- #1
happyparticle
- 442
- 20
- Homework Statement
- Derive the different forms of the continuity 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}
##
I asked this question about one year ago, but at that time I didn't really understand what I was doing.
After spending a lot of time in this problem, I still fail to get the asked answer.
Starting with ##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p## I have to derive the following expressions.
##\frac{Dp}{Dt} + \gamma p \nabla\cdot \vec{u} = 0
##
##\frac{D}{Dt} \frac{p}{\rho^{\gamma}} = 0
##
I don't know what I'm doing wrong or what I'm missing, but I couldn't get those expressions.
Here is what I did.
##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p
## (1)
## \frac{De}{Dt} = \frac{\partial}{\partial t} (\frac{P}{(\gamma - 1) \rho}) + (\vec{u} \cdot \nabla) (\frac{P}{(\gamma -1) \rho})## (2)
Replacing (2) into (1) and multiplying both side by ##(\gamma - 1) \rho##
## \frac{\partial p}{\partial t} - \frac{p \partial \rho}{\rho \partial t} + \vec{u} \cdot \nabla p - \frac{p}{\rho} \vec{u} \cdot \nabla \rho + (\gamma -1)p \nabla \cdot \vec{u} = - (\gamma -1) \vec{u} \cdot \nabla p##
Then using the definition of the material derivative.
##\frac{Dc}{Dt} = \frac{\partial c}{\partial t} + \vec{u} \cdot \nabla c##
##\frac{Dp}{Dt} - \frac{P}{\rho} \frac{D\rho}{Dt} + (\gamma -1)p \nabla \cdot \vec{u} = - (\gamma -1) \vec{u} \cdot \nabla p ##
So far there are too much terms on the left hand side. For instance, there are two terms with ##\gamma## and in the final answer there is only one.
After spending a lot of time in this problem, I still fail to get the asked answer.
Starting with ##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p## I have to derive the following expressions.
##\frac{Dp}{Dt} + \gamma p \nabla\cdot \vec{u} = 0
##
##\frac{D}{Dt} \frac{p}{\rho^{\gamma}} = 0
##
I don't know what I'm doing wrong or what I'm missing, but I couldn't get those expressions.
Here is what I did.
##\frac{De}{Dt} + (\gamma - 1)e \nabla \cdot \vec{u} = - \frac{1}{\rho} (\vec{u} \cdot \nabla)p
## (1)
## \frac{De}{Dt} = \frac{\partial}{\partial t} (\frac{P}{(\gamma - 1) \rho}) + (\vec{u} \cdot \nabla) (\frac{P}{(\gamma -1) \rho})## (2)
Replacing (2) into (1) and multiplying both side by ##(\gamma - 1) \rho##
## \frac{\partial p}{\partial t} - \frac{p \partial \rho}{\rho \partial t} + \vec{u} \cdot \nabla p - \frac{p}{\rho} \vec{u} \cdot \nabla \rho + (\gamma -1)p \nabla \cdot \vec{u} = - (\gamma -1) \vec{u} \cdot \nabla p##
Then using the definition of the material derivative.
##\frac{Dc}{Dt} = \frac{\partial c}{\partial t} + \vec{u} \cdot \nabla c##
##\frac{Dp}{Dt} - \frac{P}{\rho} \frac{D\rho}{Dt} + (\gamma -1)p \nabla \cdot \vec{u} = - (\gamma -1) \vec{u} \cdot \nabla p ##
So far there are too much terms on the left hand side. For instance, there are two terms with ##\gamma## and in the final answer there is only one.
Last edited: