What you have written down (also in your answer) is a version of the [fluid dynamic jump conditions][1]. Those result from integrating the conservation equations (for mass, momentum and energy) over 1D over an arbitrary test volume, which gives a relation of the conserved quantities at the edges of said volume. Once you have those, the relation of the  pre-to-postshock quantities (works also without shocks)  is just a matter of solving the algebra.  


In the fluid dynamics literature you can find this treated quite widely, see for example "Riemann solvers and numerical methods for fluid dynamics"  by E. Toro, or you can easily find a number of university lecture by googling "shock jump conditions".

Just to clarify, a few things that you have mentioned, put into context:  
* Venturi effect: results from the Bernoulli law, which is a 1D-integrated version of the momentum equation, aka just what you want to do  
*e.g. Weymouth and Panhandle Equations: Never heard of, probably more specialized equations with boundary conditions for tubes  
* Darcy equation: Darcy's law is used in soil physics, where the fluid is embedded in a matrix and therefore cannot build up a wave equation to communicate state changes. I would argue that using Darcy here is very wrong.   

  [1]: https://en.wikipedia.org/wiki/Rankine%E2%80%93Hugoniot_conditions