I'm taking a first course on fluid dynamics, and I have this (sort of) conceptual question that's been nagging me for a moment now. I can completely follow the mathematics behind the derivation of the time-independent plane Poiseuille flow, it's the symmetry considerations at the beginning that are giving me a headache.
Briefly, the "plane Poiseuille flow" is the steady pressure-driven laminar flow of a Newtonian fluid between two fixed parallel walls of infinite extent separated by a distance d. Most books that I've read begin by saying something in the lines of "because of the translational symmetry", the flow "cannot depend on the longitudinal coordinate". In fact, it is true that the problem looks the same if one shifts the origin an arbitrary distance along a line parallel to the walls.
What I cannot understand is how this last observation can be consistent with the fact that the pressure field does depend on that same coordinate. I know its gradient does not.
My question is: do these (so-called) symmetry considerations only apply to the velocity field? If that's the case, I cannot understand why the velocity field and the pressure field are treated differently.
I'm looking for an answer to this problem that can be extrapolated to other laminar viscous flows (such as plane and circular Couette's, etc.). I'm also interested in answers pointing to a formalization of these symmetry considerations. I've already browsed through Cantwell's Introduction to Symmetry Analysis, but right now it seems like an overkill for this problem.