What is the formula that describes the air pressure at a given radius
from the center of a rotating column of air in a centrifuge, in terms
of air pressure at the center, density at the center, radius and rpm?

For example: A closed, cylindrical vessel is rotated rapidly on its
axis. Air at  known pressure and temperature is allowed to enter at
the center. What is the gas pressure at the outside edge after the gas
is spinning at the same rate as the vessel?

If we assume that the vessle is maintained at the same temperature as
the air when it enters, and that the system is allowed to come to
thermal equilibrium, it's not necessary to consider adiabatic warming
of the air as it is pressurized.  In that case,

dP = D A dR    (P = pressure, D = density, A = acceleration, R = radius)

D = (Do/Po) P  (Do = density at center, Po = pressure at center)

A = w^2 R      (w = angular velocity)


So, dP = Do/Po P w^2 R dR    -->   dP/P = Do/Po w^2 R dR

Integrating,  ln(P) = Do/Po w^2 r^2/2 + constant 

P = Po exp(.5 w^2 R^2 Do/Po)

Some important things to remember when using this formula are that P
and Po are absolute pressure (meaning relative to a vacuum, not to
atmospheric pressure) and w is in radians per unit time.  The units I
would use are SI: pascals for pressure, kg/m^3 for density, rad/s for
w, meters for R.  To convert rpm to rad/s, multiply by 2*pi/60.  Also,
the formula treats air as an ideal gas, which is not valid if the
density gets too high.