Clepsydra shape using Fourier series

1. May 28, 2014

bermudianmango

Our Fluid Mechanics professor gave us a challenge: to find the shape of a vessel with a hole at the bottom such that the water level in the vessel will change at a constant rate (i.e. if z is the height of the water in the tank dz/dt=constant).

I presented a solution assuming that the vessel would be a 3D curve: http://imgur.com/2RhMCgD
This was correct but apparently not good enough. He responded:

"You have to show how you come up with the 1/4 power mathematically and rigorously from first principle. For instance, start with a Fourier series with a set of orthogonal functions, and take it from here."

Does anyone have any idea where to begin?

2. May 29, 2014

voko

I fail to see what that could have to do with Fourier series. You have shown that $R^2 = a \sqrt z$ if $z'(t) = \mathrm{const}$. That is all it takes.

3. May 29, 2014

AlephZero

I agree. The way to do this is set up a differential equation and solve it.

Your solution does the right sort of things, except you seem to have made some arbitrary assumptions like the first line "assume it's a parabola". The solution to the ODE will be whatever it is - you don't need to assume anything.

4. May 29, 2014

voko

It may actually be the ungrounded assumption in the beginning that made the prof unhappy. It is not necessary anyway, just drop it.

5. May 30, 2014

bermudianmango

I was unsure how the fourier series would come into it as well.
How would you drop the assumption, allowing for the possiblity that it might be a cone, or cylinder or anything? By assuming it was a curve I was able to solve for the 1/4 power.

6. May 30, 2014

voko

You used the parabolic function as a placeholder for a unknown function, and then you demonstrated that the unknown function had to satisfy a particular property, which determines it almost completely. You could have started with just an unknown function.

7. May 30, 2014

bermudianmango

OK. So like this then? http://imgur.com/zvrjcVt

But it sounds like everyone is as puzzled as me as to how one could use a Fourier series..

8. May 31, 2014

voko

You are still making ungrounded assumptions. Don't. Just use $R(z)$ and transform the equations so that you obtain $R(z) = ...$. That is all you need.

9. Jun 1, 2014

bermudianmango

OK Thanks guys. I'll rework it and have a little chat with him