Silverstone and hydrodynamical singularities

There was plenty of opportunity to watch Formula 1 cars in the wet at Silverstone last week, and it's worth noting that when a car transforms a sheet of water into a spray of droplets, each droplet has been created by a topology-changing process corresponding to a hydrodynamical singularity.

Droplet formation is a liquid version of the fracture of a solid; the continuous medium is broken into disconnected fragments. Up until the point of the singularity, droplet formation is described by a solution of the Navier-Stokes equations for free surface flows. Thus, rather than solving the equations with fixed boundary conditions, such as those provided by a pipe or a wing-surface, one must also specify how the surface of the fluid evolves in time.

Droplet formation leads to a singularity in finite-time, but how the droplet forms, whether it falls from a tap (that's a 'faucet' in the USA), or is flung from the tread of a rotating tyre, is actually irrelevant to the shape of the droplet as the water is broken. As philosopher Robert Batterman points out, the shape of a breaking droplet is largely determined by the ratio of the viscosity of the breaking fluid to the viscosity of the ambient medium.

Water, in fact, is a good deal more complex than many people imagine. Weisberg et al emphasise that "water's microstructure cannot simply be described as a collection of individual molecules...There is a continual dissociation H2O molecules into hydrogen and hydroxide ions, and a continual recombination of those ions back to H2O molecules. At the same time, the H2O molecules associate into larger polymeric species."

Philosophically speaking, this has implications for understanding the relationship between microscopic and macroscopic structure. As James Ladyman argues, "Metaphysicians expect the bridges between the ontologies of the different sciences to be synchronic but they are usually diachronic. So, for example, it is the dynamics of how hydrogen bonds form, disband, and reform that gives rise to the wateriness of water and not the mere aggregation of hydrogen and oxygen in a ratio of two to one," (Many Worlds? OUP 2010, p158).