# 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?

• yes it is possible, just use the formulas for laminar flow, probably using the properties of water. But the point of leaving this comment is to tell you that the problem is still under-determined. The force that you put on the syringe has to be distributed over some area for it to become a pressure. Mar 29, 2012 at 14:01
• @AlanSE: How would one apply these formulas to calculate it? Mar 29, 2012 at 14:03
• I'm sorry, as the answer made me realize, it's the cross-sectional area of the needle were missing, not the barrel. Mar 29, 2012 at 14:53

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.