How to show Bernoulli's equation applies on same streamline?

[member 428835]

When deriving Bernoulli's equation from Navier Stokes, how do we know it is only valid along a streamline? At the very end of my derivation, assuming Newtonian, incompressible, inviscid, irrotational flow I have $\nabla(\partial_t \phi + |\vec{u}|^2/2+p/\rho + g z) = \vec{0} \implies \partial_t \phi + |\vec{u}|^2/2+p/\rho + g z = const.$ where I think the constant implies something about which streamline you're on. Also, $\vec{u}=\nabla \phi$.

Thanks!

 

[boneh3ad]

It's probably easiest if you start from the Euler equations cast in terms of streamline variables. If you do that, then you show that integrating along a streamline results in Bernoulli's equation.

Of course, it can also be applied globally if the flow is irrotational or across multiple streamlines if all of them originate with the same total pressure.

 

[member 428835]

It's probably easiest if you start from the Euler equations cast in terms of streamline variables. If you do that, then you show that integrating along a streamline results in Bernoulli's equation.

Of course, it can also be applied globally if the flow is irrotational or across multiple streamlines if all of them originate with the same total pressure.

That's a good way to do it. But my problem is, deriving it through NS as shown above, when I arrive at $\partial_t \phi + |\vec{u}|^2/2+p/\rho + g z = const.$ how would I know, or rather where is the assumption, that only validates this equation along a streamline?

I'm not really looking at how to derive Bernoulli's along streamline, I'm wondering why that equation is only valid along one. Thanks for your reply!