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Consider the following experiment: Rising a water in a straw.

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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 1 and 2 be a point at the top and a point at the bottom that is h away from 1 respectively.

According to Bernoulli's equation, I have

$$p_1+\rho g h_1 + \frac 1 2 \rho v_1^2 =p_2+\rho g h_2 + \frac 1 2 \rho v_2^2 $$

I set

  • $p_1=p_2$ 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?

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  • $\begingroup$ How can $p_1=p_2$? If the pressures at the opposite ends of the small column of water are equal, then there is no net force due to pressure in either the upward or downward directions. All there is left is gravity, which will pull the column of water down. $\endgroup$ – Samuel Weir Mar 1 '18 at 18:04
  • $\begingroup$ @SamuelWeir: I also thought of it before. I don't quite understand the essence of Bernoulli's equation for sure. I am trying to help my brother but failed. :-) $\endgroup$ – Money Oriented Programmer Mar 1 '18 at 18:07
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    $\begingroup$ If you apply the B-equation to the column of water, you get the equation $p+\rho gh = constant$, so the difference in pressure between p1 and p2 is $\rho gh$. If on the other hand you apply it to the air coming out of the straw, you get $p+(1/2)\rho v^2 = constant$, so an increase in velocity is accompanied by a decrease in pressure. That's what causes the pressure decrease at the top of the straw which results in water coming up the straw. $\endgroup$ – Samuel Weir Mar 1 '18 at 18:18
  • $\begingroup$ @SamuelWeir: Now, the problem is which 2 points must I choose to observe. Are they both water or both air particles? Or one is water particle and the other one is air particle. It is confusing to me. $\endgroup$ – Money Oriented Programmer Mar 1 '18 at 18:23
  • $\begingroup$ I think that the idea is to first note (1) how the local pressure at the output of the straw being blown through decrease, and then note (2) how that pressure decrease causes the water column in the other straw to rise. So two separate applications of Bernoulli's equation, one for two points along the air flow, and the other for two points of different height h in the water column. $\endgroup$ – Samuel Weir Mar 1 '18 at 18:35

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