A Theoretical Model of the Surface Geometry of Laminar Fluid Chains
By Zachary Zitzewitz
During the first visit to my friend’s house since the pandemic had started, the first thing I did was wash my hands. The first thing I noticed was the eye-catching shape of the water projecting from the faucet. While the sinks I had used for the past year emitted frothy, turbulent jets, the water in this sink fell over a flat edge and created a laminar cascade of water that appeared to take the shape of mutually orthogonal chain links. I decided I had to discover what was going on. When I got home, I scoured the internet and found some qualitative explanations, but not much more. Maybe it was because I had just taken differential geometry at school and a course on planetary-scale ocean dynamics with another educational program, or that I had worked on fluid dynamics simulations for an autonomous underwater robotics project, but I really wanted to devise a mathematical description of this phenomenon. I love understanding not just the intuition behind why things happen, but the equations as well … I present a theoretical model of gravity-driven laminar fluid chains. A stream of fluid falls under this categorization based on the distinctive shape of its mutually orthogonal sheets of fluid bounded by curved jets, which will be referred to throughout this discussion as “chain links.” If the fluid falls through specific orifice geometries under the influence of gravity, the chain will be oriented downwards. The jets that bound the sheets collide over and over again, pulled towards each other by surface tension and cascading down through the system. This paper will present both a qualitative explanation of this phenomenon and simulation of it based on a mathematical model. The figure below shows an example of one such chain link. For the purposes of this paper, the flow within the sheet and jets will both be treated as laminar. The analysis will consider the case with no motion of the fluid in directions perpendicular to the direction of flow, nor eddies, vortices, or turbulence of any other kind in the stream …