Question

Find the form of the surface of fluid inside separating centrifuge while working and angular velocity of the fluid to reach the given height H.

Answer 1

Let's look at the problem in Cartesian coordinates.
Let (0, 0) be the
coordinate of the center of centrifuge, w be the angular velocity
of
centrifuge.
Any section (that crosses the origin and is perpendicular to XoY
plane) of the surface
has a form of parabola:

Z - Z0 = w^2 * x^2 /
(2g).

When we substitute x -> sqrt(x^2 + y^2) we get the equation of
the surface:

Z - Z0 = w^2 * (x^2 + y^2) / (2g).

Then

H = Z0
+ r^2 * w^2 / (2g),
where r is the radius of centrifuge.

As h = 1/2 *
(Z0 + H), so
w = 2/r * sqrt(g*(H - h)) - angular velocity.