# Derivation with Euler's Equations

I have three equations as follows (for a polytropic gas):

1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\nabla\cdot\rho \mathbf{u} = 0$

2) $\displaystyle\quad\rho \left( \frac{\partial \mathbf{u}}{\partial t}+\mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p$

3) $\displaystyle\quad \frac{\partial}{\partial t} \left(\rho \varepsilon + \frac{\rho u^2}{2}\right) = \nabla\cdot \left[\rho \mathbf{u} \left(\varepsilon\frac {u^2}{2}+p \mathbf{u}\right)\right]$

where $u$, $\rho$, and $p$ are the velocity, density, and pressure, respectively.

From these equations,

I would like to derive:

$$\left( \frac{\partial}{\partial t} + \mathbf{u}\cdot \nabla \right) p-c_s^2 \left( \frac{\partial} { \partial t}+\mathbf{u}\cdot \nabla \right) \rho = 0$$

I am not sure how to approach this problem and so I have come here for some help. Can anyone see a way forward for deriving this final equation from the ones provided?

I know that:

1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\mathbf{u}\cdot\nabla p = -\gamma p\nabla \cdot \mathbf{u}$

$\frac{\nabla P}{\rho U^2/L}$ ~ $\frac{P}{L}/\frac{\rho u^2}{L} = \frac{P}{\rho u^2}$ which is proportional to $\frac {C_s^2}{u^2}$

Note that: $\displaystyle\quad\varepsilon = \frac{p}{[\rho(\gamma -1)]}$

• So you must somehow show that the speed of sound squared is the ration of pressure over density... – 16278263789 Feb 3 '16 at 18:54
• @CarloVonSchnitzel Right, I l know that the sound speed squared is proportional to Pressure/density. I edited some more info in to address this – Jackson Hart Feb 3 '16 at 18:57
• I'm sorry, it was just me making a wild guess in order to match the final equation you need to derive. – 16278263789 Feb 3 '16 at 19:05
• @CarloVonSchnitzel Oh, no need to be sorry. I Thought your question was needed for me to clarify what I was asking. I appreciate any help you have to give. – Jackson Hart Feb 3 '16 at 19:10
• Using equation 1, it seems one must obtain $(\frac{\partial}{\partial t} + u.\nabla)\rho=-\rho \nabla.u$, does this seem right? I don't know if it helps... – 16278263789 Feb 3 '16 at 19:20