Very simple problem: Bernoulli

rsq_a · **Joined:** Mon, 26 Nov 2007 18:02:52 UTC **Posts:** 55

This was a very simple problem driving me competely insane.

The point was to derive bernoulli's equation (from the NS equations) assuming barotropic flow. That is, assuming $p=p(\rho)$ .

Now at one point in the proof, you are suppose to show,

$\frac{1}{\rho} \frac{dp}{d\rho} \nabla \rho = \nabla \int \frac{1}{\rho} \frac{dp}{d\rho} d\rho = \nabla \int \frac{dp}{\rho}$

I can do it. At least on a purely algebraic level:

Let $\frac{dQ}{d\rho} = \frac{1}{\rho}\frac{dp}{d\rho}$ . Then, the above becomes,

$\frac{1}{\rho} \frac{dp}{d\rho} \nabla \rho = \frac{dQ}{d\rho}\frac{\partial \rho}{\partial x_i} = \frac{\partial Q}{\partial x_i} = \frac{1}{\partial x_i} \int \frac{dQ}{d\rho} d\rho = \nabla \int \frac{1}{\rho} \frac{dp}{d\rho} d\rho = \nabla \int \frac{dp}{\rho}$

However, I can't understand why this works. I'm sure there is a very simple reason for all this. I just can't see it.

rsq_a · **Joined:** Mon, 26 Nov 2007 18:02:52 UTC **Posts:** 55

never mind. I figured it out.

S.O.S. Mathematics CyberBoard

Very simple problem: Bernoulli

Who is online