“The internal motion of water assumes one or other of two broadly distinguishable forms,” Osborne Reynolds reported to the Royal Society in 1883. “Either the elements of the fluid follow one another along lines of motion which lead in the most direct manner to their destination, or they eddy about in sinuous paths the most indirect possible.”

Reynolds puzzled over how to define the point at which a moving fluid (or a fluid with an object moving through it) makes the transition from stable to unstable flow. He noted that the transition depends on the ratio between inertial forces (characterized by mass, distance, and velocity) and viscous forces (characterized by the “stickiness” of the fluid). When this ratio, now termed the Reynolds number, reaches a certain critical value, the fluid shifts from orderly, deterministic behavior to disorderly, probabilistic behavior resistant to description in full detail. The two regimes are known as laminar and turbulent flow.

The Reynolds number is both non-dimensional and universal, appearing consistently across a range of phenomena as diverse as blood pumping through a heart, a fish swimming through the sea, a missile flying through the air, burning gas flowing through a jet turbine, or weather systems flowing around the Earth. It is both descriptive in the sense of capturing the characteristics of an existing flow and predictive in the sense that the Reynolds number gives a reliable indication of which regime will dominate a projected flow. Thanks to the Reynolds number, we can tackle otherwise intractable problems in fluid dynamics with scale models in a scaled flow and make predictions that hold up.

The notion of a Reynolds number and critical velocity can also be applied to non-traditional domains: for instance, the flow of information in a computer network, where velocity is represented by bandwidth and viscosity is represented by processing speed of the individual nodes, or the flow of money in an economy, where velocity is represented by how fast funds are moving and viscosity is represented by transaction costs.

Wherever things (including ideas) are either moving through a surrounding medium or flowing past a boundary of any kind, the notion of a Reynolds number can help characterize the relation between inertial forces and viscosity, giving a sense for what may happen next.

Why do things go smoothly for a while, and then suddenly they don’t?