Potentiel of a source in a corner (Fluid Mechanics)

[Nico045]

Hi, I made some calculations abiout this problem biut my results are a bit complicated :

We consider two walls in x=0 and y=0, in (a,bi) , a>0, bi>0 ,we have a source in this corner and I need to find the potential biy using the method of image. After this I need to find v, the pression and the forces applied on the walls.

So from what I know, I can add 3 images of the source in (a,-bi) ; (-a,-bi) ; (-a,bi)

Then I would have
W(z) = m/2π * (ln(z-a-bi) + ln(z-a+bi) + ln(z+a-bi) + ln(z+a+bi) )

(conjugate)v=dW(z)/dz = m/2π * (1/(z-a-bi) + 1/(z-a+bi) + 1/(z+a-bi) + 1/(z+a+bi) )

However to find the pression, from bernoulli's equation, I need to calculate v² = conjugate(v)*v and it gives me something like :

v²= (m/2π)² * (1/(x+yi-a-bi) + 1/(x+yi-a+bi) + 1/(x+yi+a-bi) + 1/(x+yi+a+bi) ) *(1/(x-yi-a-bi) + 1/(x-yi-a+bi) + 1/(x-yi+a-bi) + 1/(x-yi+a+bi) )

which is very complicated since I have to integrate this expression to get the forces.

Have I done something wrong in the beginning ?

 

[eljose79]

Hi there,

Thank you for sharing your calculations and approach to solving this problem. It seems like you have a good understanding of the method of images and have correctly added the images of the source in order to find the potential. However, I believe the issue may lie in your calculation for the pressure.

In order to calculate the pressure using Bernoulli's equation, you need to take the derivative of the potential with respect to the normal direction of the walls. This would involve taking the derivative of the potential function in terms of x and y, not just z. Additionally, you would need to take into account the distance from the source to the walls, as this would affect the pressure distribution.

I would suggest revisiting your calculation for the pressure and double-checking your derivatives. It may also be helpful to draw a diagram to better visualize the problem and make sure all distances and angles are correctly accounted for.

I hope this helps and good luck with your calculations!