Suppose we have a thermally isolated container of a homogeneous gas at temperature $T$. If, for example, the container is filled with xenon at STP, then we know that the RMS speed of each xenon particle, $S_p$, is about 240 m/s.
Now we insert a solid fan blade on a shaft into the center of the container and begin to rotate it. As it rotates it alters the velocity of each particle with which it interacts via simple, inelastic collisions.
If the fan blade speed $S_b$ is small relative to the RMS speed of the particles (i.e., $S_b \ll S_p$), then it seems like it can't increase the average RMS particle speed because on average it is just as likely to collide with a particle with a velocity component opposite the motion of the blade (reducing that particle's velocity component by $S_b$) as it is to collide with a particle to which $S_b$ is additive. Also, particle collisions are almost as likely to occur with the receding face of the blade as with the advancing face. (But that almost seems like a hint....)
However, as the fan speed becomes large relative to the RMS speed of the particles – i.e., $S_b \gg S_p$:
- The blade is much more likely to increase any particle's RMS speed, because on average even if a particle is moving opposite the direction of the advancing blade, $|S_b - S_p| > |S_p|$.
- It is much less likely for a particle to collide with the receding face of the blade.
So point #1 seems to suggest that the fan will in fact increase RMS speed of the gas by something like $|S_b - S_p|$, but point #2 seems to suggest this won't happen for long, because eventually the average velocity of the particles will match the speed of the blade.
(If necessary, we can assume that the container is cylindrical or spherical, and the blade is in the limit a point solid rotating at a fixed radius from the center. Let us further assume that external energy is added to the blade as needed to maintain its speed at $S_b$ ... although we will note that if we run the experiment over time such energy decreases rapidly as the fan "sets the gas spinning.")
What really happens in this simple model? Does the fan increase the container's temperature by a fixed amount related to $S_b$? Does the conversion of energy to heat depend on the relative velocities of the blade and the average particle, and if so what is the nature of that relationship?