Is this a theoretical or a practical question?

This is of course the theoretical question. But this is at the same time praxis-relevant. Due to the viscosity of the fluid there is alway energy dissipation which leads to the heating of the fluid. If this fluid is placed inside a thermally isolated vessel, the heat cannot be removed. So, the temperature must increase. The question is: can the temperature go to infinity or will it approach a steady-state regime?

Sorry, was just asking because of the specific nature of your propeller.....but in case of a theoretical exercise one would have to assume that the Planck Temperature would most certainly be a good guess for an upper temperature limit which would be at around 1.42 E32 K. I would assume you will need a different form of energy transport into the fluid a long time before that :) At this point either known physics fails us or you will experience a kugelblitz in which case you will experience the next K in a different universe :)

I am not so sure that the temperature would reach that theoretical limit. I would rather expect that a steady-state regime with constant but much lower temperature is reached at large time. If this is the case, the next question becomes also intriguing: what kind of fluid model admits such steady-state regime?

The first law of Thermodynamics states that energy increases if work is done to an adiabatic closed system. With U as the total internal energy, and ignoring kinetic energy of the stirred substance, we can write

delta_U = P delta_t

where P is power for driving propeller, delta_t is duration, and delta_U is change of energy. What we see is increase of energy. Energy is related to temperature and other properties of the substance in the system, how temperature changes depends on the property relations. If you start with a liquid, and the volume stays constant, you'll go towards supercritical liquid. I'd say unless a "phase change" of same kind occurs, temperature goes up, because the specifc heat must be positive for thermodynamic stability. You'll probably experience chemical changes and the like, which slow temperature increase down, or might even lead to plateaus. I am not expert for high temperatures, but in the end you'll have some kind of plasma, and still add energy (ignoring the problems with cooling container and propeller :-) ). My vote is for "T up".

I agree with Henning....since you keep adding energy which has nowhere to go there is nothing that suggests the temperature would reach a limit at any point. You will have temporary "plateaus" of temperature during phase and state changes of the fluid changes state (e.g. gas to plasma), but at the end, the basic laws apply and the energy will result in temperature increases.

Henning said only that "T up", but this does not still answer my question. It is true that there is always heating due to the energy dissipation. But the increase of temperature must not be proportional to the heat supply. The increase of temperature depends on the property of the fluid, and it might happen that the increase of temperature slows down and the temperature approaches asymptotically some high value, but not the theoretical limit.

It's a question of the specific heat, I think. If we ignore phase changes, and assume fixed volume closed system, then the first law is

m cv dT/dt = P

where P is power, m is mass in the system. For temperature to approach a limit Tlim, the specific heat, which depends on temperature, would have to go to infinity at that limit, cv(Tlim) -> infinity. Specific heat is proportional to degrees of freedom of the particles, hence this would require infinitely many degrees of freedom for finite number of particles, or infinite number of particles with finite number of degrees of freedom. Neither will be the case, specific heat is finite, there should be no limit for temperature.