Effects of Viscosity
3- 1 Introduction
Viscous flows at high Reynolds numbers constitute the most obvious example of singular perturbation problems. The viscosity multiplies the highest-order derivative terms in the Navier-Stokes equations, and these will therefore in the limit of zero viscosity degenerate to a lower order. The boundary condition of zero tangential velocity on the body (or of continuous velocity in the stream) is therefore lost. This necessitates the introduction of a thin boundary layer next to the body constituting the inner region where the inviscid equations are not uniformly valid. Unfortunately, only a very restricted class of viscous-flow problems can be analyzed by the direct use of the method of matched asymptotic expansions. First, only for a very limited class of bodies will the boundary layer remain attached to the body surface. When separation occurs, the location of the region where viscosity is important is no longer known a priori. The second difficulty is that for very high Reynolds numbers the flow in the boundary layer becomes unstable and transition to turbulence occurs. As yet, no complete theory for predicting turbulent flow exists.
Before considering some of the model problems that may be analyzed, we will give a short description of the qualitative effects of viscosity.