Constant source Poisson eq in 2D, Dirichlet BC, average value?

[david.b]

Hello,

for the Poisson problem Δu = -1 on a 2D circular disk with u = 0 on the boundary, we have

average(u) = \frac{1}{8\pi}Area(disk),

which is easy to see, as the solution is quadratic in the polar coordinate r. Does this (or a similar) relation hold for non-circular 2D domains? This problem comes up in Poiseuille fluid flow in tubes of non-circular cross section. Thanks in advance.

David

 

[Chestermiller]

Hello,

for the Poisson problem Δu = -1 on a 2D circular disk with u = 0 on the boundary, we have

average(u) = \frac{1}{8\pi}Area(disk),

which is easy to see, as the solution is quadratic in the polar coordinate r. Does this (or a similar) relation hold for non-circular 2D domains? This problem comes up in Poiseuille fluid flow in tubes of non-circular cross section. Thanks in advance.

David

Welcome to PF, David B. Nice to have you here.

Try solving this same problem for rectilinear flow between parallel plates. This is the limiting situation of a very low aspect ratio duct of rectangular cross section. I think you will find that the relationship does not hold up even in this case. However, there have been tons of papers published in the open literature for the laminar flow pressure drop/flow rate relationship in ducts of a wide range of cross sections.