Describe surfaces of equal pressure in a rotating fluid

[amrasa81]

Hi, I am trying to solve a basic question from a Fluid dynamics textbook. Could you help me with the answer? The question is as follows:

A closed vessel full of water is rotating with constant angular velocity $$\Omega$$ about a horizontal axis. Show that the surfaces of equal pressure are circular cylinders whose common axis is at a height $$g/\Omega^{2}$$ above the axis of rotation.

I don't know how to tackle this problem. Is there a technique in solving such theoretical questions?

Thanks,

P.S:- This is not a homework or coursework question. I am also new to Physics forum, and hence, my question may not be appropriate for this section. In that case please tell me in which section I should pose fluid dynamics questions.

 

[Andy Resnick]

Interesting question... I only have a partial answer, working from Tritton's 'Physical Fluid Dynamics'. In it, he starts with:

$$\frac{Du}{Dt} =\frac{1}{\rho}\nabla p -\Omega \times \Omega \times r - 2\Omega \times u + \nu \nabla^{2} u +\rho g$$

So, assuming conservation of momentum, Du/Dt = 0. Also, the second term on the rhs can be written as
$$-\nabla (\frac{1}{2}\Omega^{2}r^{2})$$

and combined to give a reduced pressure

$$p - \frac{1}{2}\Omega^{2}r^{2}$$

Then, ignoring the Coriolus term and assuming inviscid flow, I can maybe see how you get the result you mention. Maybe...

hope this helps.

 

[amrasa81]

Thanks!