I understand the idea behind a siphon, however I have never found any mathematical explaination that states how long it will take for water to flow. I've seen fluid dynamics text which talk about fluids flowing from vessels but never specifically about siphons. If anyone has any direction then I would appreciate it.

My in-expert help:

It depends upon various factors. The liquid you use matters, as some liquids have low boiling points and respond to low pressure in ways that interfere. The diameter of the tube also matters, as well as (I imagine) the material of the tube, though I doubt that would be terribly important (only important if there is some attraction of the liquid to the sides of the tubing).

A quick diagram of a siphon:

_________

[source]______] \

\_____Destination

 

The lines are the tube you siphon with

Once the tube has water in it, the water on the right flows, creating a suction. How strong that suction is depends upon the tube diameter and the distance downhill the liquid travels while moving from the source to the destination.

If you want some math, calculate the amount of vacuum that would be created if the bottom 90% or so of the downhill part of the tube (right before the destination, the two \'s stacked on each other). Find the difference between the pressure on the fluid at the source (atmospheric plus pressure from depth. The difference will yield a pressure that can lead you to the force exerted on the fluid to flow down the tube. That can then give you a general idea of the amount of flow.

 

I would bet there is some equation that simplifies this and takes into account anything I'm missing. Hopefully a physics expert will post and clarify the matter.

I understand that atmospheric pressure and the potential energy of the fluid are general what cause the siphoning effect. The reason I ask the question is that during my research project during my undergrad I used a siphon to move water to study other aspects of fluid dynamics. During this time I was never able to find a complete mathematical explanation of what is going on during siphoning.

I would think that applying Bernoulli's equation would tell you what you wanted to know.

I am talking about a mathematical equation, probably a differential equation of some kind, which tells me how long it takes and I don't mean some general high school physics equation.

If you apply the Bernoulli equation (and continuity equation, probably), you should be able to figure out a flow rate that depends on the fluid height difference and construct such a differential equation.

In most draining situations, where the height of the water is much above the drain, the velocity in the tube is proportional to the square root of the height above the drain. But, this is just a special case of the Bernoulli's equation, and if you use Bernoulli's equation you can include the friction factors from the contraction into the tube, the friction in the siphon itself (total friction depends on the total length of the tube).

 

The inviscid Bernoulli's equation can be devired by integrating the Navier-Stokes differential equations along a streamline, but energy and mass balances are the typical way it is presented.