Time taken for fluid to start flowing

  • Thread starter springwave
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In summary, we are trying to find the approximate order of magnitude of the time it takes for liquid to reach a velocity of √2gh when released from a container with a small orifice. To do this, we will need to find the initial acceleration of the liquid, which can be done by analyzing the motion of a small mass element near the bottom of the container. However, we cannot treat the column of water directly above the orifice as a freely accelerating object because the walls of the container exert an upward force on it. Finding the initial acceleration will help us find the approximate time it takes for the liquid to reach its steady flow velocity.
  • #1
springwave
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Homework Statement



A large container has of diameter D has a small orifice or diameter d at it's bottom,
When the orifice is opened, liquid of density ρ starts flowing out with an approximate velocity of √2gh, but the liquid takes some time τ to reach this velocity. Find the approximate order of magnitude of τ.

What is the acceleration of the bottom layer of the liquid, at the instant when the orifice is opened.


Homework Equations



Pascals law
Bernoullis equation





The Attempt at a Solution




If we find the initial acceleration, then it will be quite easy to find the time.
And to do this, I did the following:

Taking a small mass element Δm near the bottom, covering a of length Δx, and using force = pressure * area; I found acceleration = gh/Δx

I don't know how to proceed om here.
It would be great if someone could explain to me, how to analyze the motion of the liquid during this small interval of time, from the instant of release to start of steady flow
 
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  • #2
Treat the column of water directly above the orifice as a freely accelerating object?

F = ρvg = ρva
 
  • #3
We can't do that because, the walls of the container (bottom) exert upward force on the column
 
  • #4
springwave said:
We can't do that because, the walls of the container (bottom) exert upward force on the column

I said the column directly above the orifice. It was just for an estimate.
 
  • #5
.

I would approach this problem by breaking it down into smaller parts and using the principles of fluid mechanics to analyze the motion of the liquid.

Firstly, we can use Bernoulli's equation to determine the velocity of the liquid at the orifice when it starts to flow. We know that the pressure at the surface of the liquid in the container is atmospheric pressure, and at the orifice, it is slightly higher due to the weight of the liquid above it. Using the principle of conservation of energy, we can equate the potential energy of the liquid at the surface to its kinetic energy at the orifice. This will give us the approximate velocity of the liquid at the orifice, which is √2gh, as given in the question.

Next, we can use Newton's second law to determine the acceleration of the bottom layer of the liquid at the instant when the orifice is opened. This can be done by considering the forces acting on a small mass element near the bottom, as you have already done in your attempt. By equating the net force acting on this mass element to its mass times acceleration, we can find the approximate order of magnitude of the acceleration.

Finally, to find the time taken for the liquid to reach its steady flow velocity, we can use the equation of motion for a constant acceleration, which is v = u + at, where v is the final velocity, u is the initial velocity (which we found using Bernoulli's equation), a is the acceleration (which we found using Newton's second law), and t is the time taken. This will give us an approximate order of magnitude of τ, the time taken for the liquid to reach its steady flow velocity.

In conclusion, by breaking down the problem into smaller parts and using the principles of fluid mechanics, we can analyze the motion of the liquid and determine the approximate order of magnitude of the time taken for it to start flowing.
 

1. What factors affect the time taken for fluid to start flowing?

The time taken for fluid to start flowing can be affected by several factors, including the viscosity of the fluid, the size and shape of the container, and the pressure applied to the fluid.

2. How does the viscosity of a fluid impact the time taken for it to start flowing?

The viscosity of a fluid is a measure of its resistance to flow. The higher the viscosity, the longer it will take for the fluid to start flowing. This is because thicker fluids require more force to overcome their internal friction and move.

3. Does the size and shape of the container affect the time taken for fluid to start flowing?

Yes, the size and shape of the container can significantly impact the time taken for fluid to start flowing. Smaller containers may have higher surface tension, which can slow down the flow of the fluid. Additionally, the shape of the container can affect the pressure distribution and ultimately impact the flow rate of the fluid.

4. Can the temperature of the fluid affect the time taken for it to start flowing?

Yes, the temperature of the fluid can influence its viscosity, which in turn can affect the time taken for it to start flowing. As temperature increases, the viscosity of most fluids decreases, allowing them to flow more easily.

5. How does the pressure applied to a fluid affect the time taken for it to start flowing?

The pressure applied to a fluid can greatly impact the time taken for it to start flowing. High pressure can help overcome the resistance of thicker fluids and reduce the time it takes for them to start flowing. However, too much pressure can also cause turbulence and slow down the flow rate.

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