If I have a water pump and a fast solenoid valve (can be open or closed) and turn on my pump at a constant voltage and apply pulse width modulation to my valve. Would the resulting thrust created at the water output be constant and can it be controlled nicely by the pulse width modulation process?
What would be the equations that could model the reaction of such a setup?
It's not clear, but I think you are asking whether you can control your solenoid valve with pulses instead of steady DC.
Yes, above some frequency, the mechanical system actuated by the solenoid won't "see" individual pulses, just the average. Usually at a higher frequency, the solenoid itself will smooth out the pulses and maintain are more average steady current.
Electrically, this is basically creating a switching power supply that controls the current thru the solenoid, with the solenoid being the inductor of the switching power supply.
This is assuming your "solenoid nozzle" (do you really mean "valve"?) is intended for other than binary on/off operation, often called a proportional valve. Trying to drive a binary mechanism to in-between states may not end well. Constantly banging it between on and off may be even worse.
