Poiseuille flow
 

It seems to be a very simple question but I'm trying to answer it without results.

While performing the simple case of flow between two plates, my code doesn't work. I found that the problem comes from the boundary conditions. Because I used staggerd grid, I need the value of some fictious value of U and V. I was used to extrapolate but while doing this the continuity equation is not verified. To give you an example :

For U0 = 1 at i=0, i.e. the inflow, this is the two cases I tried :

1st case, I applied the du/dx = 0(u1,j - u-1,j = 0), but obviously d2u/dx2 !=0 (u1,j - 2*u0,j + u-1,j != 0), which is absurd mathematically !!!!!

2nd case, I extrapolate, so d2u/dx2 = 0 but du/dx !=0, it's not absurd but the continuity is not verified on the boundary.

So in the two situation, while computing the pressure boundary condition, we don't obtain dp/dx = 2*(d2u/dy2)/Re, whereas this equation is obtained analyticaly and must be verified.

So where's the problem???? Maybe I miss something but where?????