# Fluid dynamics

1. Mar 4, 2008

1. The problem statement, all variables and given/known data
Starting from the equations of conservation of mass and momentum for an inviscid compressible gas,
$$\frac{\partial \rho}{\partial t} + \bigtriangledown . (\rho {\bf u}) = 0$$

$$\rho ( \frac{\partial {\bf u}}{\partial t} + ({\bf u}. \bigtriangledown){\bf u}) = \bigtriangledown p + {\bf F}$$

derive for a fixed volume V enclosed by a surface S:

$$\frac{d}{dt} \int_V \frac{1}{2} \rho u^2 dV + \int_S \frac{1}{2} \rho u^2 {\bf u.n} dA = - \int_S p {\bf u.n} dA + \int_V (p\bigtriangledown. {\bf u} + {\bf F.u}) dV$$.

2. Relevant equations
Divergence theorem:

$$\int_V \bigtriangledown . {\bf u} dV = \int_S {\bf u.n} dA$$
where A is the surface enclosing V

3. The attempt at a solution
I started from the first equation, multiplied this through by $$\frac{u^2}{2}$$, applied the divergence theorem and looked at what the full derivative given in the question is in terms of the chain rule, and ended up with

$$\frac{d}{dt} \int_V \frac{1}{2} \rho u^2 dV + \int_S \frac{1}{2} \rho u^2{\bf u.n} dA = \int_V \rho {\bf u} . \frac{\partial {\bf u}}{\partial t} dV$$

Which looks like it might be right. However I am a bit wary though because I don't know if I should be considering the del operator acting on the $$\frac{u^2}{2}$$ that I multiplied through by. Anyway, assuming this is right (I have the alternative expression if it's not, and that also looks like it could be right) I tried to proceed by dotting the second fluid equation with $${\bf u}$$, and things were looking quite promising but it didn't quite come out. I seemed to be having particular trouble with the $$p \bigtriangledown.{\bf u}$$ term.

Am I going along the right lines here? Any hints on how I can finish this off?

Thanks.

Edit - Sorry about my lack of vectors, I can't seem to find a command that works on this forum. u and n are vectors through, and u^2 = u.u Also, I guess I should have mentioned (although the question doesn't actually specifiy) that rho, u and p are function sof position and time.