On Limits

I like to approach science by means of selected particular examples. Consider then the (two-dimensional) flow of a viscous incompressible fluid past a circular cylinder, or equivalently “flow past a circle.” If we imagine the fluid to be unbounded, the problem is characterized by a single dimensionless parameter, the Reynolds number $Re$, which is the fluid speed $U$ at infinity times the cylinder radius $R$ divided by the fluid viscosity $\nu$. Imagine a sequence of problems, at successively higher fixed values of $Re$. For $Re$ somewhat greater than 10, a thin “boundary layer” forms near the cylinder, where the tangential velocity rapidly varies from a value of magnitude $U$ to the value zero at the cylinder. The “wake” behind the cylinder has a structure similar to the boundary layer. At higher values of $Re$ instability sets in, the symmetry is broken and an array of vortices (“vortex street”) forms aft of the cylinder. For higher $Re$, the overall geometry becomes very unsteady and the details of the flow appear “random,” and is termed “turbulent.” The various italicized words are concepts with the aid of which the broad outlines of flow development can be understood. These concepts emerged from comparing experiment with analytic and numerical solutions to a well-accepted mathematical model for viscous flow, the Navier-Stokes (NS) equations. We will limit our further remarks to the behavior of relevant solutions to these equations, although it conceivable that in the future better equations will be found that will be free of the difficulties attached to solving the NS equations.