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

From Bernoulli's theorem, we know that

We know that diameter of the needle is very less. So flow velocity through the needle will be very high according to equation of continuity; Area × velocity = constant

So needle controls the velocity of the fluid flow

By using our thumb we apply more pressure to the flow of fluid . So thumb controls pressure.

The power of pressure is 1 and the power of velocity is 2. Thus size of the needle has more control over the flow rate than the thumb pressure.