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I'm playing around with the equations of fluid dynamics at the moment, and unsure about the interpretation of my findings.

I started out with the 1D system of equations for $\rho(x,t)$ and $v(x,t)$ together with a pressure term that allows sound waves, indexed is the derivative w.r.t. $x$ or $t$:
$$ \rho_t + (\rho v)_x = 0 ~~~~~~~~~~~~~~(1)$$ $$ v_t + v v_x = - \frac{c_s^2}{\rho} \rho_x ~~~~~~~~~(2)$$ and the sound speed is $c_s$.
After a bit of manipulation one can find the conservative form for the momentum density $\rho v$ that this system implies: $$ (\rho v)_t + \left[ \rho (c_s^2 + v^2) \right]_x = 0 ~~~~~~~~~~~~(3)$$ Now this is a nice-looking conservation law, but slightly confusing in it's interpretation. I'm thinking in terms of computational fluid dynamics here, where any quantity $C$ that has a conservation law $C_t + F_x = 0$ can be averaged over a spacial cell with $\int_a^b dx$, such that we get $$\bar C_t + F(a) - F(b) = 0 ~~~~~~~~~~~~~~~~ (4)$$

So in the world of CFD this would mean that the local linear momentum density is changed by its own advection, as well as by the action of sound waves that they presumably transport. Now I'm unsure if this is a subtle mistake in thinking and (3) is actually $$ (\rho v)_t + \left[ \rho v^2 \right]_x = - \left[ \rho c_s^2 \right]_x ~~~~~~~~~~~~(5)$$ so that sound waves actually act as a source term of angular momentum inside the cell. Considering that they come from pressure gradients this interpretation would make sense.

So which one is it, source or transport, or am I getting lost in details that may not have a meaning?

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  • $\begingroup$ Are you familiar with the characteristic form of the governing equations? They are the equations transformed into a diagonal form where the equations are uncoupled and each one represents a type of wave in the flow. You end up with an acoustic wave, convection wave, and entropy wave. Each of which carries with it certain information. $\endgroup$
    – tpg2114
    Commented Feb 12, 2017 at 19:43
  • $\begingroup$ @tpg2114: Yes, I'm familiar with this method. But isn't that an approach to the problem that won't help me? For what you suggest, I need a conservative form for each variable, and in the end I have eigenvectors and eigenvalues, but those don't tell me if I transport momentum (which is a combination of both variables) afaik. $\endgroup$ Commented Feb 12, 2017 at 19:55
  • $\begingroup$ @tpg2114: Anyway, thanks for the reminder, I'll try it and see what I can find. $\endgroup$ Commented Feb 12, 2017 at 19:58
  • $\begingroup$ I don't have the time to work it out now, but if I remember correctly, the governing equation for the left and right running acoustic waves do transport momentum while the entropy wave does not. I could be mistaken though. $\endgroup$
    – tpg2114
    Commented Feb 12, 2017 at 20:38
  • $\begingroup$ @tpg2114: Oh I see where this is going now. I get evolution equations for variable*eigenvalue which is $(\rho \cdot (v-c_s \sqrt{\rho} ))$ and $(v \cdot (v+c_s \sqrt{\rho} ))$ and then I need to see whatever is on the r.h.s, right? $\endgroup$ Commented Feb 12, 2017 at 22:57

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