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).

  • $\begingroup$ I don't think you can get closed form solutions for coupled problems like the one you describe. Just for testing purposes you can choose a function satisfying the boundary conditions and add source terms to the pde's so that this function is the actual solution. The problems will be coupled through the boundary conditions. $\endgroup$ – PierreCarre Feb 1 at 17:11
  • $\begingroup$ @PierreCarre Have a look math.stackexchange.com/questions/3093821/… $\endgroup$ – Indrasis Mitra Feb 1 at 18:00

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