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

[PhyIsOhSoHard]

Homework Statement

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

Homework Equations

Bernoulli's equation:
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

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?

 

[voko]

$p_2$ 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.

 

[rock.freak667]

From how it looks P1 is at one pressure like atmosphere and P2 would be where the tank is draining out to atmosphere.

 

[Chestermiller]

$p_2$ 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.

 

[voko]

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.

 

[Chestermiller]

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.