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I Variational calculus or fluid dynamics for fluid rotating in a cup

  1. Oct 9, 2016 #1
    my first post having just joined!
    Problem statement - what curve describes the surface of a rotating liquid? Stirring my cup of coffee years ago sparked this thought.
    Question - is the way to solve this problem to use variational calculus, or fluid dynamics? I have always thought the former but recently someone suggested to me that it's the latter.
    Any thoughts greatly appreciated.
  2. jcsd
  3. Oct 9, 2016 #2
    Do you have a specific, precisely stated, problem in mind?
  4. Oct 10, 2016 #3
    Hi. Here's my problem, stated with more detail.
    Imagine a cylindrical cup of coffee, being stirred and where the surface of the stirred coffee has reached some steady state. In terms of the rate of rotation of the liquid and its density, define the equation of the surface of the rotating liquid S(r,h), where r is the distance of a point on the surface of the rotating liquid from the centre of rotation and h is its height above the lowest point on the surface of the rotating liquid. Ignore friction effects of the sides and bottom of the cup which has height H and radius R. The cup is half full of coffee before being stirred.

    Hope that about does it. I'd like to know if this is a variational calculus problem or a fluid dynamics problem.

  5. Oct 10, 2016 #4


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    Is this a homework problem? If so, please post in the homework section next time.

    Some hint: In hydrostatics the equation for the pressure reads
    $$\vec{\nabla} p=\vec{F},$$
    where ##\vec{F}## is the force per volume. At the surface of the fluid you have ##p=0##, where ##p## is measured relative to the atmospheric pressure. Now you have just to specify the forces (gravity and centrifugal forces in the rotating reference frame) to get the equation of the surface. It's not too difficult!

    The variational principle for ideal-fluid mechanics can be found in

    A. Sommerfeld, Lectures on theoretical physics, vol. 2.
  6. Oct 10, 2016 #5
    Thanks for the response. It's not a homework question - just a problem I thought was interesting.

    Let me digest what you've suggested and I'll get back with some clarification questions.
  7. Oct 10, 2016 #6
    It's a fluid dynamics problem, and involves much more than just hydrostatics.
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