I am looking for some clarification on the incompressible, 2D, steady-state, cartesian Navier-Stokes equations for flow through a straight cylinder (flow aligned with the $x$-axis, so $u_y =0$), with negligible gravity:
$$ \rho u_{x}\frac{\partial u_{x}}{\partial x} = - \frac{\partial p}{\partial x} + \mu\left(\frac{\partial^{2}u_{x}}{\partial x^{2}} + \frac{\partial^{2}u_{x}}{\partial y^{2}}\right) \\ \frac{\partial p}{\partial y}=0. $$
Just to be clear, I'm not assuming fully-developed flow, so that $\partial u_x/\partial x\neq 0$.
Also, I am aware that the cylindrical Navier-Stokes equations would be more suitable in this case, but as far as I can tell, using the cartesian Navier-Stokes equations is still a valid formulation.
With that out of the way, upon looking at the cartesian Continuity equation under these same assumptions, the equation simplifies to:
$$ \frac{\partial u_x}{\partial x}=0 $$
Which seems to contradict the non-fully-developed flow assumption from the Navier-Stokes simplification.
I can't wrap my head around how Continuity implies that the flow is fully-developed, yet I never made such an assumption.
My first thoughts are that the Continuity simplification is actually a 1D simplification, not a 2D simplification like my original assumptions. However, since $u_y$ must be zero (because the flow is unidirectional), I can't see how the simplified Continuity equation would be any different from what I have written.
I then thought that this had something to do with the Continuity equation being in differential form so that it's only considering an infinitesimal point. I know that upon considering a control volume (i.e. converting Continuity to integral form), you can arrive at the steady-state conservation of mass equation (indicating that velocity in the $x$-direction does change):
$$ u_1 A_1 = u_2 A_2. $$
Though, this still doesn't really solve my problem, since the cross-sectional area of the cylinder is constant along its length. If anything, this equation just tells me that the flow is developed at every instantaneous point (i.e. it remains fully-developed even under cross-sectional changes).
This is probably the closest I feel to an actual explanation, but the problem still persists that Continuity says that $\partial u_x/\partial x=0$, meaning nothing is stopping me from substituting this into the Navier-Stokes equations, which will ultimately contradict my underlying assumptions.
The only other explanation I can seem to think of is that Continuity explains the quantity changes of the fluid in a system, whereas the Navier-Stokes equations explain the force changes (per unit volume) of the fluid in a system, so they are measuring different qualities of the fluid.
However, I'm not convinced that this has anything to do with the apparent contradiction, as I have seen many cases where a Continuity-derived expression has been directly substituted into the Navier-Stokes equations (e.g. in the derivation of the Reynolds-Averaged Navier-Stokes equations).
I'd really appreciate an explanation of what my misunderstanding is here.
I apologise if this has been asked somewhere else - I really struggled to find another question like this.
Feel free to point me in the right direction if you know where/if this has been asked before.