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

In summary, when deriving Bernoulli's equation from Navier-Stokes, it is easiest to start from the Euler equations cast in terms of streamline variables. This integration along a streamline results in Bernoulli's equation. 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. However, when deriving it through Navier-Stokes, the assumption that only validates the equation along a streamline is not explicitly stated. This is because the equation ##\partial_t \phi + |\vec{u}|^2/2+p/\rho + g z = const.## is only valid along a streamline, and the assumption is inherent in the derivation process.
  • #1
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!
 
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  • #2
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.
 
  • #3
boneh3ad said:
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!
 

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

1. How do I identify a streamline in Bernoulli's equation?

In Bernoulli's equation, a streamline is a line that represents the path of a fluid particle as it moves through a flow field. To identify a streamline, you can use a fluid flow visualization technique such as dye tracing or particle tracking.

2. What is the significance of applying Bernoulli's equation on the same streamline?

Applying Bernoulli's equation on the same streamline allows us to analyze the changes in fluid velocity, pressure, and elevation along a specific path in a flow field. This helps us understand the impact of various factors on the fluid flow.

3. Can Bernoulli's equation be applied on any streamline?

Yes, Bernoulli's equation can be applied on any streamline as long as the flow is steady, inviscid, and irrotational. These assumptions ensure that the fluid particles along the streamline experience the same conditions and follow the same path.

4. How do I solve for unknown variables in Bernoulli's equation on the same streamline?

To solve for unknown variables in Bernoulli's equation, you can use the conservation of mass and energy principles to set up a system of equations. Then, you can use algebraic or numerical methods to solve for the unknowns.

5. What are some real-life applications of Bernoulli's equation on the same streamline?

Bernoulli's equation on the same streamline is used in various engineering applications, such as designing aircraft wings, calculating flow through pipes, and analyzing air flow in HVAC systems. It is also used in the medical field for modeling blood flow in arteries.

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