Clepsydra shape using Fourier series

[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?
Thanks in advance.

 

[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.

 

[AlephZero]

I fail to see what that could have to do with Fourier series.

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.

 

[voko]

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

 

[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.

 

[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.

 

[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..

 

[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.

 

[bermudianmango]

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