# Was Rayleigh the first to derive the drag equation?

I was reading about The Drag Equation:

$$F_D = \frac{1}{2} \rho v^2 C_D A$$

where:

$$F_D$$ is the drag force

$$\rho$$ is the mass density of the fluid

$$v$$ is the flow velocity relative to the object

$$A$$ is the reference area

$$C_D$$ is the drag coefficient

It is the equation responsible for explaining the terminal velocity of a falling object within a fluid

I remember seeing it for the 1st time in a book called Fundamentals of Physics by David Halliday and Robert Resnick. I even remember solving some problems using it

I have just read that this equation is attributed to Lord Rayleigh, but I have been unable to find where did he publish it? Does anyone know where does this equation appeard for the 1st time?

## 1 Answer

Rayleigh "derived" the drag equation in On the Resistance of Fluids, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Ser.5, v.2 (1876) no. 13, 430-441. But it is hard to tell why Wikipedia decided to attribute it to him in particular, when it was an intermediate entry in a centuries long controversy over the nature of drag that focused on a side issue surrounding oblique flows, and was based on a theory of separated flow that is now discarded. This seems like a prime example of May's "priority chasing" and "the same unnatural tidiness [that] often afflicts the potted histories of textbooks, which offer students a version of the past cleansed of its natural chaos". Buchanan highlights it in The messy truth about drag:

"Consider fluid dynamics, for example, and its ‘simplest’ problem — that of finding the drag on a sphere moving at fixed speed through a viscous fluid. Many texts give the impression that George Stokes worked out a good solution in 1851, at least for slow or ‘creeping’ flow (with Reynold’s number, $$R$$, close to zero), finding that the drag coefficient goes as $$C_D\sim 1/R$$. Yet the truth about Stokes’ solution turns out to be a little messier. Physicists John Veysey II and Nigel Goldenfeld tell the surprising story of how it took more than 100 years before experiments could even measure the drag with appreciable accuracy, and 150 years of confusion before theorists learned how to calculate the drag, for a small Reynold’s number, in a systematic and accurate way.

When applied to a cylinder, rather than a sphere, his mathematical technique gives singularities. Later theorists — including Lord Rayleigh and Horace Lamb — pointed out why Stokes’ approximations broke down and recognized the importance of the boundary layer. Yet incredibly, only in the 1950s did physicists finally produce the first legitimate approximate solution to lowest order in $$R$$."

That the drag is proportional to fluid density was suggested already in the 17th century by Galileo, and that it is proportional to speed squared by Marriotte (the exponent was known to vary from under $$1.83$$ to over $$3$$ by the end of 19th century, see Forsyth, Newton's Problem). One finds a drag formula of this general form in book II of Newton's Principia (1687), where it is derived from his unrealistic model of collisions with uniformly arranged particles. It might work, at best, in a very rare medium, but is applied to ships in water nonetheless (Newton's "solid of least resistance" known from calculus of variations).

All of this happened even before the theory of continuous fluid flow was developed, and led first to the d'Alembert's paradox (1752), that a body in a potential flow of a perfect fluid encounters zero drag (because pressures on the face are canceled by equal and opposite ones on the rear), and later to Helmholtz's idea about discontinuity along a separation surface, and Stokes's "solution". A good post-Stokes account that discusses Rayleigh's 1876 contribution, and Kelvin's criticism of it based on dynamic instability of the flow near the separation surface, is Fluid Mechanics in the First Half of this Century by Goldstein.

"Kelvin seems to have been more and more unconvinced. In 1894 he published four notes on the question of resistance in Nature, which are reproduced in Volume 4 of his Mathematical and Physical Papers, with a note by the editor, Sir Joseph Larmor, that "These communications formed the subject of a prolonged playful controversy between Lord Kelvin and his intimate friend Sir George Stokes, in a series of letters which have been preserved." Kelvin showed that the results of the surfaces-of-discontinuity theory for a flat plate were not in agreement with the experiments of Dines, published in the Proceedings of the Royal Society in 1890."

Rayleigh, for his part, after discussing d'Alembert's paradox and bringing up Helmholtz's suggestion that the resistance is due to "slipping between contiguous layers", attributes the rest to "the old theory of resistances" (of separated flow, presumably) that "was not very wide of [sic!] the mark" (it is considered unsatisfactory by modern lights, except for cavity flows). He then recounts it as follows:

"Behind the lamina the fluid is at rest under a pressure equal to that which prevails at a distance, the region of rest being bounded by a surface of separation or discontinuity which joins the lamina tangentially, and is determined mathematically by the condition of constant pressure. On the anterior surface of the lamina there is an augmentation of pressure corresponding to the loss of velocity...

If $$u$$ be the velocity of the stream, the increment of pressure due to the loss of velocity is $$\frac12\rho u^2-\frac12\rho v^2$$, and can never exceed $$\frac12\rho u^2$$, which value corresponds to a place of rest where the whole of the energy, originally kinetic, has become potential. The old theory of resistances went on the assumption that the velocity of the stream was destroyed over the whole of the anterior face of the lamina, and therefore led to the conclusion that the resistance amounted to $$\frac12\rho u^2$$ for each unit of area exposed.

It is evident at once that this is an overestimate, since it is only near the middle of the anterior face that the fluid is approximately at rest; towards the edge of the lamina the fluid moves outwards with no inconsiderable velocity, and at the edge itself retains the full velocity of the original stream. Nevertheless the amount of error involved in the theory referred to is not great, as appears from the result of Kirchhoff's calculation of the case of two dimensions, from which it follows that the resistance per unit of area is $$\frac{\pi}{\pi+4}\rho u^2$$ instead of $$\frac12\rho u^2$$."