repeating the calculation for three independent incident gradients, recovering the tensor rather than just its action on $e_3$. This is a second-reflection estimate; further dipole reflections and higher multipoles have been omitted.

A literal pointwise reading of the field printed “on the surface” needs qualification. Even the isolated-sphere solution in part (i) has, on a concentric sphere of radius $R$,For $E=e_3$, its values at $n=e_3$ and $n=e_1$ are respectively $2k{r}_1^{3}E/R^3$ and $-k{r}_1^{3}E/R^3$, where $k=(a_1-a_0)/(a_1+2a_0)$. They differ whenever there is nonzero contrast. Thus a single constant matrix cannot give the total pointwise field everywhere on that sphere.

The consistent local-field interpretation is a spherical average, or equivalently the regular exciting field at the centre. Indeed $⟨n\otimes n{⟩}_S=I/3$, so the angular mean of the self-dipole is zero. The other-inclusion field is harmonic inside the surrounding sphere, and the mean value property gives its central value. With ${\Lambda }_j$ encoding this average regular correction, the calculation above is justified. If “on the surface” is instead insisted upon as a constant total pointwise field, that premise is false and the isolated-sphere values above are a counterexample. The distinction between dipole fields and local exciting fields is also treated in sections 10.2–10.3 of www.math.utah.edu/~milton/TheoryCompositesNOPRINT.pdf .

Let $k=(a_1-a_0)/(a_1+2a_0)$. Far from all the small spheres, their leading dipole contributions have the same direction and their leading $|x{|}^{-2}$ potential terms add:Translations of their centres change only higher far-field multipoles. Since the outer body is a sphere of radius $R_0$,Replacing the whole body by an isotropic effective sphere of conductivity $a^{\ast }$ gives its leading dipole coefficient $R_0^3(a^{\ast }-a_0)/(a^{\ast }+2a_0)$. Matching coefficients yieldsSolving this equation gives the Maxwell approximation for conductivity:For positive phase conductivities and $0\le p\le 1$, the denominator is positive. This rational approximation is obtained by dipole matching; its extrapolation beyond the dilute regime is a closure assumption, not an exact formula for arbitrary inclusion arrangements.

For the consistency check, $\beta =3a_{0}k$, so its dilute expansion isThe isotropic interaction relation supplied in part (iii) givesThus the Maxwell approximation and the interaction correction agree through second order in volume fraction.