The velocity potential and Laplace's equation
(a) Occurrence of irrotational flows
If a closed curve in an in viscid fluid for which ρ = f(p) has at some time no circulation round it, then by Kelvin's theorem there is never any circulation round the curve. Of course there are no such ‘ideal’ fluids around, but we have seen that at high Reynolds number and away from boundaries and other awkward regions, a fluid will behave in a near enough ideal fashion. Hence we expect that there can be large regions of flow which have no circulation round any circuit, and hence no vorticity. Thus it is well worth discussing irrotational flows, which have
∇ × v = 0.
Naturally we must not try to use irrotational flow theory in those regions where we have already seen vorticity to be inevitably developed from the no-slip condition and the diffusive action of viscosity, such as in boundary layers, wakes, eddies and enclosed regions. But away from these regions we can use irrotational flow theory provided that the flow is started, or arrives, with no vorticity. For example, the following flows are closely irrotational.
(i) Flow of air round a streamlined aeroplane wing or body. The aircraft flies into air that is effectively at rest, and the boundary layers and wake are thin enough to be neglected at a first approximation. Such flows will be discussed in detail in Chapter XVII.
(ii) Waves on the surface of reasonably deep water. The boundary condition at the surface does not bring in a noticeable boundary layer because the air is so much less dense than the water. Water waves are dealt with in Chapter XIII.
(iii) Sound waves in air (or water) are of such short time scale that diffusive effects have no time to act, and hence irrotational flow is an appropriate model in many cases. Sound waves are considered in Chapter XII.
(iv) In certain cases irrotational flow theory is useful for some regions of the flow of a uniform stream past a blunt body.
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