- #1
aliens123
- 75
- 5
In Landau-Lifshitz Volume 6 Fluid-Mechanics the following problem is given
Where the equation of continuity is given earlier:
As is Euler's equation:
And the equation of continuity for entropy:
I don't understand how this conclusion was reached. I can understand the derivation for the equation of continuity , but I have no idea how you could derive it from Euler's equation:
$$ \frac{\partial \rho}{\partial t} + \nablda \cdot ( \rho} \vec{v} )=0 $$
In 1d:
$$ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial a} ( \rho} v )=0 $$
I tried integrating this with
$$\int_{t_0}^{t} dt$$: $$\rho - \rho_0 + \frac{\partial }{\partial a} \int_{t_0}^{t} \rho \frac{\partial x}{\partial t}dt = 0$$
But I can't see any way to simplify this...
I also do not see how the next two equations were derived. Any help would be appreciated.
$$ \frac{\partial \rho}{\partial t} + \nablda \cdot ( \rho} \vec{v} )=0 $$
In 1d:
$$ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial a} ( \rho} v )=0 $$
I tried integrating this with
$$\int_{t_0}^{t} dt$$: $$\rho - \rho_0 + \frac{\partial }{\partial a} \int_{t_0}^{t} \rho \frac{\partial x}{\partial t}dt = 0$$
But I can't see any way to simplify this...
I also do not see how the next two equations were derived. Any help would be appreciated.