I have been trying to solve a heat-exchanger problem where two fluids are separated by a conducting wall between them and the fluids flow perpendicular to each other. So i need to consider two dimensional conduction in this separating wall while heat gets exchanged between the two fluids through this wall.

Are there any examples of analytic solution to a Laplace equation coupled with another ODE ? I have failed to find any such example. Most texts deal with variants of the Laplace operator with different bc(s).

The equations look like

\begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - \theta_w) &=& 0,\\ \lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} - \frac{\partial \theta_h}{\partial x} - V\frac{\partial \theta_c}{\partial y} &=& 0 \end{eqnarray} with bc(S) as

$\frac{\partial \theta_w(0,y)}{\partial x}=\frac{\partial \theta_w(1,y)}{\partial x}=0 $

$\frac{\partial \theta_w(x,0)}{\partial y}=\frac{\partial \theta_w(x,1)}{\partial y}=0 $ $\theta_h(0,y)=1$ $\theta_c(x,0)=0$

PS: *Reposted from MSE*