- #1
Alexander83
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Hi all,
I'm considering an fluids example that's giving me an apparent contradiction when I consider it from the perspective of Bernoulli's Equation vs. the Equation of Continuity.
What I'm thinking of is the common observation that putting one's thumb over a garden hose results in an increase in the velocity of the water leaving the hose. I've always understood this in terms of the Equation of Continuity under the assumption that the volume flow rate (R) is a constant.
R = v*A
Where v is the speed of the flow in the hose and A is its cross-sectional area. From this approach, the speed of water leaving the hose would increase as A is reduced by putting a thumb over the outlet of the hose.
Where I'm getting confused is in trying to apply Bernoulli's equation to explain what's going on. I'm imagining a very conceptually simple case where the water in the hose is originating from a large storage tank that is open to the atmosphere at the top. In this case, Bernouilli's equation could be used to work out the speed of the water at the hose outlet and it essentially boils down to Torricelli's theorem here and one obtains:
v = (2gh)^0.5
where h is the depth of the water in the tank feeding the hose. This equation indicates that the flow speed through the end of the hose should be a constant and independent of the diameter of the hose.
The two results clearly contradict one another, so there's clearly something I'm missing here... I suspect I'm missing something in trying to apply Bernoulli's theorem to this situation, but I can't see what it is that I'm missing.
Any help would be gratefully appreciated!
Cheers,
Alex
I'm considering an fluids example that's giving me an apparent contradiction when I consider it from the perspective of Bernoulli's Equation vs. the Equation of Continuity.
What I'm thinking of is the common observation that putting one's thumb over a garden hose results in an increase in the velocity of the water leaving the hose. I've always understood this in terms of the Equation of Continuity under the assumption that the volume flow rate (R) is a constant.
R = v*A
Where v is the speed of the flow in the hose and A is its cross-sectional area. From this approach, the speed of water leaving the hose would increase as A is reduced by putting a thumb over the outlet of the hose.
Where I'm getting confused is in trying to apply Bernoulli's equation to explain what's going on. I'm imagining a very conceptually simple case where the water in the hose is originating from a large storage tank that is open to the atmosphere at the top. In this case, Bernouilli's equation could be used to work out the speed of the water at the hose outlet and it essentially boils down to Torricelli's theorem here and one obtains:
v = (2gh)^0.5
where h is the depth of the water in the tank feeding the hose. This equation indicates that the flow speed through the end of the hose should be a constant and independent of the diameter of the hose.
The two results clearly contradict one another, so there's clearly something I'm missing here... I suspect I'm missing something in trying to apply Bernoulli's theorem to this situation, but I can't see what it is that I'm missing.
Any help would be gratefully appreciated!
Cheers,
Alex