I am trying to come up with a mathematical model so that, when the displacement of the plunger of a syringe is know, I can calculate the amount of a specific liquid in the barrel. Or the relationship between the speed of the plunger and flow rate at the tip of the needle (Again assuming that the properties of the liquid in the barrel is known). I have very limited knowledge in fluid dynamics. I made a quick research and found the Poiseuille's law for laminar flow Flowrate= π.r4 (P-P0) / 8.η.L where r is the radius of the pipe or tube, P0 is the fluid pressure at one end of the pipe, P is the fluid pressure at the other end of the pipe, η is the fluid's viscosity, and L is the length of the pipe or tube P0 will be atmospheric pressure if I assume I am injecting in air. P will be the pressure on the syringe plunger r is the radius of the barrel η is the dynamic viscosity of the fluid L is the length of the fluid inside the barrel. The flow rate unit is m3/s when I use the SI units. Also I have the Bernoulli's law to calculate the pressure difference between the syringe barrel, hub and needle, when their cross-section area is given. So, I will be converting the Pressure at the tip of the needle (which is shown as pressure P2 on the illustration) to pressure at the barrel of the syringe to use for the parameter P0. Am I correct at this point? Also, therefore, the length I will be using for the parameter L is the dashed red line, which is the distance between the plunger and end of the barrel. This is my assumption, since I already convert the pressure at the tip of the needle to barrel of the syringe. Is this correct? If I am all correct up until this point, my last question is the following; I am assuming that I can exert enough pressure for the desired flow rate. How can I convert the P parameter into speed so that I can calculate the speed of the plunger for the desired flow rate?