- #1
meldraft
- 281
- 2
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:
http://en.wikipedia.org/wiki/Potential_flow_around_a_circular_cylinder
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
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:
http://en.wikipedia.org/wiki/Potential_flow_around_a_circular_cylinder
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