Bernoulli Equation and Navier-Stokes

[member 428835]

Hi PF!

I was reading about Bernoulli's equation for steady, inviscid, incompressible flow. Now it's my understanding this equation is derived from the Navier-Stokes (momentum balance); then these two equations are identical regarding information offered. However, while thinking about applications of Bernoulli I came to a problem I couldn't reason: suppose you have a tug boat that's pulling a cone underwater, and that the process is steady and the velocity of the boat (and the cone) is constant. Let the entrance of the cone have pressure $P_1$ and velocity $V_1$ and let pressure and velocity at exit be $P_2$ and $A_2$. Bernoulli's equation gives $$P_1+\frac{1}{2}\rho V_1^2=P_2+\frac{1}{2}\rho V_2^2$$. However, Navier-Stokes is expressed $$
\partial_t\iiint_v \rho \vec{V} \, dv+ \iint_{\partial v} \rho \vec{V} (\vec{V} \cdot \vec{n}) \, dS = \sum \vec{F}\implies \\
\rho(V_2^2 A_2-V_1^2 A_1) = P_1A_1-P_2A_2+F_{boat}$$
But clearly the final expression is not the same, namely the force of the boat is present. Can someone explain why I am having a discrepancy?

 

[boneh3ad]

What are you defining as $F_{boat}$?

 

[member 428835]

What are you defining as $F_{boat}$?

$F_{boat}$ would be the tension force from the boat pulling the cone. Does that make sense or have I not described it clearly?

 

[boneh3ad]

It makes some sense but you are now essentially double-counting the tension force due to the boat. You are using a control volume analysis and you have already zeroed out the time-varying terms that are in there, which assumes the flow is steady. That assumption carries with it the constant tension force, in effect. That tension force isn't acting on the fluid (at least directly) anyway. Its effect on the fluid is essentially distributed over the surface of the cone in your example, which means it is essentially translated into the pressure gradient through the cone.

 

[member 428835]

Awesome, thanks for clearing this up! So to be clear, even if we were accelerating $F_{boat}$ would still be disregarded and the pressure forces would account for the tension force?

 

[boneh3ad]

Well you'd have to take the acceleration into account somehow. The easiest way to do it would be to take it into account via the resulting changing inflow conditions to the cone. After all, those changes are captured in the $\partial_t\int\int\int(\rho \vec{V})dv$ term, which is, at the end of the day, a force (time rate of change of momentum). The boat accelerating simply makes that term nonzero.

 

[Chestermiller]

Awesome, thanks for clearing this up! So to be clear, even if we were accelerating $F_{boat}$ would still be disregarded and the pressure forces would account for the tension force?

NO. In my judgment, your macroscopic momentum balance is perfectly valid (as reckoned by an observer traveling with the velocity of the cone). However, the Bernoulli equation tells you something else. The microscopic Bernoulli equation is obtained by dotting the Navier Stokes Eqn. (microscopic momentum balance equation) with the fluid velocity. This is usually referred to as the mechanical energy balance equation. In addition to the usual Bernoulli terms, this also includes viscous dissipation terms. The viscous dissipation terms are then neglected to obtain the Bernoulli equation. This is all addressed in Bird, Stewart, and Lightfoot.

 

[member 428835]

Chet, are you saying that $F_{boat}$ SHOULD BE considered even if we are traveling at a constant velocity?

 

[Chestermiller]

Chet, are you saying that $F_{boat}$ SHOULD BE considered even if we are traveling at a constant velocity?

Yes. F is the force that the cone exerts on the water passing through it, and it is also equal to minus the tension in the cable pulling the cone.

 

[member 428835]

Yes. F is the force that the cone exerts on the water passing through it, and it is also equal to minus the tension in the cable pulling the cone.

So in my post #1, it seems Bernoullis equation $P_1+\frac{1}{2}\rho V_1^2=P_2+\frac{1}{2}\rho V_2^2$ and this force balance give different information. But how is this possible since Bernoulli is derived from Navier-Stokes?

 

[boneh3ad]

Yes. F is the force that the cone exerts on the water passing through it, and it is also equal to minus the tension in the cable pulling the cone.

I disagree. Consider a nozzle at the end of a hose. If you want to apply Bernoulli's equation through that nozzle, you don't have to consider the tension in the hose/connection that is holding the nozzle in place in order to calculate the flow. It does not enter into the analysis of the control volume at all.

 

[member 428835]

(I'll wait on further comments until you two agree. I don't want to get in the way, and I've seen both of you in action: you both know what you're talking about. A showdown of giants)

 

[Chestermiller]

So in my post #1, it seems Bernoullis equation $P_1+\frac{1}{2}\rho V_1^2=P_2+\frac{1}{2}\rho V_2^2$ and this force balance give different information. But how is this possible since Bernoulli is derived from Navier-Stokes?

They don't. You can combine them to determine F.

I disagree. Consider a nozzle at the end of a hose. If you want to apply Bernoulli's equation through that nozzle, you don't have to consider the tension in the hose/connection that is holding the nozzle in place in order to calculate the flow. It does not enter into the analysis of the control volume at all.

The control volume analysis determines the force that the nozzle is exerting on the fluid flowing through it. This is equal an opposite to the force that the fluid is exerting on the nozzle. And the latter is equal an opposite to the tensile force that the hose exerts on the nozzle (to hold it in place). For a complete analysis of how the Bernoulli equation is properly combined with the control volume macroscopic momentum balance to determine the force that the fluid exerts on the nozzle, see the following thread: https://www.physicsforums.com/threads/nozzle-reaction-forces.888983/

 

[boneh3ad]

The control volume analysis determines the force that the nozzle is exerting on the fluid flowing through it. This is equal an opposite to the force that the fluid is exerting on the nozzle. And the latter is equal an opposite to the tensile force that the hose exerts on the nozzle (to hold it in place). For a complete analysis of how the Bernoulli equation is properly combined with the control volume macroscopic momentum balance to determine the force that the fluid exerts on the nozzle, see the following thread: https://www.physicsforums.com/threads/nozzle-reaction-forces.888983/

Right, so in applying Bernoulli's equation (or the Navier-Stokes equations) to a control volume, one need not account for the force exerted by the boat directly provided that the inlet flow field, steady or unsteady, is known (as is the case in the OP's example). Since that tension in the wire does directly equate to the force of the water on the object being dragged, you can of course then back out the tension required for a given flow condition, but that force needn't be considered in the actual analysis of the flow field given the assumptions about a known inflow.

If the inflow is not known a priori, then it's a different story. Trying to determine the flow field that results from a given tension force, for example, would be considerably more difficult. Then you would need to know the tension in the cable, but even then, you really only need to know it in order to convert that into a pressure distribution and you don't need to take it into account as an added body force on the flow.

 

[Chestermiller]

Yes. I agree. This is different from the nozzle problem because we don't know how much fluid is entering the cone. So, a much more comprehensive flow analysis would be required that takes into account the flow diverted around the cone. As a first approximation, however, I might assume that the flow through the cone is some fraction of the boat velocity times the inlet area.

In any event, the original question was whether Josh's two original equations, as written, were correct renditions of the macroscopic momentum balance and Bernoulli's equation, and whether they were independent of one another. In my judgment, the answer to this is Yes.

 

[member 428835]

Awesome, thank you both so much! The link really helped (well it was confusing but your derivation was clear!)

As a first approximation, however, I might assume that the flow through the cone is some fraction of the boat velocity times the inlet area.

Why wouldn't the flow through the cone simply be boat velocity, rather than some fraction?

 

[Chestermiller]

Awesome, thank you both so much! The link really helped (well it was confusing but your derivation was clear!)

Why wouldn't the flow through the cone simply be boat velocity, rather than some fraction?

Suppose that the cone is totally closed at the small end. Then no flow goes through, and it is just about the same as a solid being dragged through the water.

 

[boneh3ad]

Why wouldn't the flow through the cone simply be boat velocity, rather than some fraction?

That was sort of my point. If you already know the boat's velocity, then you know (at least approximately) the inflow conditions. In reality, the inflow won't exactly be the boat velocity because the cone, being an immersed body, is going to incur drag and move some of the water along with it. Some of the water will move around it instead of through it, though that will likely be a fairly small fraction. For the sake of this thought experiment, though, you are perfectly fine assuming the inflow velocity matches that of the boat, especially when neglecting viscosity. The only reason you would need to explicitly account for the tension force is if you wanted to know what sort of force (or thrust from the boat) was required to achieve such a constant speed.

 

[member 428835]

Thank you both very much!