Maximum liquid velocity in a siphon

That formula is simplified by

1. ignoring viscosity

2. ignoring that $h$ could be greater than what could be supported by atmospheric pressure (about 10 meters in the case of water), at which point the fluid separates.

Ok let me explain... First of all I regret for not being able to upload an image of siphon... So please follow my words. I hope I can clear your doubts...

So consider a working siphon. We know that the flow is very much steady in the siphon tube. Ok. Now I am asking you to consider the longer right portion of this u-tube. As we can see the diameter of the tube is constant throughout and so is the velocity according to the equation of continuity. Now tell me what conditions do you need to ensure this. Ok let us consider a ball falling at a constant rate(meaning a constant velocity). What exactly do you need for this is a force exactly equal to mg acting opposite. Now consider the case of the siphon. The pressure at the opening of the longer end is P_a. That means for a steady flow you need the pressure at the top to be equal to P_a - $rho$gh₂. Let's imagine now that you increase this height h₂ to a point where P_a is exactly equals $rho$gh₂. Ok so now the pressure at the top is zero but still the balance is well maintained. Now imagine what if you increase h₂ more. The pressure at the top can't be less than zero. So what happens now is the same as when in the case of the ball, it's weight is more than the opposing force. Yes, now the weight of the liquid column is more than the forces trying to balance it. So what will happen is an accelerated motion. The fluid stream will as a result discontinue at the top. Due to this discontinuity, more and more air will enter the tube ultimately halting the process.

So for a given h₃, because the maximum value of h₂ is $frac{P_a}{rho g}$ ,so, the maximum value of h₁ should be $frac{P_a}{rho g}$ - h₃. Input this value into the equation v = $sqrt{2gh_1}$ , and you will get v_max as $sqrt{ 2(frac{P_a}{rho} - h_3 g)}$.

Hope it's helpful...

Note,h₂ is the height of the longer right portion of the u-tube, h₃ is the shorter end, and h₁ is the difference of the two...