The Real and the Ideal

Two characteristics have now been identified in the British response to the circulation theory of lift. First, there was a desire for theories of wide scope that embrace complex viscous phenomena beyond the reach of the theory. Second, there was a tendency to read Lanchester as contributing to an inviscid theory and therefore as committed to a simplified and unreal representation of fluid flow. Both of these indicate the importance that British experts at­tached to the distinction between real fluids and ideal fluids. Taylor insisted that fluid mechanics should have a firm basis in physics and dismissed the idealizations of classical hydrodynamics. Cowley and Levy described inviscid theory as fatally flawed and spoke of the need for a theory of viscous flow that would solve the problems of aerodynamics at a stroke. Bairstow agreed that it was fundamentally impossible to represent real fluids in terms of ideal fluids and duly turned to the study of viscous flow. What Bairstow had asserted with characteristic acerbity, Lamb had hinted at with characteristic restraint. The different objections and formulations all point to one conclusion. The distinction between viscous and inviscid fluids is to be seen as the axis around which British thinking revolved.16

It is important not to view this distinction as self-evident or something that was understood in the same way by all competent operators in the field of fluid dynamics. In reality it was treated differently in different institutional settings. How then should the distinction between viscous and inviscid fluids be understood? Formally, it centers on whether p, the symbol for viscosity in the Stokes equations, is to have a value of zero or of nonzero. Was p = 0, or p Ф 0? Logically it must be one or the other and it can’t be both. Empiri­cally, whether Stokes’ equations turn out to be true, and Euler’s false, (or vice versa), is something to be settled by reference to experiment. But these tru­isms do not tell us how to interpret the difference between putting p = 0 or p Ф 0; nor do they indicate what physical meaning is to be given to the mathematical limit when p ^ 0. They do not tell us whether the distinctions involved are qualitative or quantitative or whether the boundaries under dis­cussion are strong or weak or for what purposes they might be important or unimportant. This is the point. The conceptual boundary between viscous and inviscid fluids is more than merely formal. Rehearsing the elementary mathematical properties of the distinction does not tell us what methodologi­cal implications are attached to it by the scientists concerned. I shall now illustrate the broader, methodological significance of the distinction by refer­ence to Lamb’s own discussion of viscosity.

Lamb began his account of aerodynamics, in the 1916 edition, by point­ing out that the analysis of Kirchhoff-Rayleigh flow was the first attempt, “on exact theoretical lines,” to overcome the result that a perfect fluid exerts no resultant force on a body. He added: “The absence of resistance, properly so called, in such cases is often referred to by continental writers as the ‘paradox of d’Alembert’” (664). Why did Lamb think that “absence of resistance” was the more proper description? What was wrong with talking about a “para­dox”? The reasoning behind Lamb’s remark went back to the first edition of his book, where he had originally addressed the well-known discrepancies between the empirical facts of hydraulics and the mathematical deductions of hydrodynamic theory. He traced the problem back to “the unreality of one or more of the fundamental assumptions” of the theory (244). The empirically false conclusion about resistance came from an empirically false premise, namely, the inviscid character of the postulated fluid. However, d’Alembert’s reasoning was sound, and the logic of the situation was clear. An inviscid fluid is correctly characterized by the absence of resistance. This is how ideal fluids behave or would behave. It is a simple fact about them, and there is nothing paradoxical about it.

A paradox is more than a falsehood, even a blatant falsehood. A paradox must involve a seeming contradiction. Suppose that experiments on a fluid F showed that it exerts a resultant force on a submerged body, while a mathe­matical analysis of F entails a zero resultant. Suppose, further, that the experi­ments on F seemed wholly reliable and the mathematical analysis of F seemed wholly correct. That would be paradoxical. Contradictory specifications of F have been generated from sources that seem undeniable. This is not the case if the experiments refer to a real fluid Fr, and the mathematics refers to an ideal fluid F. There is now no single point of reference as there was with the “paradoxical” fluid F. Two conditions are thus required to make d’Alembert’s result a genuine paradox: (1) there must be two plausible specifications that exclude one another, and (2) the two specifications must be applied to one and the same fluid.

Lamb avoided paradox by treating the two specifications as referring to different things. He drew a boundary between the referent of the experiment and the referent of the theory and thus rejected condition (2). In eschewing the word “paradox,” Lamb’s language was meant to carry a methodological message. It was a way of saying that viscous fluids were one thing and perfect fluids were another and never should the two be confused. This was an ad­mirably straightforward position, but was it the only tenable position? To ad­dress this question I consider a line of reasoning advanced by Ludwig Prandtl and Georg Fuhrmann in Gottingen. It will become clear that these experts did not distinguish between ideal and real fluids in precisely the same way as their British counterparts did.