Help with a mechanical comprehension/water flow problem, please

[Gaith]

I was taking a test a while ago, and came across this problem:

Water is flowing through the pipe below, left to right. If the speed of the water at both extremities is the same, what is the speed of the water in the middle? Double, the same, something else?

I guessed "double" because it seemed right, but had no formula in mind. Is there a basic formula or principle (say, along the lines of f=ma) that I should be aware of?

Thanks!
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[Amaris]

You're talking about Bernoulli's Principle.

Here's an article on it that might help you out:
http://en.wikipedia.org/wiki/Bernoulli%27s_principle

All I remember for certain is that a liquid would move through the middle pipe at a faster rate, but I don't know the specifics or the math.

 

[farmkid]

It seems to me that the speed of the water will be proportional to the cross-sectional area of the pipe. Since the diameter of the smaller pipe is 1/2 of the larger ones, and that area is proportional to the square of the radius (pi cancels out), the cross-section of the smaller one will be 1/4 of the larger ones, so the speed of the water should be 4 times that of the larger ones.

Of course, this comes from a biologist, not an engineer, so I may be completely off.

 

[Holdfast]

It's many, many (many, many!) years since I did any physics, but I think farmkid is right.

The equation you need is Q=AV (flow rate = area x velocity). If you ever forget it again in the future, it's easily derivable once you realise that a static volume of a cylinder is length times cross-sectional area, so the flow rate equation effectively divides both side by unit time, to turn it into a formula for flow rate.

Solving is now rudimentary since the only variable you're changing is A (Area of a circle = pi*r^2). A is therefore reduced by a factor of 4, so V must increase by the same to maintain the constant flow rate demanded by the rubric.

You don't need Bernouille or Poiseuille or anything like that because of the constant flow rate. ie no end to end pressure drop. There is a pressure increase in the middle (by IIRC a factor of (r'/r)^4= 16, but that's not what the question asks).

 

[Gaith]

Thanks for the replies, you three. Yes, well, I'd forgotten we were talking about volume and not area, hadn't I!

^/^^ That makes perfect sense. I'm sure you're right. Thanks again. What would I do without the TrekBBS?

 

[Amaris]

Thanks for the replies, you three. Yes, well, I'd forgotten we were talking about volume and not area, hadn't I!

^/^^ That makes perfect sense. I'm sure you're right. Thanks again. What would I do without the TrekBBS?

You'd have to actually use Google.