So the Bernoulli's EQ comes from conservation of energy. From the figure, I see that if Force 1 is greater than Force 2, the water will move to the right. The distance Force 1 travels gives work of F1d1, and the distance Force 2 is pushed back gives work F2d2, and net work on the system is F1d1 - F2d2. But then you have to equate the net work to the PE and KE change of the system, to get F1d1 - F2d2 = (0.5mv22 + mgh2) - (0.5mv12 + mgh1). But that seems to imply the volume of water displaced at the left suddenly appears at the displaced water on the right. I understand that both volumes of water are equal, but it seems as the though the water on the right side BEFORE it was pushed was already in the same pipe at the same height. So after pushing, the water at the right only changed horizontal position. Same for the left side, too. So shouldn't the differences in mgh be 0?