From fluid energy conservation equation to the continuity equation

happyparticle
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Homework Statement
Derive the continuity equation from the energy conservation equation
Relevant Equations
##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p ##
##\frac{Dp}{Dt} + \gamma p \nabla\cdot \vec{u} = 0##
##\frac{D}{Dt} \frac{p}{\rho^{\gamma}} = 0##
##e = \frac{1}{\gamma -1} \frac{p}{\rho}##
Hey there,

First of all, all energy conservation equations for a fluid I found on google hadn't the ##\gamma## coefficient. What exactly is the difference?

Secondly, by substituting e by ##e = \frac{1}{\gamma -1} \frac{p}{\rho}## in the following equation ##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p ## I got ##\frac{D}{Dt}(\frac{1}{\gamma -1} \frac{p}{\rho}) + (\gamma - 1)(\frac{1}{\gamma -1} \frac{p}{\rho}) \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p ##

=> ## \frac{D}{Dt}(\frac{1}{\gamma -1}) p + p \nabla \cdot \vec{u} = - (\vec{u} \cdot \nabla)p ##

Then, I'm wondering if in this case the pressure is constant to have the right hand side equal to 0?
Anyway, I don't get the expecting result.
 
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That's the energy-balance and momentum-balance equation for adiabatic processes. So you deal with ideal hydrodynamics. The task is to derive the continuity equation
$$\partial_t \rho + \vec{\nabla} \cdot (\rho \vec{u})=0.$$
 
If I understood
$$
\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p \hspace{10pt} (1)
$$
Is the energy-balance equation

and
$$
\frac{Dp}{Dt} + \gamma p \nabla\cdot \vec{u} = 0 \hspace{10pt} (2)
$$
is the momentum-balanced equation

I'm should be able to get (2) from (1) by replacing ##e## in (1).
 
vanhees71 said:
No, why?
I have a problem that I have get equation (2) and ##
\frac{D}{Dt} \frac{p}{\rho^{\gamma}} = 0
##

From equation (1) by replacing e with ##
\frac{1}{\gamma -1} \frac{p}{\rho}
##
 
happyparticle said:
Homework Statement: Derive the continuity equation from the energy conservation equation
Relevant Equations: ##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p ##
##\frac{Dp}{Dt} + \gamma p \nabla\cdot \vec{u} = 0##
##\frac{D}{Dt} \frac{p}{\rho^{\gamma}} = 0##
##e = \frac{1}{\gamma -1} \frac{p}{\rho}##

Hey there,

Secondly, by substituting e by ##e = \frac{1}{\gamma -1} \frac{p}{\rho}## in the following equation ##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p ## I got ##\frac{D}{Dt}(\frac{1}{\gamma -1} \frac{p}{\rho}) + (\gamma - 1)(\frac{1}{\gamma -1} \frac{p}{\rho}) \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p ##

=> ## \frac{D}{Dt}(\frac{1}{\gamma -1}) p + p \nabla \cdot \vec{u} = - (\vec{u} \cdot \nabla)p ##

Then, I'm wondering if in this case the pressure is constant to have the right hand side equal to 0?
Anyway, I don't get the expecting result.
For an ideal gas, you just need a constitutive equation, the speed of sound will do. So from:
\frac{dp}{d\rho}=a^{2}=\frac{\gamma p}{\rho}
You write:
dp=\frac{\gamma p}{\rho}d\rho\Rightarrow\frac{Dp}{Dt}=\frac{\gamma p}{\rho}\frac{D\rho}{Dt}

From here, you can manipulate it however you want.
 
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