Numerical investigations have been made on thermally, mechanically, and thermomechanically driven flows of a weakly compressible gas in a gas centrifuge assuming an axisymmetric continuum flow. The finite difference method developed for these computations is based on the Euler/upwind scheme and provides a numerically stable solution. It is found that an almost cylindrical Couette flow is established in an annulus despite the end effects of the plates and that the two mechanically driven circulations interact with the thermally driven circulations.