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We have been given the conservation of mass and conservation of momentum equations for the flow of fluid within a pipe system:

\begin{align}\frac{\partial \rho}{\partial t} + \frac{\partial\rho u}{\partial x} &= 0 \\ \frac{\partial J}{\partial t} + \frac{\partial(Ju + p)}{\partial x} &= \left(\frac{\partial p}{\partial x}\right)_f + \rho g, \end{align} where $\rho(x, t), u(x, t), J(x, t)$ and $p(x, t)$ are the one dimensional density, velocity, momentum and pressure respectively, and $(\partial p)/\partial x)_f$ is a frictional gradient term.

My questions are:

  1. We are not sure if we are assuming the fluid to be compressible or not. Most literature I find seems to assert that most fluids can safely be assumed to be incompressible. Would it be reasonable to assume this here? It seems strange to see the conservation of mass equation in this form if incompressibility is being assumed.

  2. More importantly, we need a third (constitutive?) equation to close this system. It does not seem that the conservation of energy fits, since temperature is in no way taken into account within this model. We were thinking possibly Boyle's law for ideal (and compressible) gases, but I've never seen this used anywhere before so not sure on this.

For someone that knows this seems like a pretty natural question to ask, but I cannot find anything in the literature that describes an equation to close this system. I really appreciate any advice that can be given.

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