The Hypodermic Syringe

Take the two resistances in series and compare them honestly. Flow through a smooth tube at low Reynolds number follows the Hagen-Poiseuille relation: the volumetric flow is proportional to the pressure drop times the fourth power of the radius, divided by viscosity and length. The fourth power is the entire story. A 21-gauge needle has an outside diameter of 0.819 millimetres and a bore near 0.51; a 25-gauge is 0.515 outside with a bore near 0.26. Halving the bore does not double the resistance, it multiplies it by sixteen. Now the barrel. A 10 millilitre barrel is roughly 15 millimetres across and 60 long. Its radius is thirty times the needle's, so its contribution to resistance is thirty to the fourth — about 810,000 times smaller for equal length. The barrel is, hydraulically, not there. This is why the force you feel in your thumb is set almost entirely by the needle you chose, and why the same syringe feels effortless with a wide needle and immovable with a fine one. It is also why you cannot fix a slow injection by pushing harder for long: the pressure needed rises as the fourth power falls, and glass barrels have a burst limit. Write down the two bore radii you intend to test before proceeding.

This is lens grinding applied to a cylinder, and the technique transfers directly. Take a length of thick-wall borosilicate tube for the barrel and a solid rod for the plunger, sized so the rod enters the bore with visible slop. Charge the gap with a fine abrasive slurry — 400 grit to start — and work the plunger in and out with a slow rotation, reversing direction often. The two surfaces abrade each other, and because they abrade each other they converge on the same cylinder. That is the trick: you are not grinding to a dimension, you are grinding two parts into agreement. Step down through 600 and 1000 grit, washing thoroughly between grades. One particle of coarse grit surviving into the fine stage will score the bore for its whole length and you will start again. Test by feel and by air. A finished pair, dry and clean, should slide smoothly and then stop against trapped air with no audible hiss, and should return under its own trapped pressure when released. That air seal without any elastomer is the whole achievement, and it is why nineteenth-century syringes were numbered in pairs and ruined by being mixed up. Mark both parts with the same number now, before they can be separated on the bench.

Mount the finished syringe vertically, hang known masses from the plunger, and time the delivery of a measured volume through each needle in turn. You now have flow against force, which is flow against pressure once you divide by the barrel area. Plot your points on the notebook's prediction. Expect them to sit below the ideal line, and expect the discrepancy to grow at the fine gauges. Two effects account for it, and both are real: the entrance to the needle costs pressure that Poiseuille does not model, and the ground plunger has friction that does not scale with flow. Subtract a constant friction term and the residual should collapse onto the theory — do that fit, because it is how you separate a fluid effect from a mechanical one in any pump. The limits are worth stating plainly. This is a bench pump for water and glycerol. A real hypodermic is a sterile single-use device, and its whole safety case rests on things this build does not have: certified needle sharpness and grind geometry, guaranteed particulate-free fluid path, sterility at point of use, and a disposal route that stops the sharp from reaching anybody else. None of those are grinding problems, which is precisely why the disposable plastic syringe replaced the beautiful matched glass pair everywhere by the 1970s. The ground fit was not beaten on performance. It was beaten on sterility.