1 Introduction

In Chapters 6 and 7 we studied inviscid flows in which the viscous terms in the Navier Stokes equations were dropped. The underlying assumption was that the vis-cous forces were confined to thin boundary layers near solid surfaces, so that the bulk of the flow could be regarded as inviscid (Figure 6.1). We shall see in the next chapter that this is indeed valid if the Reynolds number is large. For low values of the Reynolds number, however, the entire flow may be dominated by viscosity, and the inviscid flow theory is of little use. The purpose of this chapter is to present certain solutions of the Navier Stokes equations in some simple situations, retaining the viscous term ?? ² u everywhere in the flow. While the inviscid flow theory allows the fluid to "slip" past a solid surface, real fluids will adhere to the surface because of intermolecular interactions, that is, a real fluid satisfies the condition of zero relative velocity at a solid surface. This is the so-called no-slip condition.

Before presenting the solutions, we shall first discuss certain basic ideas about viscous flows. Flows in which the fluid viscosity is important can be of two types, namely, laminar and turbulent. The basic difference between the two flows was dramatically demonstrated in 1883 by Reynolds, who injected a thin stream of dye into the flow of water through a tube (Figure 9.1). At low rates of flow, the dye stream was observed to follow...