One approach to convert this into an ODE is with index reduction methods. These allow you to convert high-index DAEs into low-index DAEs or ODEs. See section VII.2 of "Solving Ordinary Differential Equations II" by Hairer and Wanner.
Consider a generic, Hessenberg index-2 DAE
y' &= f(y, z) \\
0 &= g(y)
I have a paper on this which, as long as your boundary is flat and you don't expecct it to pass vortexes, should do you well: https://doi.org/10.1002/fld.1427.
The gist of it is that as long as the boundary is flat and you enforce no tangential flow, the normal traction is equal to the pressure in an incompressible flow. This is an OK BC for Poiseuille-like ...