In a cylindrical container (with radius R) there are 2 fluids (separated like water and oil, fluid 1 lies under fluid 2) with given volumes V_i, given densities ρ_i.
You let them rotate with respective angular frequencies ω_i.
There is no friction.
Find the functions of form [itex]y_i=a_i x^2+b_i[/itex] (cylindrical coordinates) that describe the surface (y_i) of each of the 2 fluids.
[Ignore cases of the surfaces touching each other or the bottom of the container]
2. The attempt at a solution
I was thinking about finding the functions y_i that would minimize the overall energy for the given values.
I wrote down the following:
This is a function of ρ, ω, V, a_1, b_1, a_2, b_2
But because Volume is conserved, b_i depends on a_i so we can leave out b_i as an argument of the function.
Now we want to minimize the energy by solving the following two equations for the 2 unknown a.
By solving this we obtain a_i --> b_i --> y_i(r)
Is "minimizing the energy" a legitimate way of finding a solution for this problem?
I'm being careful, because I tried this on the same problem but exclusively for 1 fluid and it didn't work, because ∂E/∂a was independent of a.