New Approaches to Ideal Fluid Theory

Something was badly wrong with the picture of air behaving and moving as an ideal fluid. It was mathematically impressive but empirically defective. What exactly was wrong? Was it the assumption of zero viscosity itself that should be dropped or were there perhaps other, unnoticed, assumptions at work in the picture of the flow that might be the cause of the trouble? What about Laplace’s equation and the assumption of irrotational motion?

This question was addressed by a number of late nineteenth-century ex­perts whose investigations greatly deepened the understanding of ideal fluid theory. They began to explore some possibilities that previously had been neglected. But why did they not simply abandon ideal-fluid theory as em­pirically false and turn directly to the analysis of viscous fluids? Attempts were made to do this but with very limited success. The reason was that the mathematics of viscous fluids was so difficult. It was possible to write down the equations of motion of a viscous fluid by taking into account the trac­tion forces along the surface of the fluid element, but it was another matter to solve the equations. The full equations of viscous flow are now called the Navier-Stokes equations, though in Britain they used to be called just the Stokes equations. Of course, they contain a term involving the symbol Ц standing for the coefficient of viscosity. If the value of this coefficient is set at zero to symbolize the absence of viscosity, that is, Ц = 0, the Navier – Stokes equations turn back into the Euler equations that have been described earlier in this chapter. In a later chapter I look more closely at the status of the Navier-Stokes equations and the different responses to their seemingly intractable nature. For the moment it is only necessary to appreciate the problem they posed. No one could see how to solve and apply the equations except in a few simple cases. Because of their intractability, any attempt to avoid the impasse thrown up by the zero-resultant theorem had to be one that stayed with the Euler equations and thus within the confines of ideal- fluid theory.

The crucial insight that permitted the further development of ideal-fluid theory was provided by Helmholtz and Kirchhoff in Germany and Rayleigh in Britain. These men realized that the solutions to the Euler equations that gave the streamlines around an obstacle were not unique. More than one set of streamlines were possible and consistent with the equations. More than one kind of flow could satisfy the equations and meet the given boundary conditions. What is more, some of these flows could generate a resultant force. There were in fact two very different kinds of flow that might have this desired effect and, in principle, allow the zero-resultant outcome to be evaded. Rayleigh contributed to the study of both. Both approaches involved the limited introduction of fluid elements that possessed rotation and vor – ticity. The strict condition of irrotational motion was dropped. On one ap­proach this involved the introduction of just one singular point in the flow that rotated and constituted the center of a vortex. On the other approach a sheet or surface of vorticity was postulated. In both cases the remainder of the flow was still irrotational. These two approaches provided, respectively, the basis for the two different theories of lift that I mentioned in the introduction and called the circulatory or vortex theory of lift and the discontinuity theory of lift. Historically, the first of the two approaches to be developed in detail was the one that led to the discontinuity theory. I now introduce the ideas underlying this approach. The other approach and the other theory of lift are introduced in chapter 4.