# Answer on Differential Equations Question for Julia

Question #14968

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.

Expert's answer

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.

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.

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