Explain why The size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection

#### Solution 1

The small opening of a syringe needle controls the velocity of the blood flowing out. This is because of the equation of continuity. At the constriction point of the syringe system, the flow rate suddenly increases to a high value for a constant thumb pressure applied.

#### Solution 2

For a constant height, the Bernoulli’s theorem is expressed as P +1/2 ρ v^{2 }= Constant

In this equation, the pressure P occurs with a single power whereas the velocity occurs with a square power. Therefore, the velocity has more effect compared to the pressure. It is for this reason that needle of the syringe controls flow rate better than the thumb pressure exerted by the doctor.