a boundary layer

Hold your hand out of a fast-moving car window. The air rushes past, yet a microscopically thin film of air right against your skin is barely moving at all. That thin region where the fluid speed climbs steeply from zero (at the surface) to the full free-stream speed is a boundary layer. It is where all the friction, drag, and heat exchange between a body and a flowing fluid actually happen.

The puzzle the boundary layer resolves is this: for fast flow the viscosity mu is tiny, so it is tempting to drop the viscous term and use the inviscid Euler equations. But viscosity, however small, must enforce the no-slip condition — the fluid touching a solid surface moves with the surface, i.e. is at rest relative to it. A tiny coefficient on the highest-derivative term mu Laplacian u that you cannot ignore near the wall is the hallmark of a singular perturbation. Prandtl's insight (1904) was that the viscous term matters only in a thin layer of thickness scaling like 1/sqrt(Re); inside it you keep a simplified set of equations (the boundary-layer equations, a parabolic-like reduction of Navier-Stokes), while outside it the inviscid Euler description is fine. You solve the two and match them.

Boundary layers govern aerodynamic drag, heat transfer from chips and engines, and — crucially — flow separation: when the boundary layer detaches from a surface, you get stall on a wing, the wake behind a car, and a sudden jump in drag. The whole discipline of aerodynamic design is largely the art of keeping boundary layers attached. The boundary layer is the canonical example of why you cannot always just set a small parameter to zero in a PDE.