[Fluid Mechanics] How is the pressure at 2 different heights the same?

In summary: Assumption (5) is acceptable because it simplifies the problem. By assuming that the pressure just outside the tank is equal to atmospheric pressure, we can solve for P2 without having to worry about the pressure inside the tank.
  • #1
PhyIsOhSoHard
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Homework Statement


Lzcy4z5.gif


A tank containing water with a small nozzle at the bottom right where the water flows out.


Homework Equations


Bernoulli's equation:
[itex]p_1+\frac{1}{2}\rho v^2_1+\rho g h_1=p_2+\frac{1}{2}\rho v^2_2+\rho g h_2[/itex]

3. Assumptions
(1) Quasi-steady flow
(2) Incompressible flow
(3) Neglect friction
(4) Flow along a streamline
(5) ##p_1=p_2##

4. The attempt at a solution
Can somebody explain to my why assumption (5) is acceptable? My intuition tells me that the lower you dive into water the more does the pressure rise.

How can my textbook make this assumption for Bernoulli's equation?
 
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  • #2
##p_2## here means the pressure "just outside the tank". Which would be atmospheric. Just inside the tank, at the opening, the pressure will indeed be greater. You can set up the Bernoulli equation for the "just inside" and "just outside" points, you should get the same exit velocity.
 
  • #3
From how it looks P1 is at one pressure like atmosphere and P2 would be where the tank is draining out to atmosphere.
 
  • #4
voko said:
##p_2## here means the pressure "just outside the tank". Which would be atmospheric. Just inside the tank, at the opening, the pressure will indeed be greater. You can set up the Bernoulli equation for the "just inside" and "just outside" points, you should get the same exit velocity.
I don't think that this is quite correct. Just inside the tank adjacent to the exit, the fluid velocity is already essentially at the exit velocity, and the pressure at that location is thus also very close to atmospheric. The acceleration as a result of the decrease in potential energy has mostly taken place by the time the fluid parcels reach the location "just inside" the exit.
 
  • #5
I do not see any disagreement with what I wrote, Chestermiller. "Just inside" the pressure may be very close to atmospheric, but still it is a tad greater. I did not mean to imply that there is a major pressure gradient between "just inside" and "just outside". Perhaps that should have been stated explicitly, though.
 
  • #6
voko said:
I do not see any disagreement with what I wrote, Chestermiller. "Just inside" the pressure may be very close to atmospheric, but still it is a tad greater. I did not mean to imply that there is a major pressure gradient between "just inside" and "just outside". Perhaps that should have been stated explicitly, though.
Then, yes, we are in perfect agreement.
 

1. How does fluid pressure change with height?

The pressure in a fluid increases with depth or height. This is because the weight of the fluid above exerts a force on the fluid below, increasing the pressure.

2. Why is the pressure at the bottom of a container with fluid higher than at the top?

This is due to the weight of the fluid above. The fluid at the bottom of the container has more fluid above it, so it experiences a greater force and therefore a higher pressure.

3. How is pressure at 2 different heights the same in a fluid?

In a stationary fluid, the pressure at any given horizontal level is the same. This is because the weight of the fluid above that level is evenly distributed, resulting in an equal force on all points at that level.

4. Does the shape of the container affect the pressure at different heights?

No, the shape of the container does not affect the pressure at different heights. As long as the fluid is in a stationary state, the pressure will be the same at any given horizontal level regardless of the container's shape.

5. Can the pressure at 2 different heights ever be different in a fluid?

Yes, if the fluid is in motion, the pressure at different heights can vary. This is due to the difference in velocity and the Bernoulli's principle, which states that as the velocity of a fluid increases, the pressure decreases. In this case, the pressure at the bottom of the container may be higher than the pressure at the top.

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