Boundary conditions for known pressure drop
 

The question is very simple, but it is simply asked and answered in any thread I could find:

I am solving for pressure and velocity along a cylindrical tube. The boundary is divided in three patches: inlet, outlet and wall. Pressure is imposed at the inlet and outlet. No-slip condition is imposed on the wall.

What are the appropriate boundary conditions for pressure at the wall and for velocity on inlet and outlet?

(I have my opinion on that, but the numerical results seems not to support it.)