I can't wrap my head around one detail in pressure-correction method for the Navier-Stokes equations. Generally, the no-slip boundary condition implies no pressure drop at the boundary, so $\vec{n} \cdot \vec{\nabla} p = 0$. However, textbooks (computational softwares, engineers, ...) claim, that even at the inlet with prescribed ideal parabolic velocity profile (let's say the 2D inlet of width $L$ is in the $+x$ direction) $$ \vec{v} = v_0 \left( 1 - \left( \frac{2 y}{L} \right)^2 \right) \vec{e}_x $$ the pressure BC for this part of the boundary is still $\vec{n} \cdot \vec{\nabla} p = 0$. However, when I take N-S equations and figure out the pressure that makes the momentum equation consistent, it turns out to be $$ p = - \frac{8 \nu \rho v_0}{L^2} x $$ (so there is a constant drop in pressure in the pipe if the flow is ideally parabolic)
I imagine having an infinitely long pipe filled up with liquid which travels with perfect, laminar, parabolic profile. Now if we suddenly say "this here is our inlet" and treat the next piece of the pipe as our domain of interest, I see no reason why should pressure suddenly stop dropping, just because we called it an "inlet". So, more correctly, I would say the correct BC for pressure at the inlet with parabolic profile is $$ \vec{n} \cdot \vec{\nabla} p = - \frac{8 \nu \rho v_0}{L^2} $$ The question is: why don't we impose this kind of boundary condition for pressure at the inlet boundary? Why do we still demand the homogenous Neumann BC?