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Suppose that a uniform thermal gradient in the +x direction exists in a very large (i.e. effectively infinite) domain of conductivity $k_2$ such that the temperature field $u_{\infty}(r,\theta)$ can be represented by

$$

u_{\infty}(r,\theta) = Ar\cos\theta

$$

where $A$ is a constant which reflects the gradient magnitude.

Suppose that a circular region of material of radius a is removed an replaced with a new material with conductivity $k_1$.

The most general version

of this problem is when the inner and out conductivities are of unequal, but comparable, magnitudes.

In this case, the steady temperature field inside the disk and outside the disk must be solved separately to obtain the inner and outer solutions, $u_1$ and $u_2$ respectively.

The constraint on the inner solution is boundedness at the origin

$$

\lim_{r\to 0}|u_1(r,\theta)| < \infty.

$$

The outer solution must asymptotically approach the undisturbed temperature field at large distances:

$$

\lim_{r\to\infty}u_2(r,\theta)\to u_{\infty}(r,\theta).

$$

Each of these solutions will contain series coefficients that must be determined by jointly imposing continuity of temperature and heat flux at the boundary $r = a$:

\begin{alignat*}{3}

u_1(a,\theta) & = & u_1(a,\theta)\\

k_1u_{1_r}(a,\theta) & = & k_2u_{2_r}(a,\theta)

\end{alignat*}

Obtain the solutions for the temperature fields $u_1$ and $u_2$ with series coefficients expressed terms of $a$, $k_1$ and $k_2$.