# Flow Rate and Area of Hole

1. Apr 10, 2014

### chem1995

Let's say that you have an open tank of water and a hose connected to the bottom of it. Water is flowing out of the hose. You then cover half of the hose with your thumb. Will the flow rate (liters/second) right before you cover the hole be the same, less, or greater than right after you cover it?

I realize that as you let water flow out, the flow rate is constantly decreasing because the water level is constantly decreasing and thus the water pressure is also decreasing, which decreases the flow rate. DO NOT take this into account.

Thanks!

2. Apr 10, 2014

### dauto

What do you think?

3. Apr 10, 2014

### chem1995

I'm not sure. I think that the flow rate might decrease because there's less of an area for the water to flow out, but I'm not sure if the velocity of the water increases as you decrease the area. If it does, then I could also see it being that the increase in velocity of the water in essence cancels out the decrease in area (in terms of their effects on flow rate), thus making it so that the flow rate is the same in both scenarios.

Part of me goes back to a real-life situation using a garden hose - I know from personal experience that if you cover half the nozzle, the water sprays out at a faster velocity. But I don't know if that's applicable or not here because if I'm not mistaken, in a garden hose the source of water is so massive and its pressure so great that the water going into the hose is nearly constant regardless of any changes in hole size, so in order to compensate for a decrease in area, the velocity of the water has to increase. In other words, if the source of water wasn't so large (like it is in this case), the amount of water flowing into the hose would change as the area of the nozzle changes, whereas if the source of water is extremely large, the amount of water flowing into the hose doesn't change regardless of the change in area. Does the size of the reservoir matter, and if it does, did I correctly analyze its impact on the change in flow rate as the area of the nozzle decreases?

(As you can probably tell, my understanding of fluid mechanics is pretty poor!)

Last edited: Apr 10, 2014
4. Apr 11, 2014

### chem1995

Does anyone have any ideas?

5. Apr 11, 2014

### Jano L.

By covering the hose, you will force the fluid to flow through tighter space - smaller effective orifice. Impact of the friction on the flow will be higher and the velocity of the jet will decrease. The area of the jet decreases as well, so you will almost certainly decrease the total flow rate (liters/second).

6. Apr 11, 2014

### chem1995

Won't the velocity of the water increase though as you cover half the nozzle? With a garden hose, covering half the nozzle makes the water spray out to a farther distance (i.e. with a greater velocity).

7. Apr 12, 2014

### enochnotsocool

By saying that the velocity of the flow would increase, you have made the assumption that there is a force (the pump behind you tap) forcing the flow rate to be (approximately) constant.
This is not the case here, just having a tub of water with a hole on the bottom in just maintaining the pressure constant.

8. Apr 12, 2014

### chem1995

In a garden hose, is there a pump that is forcing the water into the hose at a constant rate? I was under the impression that it worked in a similar fashion to this problem but on a much larger scale - that is, instead of a small tank, there's a water tower.

In the example of a garden hose, is it literally the exact same amount of water that flows out when you cover half the nozzle, or is it slightly less?

9. Apr 12, 2014

### tennispro1213

I think that the velocity of the water would increase regardless whether it passes through a half-covered hole in a hose or at the bottom of a tub/tank/water-tower. In each scenario, the water is being acted upon by a force in the direction of the (half-covered) hole. So if you can assume that the velocity of water increases when it passes through a half-covered hole in a hose due to the constant pressure being supplied at one end, you must also assume that the water in a tub/tank/water-tower also increases it's velocity by passing through a half-covered hole. The pressure acting on it in the latter case is caused by the downward force of gravity which is strongest in the downward direction. If the hole in the tub/tank/water-tower is at the bottom then, considering the downward force of gravity would lead you to assume that the water is acted upon by a force, which causes it's velocity to increase.

I'm no physicist, but I guess that's what would happen. Tell me what you think...

10. Apr 12, 2014

### chem1995

That's what I initially thought would happen, tennispro1213. I think the difference (or the supposed difference) between this example and a garden hose is that in the case of the garden hose, the amount of water being pumped in is constant. That means that in order to counteract the smaller area, the water must move faster. In the case of the tank, the velocity of the water is allegedly constant (although that admittedly seems counterintuitive to me) because there is nothing forcing the same amount of water to be pumped in as you decrease the size of the nozzle.

11. Apr 12, 2014

### tennispro1213

There's always gravity...and you should also consider the pressure in a liquid. (There's a formula for it which depends on depth) so the deeper the hole in the tank, or higher the level of water in the tank, the more pressure will be acting at a certain depth (with the most at the bottom , where the hole will most probably be located).

12. Apr 13, 2014

### Jano L.

That is true, I was wrong in the previous post. What I was thinking is the situation when you're closing a water tap, the velocity of the water coming out decreases.

I think that the answer to your question is still "flow rate decreases", for the following reason.

Covering the nozzle increases the velocity of the water spraying out, but it decreases velocity of the water inside the hose, which is the one determining the flow rate. Why does the water inside get slower?

Water sources (whether it is a tank or a pump) usually approximately maintain constant pressure difference. Since the water sprays out more quickly under the same pressure difference when you cover the nozzle, from the Bernoulli law we know the water pressure in the hose just before the nozzle is greater than before. This can only happen if the frictional losses of pressure along the hose are lower. Since the frictional losses of the pressure are propotional to transverse velocity gradient in the hose and the velocity at the wall is always 0, these losses of pressure will be lower only if the velocity of the water inside the hose gets lower. In other words, covering the nozzle decreases the effect of the friction along the hose and more energy of the source goes into the water sprinkling out of the nozzle.

13. Apr 13, 2014

### jbriggs444

The key part of the original post is "right after you cover it". We are not talking about long term stable state. We are talking about short term effects.

How short term? And with what simplifying assumptions?

The water in the hose has momentum. The pressure drop at the nozzle is finite, even if the original flow rate is preserved. Accordingly, it will take time for the momentum of the water in the hose to be reduced. Equivalently, the flow rate of the water in the hose will take time to reduce. That is, it must vary smoothly.

If one assumes that water is incompressible and that the hose does not swell under pressure then it is clear that the flow rate at the nozzle must match the flow rate in the hose. Since the flow rate in the hose varies smoothly, it follows that the flow rate at the nozzle must vary smoothly.

It then follows that the flow rate at the nozzle will not change appreciably "right after" you put your thumb over the end.

14. Apr 13, 2014

### Jano L.

It will change, because covering the nozzle will increase pressure in the hose and slow down the water in it. This will happen very fast, as the water is badly compressible and the speed of propagation of pressure changes is kilometers per second.

15. Apr 13, 2014

### jbriggs444

The pressure changes will propagate at the speed of sound. The flow rate changes will take place much more slowly. Have you ever heard of a "water hammer"?

16. May 6, 2014

### chem1995

Is this how garden hoses work? Is there a pump somewhere that forces the flow rate to remain constant? enochnotsocool says that it is 'approximately' constant: does this mean that the flow rate will decrease in a garden hose if you cover half of it, but it will only decrease by a very small amount?

I was under the impression that a garden hose just works on a much bigger scale than the problem I presented with the tank of water. That is, the tank of water is larger with the garden hose because the water is coming from the water tower. Is this wrong?

17. May 7, 2014

### jbriggs444

A tap on the side of your house that is turned on all the way approximates an ideal constant pressure source (a big bucket of water) better than it approximates an ideal constant flow source (a pump).

The fact that you can turn off the tap without causing the pipes to burst is evidence for this.