Heat transfer between two fluids through a sandwiched solid (coupled problem) Two fluids ($t_h,t_c$) flow opposite to each other on either side of a solid ($T$), while exchanging heat among themselves. In such a scenario, the conduction in the solid is governed by:
$$x\in[0,1], y\in[0,1]$$
$$\kappa \frac{\mathrm{d}^2 T}{\mathrm{d} x^2} + \mu b_h(t_h-T) - \nu b_c(T-t_c)=0 \tag1$$
with boundary condition as $T'(0)=T'(1)=0$.
The fluids are governed by the following equations:
$$\frac{\mathrm{d} t_h}{\mathrm{d} x}+b_h(t_h-T)=0\tag2$$
$$\frac{\mathrm{d} t_c}{\mathrm{d} x}+b_c(T-t_c)=0\tag3$$
The hot fluid initiates at $x=0$ and the cold fluid starts from $x=1$. The boundary conditions are $t_h(x=0)=1$ and $t_c(x=1)=0$.
Equation $(1),(2)$ and $(3)$ form a coupled system of ordinary differential equations.
It is pretty evident that using $(2)$ and $(3)$, Equation $(1)$ can be re-written as:
$$\kappa \frac{\mathrm{d}^2 T}{\mathrm{d} x^2} - \mu \frac{\mathrm{d} t_h}{\mathrm{d} x} + \nu \frac{\mathrm{d} t_c}{\mathrm{d} x}=0 \tag4$$
However, I have not been able to proceed further.
Some parameter values are $b_c=12.38, b_h=25.32, \mu=1.143, \nu=1, \kappa=2.16$.

Hint.
Calling $T_1 = T$ and $T_2=T'_1$ we have
$$
\left(
\begin{array}{c}
T'_1\\
T'_2\\
t'_h\\
t'_c
\end{array}
\right) = 
\left(
\begin{array}{cccc}
0 & 1 & 0 & 0\\
\frac{\mu b_h}{\kappa}+\frac{\nu b_c}{\kappa} & 0 & -\frac{\mu b_h}{\kappa} & -\frac{\nu b_c}{\kappa}\\
b_h & 0 & -b_h & 0\\
-b_c & 0 & 0 & b_c
\end{array}
\right)
\left(
\begin{array}{c}
T_1\\
T_2\\
t_h\\
t_c
\end{array}
\right)
$$
