Bernoulli’s Equation for plugging finger into dike under water

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In summary, the author assumes that the water outside the hole is stationary and that the pressure inside the Holland is 1 atm. Additionally, the author explains that the water surface can be used to find v1 and p1, which both give the same result as if the water outside the hole was stationary.
  • #1
ChiralSuperfields
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Homework Statement
A legendary Dutch boy saved Holland by plugging a hole
of diameter 1.20 cm in a dike with his finger. If the hole was 2.00 m below the surface of the North Sea (density 1030 kg/m^3) (b) If he pulled his finger out of the hole, during what time interval would the released water fill 1 acre of land to a depth of 1 ft? Assume the hole remained constant in size.
Relevant Equations
Bernoulli’s Equation
For this part (b) of this problem
1670119512790.png
,
The solution is,
1670119564095.png

1670119694881.png

However, why are they allowed to assume for the Bernoulli's equation for the water outside the hole, that the water is stationary (i.e v_1 = 0)?

It also appears that they assume that the pressure inside the Holland is 1 atm, which is only valid when the water level is below the hole.

Many thanks!
 
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  • #2
Callumnc1 said:
, why are they allowed to assume for the Bernoulli's equation for the water outside the hole, that the water is stationary (i.e v_1 = 0)?
On the sea side, I assume you mean.
You are right that immediately in front of the hole the water will be moving some, but also the pressure will have dropped a little in consequence. As you move a bit further back into the water, both of those effects diminish.
You could also forget about that region and instead consider the water surface. That certainly gives v1=0, z1=2m, p1=atm.
 
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  • #3
haruspex said:
On the sea side, I assume you mean.
You are right that immediately in front of the hole the water will be moving some, but also the pressure will have dropped a little in consequence. As you move a bit further back into the water, both of those effects diminish.
You could also forget about that region and instead consider the water surface. That certainly gives v1=0, z1=2m, p1=atm.
Thanks for your reply @haruspex ! I was assuming Holland was 2 m below the water as the problem says that. Were you assuming Holland was at sea level?

Is that why you said that I would be allowed to forget about that region and instead consider the water surface? The water surface is at 1 atm, while the region at the same height as the hole (2m below the surface) is at 1 atm + 20.2 kPa.

Many thanks!
 
  • #4
Callumnc1 said:
Were you assuming Holland was at sea level?
No.
In using Bernoull's principle you pick any two points in the 'streamline'. You have the exit point on the Holland side of the hole, but on the water side you do not have to choose a point right in front of the hole. For the reasons I gave, it is not entirely clear what the velocity and pressure are there. Instead, you can choose a point somewhat further away, on the same level, or a point on the surface. The first will give a pressure patm+gρz (z=2m) and a height difference zero, the second will give a pressure patm and a height difference z=2m. Either way, it adds up to the same.
 
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  • #5
haruspex said:
No.
In using Bernoull's principle you pick any two points in the 'streamline'. You have the exit point on the Holland side of the hole, but on the water side you do not have to choose a point right in front of the hole. For the reasons I gave, it is not entirely clear what the velocity and pressure are there. Instead, you can choose a point somewhat further away, on the same level, or a point on the surface. The first will give a pressure patm+gρz (z=2m) and a height difference zero, the second will give a pressure patm and a height difference z=2m. Either way, it adds up to the same.
Oooh thanks so much @haruspex , I understand now!
 

1. What is Bernoulli’s Equation for plugging finger into dike under water?

Bernoulli’s Equation is a fundamental principle in fluid dynamics that describes the relationship between pressure, velocity, and elevation in a fluid flow. It states that as the velocity of a fluid increases, the pressure decreases, and vice versa.

2. How does Bernoulli’s Equation apply to plugging a finger into a dike under water?

In this scenario, the finger acts as an obstruction to the fluid flow, causing a decrease in velocity and an increase in pressure. According to Bernoulli’s Equation, this increase in pressure will help to prevent water from passing through the hole in the dike.

3. What are the assumptions made in Bernoulli’s Equation?

Bernoulli’s Equation assumes that the fluid is incompressible, inviscid (no friction), and that the flow is steady and laminar. It also assumes that there is no external work being done on the fluid.

4. Can Bernoulli’s Equation be used for any type of fluid?

No, Bernoulli’s Equation is only applicable to incompressible fluids, such as water or air at low speeds. It does not hold true for compressible fluids, such as gases, or for high-speed flows where turbulence is present.

5. Is Bernoulli’s Equation for plugging finger into dike under water always accurate?

No, Bernoulli’s Equation is a simplified mathematical model and may not always accurately predict the behavior of fluids in real-world situations. Factors such as turbulence, viscosity, and external forces can affect the accuracy of the equation.

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