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How to derive the solution for potential flow around a circular cylinder

  1. Feb 21, 2012 #1
    Hey all,

    I've been trying for a while now to derive the following solution, for a circular cylinder under uniform flow:

    [tex]φ(r,θ)=U(r+\frac{R^2}{r})cos θ[/tex]

    where φ is the flow potential that satisfies Laplace's equation, as defined in this article:


    I know how to solve laplace's equation in a rectangular domain, using separation of variables, but here I am at a loss. I simply can't figure out how to implement the circular geometry into the rectangular domain.

    To make it more clear, I am assuming a rectangular domain with a circle inside. The domain has a Dirichlet condition on two opposite sides (flow velocity), and a Neuman condition on the surface of the sphere and on the other two sides of the rectangle.

    Since this solution is on wikipedia, I figured that it would be well documented, but, after scouring the internet and my books for days, I simply can't find how it's derived anywhere. If someone could provide a link or some help in deriving the solution, I would be grateful :biggrin:
  2. jcsd
  3. Feb 22, 2012 #2
    shameless bump :tongue: Searching around the internet I found a proof that basically uses the potential at infinity to formulate the solution, so there was no actual solution of the PDE.

    An idea a colleague had was that maybe it is possible to solve the PDE for the cylinder with boundaries at infinity, and then solve the rectangle with the boundaries of my problem, and superpose the solutions? This seems likely in principle, but something doesn't really sit right with this approach. Namely, since the flow around the cylinder is generated by the boundary conditions on the rectangular domain, it is pretty unlikely that I will get any meaningful results without properly implementing that boundary condition in the cylindrical problem.
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