Noisy computer fan

This would indicate that the computer is overheating.

I would look at the cooler first.

It does sound like an overheating computer they are designed nowadays to shut themselves down if there is a problem

What you should do is check the fan assembly If you see a build up of gunk and debris then get yourself a can of compressed air used for cleaning computers and take the unit outside.

Whilst wearing glasses use the nozzle on the can to blow out any debris from the CPU and power supply.
If everything went to plan you should now be back in business.

Another consideration is that many cases have places in them for extra fans and usually extra connections on the motherboard too.

They are quick and easy to fit.

Quite often I will fit TWO in what is called a push/pull configuration. One pulls cool air in while the other pushes hot air out.

These will not only help keep the computer cooler but also cleaner

Open the case and see if dust is around the fan. If it is, then get a can of air and gently blow the dust off. You can also replace the fan. These go out quite often due to them being on all the time. If you have an old computer, you may be able to use the fan from it, if it will fit. Or you can buy one cheap off eBay. Be sure you are grounded when opening the computer case so you will not get static electricity in the computer and make sure it is unplugged.

There are two different types of fans in a computer.

Open the computer and look at the CPU fan. Is it turning. Also, if you hear a grinding sound, it is the power supply fan and you might as well just get another power supply.

Also, if the system is pretty dusty, used compressed air to clean out the system.

As more dust gets into a system it actually dries out the bearings.