Heat transfer between two fluids through a sandwiched solid (coupled problem):

Two fluids $({t}_{h},\text{}{t}_{c})$ 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:

$x\in [0,1],y\in [0,1]$

1) $\frac{dT}{d{x}^{2}}+\mu {b}_{h}({t}_{h}-T)-\nu {b}_{c}(T-{t}_{c})=0$

with boundary condition as ${T}^{\prime}\left(0\right)={T}^{\prime}\left(1\right)=0$

The fluids are governed by the following equations:

2) $\frac{d{t}_{h}}{dx}+{b}_{h}({t}_{h}-T)=0$

3) $\frac{d{t}_{c}}{dx}+{b}_{c}(T-{t}_{c})=0$

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:

4) $\frac{{d}^{2}T}{d{x}^{2}}-\mu \frac{d{t}_{h}}{dx}+\nu \frac{d{t}_{c}}{dx}=0$

However, I have not been able to proceed further.

Some parameter values are

${b}_{c}=12.38,\text{}{b}_{h}=25.32,\text{}\mu =1.143,\text{}\nu =1,\text{}\kappa =2.16$