I want to solve a system of partial differential equations (Navier-Stokes with continuity and momentum equations in one-dimension) using a finite element routine. I have several questions:

1. How to couple the two equations (continuity and momentum) into a single system with a single solution.

2. Apart from time, I have two unknowns: volumetric flow rate (Q) and pressure (p). Should I consider these as two degrees of freedom on each node in the domain.

3. How to account for time evolution. From my reading, it seems as if the time dependency is accounted for not by finite element, but by coupling a series of steady state finite element solutions through the use of time stepping methods such as Adams-Bashforth and Adams-Moulton methods. Can you clarify how to do this in a finite element computer program.

Any advice or direction to some materials (books, papers, internet, etc.) that help in this specific problem will be highly appreciated.