Can anyone suggest any methods to measure the RPM of a magnetic stirrer that does not have any scales written on the instrument itself, nor in its operation manual?
Consider using a stroboscopic system to "freeze" the stirrer and relate it to the image capture frequency. Stroboscopic illumination, or cameras used in machining mechanics may be useful.
Inductive hour meters often double as tachometers and are very affordable. Place the meter's probe against the motor to obtain an RPM reading.
Here's an example of such a product: https://www.run-leader.com/index.php/product/info/112.html
Set it to 1P1R mode (one pulse per revolution) and divide the RPM reading in the display by the number of windings of the motor. It will be indicated in its datasheet.
From what I can see on the web, the magnetic stirrer has a magnet inside the stirred liquid. A rotating magnetic field drives this magnet. The magnet is then synchronous with the field, but has a phase shift. Put a test coil close to the flask and the field pickup can be monitored on an oscilloscope to measure the frequency. Some oscilloscopes would have scaling facilities to convert Hz into RPM.
There is a possible scaling error if the rotating magnetic field has multiple poles. So you may need an integer factor such as 2 or 4 to scale your answer.