Explanation for breaking up of a stream of water into droplets

Question

Water falling from a tap eventually breaks into droplets at a particular distance from the tap. The distance(from the tap) at which it breaks into droplets is observed to be an increasing function of the volume flux of the water flowing out of the tap. What is the valid explanation for this phenomenon?

My own line of reasoning is as follows : let $D$ be the critical distance measured from the tap, i.e. the distance at which the water breaks into droplets. At a distance $d<D$ the flow can be assumed to be streamlined to a good approximation; in which case the cross-sectional area of the stream goes on steadily decreasing down the flow. Due to this, the surface energy per unit volume associated with the surface of the water should go on increasing with the distance from the tap until it reaches a point where it becomes greater than/equal to the surface energy of the same volume of water as a water drop. At this distance, by minimisation of potential energy the water should break into droplets to minimise the surface energy. Let's say this happens when the cross section of the flow reaches a threshold value $A$. Now if the volume flux is low; it would mean that the cross sectional area at the tap is itself low. Due to that, the threshold area $A$ would be attained at a lesser distance from the tap. This probably explains the observation that the distance(from the tap) at which it breaks into droplets is observed to be an increasing function of the volume flux of the water flowing out of the tap. Am I right? Please correct/add to my explanation.

Answer 1

Your description of the situation sounds like an example of the Plateau-Raleigh instability of fluid jet. You should be able to find several sources for the detailed analysis if you search on the web under that name. High-speed photography suggests that the phenomenon may be more dynamic than you have assumed.

As a next step for your thinking, perhaps try writing down an equation with all the variables you have identified and check whether your dimensions work out. This could suggest something you have left out, or give you a clue of what type of experimental evidence might disprove your hypothesis. For example, after reading your question I wrote down the following equation, the two constants k are dimensionless, but I had to include water density to make the unit dimensions work out. I'm not claiming this is a solution, just a way to continue thinking. You might write something different, but the dimensions must always work out as you intend.

$ D = k_{1}\left (\frac{F^{2}\rho _{w}}{A\gamma_{s}} \right )^{k_{2}} $

This problem has gotten the attention of some of the giants of physics, so I wouldn't expect the answer to be too trivial.