Pressure gradient causes conservative force?

In summary, the question of whether the force that accelerates a fluid between two points of different pressure is conservative is a complex one. While the force per unit volume can be calculated using the vector identity ## \vec{f}_v=-\nabla P ##, there are also viscous damping forces that must be taken into account in a real fluid. Thus, the forces in a liquid are ultimately non-conservative, despite the conservative nature of the pressure gradient force. Additionally, the force of gravity on an object in the fluid must also be considered. Archimedes' principle can be derived from the pressure gradient force using the vector identity ## \int \nabla P \, d^3 x=\int P \,
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Alkmini Moschoviti
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Is the force that accelerates afluid betwen two points of different pressure conservative?
 
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A very interesting question. Since the force per unit volume is ## \vec{f}_v=-\nabla P ## , and ## \nabla \times \nabla P=0 ##, (a vector identity that is commonly known from E&M theory), you might think that could be possible. In a real fluid however, there will also be viscous damping forces on the volume that are part of the total force equation that are often left out of the equation ## \vec{f}_v=-\nabla P ##. ## \\ ## So that the answer is the forces in a liquid are non-conservative, even though the part that comes from the minus of the pressure gradient could be considered to be conservative. ## \\ ## Note: This is my own analysis=others may agree or disagree=this is the first time I have encountered this question. ## \\ ## Additional note: You could think of these forces as acting on a small object of finite volume that is placed in the fluid. (And of course gravity will also act on the object, which is a conservative force).
 
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Thank you so much for your answer
It is clear
 
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Alkmini Moschoviti said:
Thank you so much for your answer
It is clear
I added one more part above, about the force of gravity on an object in the fluid=be sure and see that. ## \\ ## Additional item: There is a vector identity, ## \int \nabla P \, d^3 x=\int P \, \hat{n} dA ##, (where ## \hat{n} ## is the outward pointing normal), and to have equilibrium, ## -\delta g \, \hat{z}+-\nabla P= 0 ##, (where ## \delta ## is the density (mass per unit volume)), so that you can actually derive Archimedes' principle from the pressure gradient force.
 
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FAQ: Pressure gradient causes conservative force?

What is a pressure gradient?

A pressure gradient is the change in pressure over a given distance. It is represented by the symbol ∇P.

How does a pressure gradient cause a conservative force?

A pressure gradient causes a conservative force by creating a difference in pressure between two points. This difference in pressure results in a net force that acts in the direction of decreasing pressure, leading to the movement of fluids or particles.

Can you give an example of how a pressure gradient causes a conservative force?

One example of this is wind. Wind is caused by a pressure gradient between high and low pressure systems. The air moves from high pressure to low pressure, creating a force that causes wind to blow.

What is the relationship between pressure gradient and conservative force?

The relationship between pressure gradient and conservative force is that a pressure gradient creates a conservative force. This force acts in the direction of decreasing pressure and is responsible for the movement of fluids or particles.

How does the strength of a pressure gradient affect the magnitude of a conservative force?

The strength of a pressure gradient directly affects the magnitude of a conservative force. A larger pressure gradient leads to a stronger force, while a smaller pressure gradient results in a weaker force. This is because a larger pressure gradient creates a larger difference in pressure, resulting in a stronger force.

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