Fluid Mechanics: Momentum Equation- When to include p_atm in equation?

• Master1022
In summary: Again, thank you for your interest in the Bernoulli and momentum equations.In summary, the conversation discusses the use of atmospheric pressure in the Bernoulli and momentum equations. It is mentioned that in the Bernoulli equation, it doesn't matter if you use gauge pressure or absolute pressure as long as you are consistent, but in the momentum equation, it is important to be aware of which one you are using. It is also noted that in the momentum equation, the net force exerted on the control volume needs to be carefully considered when taking into account the combination of fluid and surrounding atmosphere.
Master1022
Homework Statement
General problem. For example, air enters a wind tunnel (or a jet; basically something that has a larger area of entry than exit) and the total pressure is atmospheric pressure. Write out the momentum equation for this control volume.
Relevant Equations
Bernoulli's Equation
Momentum Equation
So if we define point 1 at the entrance and point 2 at the exit, then we can write out Bernoulli's equation along a horizontal streamline as such: $$p_1 + \frac{1}{2}\rho v_{1}^2 = p_2 + \frac{1}{2}\rho v_{2}^2 = p_{atm}$$

One question is: won't there be p_atm also contributing to the static pressure at p1 and p2 (i.e. p_atm + p_1 or p_atm + p_2)? Mathematically I see that there would be problems if we had this in the equation, but I cannot understand why. Does atmospheric pressure not contribute to the static pressure inside an object (e.g. this wind tunnel or a pipe)?

Then later on, when we write the momentum equation for our control volume, we get (Force = Change in Momentum Flux R(-->)):
$$p_1 A_1 - p_2 A_2 - p_{atm}(A_1 - A_2) + F = \rho u_2 ^ 2 A_2 - \rho u_1 ^ 2 A_1$$

However, once again, I cannot understand why there isn't a p_atm term for A1 & A2?

Apologies for this, as I understand there is some basic error.

In the Bernoulli equation, it doesn't matter (since it cancels).

It isn't clear to me what you are doing in the momentum equation. You can work in terms of absolute pressures or you can work in terms of gauge pressures. But your choice dictates your interpretation of the force F.

Chestermiller said:
In the Bernoulli equation, it doesn't matter (since it cancels).

Thank you for your response. So if we had a $p_{atm}$ term, the equation would read $$p_{atm} = p_{atm} + \frac{1}{2}\rho u^2 + p_1$$ If we were to consider pressure this way, would this mean that the $p_1$ is negative if $u > 0$?

Master1022 said:
Thank you for your response. So if we had a $p_{atm}$ term, the equation would read $$p_{atm} = p_{atm} + \frac{1}{2}\rho u^2 + p_1$$ If we were to consider pressure this way, would this mean that the $p_1$ is negative if $u > 0$?
Please tell me what you mean by this equation. You seem to be using gauge pressures. Is that correct. If so, then ##\p_1## is sub-atmospheric (i.e., partial vacuum).

So, in the Bernoulli equation, you can use either gauge pressures or absolute pressures as long as you are consistent.

In the momentum equation, you need to be careful what you are doing, particularly if you are trying to find the net force exerted on the control volume (by the combination of fluid and surrounding atmosphere).

Master1022 said:
Thank you for your response. So if we had a $p_{atm}$ term, the equation would read $$p_{atm} = p_{atm} + \frac{1}{2}\rho u^2 + p_1$$ If we were to consider pressure this way, would this mean that the $p_1$ is negative if $u > 0$?
In your equation, I'm guessing that ##p_1## is gauge pressure. If that interpretation is correct, then the equation is telling us that ##p_1## is less than atmospheric pressure (negative gauge pressure).

In the Bernoulli equation, it doesn't matter if you use gauge pressure or absolute pressure, as long as you are consistent.

In the momentum equation, you need to be careful and be aware of which you are using, particularly if you want to use the equation to determine the net force of the atmosphere and the fluid on the body of the control volume.

Chestermiller said:
Please tell me what you mean by this equation. You seem to be using gauge pressures. Is that correct. If so, then ##\p_1## is sub-atmospheric (i.e., partial vacuum).

So, in the Bernoulli equation, you can use either gauge pressures or absolute pressures as long as you are consistent.

In the momentum equation, you need to be careful what you are doing, particularly if you are trying to find the net force exerted on the control volume (by the combination of fluid and surrounding atmosphere).
Thank you for your response. Yes, you are correct, that is what I meant. It now makes more sense.

Master1022 said:
Thank you for your response. Yes, you are correct, that is what I meant. It now makes more sense.

1. What is the momentum equation in fluid mechanics?

The momentum equation in fluid mechanics is a fundamental equation that describes the relationship between the forces acting on a fluid and the resulting motion of the fluid. It is typically written as F = ma, where F is the sum of all external forces, m is the mass of the fluid, and a is the resulting acceleration.

2. When do we need to include p_atm in the momentum equation?

The atmospheric pressure, denoted as p_atm, should be included in the momentum equation when dealing with fluid flow in open systems such as pipes, channels, and open channels. This is because the atmospheric pressure acts as a boundary condition on the fluid and affects its behavior.

3. Can we neglect p_atm in the momentum equation for closed systems?

In closed systems, where there is no interaction with the atmosphere, p_atm can be neglected in the momentum equation. This is because there is no external atmospheric pressure acting on the fluid and thus it does not need to be considered in the calculations.

4. How does p_atm affect the momentum equation?

The atmospheric pressure, p_atm, acts as a reference point for fluid pressure in open systems. It affects the momentum equation by adding a constant value to the fluid pressure, which can alter the resulting acceleration of the fluid. In closed systems, where p_atm is not considered, it does not affect the momentum equation.

5. What are some common applications of the momentum equation in fluid mechanics?

The momentum equation has a wide range of applications in fluid mechanics, including analyzing the flow of fluids in pipes, channels, and pumps, as well as in predicting the behavior of objects moving through fluids, such as ships and airplanes. It is also used in the design of hydraulic systems and in the study of fluid dynamics in natural phenomena such as ocean currents and weather patterns.

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