Consider the following experiment: Rising a water in a straw.
For the sake of simplicity, let me assume that there is no air in the straw by redefining the height
h as the height of the top tip from the water surface in the container.
Now let subscript
2 be a point at the top and a point at the bottom that is
h away from
According to Bernoulli's equation, I have
$$p_1+\rho g h_1 + \frac 1 2 \rho v_1^2 =p_2+\rho g h_2 + \frac 1 2 \rho v_2^2 $$
- $p_1=p_2$ to $p_0$ because both are atmosphere pressure,
- the vertical component of $v_1$ and $v_2$ to zero because the water flows slowly in the pipe,
- the horizontal component of $v_2$ to zero but the horizontal component of $v_1$ to $v$.
Finally I have
$$ 2 g h + v^2 =0 $$
Mathematically this equation has no real solution. What is wrong?