Pressure gradient causes conservative force?

[Alkmini Moschoviti]

Is the force that accelerates afluid between two points of different pressure conservative?

 

[Charles Link]

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).

 

[Alkmini Moschoviti]

Thank you so much for your answer
It is clear

 

[Charles Link]

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.