I am trying to solve a physical problem on a rectangular domain (thin film flow), which is modeled by the Laplace equation $$ \nabla^2 p=0 $$ up to now, I've generally solved PDEs using some sort of numerical solver and I am generally pleased with the results. But now I need to do an optimization study and obviously the numerical solver is a huge setback since it takes too much time. I know from several books and online resources that it is possible to use conformal maps to solve two dimensional PDEs.
So I guess to formulate my question clearly I would like to first describe the problem. Basically I have a rectangular domain of length $l$ and width $w$ that represent a face in which holes of diameter $d$ are drilled. These holes will be transporting pressurized fluid to generate a pressure distribution $p(x,y)$. Here's a numerically solved problem.
Link, since I'm now allowed to embed them yet.
When I solve this, I generally provide Dirichlet conditions to the solver, since the holes are round I give something like this for each hole
$$ p(x,y)=p_s \\(x-x_i)^2+(y-y_i) = r_i^2$$
where $p_s$ is the source pressure and of course a Dirichlet boundary condition for the rectangular border
$$ p(x,y) = p_a \\ 0 \geq x \geq l \\ 0 \geq y \geq w $$
where $p_a$ is the ambient pressure.
I have very little formal mathematical knowledge, so my question is two fold.
- Is it possible to generate a solution using conformal maps for this problem and derivatives (holes in different locations, different diameters)
- If yes, how would I go about it? Could anyone provide an example which I guess I could work from and solve for other problems?