Does a fan rotating with a uniform angular velocity consume electrical energy? Work done on a rotating body is equal to the change in its kinetic energy. When an electric fan rotates with a constant angular velocity, then its kinetic energy doesn't change. Does it mean that it doesn't consume electrical energy?
 A: Indeed a rotating fan does not consume any energy to maintain the same angular velocity... in a vacuum. 
But if a medium is preset (eg. air, water...), its kinetic energy is increasing (that is the scope of a fan!)
A: Does a car need fuel to drive on a motorway?
You already know what happens when you let go of the gas:
friction, mostly from the air.
This is exactly the same situation.
It makes no difference whether the momentum is linear or angular.
A: It definitely does consume electrical energy. Why? Because there's some opposing force faced by it while it rotates, and this force is often known as air drag/air resistance. You can see the effect of air drag once you switch off the fan. The fan decelerates from its original angular velocity until it stops completely. This deceleration is due to the motion opposing air drag. And thus while rotating, the fan continually loses it's kinetic energy (due to the air drag) and this lost energy is primarily converted to heat energy Thus you don't need electricity to change the kinetic energy, rather you need electrical energy to compensate for the energy lost due to the air drag acting on the fan.
Also the air drag is the most common, most general and easy to understand among all the losses experienced by a fan. However there are many other factors which also increase the loss of energy in a fan. Here's an extremely nice flowchart/Sankey diagram showing this:

Source (PDF)
A: when rotating at constant speed, the fan disc is continuously performing work on the air drawn through it by imparting momentum to it. To perform that work requires a constant input of energy from the motor, and therefore the motor is continually absorbing electrical power while the fan is running. 
Some of that work is wasted in overcoming drag, but most of it is consumed in accelerating air. 
A: But that constant angular velocity is not cost free. It has to be maintained. As others have already mentioned. Air drag of the spinning fan, mechanical friction all tend to slow down the fan. You need power to maintain the velocity. That is why most fans are connected directly or indirectly to electrical power source. 
A: I assume you mean an indoor cooling fan (either stand-alone or a ceiling fan).
You're right that total energy is always conserved but:


*

*Kinetic energy is just one type of energy. Heat is relevant here too. Note that heat is basically the kinetic energy of randomly moving particles.

*The fan is not a closed system. It's connected to the power supply (as you correctly mentioned) and the air in the room (which you appear to forget).


Recall that the purpose of a fan is to accelerate air in the room to provide a draft. This process requires energy.
Specifically, with a fan rotating at constant speed (i.e. most fans that have been powered on for more than ten seconds), where does the power coming from the power supply go?


*

*Some of it goes to losses (in the electric motor, due to friction, and acoustic losses). In all of these cases the lost energy is eventually converted to heat. The fan heats up.

*Some of the power is needed to accelerate the air in the room, increasing the kinetic energy of the air.


Then, what happens to the kinetic energy in the air?


*

*Most of it will get dissipated due to the air's internal friction. This heats up the air. Simply put, when the air comes from the fan, the molecules roughly move in one direction (away from the fan). Then as they encounter other air molecules and obstacles, the motion of the individual molecules becomes increasingly unordered until it is completely random. Random motion of molecules is what we call heat.

*As a side effect the fast air stream of the fan takes away lots of heat from warmer objects it encounters such as people. That might be the reason you turned on the fan in the first place. More on this at livescience.com
Eventually almost all the energy from the fan gets dissipated as heat one way or another. So if your room is well insulated, the temperature will rise.
A: As stated in the other answers, it is true that a fan rotating with a uniform angular velocity consumes electric energy due to the presence of energy dissipation. But it's not only due to the energy transferred to the air molecules (as others state as "air drag"), but also due to other factors like - friction in the bearing, Joule heating and electromagnetic damping in the motor's coil. Electromagnetic damping also has some useful applications (see Eddy current brake). 
It must be noted that when we turn off the fan, electromagnetic damping is present only when the AC motor has a permanent magnet. If the magnetic field for the rotation of the shaft is produced by loops of wires instead of a permanent magnet, it would also become zero. In this case the fan is brought to rest solely by friction and loss of energy to air molecules. 
