Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 39C / a / Solution

The Reynolds number isWhen it is large, viscosity is negligible in most of the flow, but the inviscid outer solution cannot generally satisfy the no-slip boundary condition at a rigid wall. A thin boundary layer resolves this mismatch: tangential gradients remain of outer scale $\mathcal{L}$, normal gradients have the much shorter scale $\delta$, and the small viscous coefficient is offset by $u_{yy}=O(\mathcal{U}/{\delta }^2)$. Matching a viscous inner solution to an inviscid outer solution retains the leading viscous effects without solving the full equations uniformly everywhere.