- #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: