1 Relevance of Irrotational Flow Theory

The vorticity equation given in the preceding chapter implies that the irrotational flow(such as the one starting from rest) of a barotropic fluid observed in a nonrotating frame remains irrotational if the fluid viscosity is identically zero and any bodyforce are conservative. Such an ideal flow has a nonzero tangential velocity at a solid surface (Figure 6.1a). In contrast, a real fluid with a nonzero ? must satisfy a no-slip boundary condition. It can be expected that viscous effects in a real flow will be confined to thin layers close to solid surfaces if the fluid viscosity is small. We shall see later that the viscous layers are thin not just when the viscosity is small, but when a non-dimensional quantity Re = UL/ ?, called the Reynolds number, is much larger than 1. (Here, U is a scale of variation of velocity in a length scale L.) The thickness of such boundary layers, within which viscous diffusion of vorticity is important, approaches zero as Re ? ? (Figure 6.1b). In such a case, the vorticity equation implies that fluid elements starting from rest, or from any other irrotational region, remain irrotational unless they move into these boundary layers. The flow field can therefore be divided into an "outer region" where the flow is inviscid and irrotational and an "inner region" where viscous diffusion of vorticity is important. The outer flow can be approximately predicted by...