Hydrostatic Equilibrium In Centrifugal Field

Centrifugal field is also an application of Hydrostatic Equilibrium it works on the fact that the more the fluid dense, more it farther from the axis of rotation .The lighter (with comparatively low density) liquid will form a layer on heavy liquid.

In a rotatory centrifuge the liquid is thrown outward from the axis of rotation. Surface above the liquid take a shape of a curve or parabolic shape when the centrifuge is rotated about its axis.But at very high speed(In industrial centrifuge) it is rotated at very high speed & surface above liquid take a shape of cylindrical layer.
                                     
                                      r1 is the radial distance from the axis of rotation to the free liquid surface.

                                     r2 is the radius of the centrifugal bowl.

 The whole mass of the liquid rotates like a rigid body.

 F = ma.........(1)

                         Taking Differential

 dF = adm......(2)

  The acceleration of the fluid is given as; a = ω 2 r

Substituting  a = ω 2 r in eq (1) becomes

dF = ω 2 rdm......(3)

If ρ is the density of the given liquid, and b is the breadth of the ring, thus, the mass of the element can be written as; m = πr2 b 

 differentiating equation (3)  leads to  dm = d(πr2 b · ρ) = 2πrbρ · dr 

 Final equation  of  Centrifugal field become