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?