  We present a finite-element analysis of the fluid dynamics, thermal dynamics,
and alkali diffusion in a common cell geometry used for spin-exchange optical
pumping of $^{129}$Xe using a flow-through polarizer design. The analysis is
the first to simulate aspects of the laser run-away effect observed in some
optical pumping cells. The analysis further suggests that high-temperature gas
in the outlet tube may give rise to rapid depolarization due to high
concentrations of alkali vapor. Finally, the analysis indicates that the alkali
number density distribution and the specifics of the dynamics are highly
dependent upon the the distribution of the alkali metal in the optical pumping
portion and outlet tube of the cell.
