Flow in a circular duct: Simsons Rule

[mcooper]

I want to integrate log(r_i)log(r_j) over the unit circle where i,j=1,2,...,N and r_i=distance between a point (x,y) and a singularity outside the domain. The answers will form the a_ij of a matrix.

The method I am using is to pattern the singuarities so that they form a circle around the domain. Therefore every singularity is a distance 2 from the centre of the domain. So does every log(r_i/j)=2?

Also what are the limits of intergration? Are they -1 and 1 for dy and -sqrt(1-y²) and sqrt(1-y²) for dx?

I understand this is not specifically a Simpsons Rule question but once these confusions have been sorted I will be going on to solve the integral using Simpsons on Matlab.

Thank you

 

[LCKurtz]

Just trying to understand your question. Here's my take on it for you to clear up:

You apparently have N singularity points:

P₁(x₁,y₁),P₂(x₂,y₂)...P_N(x_N,y_N)

For each a_ij you want to do the following integral, where P(x,y) represents a point in the unit circle and || || is the Euclidean norm?

$$a_{ij} = \int \int_C \log(||P-P_i||\cdot \log||P-P_j||\ dA$$

$$= \int \int_C \log(\sqrt{(x-x_i)^2+(y-y_i)^2})\cdot \log(\sqrt{(x-x_j)^2+(y-y_j)^2})\ dA$$

To answer your question about the limits, yes, you have them correct for a dx dy integral

 

[mcooper]

Correct. I am unsure whether I have to integrate for different points (x,y) in the unit circle (domain) or just choose one, in which case the easiest point to choose would be (0,0) and log(r_i) would be 2 for every i=1,...n.

Hopefully I have managed to attach the report that I am basing my problem on. Mine is a more simple problem of the circular pipe but the calculations are similar. The question I have asked is regarding the matrix A at the top of page 3. I am trying ot find the unknowns c_j.

Thanks