Fluids - Bernoulli's equation - (First and Third Terms)

In summary: So if we have a system in water with air pressure outside of the system, then the pressure from the atmospheric pressure will act on the system and be in the second term of the equation. However, if we remove the atmospheric pressure and only have the water pressure on the system, then the first term would be zero because the pressure from the water would be the same on both sides.
  • #1
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Hello,

I am having some difficulty in understanding the terms in Bernoulli's equation.

Bernoulli's equation is given as follows:
P + 0.5(rho)(V^2) + (rho)gh = constant (along a streamline)
  • The first term (P) is the Static Pressure.
  • The second term (0.5(rho)(V^2)) is the Dynamic (Kinetic) Pressure.
  • The third term ((rho)gh) is the Hydrostatic Pressure.
Bernoulli's equation is only applicable along a streamline in steady state flow, inviscid flow (shear stress = 0), and incompressible fluids.

I am having difficulty in understanding the difference between the first term (Static Pressure) and third term (Hydrostatic Pressure) in Bernoulli's equation. Here's what I understand so far.

===

The Static Pressure is the pressure (force/area) that the fluid molecules exert on each other and the walls of the container. Can I think of this as the internal energy of the fluid molecules? Meaning that this is the energy that the fluid molecules possesses to do something (work?). Can I think of Static Pressure as a Pressure (Force/Area) which has an ability to do work?

If so, then the Static Pressure is the same in the entire fluid at a given steady state. If the force exerted by the fluid molecules increases, the static pressure of the system also increases. This increase in pressure can be created with a differential pressure, hence you get a moving fluid, hence you get a velocity and hence this is where the Dynamic (Kinetic) Pressure comes into play.

The Hydrostatic Pressure is the pressure at a depth felt by the weight of the fluid above. This pressure increases with depth as there is more fluid weight above.

===

Is my understanding correct? Please feel free to correct anything I have written above in terms of terminology, description, and meaning. Any analogies are also welcome in the understanding of the three terms in Bernoulli's equation.
 
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  • #2
Its an energy conversion equation. Maybe is better to look in this form to beter understand the terms.
E= En+Ek+Ep = P1V1+1/2mv2+mgh

En- external work applied on the system
Ek- kinetic energy of the system
Ep- potential energy of the system

And your equation is represents the Bernoulli equation in terms of energy per unit volume.

- So the first term is hidrostatic pressure, but hidrostatic pressure of the environment outside the system. If the system on both sides is open to the atmospheric pressure, like a bucket full of water with a hole in the bottom, and you are watching the two open sections then this term will be zero because there will be no pressure difference (no work from outside) acting on the system.
- The second term as you said is the dynamic presure which depends of the square of the speed of the fluid.
- The third term represents the potential difference between the two sections of the system and essentially is the pressure at a depth felt by the weight of the fluid above. If you have a horizontal pipe, this member will disappear because there will be no hight deference between the sections.
 
  • #3
That isn't entirely correct either. For the most part, these explanations are correct with a few details off.

It does represent conservation of energy as previously stated.

So let's rewrite the equation:
[tex]
P + \frac{\rho V^2}{2} + \rho gz = Const.
[/tex]

In general, conservation of energy says that the total energy is equal to potential energy plus kinetic energy. We all know the form of the equation for kinetic energy, and [itex]\frac{\rho V^2}{2}[/itex] looks an awful lot like it. In fact, that term represents the fluid kinetic energy per unit volume. The remaining two terms are forms of potential energy. [itex]\rho gz[/itex] is obviously the gravitational potential energy (again, per unit volume). Those terms are easy.

Pressure isn't quite as clear. The force exerted on a fluid by pressure is defined as [itex]F=-\nabla P[/itex]. That implies that [itex]-P[/itex] can be treated as a potential and the [itex]P[/itex] term is the potential energy per unit volume in the system due to the pressure force.
 
  • #4
As I said, that form that you wrote comes from the conservation energy law and it is in units energy per volume, I clearly stated that above.

The pressure represents the pressure that acts from the environment on the boundaries of the system that we chose.
 

Related to Fluids - Bernoulli's equation - (First and Third Terms)

What is Bernoulli's equation and how is it used?

Bernoulli's equation is a fundamental principle in fluid dynamics that describes the relationship between the pressure, velocity, and height of a fluid. It is used to calculate the pressure or speed at any point in a fluid flow, as well as to analyze the effects of different conditions on the fluid.

What are the first and third terms in Bernoulli's equation?

The first term in Bernoulli's equation is the static pressure, which represents the pressure exerted by the fluid at a specific point. The third term is the potential energy term, which represents the height of the fluid relative to a chosen reference point and its effect on the fluid's energy.

What is the significance of the first term in Bernoulli's equation?

The first term, or static pressure, is important because it represents the pressure exerted by the fluid at a specific point. This pressure is caused by the collisions of the fluid particles with the walls of the container or any objects in the fluid's path. It also plays a role in determining the direction and speed of the fluid flow.

How does Bernoulli's equation relate to the conservation of energy?

Bernoulli's equation is a manifestation of the principle of conservation of energy. It states that the sum of the kinetic energy, potential energy, and internal energy of a fluid must remain constant as it flows from one point to another. This means that as the fluid speeds up, its potential energy decreases and its kinetic energy increases to maintain the overall energy balance.

Can Bernoulli's equation be applied to all types of fluids?

Bernoulli's equation can be applied to all types of fluids, including liquids and gases, as long as certain assumptions are met. These assumptions include the fluid being incompressible, the flow being steady, and the fluid particles being non-viscous (i.e. not sticking to each other or the container walls). If these conditions are met, then Bernoulli's equation can accurately describe the behavior of the fluid flow.

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