Consider a pipe with length $L$ and uniform radius $A$ is held vertically. According to the continuity equation, the velocity of water going into the pipe seems to be the same as the velocity of water coming out. But according to Bernoulli's equation:
$$P_{atm}+\frac{1}{2}\rho v_1^2+\rho gL=P_{atm}+\frac{1}{2}\rho v_2^2$$
$$v_2=\sqrt{v_1^2+2gL}$$
Which means that the e water would come out faster, which makes much sence. What is wrong with my equations?