Pressure Gradient in Astronaut’s Centrifuge? Status is no progress yet
Intro
Recent questions mentioned pressure gradients in a box. If you weigh a box in a vacuum, the gas contributes its weight with a pressure gradient inside the box. If you weigh something that has vacuum chamber inside it, you will under-weigh it.
But what if it’s moving?
Question
We have a NASA training centrifuge, and it starts up with constant angular acceleration.
What is the inside pressure at different times and different locations?
Assume arm length $l$ is large compared to radial dimension of the enclosed space X ($0 \le x \le X$).
That alone will suffice as an answer.

Challenge
Relax the assumption that X is small. Did anything change? (I honestly don’t know).

Super Challenge:
Answer or explain why the following cannot be answered analytically no matter what assumptions (ideal gas, inviscid, etc): We reach and maintain maximum speed $\omega= \Omega$, the hatch is opened, facing radially outward. What is the pressure? Can we even guesstimate anything with a model analytically?
 A: This can be addressed (to some extent) by looking at the centrifuge chamber in a rotating reference frame.  Assume that the gas is in mechanical equilibrium in the rotating reference frame (i.e., there are no bulk motions of the gas within the chamber).  Then a parcel of air within the centrifuge will experience three forces that must be balanced by the pressure gradient:  gravity, the centrifugal force, and the Euler force.  Notably, all three of these forces are proportional to the mass of the air parcel (they're all fictitious forces, after all):
$$
\vec{F} = m \underbrace{\left[ \vec{g} + \vec{\omega} \times (\vec{\omega} \times \vec{r}) + \dot{\vec{\omega}} \times \vec{r} \right]}_{\equiv \vec{g}_\text{eff}}
$$
This means that as far as the gas in the box is concerned, it is experiencing a "weight" equal to its mass times the above vector. However, the last term is does not correspond to a conservative force (its curl is non-zero), so I am not immediately sure how to proceed.
This assumes that any changes in $\vec{g}_\text{eff}$ are slow enough that the gas can come to equilibrium in response to them.  A sudden burst of $\dot{\vec{\omega}}$, for example, would set up pressure waves inside the chamber, which would reverberate back & forth inside the chamber for a while before dissipating.  As a rule of thumb, I would expect that the above approximation would be valid if the time-scale of any changes in $\vec{\omega}$ or $\vec{\alpha}$ was large compared to the reverberation time of the chamber.
None of the above discussion requires the chamber's size to be small compared to the radius of the centrifuge arm, or for it to have any particular shape.
