Hi Bob
Understood. There are differences in the two applications. Forgive me if I go into lecture mode – force of habit.
The micro-metering valves are designed to provide a continuous flow at a certain rate. This could be something like 1 uL/min, but the flow might last an hour or a day. This is done by having a very small aperture and relying on standard fluid impedance theory – if anyone really understands the flow regime, which they don't. Often the flow through the hole is actually limited by obscure hydrodynamic factors as well, which is why they specify the valves for high-pressure use. But the flow is continuous. The rate then depends on the effective cross-section of the hole.
Yes, there are PhD theses and published research papers on how fluid flow through an orifice works. The maths is horrible.
The injet printer does not have continuous flow at all. It operates by 'drop on demand' instead. Typically it does this by exploiting some other physics. What happens is as follows.
First, the ink fills up a tiny chamber by capillary action. Obviously, the lining of the tubes carrying the ink must be wettable for this to happen. Then, when the printer wants one drop of ink, a tiny heater at the back of the tiny chamber glows for a microsecond or two, turning the immediately adjacent water in the ink into steam. That might be 1% of the ink volume in the tiny chamber. You will have seen all those gold-plated electrical contacts the the head of the printer: they are for the heaters.
The pressure inside the tiny chamber rockets up and the ink 'tries to escape'. Some of the ink does actually go back up the tube by which it came in, but that tube is very small so what escapes there is very little. The rest of the ink in the tiny chamber finds an escape via the small hole at the front of the chamber. Translation: a tiny drop of ink is spat out forwards, to hit the paper and be absorbed there. Now repeat 10 or 20 times in a vertical row, and at a kHz rate or more, and you have a B&W ink jet printer. Have 5 o6 columns of these holes and you have a colour ink jet printer. Yes, the print head is a marvel of modern micro-engineering – but if you want a million of them, technology can provide.
Why does the ink not dribble out of the holes at the front of the chamber? Surface tension keeps it in, just like with DWR-coated fabric. Obviously, the hole in a stove valve is much bigger.
HTH
Cheers