homework and exercises - Heating transient of a composite rod?

A straight rod is made of two parts, $[0,x_1]$ (green in the figure) with thermal diffusivity $\kappa_1$ and $[x_1,x_2]$ (blue) with thermal diffusivity $\kappa_2$. The rod is perfectly insulated. Zero $y$ and $z$ temperature gradients are assumed.

At $x=0$ temperature is maintained at constant $T_0$. At $x=x_2$ the rod is embedded into a perfect insulator ($\kappa=0$). At $t=0$ the rod has a uniform temperature $T(x,0)=T_i$.

Question: what is the temperature evolution of the rod?

1. The simple case where $\kappa_1=\kappa_2=\kappa$:

Let $u(x,t)=T(x,t)-T_0$.

Then Fourier's equation tells us:

$$u_t=\kappa u_{xx}$$

Boundary conditions:

$$u(0,t)=0$$ $$u_x(x_2,t)=0$$

Initial condition:

$$u(x,0)=u_i=T_i-T_0$$

Using the Ansatz $u(x,t)=X(x)\Gamma(t)$, separation constant $-k^2$ and the boundary conditions above, this solves easily to:

$$\Large{u(x,t)=\displaystyle \sum_{n=1}^{+\infty}B_n\sin\Bigg(\frac{n\pi x}{2x_2}\Bigg)e^{-\kappa \Big(\frac{n\pi }{2x_2}\Big)^2t}}$$

(for $n=1,3,5,7,...$)

The $B_n$ coefficients can easily be obtained from the initial condition with the Fourier sine series:

$$B_n=\frac{4u_i}{n\pi}$$

Back-substituting we get:

$$T(x,t)=T_0+\frac{4(T_i-T_0)}{\pi}\displaystyle \sum_{n=1}^{+\infty}\frac{1}{n} \sin\Bigg(\frac{n\pi x}{2x_2}\Bigg)e^{-\kappa \Big(\frac{n\pi }{2x_2}\Big)^2t}$$

(for $n=1,3,5,7,...$)

A plot for the first three terms at $t=0.1$:

2. The case where $\kappa_1\neq\kappa_2$:

We define two functions $u_1(x,t)$ for $[0,x_1]$ and $u_2(x,t)$ for $[x_1,x_2]$. We use the same Ansatz as under $1.$ We'll assume both functions have their own eigenvalues.

Boundary conditions:

$$u_1(0,t)=0\implies X_1(0)=0\tag{1}$$ $$\frac{\partial u_2(x_2)}{\partial x}=0\implies X_2'(x_2)=0\tag{2}$$

In addition (continuity):

$$u_1(x_1,t)=u_2(x_1,t)\tag{3}$$

With Fourier, the heat flux is the same at $x=x_1$:

$$\alpha_1\frac{\partial u_1(x_1)}{\partial x}=\alpha_2\frac{\partial u_2(x_1)}{\partial x}\tag{4}$$

Where $\alpha_i$ are the thermal conductivities.

a. for $u_1(x,t)$:

$$X_1(x)=c_1\cos k_1x+c_2\sin k_1x$$ $$X_1(0)=0\implies c_1=0\implies X_1(x)=c_2\sin k_1x\tag{5}$$

b. for $u_2(x,t)$:

$$X_2(x)=c_3\cos k_2x+c_4\sin k_2x$$ $$X_2'(x_2)=0\tag{2}$$ $$\implies -c_3k_2\sin k_2x_2+c_4k_2\cos k_2x_2=0\tag{6}$$ Using the additional conditions $(3)$ and $(4)$:

$$c_2\sin k_1x_1=c_3\cos k_2x_1+c_4\sin k_2x_1\tag{7}$$ $$c_2\alpha_1k_1\cos k_1x_1=-c_3\alpha_2k_2\sin k_2x_1+c_4\alpha_2k_2\cos k_2x_1\tag{8}$$

Problem:

$(6)$, $(7)$ and $(8)$ form a system of three simultaneous equations but with five unknowns: $c_2$, $c_3$, $c_4$, $k_1$ and $k_2$.

I'm tempted to set $c_3=0$ as it would yield $k_2$ from $(6)$. I think this would yield also the remaining unknowns. But can I a priori assume $c_3=0$? Or is there another approach possible?

I'm also left wondering whether perhaps $k_1=k_2$. The eigenvalues do not depend on $\kappa$, so perhaps the eigenvalues $k$ are common to both functions. Due to $(4)$, $u_1$ and $u_2$ would then still be distinct.