A question regarding Bernoulli's equation

[Volta]

P₁ + ρgh₁ + 0.5ρv²₁ = P₂ + ρgh₂ + 0.5ρv²₂
In the derivation of this equation from the theorem of Work-Kinetic Energy, pressures ( P₁ and P₂) represent are derived from F = PA, forces affected by other portions of fluid upon the fluid in the middle (which is our concern) at 2 different points. So do we consider them the pressure of the fluid itself or the pressure exerted by surroundings on the fluid?

 

[BvU]

The pressure in the fluid which is 'our concern'

 

[Mister T]

So do we consider them the pressure of the fluid itself or the pressure exerted by surroundings on the fluid?

The pressure exerted by the fluid element on its surroundings is equal to the pressure exerted on the fluid element by its surroundings.

 

[Volta]

Ah I see. But is this always the case?

 

[BvU]

Can you think of a possible exception ?

 

[Mister T]

Ah I see. But is this always the case?

It's a consequence of Newton's Third Law.

 

[Volta]

Sorry for the late reply.

Like, this applies when the fluid is in dynamic equilibrium right?

For example, consider the glucose solution plastic bag attached to the veins in the hand of a patient, if we set the bag at some specific height, glucose will reach the veins but won't be able to get inside since blood pressure is higher, so I think blood will get out rather - until- the pressure of the glucose solution is equal to the blood pressure, then we have a state of dynamic equilibrium right? But if we increase the pressure of the glucose by increasing the height of the bag, we would create a constant difference in pressure, glucose pressure will be higher and thus glucose will flow inside the veins, here both substances don't have equal pressures

 

[russ_watters]

Like, this applies when the fluid is in dynamic equilibrium right?

For example, consider the glucose solution plastic bag attached to the veins in the hand of a patient, if we set the bag at some specific height, glucose will reach the veins but won't be able to get inside since blood pressure is higher, so I think blood will get out rather - until- the pressure of the glucose solution is equal to the blood pressure, then we have a state of dynamic equilibrium right? But if we increase the pressure of the glucose by increasing the height of the bag, we would create a constant difference in pressure, glucose pressure will be higher and thus glucose will flow inside the veins, here both substances don't have equal pressures

That all sounds fine, but I don't see how it is related to what you were asking before. Force (pressure) applied to something always equals force (pressure) returned from it. That's Newton's 3rd law and it is always true.

What may be your confusion is that Newton's 3rd Law applies all the time, at every point in space and time, but that doesn't mean every point in space and time is the same. You may not be able to compare different points at different places and times unless certain constraints exist. And That's what Bernoulli's principle/equation does; it enables the comparison, with certain defined constraints.

 

[A.T.]

So do we consider them the pressure of the fluid itself or the pressure exerted by surroundings on the fluid?

The pressure of the fluid itself doesn't have to be constant. There can be a pressure gradient across the fluid. And If there is a net force, it will start to flow. But at each interface the pressure by the fluid equals the pressure on the fluid.

 

[Mister T]

But if we increase the pressure of the glucose by increasing the height of the bag, we would create a constant difference in pressure, glucose pressure will be higher and thus glucose will flow inside the veins, here both substances don't have equal pressures

What you describe is essentially the increase in pressure with depth, $\rho gh$. But imagine any horizontal plane within the fluid, the pressure exerted by the fluid above the plane is equal to the pressure exerted by the fluid below the plane at the plane.