Flow rate of a syringe Suppose a syringe (placed horizontally) contains a liquid with the density of water, composed of a barrel and a needle component. The barrel of the syringe has a cross-sectional area of $\alpha~m^2$, and the pressure everywhere is $\beta$ atm, when no force is applied. 
The needle has a pressure which remains equal to $\beta$ atm (regardless of force applied). If we push on the needle, applying a force of magnitude $\mu~N$, is it possible to determine the medicine's flow speed through the needle? 
 A: The appropriate equation for laminar flow (i.e., not turbulent) of a 
liquid through a straight length $l$ of pipe or tubing is:
$$Flowrate = \frac{\pi r^4 (P - P_0)}{8 \eta l}$$
where $r$ is the radius of the pipe or tube, $P_0$ is the fluid 
pressure at one end of the pipe, $P$ is the fluid pressure at the other end of 
the pipe, $\eta$ is the fluid's viscosity, and $l$ is the length of the pipe or 
tube. In your case $P$ is presumably $\mu$ divided by $\alpha$ and $P_0$ is $\beta$. Make sure you keep the units consistent - your question gives $\beta$ in atmospheres.
The equation is called Poiseuille’s law. Google for this for more details.
A: I've already modelled this case and you'll find that the flow is indeed laminar and for a medical syringe (say 5ml) with a 26 or 27G needle you'll get a Re value of under 100. This situ changes if the liquid is more or less viscous e.g. due to temperature.
Typically forces at the plunger are between 2 to 20N.
When using the Poiseuille formula remember that the Po (when you action the syringe in air) will be atmospheric pressure but when injected in real conditions it will be the blood stream pressure or dermis. The P value is the pressure you obtain by applying a force to the syringe plunger.
Also the viscosity is dynamic not kinematic viscosity.
Initially I would neglect the friction effects in the needle and focus more on the real internal diameter and shape of the needle, hence the gauge value and needle length are more important.
