Is sedimentation coefficient constant and why?
Sedimentation due to Gravity

The sedimentation coefficient of a particle falling in a fluid due to gravity is easier to understand. It is the ratio of the settling velocity ($v_{t}$) to the acceleration causing the settling($a_{g}$).

$$s=\frac{v_{t}}{a_{g}}$$  

In fluids on earth $a_{g}$ is acceleration due to gravity which is known to be constant. 

$v_{t}$ , which is also referred to as terminal velocity, is constant.  It is constant because the force driving the acceleration of the particle through a fluid ($F_{g}$) is equal in magnitude and in an opposite direction to the drag force ($F_{d}$).  Since these two forces cancel out the net force ($F_{n}$) is 0.

$$F_{g} - F_{d} = F_{n} = 0$$  

There is no net acceleration $a_{n}$ since $F_{n}=m_{p}a_{n}=0$.  The particle mass ($m_{p}$) cannot be zero so there is no net acceleration of the particle, thus the terminal velocity is constant.  

A ratio of two constants are constant so the sedimentation coefficient is constant.

Sedimentation due to Centrifugal Acceleration

Radial acceleration, $a_{r}$, equals the angular velocity $\omega$ squared times the radius, $r$.

$$a_{r} =\omega^{2}r$$

If the angular velocity is constant, which it is for most of centrifugation, and the radius is constant, then the radial acceleration, like the acceleration due to gravity, is constant.

The sedimentation velocity in centrifugation, just as in gravity settling, is constant.  It is constant because the drag force of the particle in the fluid is balanced by the centrifugal forces generated by the centrifuge. A centrifuge can generate forces several times higher than gravity so settling occurs much faster during centrifugation.

Again, the ratio of two constants are constant so the sedimentation coefficient is constant. 

The other equation looks like it is a derivative of Stoke's law, which is used to determine the drag force of a particle in a fluid.