Equation of continuity says us that if we insert some fluid in a tube, the same amount of fluid will come out from the other end. If we make a small hole in a hose pipe, water will come out with a great speed. The bigger the hole, the slower the speed. This is a direct consequence of the equation of continuity.
But at the case of water tap, when we start to turn on the tap slightly, the velocity of water is slow. As we turn on the tap more, the speed increases. This is contradictory with equation of continuity.
Best Answer
The continuity equation says that IF VOLUMETRIC FLOW IS FIXED, e.g. "always XYZ liters per second", then a smaller pipe or smaller hole causes higher fluid velocity.
The water faucet is not like that. The flow is not fixed. When the faucet is slightly open it is "X liters per second" flowing out ... when the faucet is fully open it is "Y liters per second" flowing out. Y is bigger than X. It's NOT a constant fixed flow rate. Therefore the continuity equation does NOT imply that a narrower opening means higher fluid velocity.