Why Does Bernoulli's Equation Seem Incorrect for a Rotating Fluid in a Bucket?

[jbar18]

Homework Statement

What is wrong with the following argument from Bernoulli's equation?

Suppose a fluid in a bucket is rotating under gravity with constant angular velocity W so that velocity is:

u = (-\Omega y,\Omega x, 0).

Then:

\frac{P}{\rho} + \frac{u^2}{2} + gz = constant,

z = constant - \frac{(\Omega)^2}{2g} (x^2+y^2)

But this implies that the highest point of the water is in the middle, which is obviously not true.

2. The attempt at a solution

I was wondering if perhaps it might have something to do with P or rho (or both) being a function of x and y? In the problem the whole pressure term seems to have been grouped with the constant, and I'm wondering if that is justifiable. Beyond that I don't know, it looks like Bernoulli's equation is just not appropriate for this situation for some reason (or else it has been applied incorrectly, but I am not sure why).

I've put this in the homework section, but I should mention that it is not assessed and I am unable to check my answer, so this is just out of interest.

 

[TSny]

Hello, jbar18

Bernoulli's equation \frac{P}{\rho} + \frac{u^2}{2} + gz = constant generally holds only along a streamline. The "constant" on the right hand side can be different for different streamlines. Does Bernoulli's equation hold along a steamline for the rotating fluid in the bucket?

[EDIT: Just found this link: Bernoulli’s Equation for a Rotating Fluid ]

 

[jbar18]

Hi TSny,

I imagined those streamlines would exist in a plane of constant z. I wondered about the constants, is that the flaw in the logic? Given that this method doesn't work, how might we go about finding the shape of the fluid surface?

Thanks for your reply

 

[TSny]

how might we go about finding the shape of the fluid surface?
Thanks for your reply

Look at the derivation of equation (4) in the link and then see how it's applied to the upper surface of the rotating fluid to get equation (6).

 

[jbar18]

Thanks TSny, I'll take a look at that.