## Analogy Microfluidics

The sandpile analogy suggests that in the smallest systems micro-fluidics, the study of microflows, must be central to any ideas we might have for shrinking complex insect-like systems (Ref: Micro-fluidics and Drops). A dripping tap reveals just how complex even the simplest microfluidic system is. Singularities in patterns of flow occur almost everywhere, even in free-surface flows.

A drop hanging and then falling from a faucet models a liquid separating into two or more pieces. The change in structure of the drop over time reveals that as the drop falls, it first tugs out a long neck between two masses of fluid. One mass will remain on the tap; the other will fall as a new drop. The neck thins out elongating until it breaks. What is the shape of the drop as the neck fractures? Something complex happens in the mathematical description of the liquid at this critical point, because the drop undergoes a transition in its topology. A drop starts as a single, connected fluid but ends as two or more separate drops. Separation is just one example of a finite-time singularity, because a drop's breakup happens just after the drop becomes unstable enough to fall. A singularity arises for the topological transition because the radius of the neck holding the drop to the larger mass of fluid gradually shrinks and disappears. As the radius of the neck goes towards zero, the drop's curvature diverges, so the forces of surface tension become infinite. How do such dramatic dynamics develop in a system that has smooth initial conditions and forcing terms? Similar transitional singularities are common in many diverse systems, from stellar structures to turbulent air and water to a bacterial colony's growth. Now imagine what happens to hemolymph in the hemocoel of a bee on the wing!

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