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So say we have a syringe filled with liquid and we apply a force to dispense it. As the liquid is dispensed the volume of liquid in the syringe would decrease and therefore the pressure would increase and therefore the force required would increase?

Is this correct? Intuitively it doesn't seem right to me but seems to be what Bernoulli's equation suggests.

As Bernoulli states:

$$Pressure∝\frac{1}{Volume}$$

and

$$Pressure=\frac{Force}{Area}$$

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    $\begingroup$ I'd suggest looking into the assumptions that your equations are based on. Do they assume constant mass, for example? That's certainly not the case when you're ejecting liquid. $\endgroup$ – Chemomechanics Dec 23 '20 at 21:19
  • $\begingroup$ If you had a syringe filled with a compressible substance (air) then pressure / volume relationship would hold: higher finger pressure makes higher syringe pressure with less volume (equation 1) and higher flow rate from needle. But for non-compressible liquid the flow volume through the needle per unit time = volume being dispensed from the syringe. Delta volume/delta time is the same full, half full, or nearly empty. $\endgroup$ – Robert DiGiovanni Dec 24 '20 at 10:52
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The cross section area of the needle is flow rate limiting and is constant.

For a given finger pressure, the flow rate theoretically will be the same until the syringe empties. In practice, residual liquid on the walls of the syringe act as a lubricant, making it easier to push the plunger once started, resulting in a higher flow rate.

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Bernoulli's equation states that $$p\propto \rho$$ and this assuming that a conservative force is being applied. Your first equation is, therefore, incomplete, missing a direct proportionality with mass. So you see that the pressure should stay the same (as long as you don't push too hard, because that can change the fluid density, $\rho$). However, the validity of Bernoulli's equation is questionable in this situation, since you would have to apply a conservative force.

Like @Chemomechanics said, you should consider the assumptions made to arrive at this equation.

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