Answer to Question #14968 in Differential Equations for Julia

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.