No Slip Condition

We saw previously (for instance, in Section 5.8) that when an inviscid fluid flows around a rigid stationary obstacle then the normal fluid velocity at the surface of the obstacle is required to be zero. However, in general, the tangential velocity is non-zero. In fact, if the fluid velocity field is both incompressible and irrotational then it is derivable from a stream function that satisfies Laplace's equation. (See Section 5.2.) It is a well-known property of Laplace's equation that we can either specify the solution itself, or its normal derivative, on a bounding surface, but we cannot specify both these quantities simultaneously (Riley 1974). The constraint of zero normal velocity is equivalent to the requirement that the stream function take the constant value zero (say) on the surface of the obstacle. Hence, the normal derivative of the stream function, which determines the tangential velocity, cannot also be specified at this surface, and is, in general, non-zero.

In reality, all physical fluids possess finite viscosity. Moreover, when a viscous fluid flows around a rigid stationary obstacle both the normal and the tangential components of the fluid velocity are found to be zero at the obstacle's surface. The additional constraint that the tangential fluid velocity be zero at a rigid stationary boundary is known as the no slip condition, and is ultimately justified via experimental observations.

The concept of a boundary layer was first introduced into fluid mechanics by Ludwig Prandtl (1875-1953) in order to account for the modification to the flow pattern of a high Reynolds number irrotational fluid necessitated by the imposition of the no slip condition on the surface of an impenetrable stationary obstacle. According to Prandtl, the boundary layer covers the surface of the obstacle, but is relatively thin in the direction normal to this surface. Outside the layer, the flow pattern is the same as that of an idealized inviscid fluid, and is thus generally irrotational. This implies that the normal fluid velocity is zero on the outer edge of the layer, where it interfaces with the irrotational flow, but, in general, the tangential velocity is non-zero. However, the no slip condition requires the tangential velocity to be zero on the inner edge of the layer, where it interfaces with the rigid surface. It follows that there is a very large normal gradient of the tangential velocity across the layer, which implies the presence of intense internal vortex filaments trapped within the layer. Consequently, the flow within the layer is not irrotational. In the following, we shall attempt to make the concept of a boundary layer more precise.