Centripetal force equation doubt 
In a centrifuge, $a_c$ should be constant. If $m$ increases, the $r$ will increase in order to maintain a constant $a_c$.

Constant centrieptal acceleration is given by
$a_c={ v^2 \over r}$
and $a_c = \omega^2 * r$
But the conflict between these two equations arises when we increase $r$.
In the first equation - if we increase $r$, $v$ should increase so that $a_c$ remains constant.
But in the second equation - since $\omega$ is constant in a centrifuge, if we increase $r$ then $a_c$ is no longer constant.
I dont understand how this is happening and would appreciate any help.I have also looked at other similar questions on this site but they do not answer to this conflict. This is not a duplicate.
Also, does anyone know how to mathematically show that radius increases when mass increases?
 A: Both the equations are equivalent. By definition of angular velocity,
$$\omega=v/r$$
You substitute this in $m \omega^2 r$ to get $m v^2/r$.
NOTE THAT $\omega$ is not constant if you change radius, as you claim!

And for the mass to radius relation, its straight forward. Take,
$$F_c = m v^2 /r$$ and ask what you get if you keep $F_c$ constant while increasing the radius.
A: If $a_c$ really means the magnitude of centripetal acceleration and $m$ means mass then the sentence

If $m$ increases, the $r$ will increase in order to maintain a constant $a_c$.

cannot be correct, since neither expression for $a_c$ involves $m$ at all; $a_c$ is independent of $m$.
If $\omega$ were changing over time then $r$ could also change to maintain a constant $a_c$. But in a centrifuge (in its steady state) the angular speed $\omega$ is constant, so $a_c$ is also constant for a given value of $r$. If $r$ changes then $a_c \propto r$.
A: The misconception lies on for what you define $a_c$. Imagine that you put two balls with mass $M$ and $m$ in the centrifuge ($M\gt m$). When the centrifuge reaches constant angular velocity $\omega$, let's say those balls are respectively $R$ and $r$ distance away from the centre of rotation. From the observation we know that $R\gt r$.* Now let's calculate the centripetal acceleration of two masses. $$M\Rightarrow a_c=R\omega^2$$ $$m\Rightarrow a_c=r\omega^2$$ This yields, centripetal acceleration of $M$ is greater than that of $m$. Which means that the sentence

In a centrifuge, $a_c$ should be constant

makes no sense.

*Physics concepts are based on observations. Not observations are built by physics concepts.
A: I do not see what is the conflict here .
If you increase the radius while also increasing the velocity appropriately, then  acceleration will stay constant.
But if you increase the radius, while keeping the angular velocity constant, then acceleration will increase.
What is the conflict in that ?
