Consider a one dimensional model (tube with diamter $D$) with a compressible viscose ($\mu$) fluid (e.g. air). further assumptions are

• conduction and radiation are negligible
• gravity's effect is negligible
• flow is one dimensional and uniform
• the flow is not steady-state so it is changing rapidly by time (e.g. Sod shock tube)
• gas is ideal $p=\rho \mathring{R} T$

based on this assumptions it is possible to write conservation equations of mass, momentum and heat. However, solving those PDEs is very difficult. I'm looking for empirical equations to approximate the mass flow dependency on pressure gradient:

$q= \rho \nu =f\left( \frac{\partial p}{\partial x},D,... \right)$

I have looked into the literature but what I have found is mostly for steady state flow in long tubes (e.g. Weymouth and Panhandle Equations). I would appreciate if you could help me find such an approximate solutions for the said assumptions.

P.S.1. I'm not interested in pressure loss. I have Darcy equation and its derivatives for that reason.

P.S.2. I was not sure what tags to attache, so let me know please if they are irrelevant.

P.S.3. I have found these equations from one article I'm reading:

But the paper has provided no reference or explanation where these equation came from, nor any name is mentioned.

Consider a one dimensional model (tube with diamter $D$) with a compressible viscose ($\mu$) fluid (e.g. air). further assumptions are

• conduction and radiation are negligible
• gravity's effect is negligible
• flow is one dimensional and uniform
• the flow is not steady-state so it is changing rapidly by time (e.g. Sod shock tube)
• gas is ideal $p=\rho \mathring{R} T$

based on this assumptions it is possible to write conservation equations of mass, momentum and heat. However, solving those PDEs is very difficult. I'm looking for empirical equations to approximate the mass flow dependency on pressure gradient:

$q= \rho \nu =f\left( \frac{\partial p}{\partial x},D,... \right)$

I have looked into the literature but what I have found is mostly for steady state flow in long tubes (e.g. Weymouth and Panhandle Equations). I would appreciate if you could help me find such an approximate solutions for the said assumptions.

P.S.1. I'm not interested in pressure loss. I have Darcy equation and its derivatives for that reason.

P.S.2. I was not sure what tags to attache, so let me know please if they are irrelevant.

P.S.3. I have found these equations from one article I'm reading:

But the paper has provided no reference or explanation where these equation came from, nor any name is mentioned.