6.1 Introduction to Boundary Layers

As we saw in Chapter 5, solutions to the full Navier-Stokes equations are few in number and difficult to obtain. In the exact solutions of the Navier-Stokes equations, it was repeatedly seen that when a local Reynolds number was large, viscous effects are felt mainly in the immediate vicinity of a solid boundary. In 1904 Prandtl introduced an approximate form of the Navier-Stokes equations that holds in the thin boundary layer near the wall, where viscous effects are comparable to the inertia effects. Here, the relationship between the boundary layer equations and the full Navier-Stokes equations is investigated by demonstrating how the boundary layer equations can be derived as a limiting form of the Navier-Stokes equations.

To understand how inviscid flow theory and boundary layer theory fit with the Navier-Stokes equations, consider the following example, first presented by Friedrichs in 1942. His model equation is

The second-order derivative can be thought of as the viscous terms with a small viscosity ?, the first derivative as a momentum, and the constant a as a pressure force. When the highest-order derivative is neglected ( inviscid approximation), the first-order equation that remains has the solution f = ax+ c, c being a constant of integration. Clearly, only one of the boundary conditions can be satisfied. Our version of relaxing the no-slip condition would be to impose f(1) = 1, giving c = 1 ? a. Thus, our inviscid solution is

To take care...