Because friction does not vary linear with speed in this system, you are going to have a tough time getting a consistent speed with an open-loop system.
To a first order approximation, motor torque is proportional to current, and motor speed is proportional to voltage. In fact, in an ideal motor which has no winding resistance and is driven by an ideal voltage source, this is exactly true. So, you might try selecting a motor with a low winding resistance, and assuring that your motor driver is as low impedance as possible. This will give you better speed regulation in an open-loop system.
It will probably also help to select a motor with a high quality transmission with low backlash. Any slop in the transmission is really going to make things hard. Perhaps a high quality ball screw will give you a more linear and predictable transmission.
Another mechanical solution would be a flywheel, which if massive enough, would smooth out the torque variations from the slip-stick action to a manageable level.
However, if you really need a predictable speed, I bet you are going to need a closed loop system. You have a couple options here. A tachometer as you mention would be great, but probably you'd want it to be on the motor shaft, before the mechanical reduction. Here it becomes all the more important to have a good transmission, otherwise all the transmission slop will show up as error in your measurements, even if the motor speed regulation is perfect.
A somewhat less ideal solution is to measure the back-EMF of the motor. You may not get the same precision and responsiveness as with a tachometer, but it doesn't require any additional parts. See How can I measure back-EMF to infer the speed of a DC motor?
A final approach you may consider is using a stepper motor. Provided you can select a motor and transmission with enough maximum torque that steps aren't missed, then you can get a consistent speed in an open-loop system. An issue here may be the high torque ripple of such motors, which might be a problem for your experiment.
Does the speed of rotor changes to bring the armature current back to
the constant value?? What causes the armature current to not change
from its constant value?
If you have an external field winding that controls the magnetic field of a DC motor, then a higher magnetic field means that the armature's back emf equalizes when the speed is lower. This is the criteria for stabilization of speed.
If the back-emf is too low then the motor armature speeds up until the constant applied armature voltage and the back-emf are at the right level to permit the right amount of armature current needed to generate mechanical power to the load and against friction.
It's all embedded in Faraday's induction equation: \$\text{emf} = -N\dfrac{d\Phi}{dt}\$. In other words, to make emf the right value to drive load and friction, the motor can run slower when the flux is higher because, what is lost in the rate of change of flux is regained by the flux actually being higher.
Best Answer
The instant flux is reduced, the back emf reduces and this causes the armature to increase current. More current means a higher driving torque and this accelerates the armature to run at a higher speed until the speed equation is in balance again.
If there is a significant mechanical load this may not happen of course.